Signal restoration through deconvolution applied to deep ...rosie/mypresentations/ASUSensip.pdf ·...

OutlineIntroductionGeophysics

Mathematical ModelExperiments

Conclusions and AcknowledgmentSignificant References

Signal restoration through deconvolution appliedto deep mantle seismic probes

Wolfgang Stefan Rosemary Renaut Ed Garnero

Arizona State University

March 31 2006

Wolfgang Stefan Rosemary Renaut Ed Garnero Signal restoration through deconvolution applied to deep mantle seismic probes




IntroductionThe Physical ProblemThe Mathematical Problem

GeophysicsEarthDatasetDrdquo Evidence

Mathematical ModelIntegral EquationsStructured Inverse ProblemRegularization

Experiments1D Synthetic DataReal Data

Conclusions and AcknowledgmentRegularizationSolution Techniques

Significant References





The Physical ProblemThe Mathematical Problem

The Physical Problem

I Image Earthrsquos interior Deep EarthI interface layer of the liquid core and solid mantle

I Earthrsquos interior is inaccessible remote sampling needed

I Important steps prior to data inversion include

(a) accurate characterization of seismic energy(b) reliable measurement of seismic wave timing


























The Mathematical Problem

I Topographic inversion with extreme sparse data coverage

I Poor signal to noise ratio (SNR)

I Extremely Ill-posed

GOAL improve data set prior to data inversion







II Topographic inversion with extreme sparse data coverage




























EarthDatasetDrdquo Evidence

I











The Data Set

I Deep focus earthquakes (events) in South America

I Picked up at broad band stations in Europe

I Seismic energy reflects at the core-mantle boundery under theatlantic ocean

I Each event-station pair produces a seismogram






The Data Set










The Data Set










The Data Set










Drdquo Evidence

I Seismograms sometimes show evidence a reflecting layer atthe core-mantle boundary

I An additional seismic phase is visible

I The additional phase is usually very weak

I And only visible in very view traces (due to locality of the Drdquolayer)

I ie even traces with very poor SNR have to be considered






Drdquo Evidence











Drdquo Evidence











Drdquo Evidence
















Problem Formulation

I Invert the blurring Effect (attenuation) of Earthrsquos mantleand core to

I (a) get clearer evidence of the existence of structures likethe ultra low velocity zone (ULVZ)

I (b) get accurate timing information for data inversionprecise information provides more information on structuresizeheight Is the ULVZ 20km or 400km deep






Problem Formulation









Problem Formulation








Integral EquationsStructured Inverse ProblemRegularization

Integral Equations

intΩ

input X system dΩ = output

bull Given noisy output determine input orand the system

bull General Applications Image Processing SignalIdentification

bull HERE Seismic Signal Restoration






Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Invariant Model

bull Signal degradation is modeled as a convolution

g = f otimes h + n

bull where g is the blurred signalbull f is the unknown signalbull h is the point spread function (PSF)- knownbull n is noise

bull Matrix Formulation H is Toeplitz (structured)

g = Hf + n






Invariant Model


g = f otimes h + n



g = Hf + n






Example of Convolution

g = f otimes h + n






Estimation of the PSF

I Ideally seismic deconvolution yields a spike train- so timing isacccurate

I The corresponding PSF is unknown (if it exists)I Estimations of this PSF (in seismology wavelet) come from

I stacking traces (problem traces are very different)I estimating Earthrsquos filter (basically a low pass filter very

difficult due to inhomogeneities)I use a very basic (common) shape like a Gaussian (very rough

estimate)








I The corresponding PSF is unknown (if it exists)

I Estimations of this PSF (in seismology wavelet) come fromI stacking traces (problem traces are very different)I estimating Earthrsquos filter (basically a low pass filter very


estimate)











estimate)






Gaussian Wavelet

I also called a Ricker Wavelet

I h(t) = 1σradic

2πeminus

t2

2σ2

I σ is a width parameter chosen such that the waveletapproximates the phase of interest






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Inverse with Known PSF

bull Find f from g = f otimes h + n given g and h with unknown n

bull Assuming normal distributed n yields the estimator

f = arg minfg minus f otimes h2

2

bull Reconstruction with n isin N(0 10minus7)







bull Find f from g = f otimes h + n given g and h with unknown nbull Assuming normal distributed n yields the estimator


2










2












Ill-posedness

bull Add more information about the signal

bull eg statistical properties or information about the structure(eg sparse decon or total variation decon)

bull Regularize


2 + λR(f )

where R(f ) is the penalty term and λ is a penalty parameter






Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Regularization Methods

bull Tikhonov (TK)

R(f ) = TK(f ) =

intΩ|nablaf (t)|2dt

bull Total Variation (TV)

R(f ) = TV(f ) =

intΩ|nablaf (t)|dt

bull Sparse deconvolution (L1)

R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )

I The problems are very large (n order of 10000)I TK is a linear least squares (LS) problem

f = arg minfg minus Hf 2

2 + λLf 22)

I Standard Solution f solves normal equations

(HTH + λLTL)f = HTg

I But problem is large and structured use iterative methodsI Evaluation of the OF and its gradient is cheap (some FFTs

and sparse matrix-vector multiplications)






Cost Functional


2 + λR(f )

I The problems are very large (n order of 10000)

I TK is a linear least squares (LS) problem


2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)



I But problem is large and structured use iterative methods

I Evaluation of the OF and its gradient is cheap (some FFTsand sparse matrix-vector multiplications)






Cost Functional


2 + λR(f )



2 + λLf 22)




and sparse matrix-vector multiplications)Wolfgang Stefan Rosemary Renaut Ed Garnero Signal restoration through deconvolution applied to deep mantle seismic probes





Making the TVOF differentiable

R(f ) =sum

i

fi+1 minus fi

I The TV objective function is non differentable

J(f ) = g minus Hf 22 + λLf 1

I for a small β isin 10minus5 to 10minus9 define

Rβ =sum

i

radic(fi+1 minus fi )2 + β

Rudin Osher amp Fatemi (1992)







R(f ) =sum

i

fi+1 minus fi

II The TV objective function is non differentable



Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i








Regularization Notes


2 + λR(f )

Ibull λ Governs the trade off between the fit to the data and thesmoothness of the reconstruction

I small λ puts weight on the data fit term

I large λ puts weight on the smoothness of the solution

I Idea of L-curve vary λ (from small to large) and observe howboth terms vary Pick corner point








2 + λR(f )

bull λ Governs the trade off between the fit to the data and thesmoothness of the reconstruction











2 + λR(f )












2 + λR(f )










L-curve (cont)






Parameter choice β and λ ILLUSTRATES TV






Comparing TV and TK Regularization






Observations

bull TV yields a piece wise constant reconstruction and preservesthe edges of the signal

bull TK yields a smooth reconstruction

bull To find the minimum we use a limited memory BFGS method

I Involves approximating the inverse Hessian using rank oneupdates starting with initial steepest descent






Observations










Observations










Observations



bull To find the minimum we use a limited memory BFGS methodI Involves approximating the inverse Hessian using rank one

updates starting with initial steepest descent






Water level deconvolution

I Currently used in deep mantle seismology

I Transform problem in fourier space ie forward model is

g(ξ) = f (ξ) middot h(ξ) + noise

I Observe the unregularized solution g(ξ) = f (ξ)

h(ξ)has a problem

if h(ξ) is small

I Idea Threshold fourier coefficients of h ie

ˆh(ξ) =

h(ξ) if |h(ξ)| gt th

th middot h(ξ)

|h(ξ)|if |h(ξ)| le th











h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)







Water level deconvolution (cont)

I Solution is found as

g(ξ) =f (ξ)ˆh(ξ)

I Thresholding introduces Gibbs phenomenon

I Solution is then convolved (smoothed) with a Gaussian toeliminate unwanted high frequency oscillations

























1D Synthetic DataReal Data

I Data taken at critical angle asymp 110 deg SKS starts to diffractalong the core






SKS at 112 deg deconvolved with SKS from 99 deg






SKS at 112 deg deconvolved with a Gaussian






Figure Note alignment to first SKS in each signal






Error estimates

I Arrival time from an edge detection vs ray theory prediction






Real Data (SV) from an earthquake in South America






Real Data (SH) from an earthquake in South America






Evidence of the ultra low velocity zone (ULVZ)





RegularizationSolution Techniques

Conclusions

I TV regularized deconvolution is more robust then establishedmethods

I Automatic travel time picking is more accurate then handpicking

I TV deconvolution yields usable results even for roughestimates of the wavelet






Conclusions









Conclusions









Future Work

I Better estimates of the wavelet eg two-sided Gaussian willimprove results further

I Blind Deconvolution techniques

I Total Least Squares with Regularization






Future Work









Future Work









Matrix Formulation H not Toeplitz

bull Consider g = Hf + n where either H is structured but notknown or H is estimated and unstructured

bull Can be used to allow spatially variable H or two sidedGaussian














min(E f )F subject to (A + E )x = b + f

bull Classical Algorithm Golub Reinsch Van-Loan

Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f

bull SVD Solution uses the SVD of augmented matrix [A|b][xTLS

minus1

]=

minus1

vn+1n+1vn+1

I Direct Compute the SVD and solve








Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1







Rayleigh Quotient Formulation for TLS

bull An Iterative Approach

xTLS = argminxφ(x) = argminxAx minus b2

1 + x2

bull φ is the Rayleigh quotient of matrix [A b]

bull TLS minimizes the sum of squared normalized residuals









1 + x2











1 + x2








An eigenvalue problem for TLS

bull Bjorck Hesternes and Matstoms(BHM) 2000 Eigenvalue equationfor TLS (

ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)

bull σ2 is to be minimized System matrix is constant

bull First block equation gives the normal equations for TLS

(ATAminus σ2I )x = ATb

bull Use Rayleigh quotient iteration to find the minimal eigenvalue

bull Solve shifted normal equations by Preconditioned ConjugateGradients each iteration








ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)bull σ2 is to be minimized System matrix is constant












ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS

Regularization Stabilize TLS with realistic constraint on data Lx le δ δ is prescribed from known information on the solutionand L is typically a differential operator

Reformulation with Lagrange Multiplier If the constraint is active thesolution xlowast satisfies normal equations

(ATA + λI I + λLLTL)xlowast = ATb (xlowast)TLTLxlowast minus δ2 = 0

λI = minusAxlowast minus b2

1 + xlowast2 λL = micro(1 + xlowast2)

micro = minus 1

δ2

(bT (Axlowast minus b)

1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)

Golub Hansen and OrsquoLeary 1999 Solution technique isparameter dependent λL λI δ micro






Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Rayleigh Quotient Formulation for RTLS

bull Recall

xTLS = argminxφ(x) = argminx

Ax minus b2

1 + x2

bull Formulate regularization for TLS in RQ form

minxφ(x) subject to Lx le δ

bull Augmented Lagrangian

L(x micro) = φ(x) + micro(Lx2 minus δ2)







bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2










Eigenvalue Problem RTLS Renaut amp Guo (SIAM 2004)

B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x

lowast)LTL ATbbTA minusλL(x

lowast)γ + bTb

)

λL = micro(1 + xlowast2)

bull Seek the minimal eigenpair for B Note B is not constant

bull Specify constraint Lxlowast2 le δ2 use γ = δ2 or Lxlowast2







B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development

bull At a solution constraint is active measure normalized discrepancy

g(x) = (Lx2 minus δ2)(1 + x2)

bull Denote B(θ) = M + θN N =

(LTL 00 minusδ2

)

bull Denote eigenpair for smallest eigenvalue of B(θ) as n+1 (xTθ minus1)T

bull Reformulation

For a constant δ find a θ such that g(xθ) = 0






Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Regularized Total Least Squares

Numerical experiments verify the theory

bull A new algorithm provides an efficient and practical approachfor the solution of the RTLS problem in which a goodestimate of the physical parameter is provided

bull If blow up occurs bisection search may yield a better solutionsatisfying the constraint condition

bull If no constraint information is provided the L-curve techniquecan be successfully employed

bull Algorithm needs to be extended for TV case to be used inseismic signals







































Thanks to

I Sebastian Rost and Matthew Fouch for discussions and data

I This study was supported by the grant NSF CMG-02223





Lay T Garnero E J and Russell SA 2004 Lateral Variation of the DrdquoDiscontinuity Beneath the Cocos Plate Geophys Res Lett 31doi1010292004GL020300

Rudin L I Osher S amp Fatemi E 1992 Nonlinear total variation based noiseremoval algorithms Physica D 60 259-268

Vogel C R amp Oman M E 1996 Iterative methods for total variationdenoising SIAM J Sci Statist Comput 17 227-238

Vogel C R 2002 Computational Methods for Inverse Problems SIAMFrontiers in Applied Mathematics

Zhu C Byrd R Lu P amp Nocedal J 1997 Software for Large-scale Bound-constrained or Unconstrained Optimizationhttpwww-fpmcsanlgovotcToolsLBFGS-B


Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations

Structured Inverse Problem

Regularization

Experiments

1D Synthetic Data

Real Data

Conclusions and Acknowledgment

Regularization

Solution Techniques





IntroductionThe Physical ProblemThe Mathematical Problem

GeophysicsEarthDatasetDrdquo Evidence

Mathematical ModelIntegral EquationsStructured Inverse ProblemRegularization

Experiments1D Synthetic DataReal Data

Conclusions and AcknowledgmentRegularizationSolution Techniques













































































I











The Data Set










The Data Set










The Data Set










The Data Set










Drdquo Evidence











Drdquo Evidence











Drdquo Evidence











Drdquo Evidence
















Problem Formulation









Problem Formulation









Problem Formulation









Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Invariant Model


g = f otimes h + n



g = Hf + n






Invariant Model


g = f otimes h + n



g = Hf + n







g = f otimes h + n











estimate)











estimate)











estimate)






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques












































































I











The Data Set










The Data Set










The Data Set










The Data Set










Drdquo Evidence











Drdquo Evidence











Drdquo Evidence











Drdquo Evidence
















Problem Formulation









Problem Formulation









Problem Formulation









Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Invariant Model


g = f otimes h + n



g = Hf + n






Invariant Model


g = f otimes h + n



g = Hf + n







g = f otimes h + n











estimate)











estimate)











estimate)






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques











The Data Set










The Data Set










The Data Set










The Data Set










Drdquo Evidence











Drdquo Evidence











Drdquo Evidence











Drdquo Evidence
















Problem Formulation









Problem Formulation









Problem Formulation









Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Invariant Model


g = f otimes h + n



g = Hf + n






Invariant Model


g = f otimes h + n



g = Hf + n







g = f otimes h + n











estimate)











estimate)











estimate)






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






The Data Set










The Data Set










The Data Set










Drdquo Evidence











Drdquo Evidence











Drdquo Evidence











Drdquo Evidence
















Problem Formulation









Problem Formulation









Problem Formulation









Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Invariant Model


g = f otimes h + n



g = Hf + n






Invariant Model


g = f otimes h + n



g = Hf + n







g = f otimes h + n











estimate)











estimate)











estimate)






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






The Data Set










The Data Set










Drdquo Evidence











Drdquo Evidence











Drdquo Evidence











Drdquo Evidence
















Problem Formulation









Problem Formulation









Problem Formulation









Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Invariant Model


g = f otimes h + n



g = Hf + n






Invariant Model


g = f otimes h + n



g = Hf + n







g = f otimes h + n











estimate)











estimate)











estimate)






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






The Data Set










Drdquo Evidence











Drdquo Evidence











Drdquo Evidence











Drdquo Evidence
















Problem Formulation









Problem Formulation









Problem Formulation









Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Invariant Model


g = f otimes h + n



g = Hf + n






Invariant Model


g = f otimes h + n



g = Hf + n







g = f otimes h + n











estimate)











estimate)











estimate)






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Drdquo Evidence











Drdquo Evidence











Drdquo Evidence











Drdquo Evidence
















Problem Formulation









Problem Formulation









Problem Formulation









Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Invariant Model


g = f otimes h + n



g = Hf + n






Invariant Model


g = f otimes h + n



g = Hf + n







g = f otimes h + n











estimate)











estimate)











estimate)






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Drdquo Evidence











Drdquo Evidence











Drdquo Evidence
















Problem Formulation









Problem Formulation









Problem Formulation









Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Invariant Model


g = f otimes h + n



g = Hf + n






Invariant Model


g = f otimes h + n



g = Hf + n







g = f otimes h + n











estimate)











estimate)











estimate)






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Drdquo Evidence











Drdquo Evidence
















Problem Formulation









Problem Formulation









Problem Formulation









Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Invariant Model


g = f otimes h + n



g = Hf + n






Invariant Model


g = f otimes h + n



g = Hf + n







g = f otimes h + n











estimate)











estimate)











estimate)






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Drdquo Evidence
















Problem Formulation









Problem Formulation









Problem Formulation









Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Invariant Model


g = f otimes h + n



g = Hf + n






Invariant Model


g = f otimes h + n



g = Hf + n







g = f otimes h + n











estimate)











estimate)











estimate)






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques











Problem Formulation









Problem Formulation









Problem Formulation









Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Invariant Model


g = f otimes h + n



g = Hf + n






Invariant Model


g = f otimes h + n



g = Hf + n







g = f otimes h + n











estimate)











estimate)











estimate)






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Problem Formulation









Problem Formulation









Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Invariant Model


g = f otimes h + n



g = Hf + n






Invariant Model


g = f otimes h + n



g = Hf + n







g = f otimes h + n











estimate)











estimate)











estimate)






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Problem Formulation









Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Invariant Model


g = f otimes h + n



g = Hf + n






Invariant Model


g = f otimes h + n



g = Hf + n







g = f otimes h + n











estimate)











estimate)











estimate)






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Invariant Model


g = f otimes h + n



g = Hf + n






Invariant Model


g = f otimes h + n



g = Hf + n







g = f otimes h + n











estimate)











estimate)











estimate)






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Integral Equations

intΩ










Integral Equations

intΩ










Integral Equations

intΩ










Invariant Model


g = f otimes h + n



g = Hf + n






Invariant Model


g = f otimes h + n



g = Hf + n







g = f otimes h + n











estimate)











estimate)











estimate)






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Integral Equations

intΩ










Integral Equations

intΩ










Invariant Model


g = f otimes h + n



g = Hf + n






Invariant Model


g = f otimes h + n



g = Hf + n







g = f otimes h + n











estimate)











estimate)











estimate)






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Integral Equations

intΩ










Invariant Model


g = f otimes h + n



g = Hf + n






Invariant Model


g = f otimes h + n



g = Hf + n







g = f otimes h + n











estimate)











estimate)











estimate)






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Invariant Model


g = f otimes h + n



g = Hf + n






Invariant Model


g = f otimes h + n



g = Hf + n







g = f otimes h + n











estimate)











estimate)











estimate)






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Invariant Model


g = f otimes h + n



g = Hf + n







g = f otimes h + n











estimate)











estimate)











estimate)






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques







g = f otimes h + n











estimate)











estimate)











estimate)






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques











estimate)











estimate)











estimate)






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques











estimate)











estimate)






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques











estimate)






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2







Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Gaussian Wavelet


I h(t) = 1σradic

2πeminus

t2

2σ2











2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques










2










2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques









2










2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques









2












Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques











Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Ill-posedness



bull Regularize


2 + λR(f )







Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Ill-posedness



bull Regularize


2 + λR(f )








bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques







bull Tikhonov (TK)

R(f ) = TK(f ) =



R(f ) = TV(f ) =

intΩ|nablaf (t)|dt


R(f ) = f 1 =

intΩ|f (t)|dt






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Cost Functional


2 + λR(f )




2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Cost Functional


2 + λR(f )



2 + λLf 22)










Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Cost Functional


2 + λR(f )



2 + λLf 22)










R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques







R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques







R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i









R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques







R(f ) =sum

i

fi+1 minus fi




Rβ =sum

i










2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques








2 + λR(f )












2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques








2 + λR(f )












2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques








2 + λR(f )












2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques








2 + λR(f )










L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






L-curve (cont)


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques


















Observations










Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Observations










Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Observations










Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Observations















h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques











h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques











h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques











h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)












h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques











h(ξ)has a problem

if h(ξ) is small


ˆh(ξ) =


th middot h(ξ)





























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques




























































Error estimates

























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques
























Conclusions









Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Conclusions









Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Conclusions









Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Future Work









Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Future Work









Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Future Work



























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques
























Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques








Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1









Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques








Solve [A|b]

[xTLS

minus1

]= 0 A = A + E b = b + f


minus1

]=

minus1

vn+1n+1vn+1










1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques









1 + x2











1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques









1 + x2











1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques









1 + x2










ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques








ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1

)













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques








ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques








ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques








ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1













ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques








ATA ATbbTA bTb

) (xminus1

)= σ2

(xminus1











Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)







Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Tikhonov TLS






micro = minus 1

δ2


1 + xlowast2+

Axlowast minus b2

(1 + xlowast2)2

)








bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques







bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques







bull Recall


Ax minus b2

1 + x2











bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques







bull Recall


Ax minus b2

1 + x2











B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques







B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques







B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)










B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques







B(xlowast)

(xlowast

minus1

)= minusλI

(xlowast

minus1

)

B(xlowast) =

(ATA + λL(x


lowast)γ + bTb

)









Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation







Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques






Development


g(x) = (Lx2 minus δ2)(1 + x2)


(LTL 00 minusδ2

)


bull Reformulation



















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques


















































Thanks to













Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques











Outline

Introduction



Geophysics

Earth

Dataset

D Evidence

Mathematical Model

Integral Equations


Regularization

Experiments

1D Synthetic Data

Real Data


Regularization

Solution Techniques


Signal restoration through deconvolution applied to deep ...rosie/mypresentations/ASUSensip.pdf ·...

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Transcript of Signal restoration through deconvolution applied to deep ...rosie/mypresentations/ASUSensip.pdf ·...