Reconstruction of Chlorophyll Concentration Pro le in O ... fileReconstruction of Chlorophyll...

Reconstruction of ChlorophyllConcentration Pro�le

in O�shore Ocean Water

Roberto Pinto Souto - [email protected]

3rd Workshop on Modeling andSensing Environmental Systems (MoSES-III)

MoSES-III - Petrópolis - August 8th, 2011LNCC () MoSES-III August 8th, 2011 1 / 79

Motivation

Outline

1 Motivation

2 Hydrologic OpticsRadiative Transfer Equation - RTEInverse Problem - IPIP: radiances in di�erent depthsIP: multispectral radiances only in surface

3 Applying Data Assimilation

4 Final Remarks

LNCC () MoSES-III August 8th, 2011 2 / 79

Motivation

Phytoplankton

Oceanic micro-organisms with chlorophyll-a pigments(Chl-a);

They are the base of the ocean's life;

About half of the earth oxygen comes fromphotosynthesis made by phytoplankton;

They are responsible by nearly 40% of the carbonsequestration.


Motivation

Chlorophyll Concentration


Motivation

Satellite Ocean Color


Motivation

Climate Changes Impact


Hydrologic Optics

Outline

1 Motivation



4 Final Remarks


Hydrologic Optics Radiative Transfer Equation - RTE

Outline

1 Motivation



4 Final Remarks



(ζ,µ,ϕ)

(0,−µ,ϕ)(0,µ,ϕ)

(ζ,−µ,ϕ)

b

p( )

a

Θ

LL

θ

0

internalsources

air

Sun

ζ

LL

ϕ

water

Θ

radiance: L

τ=

=τ

bottom



Inherent Optical Properties - IOPs

optical depth: τ = ζ

τ is the optical variable:dτ = c(z)dzτ =

∫z

0c(z ′)dz ′

z =∫

τ

0

1

c(τ ′)dτ ′

scattering coe�cient: b(τ,λ )

absorption coe�cient: a(τ,λ )

attenuation coe�cient: c(τ,λ ) = a(τ,λ )+b(τ,λ )



Inherent Optical Properties - IOPs

single scattering albedo : ϖ0(τ,λ ) = b(τ,λ )/c(τ,λ )

scattering phase function : p(Θ) = p(µ,ϕ ;µ ′,ϕ ′)

*µ = cosθ and ϕ are polar and azimuthal directions



Radiative Transfer Equation (RTE)

µ∂

∂τLλ (τ,µ,ϕ) =−Lλ (τ,µ,ϕ)

ϖ0(τ,λ )

4π

∫1

−1

∫2π

0

p(µ,ϕ ;µ ′,ϕ ′)Lλ (τ,µ′,ϕ ′)dϕ

′dµ′

+Sλ (τ,µ,ϕ)

subjects to

Lλ (0,µ,ϕ) = Fδ (µ−µ0)δ (ϕ−ϕ0)

Lλ (ζ ,−µ,ϕ) = 0,

for µ ∈ (0,1] e ϕ ∈ [0,2π].


Hydrologic Optics Inverse Problem - IP

Outline

1 Motivation



4 Final Remarks



Inverse Problems

BOUNDARY

PARAMETERS

CONDITIONS

+

DATA

OBSERVED

DIRECT MODEL

INVERSE MODEL

Causes Effects



E�ect:Radiance

Causes:Boundary ConditionsScattering phase functionAbsorption coe�cientScattering coe�cientInternal source



Plane-parallel geometry

0

τ1

τ

τr

τ

τR−1

R

τ = 0

τ = ζ

R−2

τ2

r−1

τ

. . . .

. . . .

region 2

region 1

region r

Superface

region R−1

region R



Bio-optical models

ϖr ,λ =br ,λ

ar ,λ +br ,λ

ar ,λ =[aw

λ+0.06 ac

λC 0.65r

]×[1+0.2 e−0.014(λ−440)

]

br ,λ =

(550

λ

)0.30 C 0.62

r

*Typically, in o�shore oceanic waters, most of attenuationis due to chlorophyll pigments found in phytoplankton, asis required for the above bio-optical models.



The original inverse problem of absorption and scatteringcoe�cients estimation yields to recovering chlorophyllvertical pro�le, discretized in R+1 depths:

C = [ C (τ0) C (τ1) C (τ2) · · · C (τR) ]= [ C1 C2 C3 · · · CR+1 ]



Real chlorophyll pro�le

0 0.2 0.4 0.6 0.8 1 1.2 1.4−80

−70

−60

−50

−40

−30

−20

−10

0

Clorofila mg/m3

Pro

fund

idad

e [m

etro

s]



The real pro�le can be modeled with a Gaussian model:

C (z) = C0+h

s√2 π

e−1

2( z−zmax

s)2

where z is the geometric depth given in meters.



C0 h s zmax0.2 144 9 17

0 1 2 3 4 5 6 7 8−40

−35

−30

−25

−20

−15

−10

−5

0

C (mg/m3)

geo

met

ric

dep

th z

(m

)


Hydrologic Optics IP: radiances in di�erent depths

Outline

1 Motivation



4 Final Remarks



Objective Function

Radiances observed in each depth

J(C ) =Nµ

∑i=1

R

∑r=0

[Lobs(τr ,µi)−L

C(τr ,µi)]2

+ γ Γ(C )

Regularization Term:

γ Γ(C ) = γ

R

∑r=2

(Cr−1−2Cr +Cr+1)2



Ant Colony Optimization (ACO)

It is a metaheuristic based on collective behavior of ants,searching the smallest path between the ant colony andthe source of food [Dorigo et al., 1996].



ACO parameters

ns: number of parameter (assigned to points in agraph) to �nd.

np: number of paths between two parameters (twopoints).

na: number of ants

Tij = To : initial amount of pheromone in all pathsi = 1, . . . ,ns e j = 1, . . . ,np

ρ : pheromone evaporation rate Tij = (1−ρ)Tij

mit : maximum number of iterations

qo : decision threshold



ACO scheme

At each iteration the smallest objective function costsolution {ijmin} is chosen.The path traveled by the ant is assigned as thesmallest cost solution, and it is marked withpheromone:

Tijmin= (1−ρ)Tijmin

+To



ACO - 90 ants

-40

-35

-30

-25

-20

-15

-10

-5

0

0 2 4 6 8 10

De

pth

z (

m)

C (mg/m3)

EXACTAVG. SOL.

SEED SOL.



Original upward radiances

−40

−30

−20

−10

0

−1

−0.8

−0.6

−0.4

−0.2

00

0.05

0.1

0.15

0.2

noiselessLNCC () MoSES-III August 8th, 2011 28 / 79


Upward radiances with depth correction

−40

−30

−20

−10

0

−1

−0.8

−0.6

−0.4

−0.2

00

0.05

0.1

0.15

0.2



Original downward radiances

−40

−30

−20

−10

0

0

0.2

0.4

0.6

0.8

10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4



Downward radiances with depth correction

−40

−30

−20

−10

0

0

0.2

0.4

0.6

0.8

10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4



ACO - 90 ants

-40

-35

-30

-25

-20

-15

-10

-5

0

0 2 4 6 8 10

De

pth

z (

m)

C (mg/m3)

EXACTAVG. SOL.

SEED SOL.




−40

−30

−20

−10

0

−1

−0.8

−0.6

−0.4

−0.2

00

0.05

0.1

0.15

0.2

noise 5%LNCC () MoSES-III August 8th, 2011 33 / 79



−40

−30

−20

−10

0

−1

−0.8

−0.6

−0.4

−0.2

00

0.05

0.1

0.15

0.2




−40

−30

−20

−10

0

0

0.2

0.4

0.6

0.8

10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4



ACO (5% noisy data) - 90 ants

0 1 2 3 4 5 6 7 8−40

−35

−30

−25

−20

−15

−10

−5

0Identificacao do perfil de concentracao de clorofila

C (mg/m3)

prof

undi

dade

geo

met

rica

z (m

)realestimado




−40

−30

−20

−10

0

−1

−0.8

−0.6

−0.4

−0.2

00

0.05

0.1

0.15

0.2




−40

−30

−20

−10

0

0

0.2

0.4

0.6

0.8

10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4



ACO (10% noisy data) - 90 ants

0 1 2 3 4 5 6 7 8−40

−35

−30

−25

−20

−15

−10

−5

0Identificacao do perfil de concentracao de clorofila

C (mg/m3)

prof

undi

dade

geo

met

rica

z (m

)realestimado



ACO with Intrinsic Regularization -ACOwIR

It uses a pre-selection of candidate solutions in eachACO iteration [Preto et al., 2004][Souto et al., 2006],according 2nd -order Tikhonov criterion.

For instance, 15 of 90 ants (solutions) are chosenaccording their smoothness.

It is reached a gain of performance.



ACO - 90 ants

-40

-35

-30

-25

-20

-15

-10

-5

0

0 2 4 6 8 10

De

pth

z (

m)

C (mg/m3)

EXACTAVG. SOL.

SEED SOL.



ACO - 15 ants

-40

-35

-30

-25

-20

-15

-10

-5

0

0 2 4 6 8 10

De

pth

z (

m)

C (mg/m3)

EXACTAVG. SOL.

SEED SOL.



ACOwIR - 15 of 90 ants

-40

-35

-30

-25

-20

-15

-10

-5

0

0 2 4 6 8 10

De

pth

z (

m)

C (mg/m3)

EXACTAVG. SOL.

SEED SOL.



ACO Fuzzy

An amount of pheromone is also deposited in theneighbor paths, with weigths w1 and w2.

Tijmin−2 = (1−ρ)Tijmin−2+w2To

Tijmin−1 = (1−ρ)Tijmin−1+w1To


+To

Tijmin+1= (1−ρ)Tijmin+1

+w1To


+w2To

where w2 < w1 < 1.0



ACO Fuzzy: Example

w = [w1 w2]T = [0.4 0.1]T

Tijmin−2 = (1−ρ)Tijmin−2+0.1To

Tijmin−1 = (1−ρ)Tijmin−1+0.4To


+To


+0.4To


+0.1To



ACOwIR - 15 of 90 ants

-40

-35

-30

-25

-20

-15

-10

-5

0

0 2 4 6 8 10

De

pth

z (

m)

C (mg/m3)

EXACTAVG. SOL.

SEED SOL.



ACOwIR Fuzzy - 15 of 90 ants

-40

-35

-30

-25

-20

-15

-10

-5

0

0 2 4 6 8 10

De

pth

z (

m)

C (mg/m3)

EXACTAVG. SOL.

SEED SOL.



ACO - 15 ants

-40

-35

-30

-25

-20

-15

-10

-5

0

0 2 4 6 8 10

De

pth

z (

m)

C (mg/m3)

EXACTAVG. SOL.

SEED SOL.



ACO Fuzzy - 15 ants

-40

-35

-30

-25

-20

-15

-10

-5

0

0 2 4 6 8 10

De

pth

z (

m)

C (mg/m3)

EXACTAVG. SOL.

SEED SOL.


Hydrologic Optics IP: multispectral radiances only in surface

Outline

1 Motivation



4 Final Remarks



Objective Function

Multispectral radiances observed in water surface (z=0)

J(C ) =Nλ

∑j=1

Nµ/2

∑i=1

[Lobs(0,−µi)−L

C

λj(0,−µi)

]2+ γ Γ(C ).



Noiseless multispectral radiancesementes ns np na nap mit ρ q0

{03,15,21,31,45} 10 3000 360 12 500 0.03 0.0{63,77,81,95,99}

-40

-35

-30

-25

-20

-15

-10

-5

0

0 2 4 6 8 10

Dep

th z

(m

)

C (mg/m3)

EXACTAVG. SOL.

SEED SOL.



Negative Concavity: step-1sementes ns np na nap mit ρ q0

{03,15,21,31,45} 10 3000 360 12 500 0.03 0.0{63,77,81,95,99}

-40

-35

-30

-25

-20

-15

-10

-5

0

0 2 4 6 8 10

Dep

th z

(m

)

C (mg/m3)

EXACTAVG. SOL.

SEED SOL.




{03,15,21,31,45} 10 3000 360 12 500 0.03 0.0{63,77,81,95,99}

-40

-35

-30

-25

-20

-15

-10

-5

0

0 2 4 6 8 10

Dep

th z

(m

)

C (mg/m3)

EXACTAVG. SOL.

SEED SOL.



Negative Concavity: step-1, noise 1%sementes ns np na nap mit ρ q0

{03,15,21,31,45} 10 3000 360 12 500 0.03 0.0{63,77,81,95,99}

-40

-35

-30

-25

-20

-15

-10

-5

0

0 2 4 6 8 10

Dep

th z

(m

)

C (mg/m3)

EXACTAVG. SOL.

SEED SOL.



Negative Concavity: step-2 noise 1%sementes ns np na nap mit ρ q0

{03,15,21,31,45} 10 3000 360 12 500 0.03 0.0{63,77,81,95,99}

-40

-35

-30

-25

-20

-15

-10

-5

0

0 2 4 6 8 10

Dep

th z

(m

)

C (mg/m3)

EXACTAVG. SOL.

SEED SOL.




{03,15,21,31,45} 10 3000 360 12 500 0.03 0.0{63,77,81,95,99}

-40

-35

-30

-25

-20

-15

-10

-5

0

0 2 4 6 8 10

Dep

th z

(m

)

C (mg/m3)

EXACTAVG. SOL.

SEED SOL.



Negative Concavity, noiselesssementes ns np na nap mit ρ q0

{03,15,21,31,45} 10 3000 300 12 500 0.03 0.0{63,77,81,95,99}

-80

-70

-60

-50

-40

-30

-20

-10

0

0 0.1 0.2 0.3 0.4 0.5

Dep

th z

(m

)

C (mg/m3)

EXACTAVG. SOL.

SEED SOL.



Negative Concavity, noise 1%sementes ns np na nap mit ρ q0

{03,15,21,31,45} 10 3000 300 12 500 0.03 0.0{63,77,81,95,99}

-80

-70

-60

-50

-40

-30

-20

-10

0

0 0.1 0.2 0.3 0.4 0.5

Dep

th z

(m

)

C (mg/m3)

EXACTAVG. SOL.

SEED SOL.


Applying Data Assimilation

Outline

1 Motivation



4 Final Remarks



Data Assimilation

Elements description

Ce Exact Chl-a pro�leCb Recovered Chl-a pro�le (background)N Number of recovered Chl-a pro�les

(number of observations/measurements/samples)Nb Number of components of Chl-a pro�leεb absolute error between Cb e Ce

Pb Covariance error matrix (Nb×Nb) of background



Data Assimilation

Elements description

Le Exact radiance values due to Ce

Lo Observed radiance dataNo Number of observed radiance dataεo error between Lo e LePo Covariance error matrix (No×No) of observationsH non-linear RTE modelH Jacobian matrix (No×Nb) of model H

due to small variations of Cb

W gain matrix (Nb×No)Ca Chl-a pro�le analysis (result of assimilation)



Data Assimilation

Formulation

Ca = Cb +W [Lo−H (Cb)]

W = PbHT (HPbH

T +Po)−1

Pb = E{εbεTb }

Po = E{εoεTo }



Data Assimilation

Case Study

Radiances in each depthNb 10 depthsNo 100 (10 polar directions × 10 depths)N 200 samples



Best (but not typical) result of 200 samples

-80

-70

-60

-50

-40

-30

-20

-10

0

0 0.1 0.2 0.3 0.4 0.5

Dep

th z

(m

)

C (mg/m3)

SIZE:200 SAMPLE #033 SQUARE ERROR: BKGD=0.047239 ANLS=0.0090695EXACT

BKGD.

ANLR.



Worst result of 200 samples

-80

-70

-60

-50

-40

-30

-20

-10

0

0 0.1 0.2 0.3 0.4 0.5

Dep

th z

(m

)

C (mg/m3)

SIZE:200 SAMPLE #128 SQUARE ERROR: BKGD=0.043895 ANLS=0.056691EXACT

BKGD.

ANLR.



0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1

BKGD/ANLS SQUARE ERROR RATIO20 OF SAMPLE SIZE 20

20 OF SAMPLE SIZE 200200 OF SAMPLE SIZE 200

BKGD SQR ERROR = ANLS SQR ERROR


Final Remarks

Outline

1 Motivation



4 Final Remarks


Final Remarks

Data Assimilation

It is still necessary further studies about the use of dataassimilation:

How to best estimate covariance error matrices?

How many samples is need to reach a better analysis?

Apply it to surface multispectral radiances.


Final Remarks

Computational Performance

Another important issue is computational performance:

Case study was shown for radiances with azimuthalsymmetry (isotropic medium);

In anisotropic medium the computational cost ismuch higher;

Strategies of parallelism with MPI: ACO hasindependent ants, and RTE solver performsindependent azimuthal modes;

Strategies of parallelism with OpenMP and CUDA:exploring multi and manycore architectures to solvelinear systems and eigenvalues routines.


Final Remarks

Further Case Studies

Besides to employ an anisotropic medium, it is possible:

Increase the level of discretization of Chl-a. At leastone value of chlorophyll pro�le in each meter of deph.

Use of real, not synthetic, remote sensing data, fromocean optics spectrometers and satellites.


Final Remarks

Acknownledgements

PCI/CNPq/LNCC - processes number 381243/2010-9 and 300338/2011-2


Final Remarks

Thank you


Final Remarks

Carvalho, A., de Campos Velho, H., adn R.P. Souto,S. S., Becceneri, J., and Sandri, S. (2008).Fuzzy ant colony optimization for estimatingchorophyll concentration pro�le in o�shore sea water.Inverse Problems in Science and Engineering,16(6):705�715.

Dorigo, M., Maniezzo, V., and Colorni, A. (1996).The ant system: optimization by a colony ofcooperating agents.IEEE Transactions on Systems, Man, andCybernetics�Part B, 26(2):29�41.

Preto, A. J., CamposVelho, H. F., Becceneri, J. C.,Arai, N. N., Souto, R. P., and Stephany, S. (2004).


Final Remarks

A new regularization technique for an ant-colony basedinverse solver applied to a crystal growth problem.In Anais..., pages 147�153, Cincinnati. Inverse Problemin Engineering Seminar.

Souto, R. P., CamposVelho, H. F., Stephany, S., andSandri, S. (2006).Estimating vertical chlorophyll concentration ino�shore ocean water using a modi�ed ant colonysystem.Journal of Mathematical Modelling and Algorithms(JMMA).


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