Skeletonized wave-equation Qs tomography using surface-wavesJing Li∗, Gaurav Dutta†, Gerard T. Schuster∗.∗ Department of Earth Science and Engineering, King Abdullah University of Science and Technology,Thuwal, Saudi Arabia, 23955-6900.† CGG, Houston, TX, USA.

SUMMARY

We present a skeletonized inversion method that invertssurface-wave data for the Qs quality factor. Similar to the in-version of dispersion curves for the S-wave velocity model, thecomplicated surface-wave arrivals are skeletonized as simplerdata, namely the amplitude spectra of the windowed Rayleigh-wave arrivals. The optimal Qs model is then found that min-imizes the difference in the peak frequencies of the predictedand observed Rayleigh wave arrivals using a gradient-basedwave-equation optimization method. Solutions to the vis-coelastic wave-equation are used to compute the predictedRayleigh-wave arrivals and the misfit gradient at every itera-tion. This procedure, denoted as wave-equation Qs tomogra-phy (WQs), does not require the assumption of a layered modeland tends to have fast and robust convergence compared to Qfull waveform inversion (Q-FWI). Numerical examples withsynthetic and field data demonstrate that the WQs method canaccurately invert for a smoothed approximation to the subsur-face Qs distribution as long as the Vs model is known withsufficient accuracy.

INTRODUCTION

Surface waves play an important role in the characterizationof the near surface for earthquake, engineering and environ-mental studies. Inverting and imaging surface waves can be aneffective means for characterizing the subsurface at differentscales (Dasgupta and Clark, 1998; Xia et al., 1999; Lin et al.,2008; Li and Hanafy, 2016), and can be important for site re-sponse and seismic hazard studies. Since the surface wavesare sensitive to the near-surface elastic properties, estimationof the surface-wave velocity Vs and quality factor Qs are ofsignificant interest in exploration and earthquake seismology(Xia, 2014).

The near-surface S-wave velocity model is usually estimatedfrom the dispersion curves of the recorded surface-wave ar-rivals (Park et al., 1998). However, in a real dissipative media,the propagation of the surface waves is strongly influenced byelastic damping in the near-surface that results in increasingamplitude loss and attenuation of high frequencies with dis-tance travelled. As a result, the dispersion curves are alsosensitive to the near-surface Qs variations. He et al. (2015)showed that the phase velocity of the fundamental mode of theRayleigh waves increases with decreasing values of Qs. Thus,inverting dispersion curves for the S-wave velocity withouttaking into consideration the effect of Qs can lead to erroneousestimates of the near-surface S-velocity distribution. Grooset al. (2014) showed that the S-wave velocity tomograms ob-

tained from elastic full waveform inversion (FWI) of Rayleighwaves have lower resolution when compared to the tomogramsobtained by viscoelastic FWI when the near-surface is stronglyanelastic. The increased resolution in S-wave tomograms ob-tained by taking into account the effect of Qs can be usefulin delineating near-surface faults or local anomalies (Pinsonet al., 2008).

In this paper, we present a novel wave-equation Qs tomogra-phy method (WQs) that inverts skeletonized surface waves forthe quality factor Qs. The skeletonized data are the frequencyshifts of the observed and predicted surface-wave’s spectralpeaks. The Qs method is similar to the wave-equationQp

inversion developed by Dutta (2016) and Dutta and Schuster(2016), except it inverts for Qs from surface waves rather thanQp from body-wave arrivals. The Qs model is then found thatminimizes the squared differences between the predicted andobserved peak-frequency shifts associated with the Rayleighwave arrivals. For this method, the isotropic viscoelastic waveequation based on the standard linear solid model (Robertssonet al., 1994) is used to generate the predicted Rayleigh-wavearrivals. The adjoint viscoelastic wave-equation is then used tobackpropagate the residual traces that are obtained by weight-ing the observed Rayleigh-wave arrivals with their correspond-ing frequency shifts. The gradient for WQs can be interpretedas the zero-lag cross-correlation between the forward propa-gated source wavefield and the backprojected weighted resid-ual wavefield. Numerical examples on synthetic and field datavalidate the effectiveness of the proposed approach.

THEORY OF WAVE-EQUATION Q S SKELETONIZEDINVERSION

The theory of WQs is derived in a manner that is similar tothat for wave-equation traveltime inversion (Luo and Schuster,1991) and surface-wave dispersion inversion (Li and Schus-ter, 2016; Li et al., 2017b). These steps include: (1) define afrequency-shift misfit function, (2) define a connective func-tion that connects the frequency-shift residual of the Rayleigh-wave arrivals with the particle velocity seismograms, and (3)derive the gradient of the misfit function with respect to Qs us-ing the isotropic viscoelastic wave equation and the connectivefunction in step (2).

In our analysis, we assume that the wave propagation hon-ors the 2D isotropic viscoelastic equations of motion based onthe standard linear solid (SLS) mechanism (Robertsson et al.,

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WQs Tomography

1994):

ρ∂u∂ t

=∂σxx

∂x+

∂σxz

∂ z,

ρ∂w∂ t

=∂σzx

∂x+

∂σzz

∂ z,

∂σxx

∂ t= π

τ pε

τσ(

∂u∂x

+∂w∂ z

)−2µτs

ετσ

∂w∂ z

+ rxx +Sxx,

∂σzz

∂ t= π

τ pε

τσ(

∂u∂x

+∂w∂ z

)−2µτs

ετσ

∂u∂x

+ rzz +Szz,

∂σxz

∂ t= µ

τsε

τσ(

∂w∂x

+∂u∂ z

)+ rxz,

∂ rxx

∂ t=−

1τσ

(rxx +π(τ p

ετσ

−1)(∂u∂x

+∂w∂ z

)−2µ(τs

ετσ

−1)∂w∂ z

),

∂ rzz

∂ t=−

1τσ

(rzz +π(τ p

ετσ

−1)(∂u∂x

+∂w∂ z

)−2µ(τs

ετσ

−1)∂u∂x

),

∂ rxz

∂ t=−

1τσ

(rxz +µ(τs

ετσ

−1)(∂u∂ z

+∂w∂x

)). (1)

Here,u andw are the horizontal- and vertical-particle-velocitycomponents,respectively,σi j denotes thei j-th component ofthe symmetric stress tensor,ri j is the memory variable,τ p

ε andτs

ε are the strain relaxation times for the P and SV waves, re-spectively, andτσ is the stress relaxation time for both the Pand SV waves.Sxx andSzz denote the source wavelets for thespecial case of an explosive source. The variableπ = λ +2µ isrelated to the Lame parametersλ andµ whereas the stress andstrain relaxation times are related to the quality factors Qp andQs and the reference angular frequencyω as (Carcione et al.,1988):

τσ =

√

1+ 1Q2

p−

1Qp

ω,

τsε =

1+ωQsτσωQs −ω2τσ

,

τ pε =

1ω2τσ

. (2)

Equation 1 is solved for a point source at each shot point byan O(2,4) time-space domain staggered grid finite-differencealgorithm. In order to generate surface waves, an explicit free-surface boundary condition is implemented by using the mir-roring technique proposed by Levander (1988).

It is possible to approximate a frequency independent seis-mic quality factor in a limited frequency range. We define thefrequency-independent parameterη as:

η =τε s

τσ−1=

1+(√

1+ 1Q2

p−

1Qp

)2

(√

1+ 1Q2

p−

1Qp

)[Qs − (√

1+ 1Q2

p−

1Qp

)].

(3)

For the parametrization in WQs, η is used because it is quitesensitive to small changes in Qs. The relaxation ratioη is in-verted at each iteration and the updates inη are then mappedto Qs using equation 3.

Misfit FunctionWe denote the Rayleigh-wave arrivals that are extracted from

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

Frequency (Hz)

Am

plit

ud

e

Frequency Shift

Recorded TracePredicted Trace

0 0.1 0.2 0.3 0.4 0.5−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Time (s)

Am

plit

ude

Single Surface Wave Trace

Recorded TracePredicted Trace

b) c)

Tim

e (

s)

0

0.05

0.1

0.15

0.25

10 20 30 40 50 60

Offset (m)

a) Windowed Rayleigh Wave Arrivals

Red Line: Normal

Black Line: With Qs

Figure 1: (a) A common shot gather (CSG) comparing theRayleigh-wave arrivals with and without Qs. The blue dashedline shows the window used to extract these arrivals for WQs,(b) comparison between a predicted and an observed surface-wave arrival, and (c) comparison between their amplitudespectra.

the recorded data asD f obsg(g, t;s)obs for a vertical-component

point source ats and a vertical-component receiver atg (theblack curve in Figure 1a). The predicted Rayleigh-wave ar-rivals for the same source-receiver pair are denoted byD f pred

g(g, t;s)pred

(the red curve in Figure 1a). The peak frequency of the ob-served and predicted spectra are denoted asf obs

g and f predg , re-

spectively. Here,f predg = f and f obs

g = f − f1, wheref is the peakfrequency of the event andf1 is the shift between the peak fre-quencies of the predicted and the observed traces due to Qs.A comparison between the windowed observed and predictedRayleigh-wave arrivals for a given Qs model is shown in Fig-ure 1b. The amplitude spectra of these arrivals are plotted inFigure 1c, where it is evident that the observed spectrum has alower peak frequency than the predicted spectrum. The peakfrequencies in these spectra are denoted as the skeletonizedsurface-wave data.

The goal of WQs is to find the attenuation model Qs =F(η(x))−1

so that f predg ≈ f obs

g for all the traces. In our case, we use the

frequency-shift residual∆ f = f predg − f obs

g to form the skele-tonized misfit function:

ε =12

∑

s

∑

g

∆ f (g,s)2. (4)

The gradientγ(x) is given by

γ(x) =∂ε

∂η(x)=

∑

s

∑

g

∂∆ f∂η(x)

∆ f (g,s). (5)

The η model is updated at each iteration using the iterativesteepest descent method:

η(k+1) = η(k)−α

γ(x)=gradient︷ ︸︸ ︷∑

s

∑

g

∂∆ f∂η(x)

∆ f (g,s), (6)

whereα is the step-length at thek-th iteration (Nocedal andWright, 1999). The update for the relaxation ratioη is thenmapped to Qs using equation 3.

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WQs Tomography

Connective FunctionTo find an analytic expression for the Frechet derivative∂ ∆ f

∂ η(x)in equation 6, we define the connective functionΦ(g, t;s) thatconnects the change in peak frequency of an arrival with theobserved and the predicted Rayleigh-wave arrivals in Figure 1bas:

Φ f1(g, t;s) =∫

D f (g, t;s)D f− f1(g, t;s)obsdt. (7)

We seek an optimal Qs model that minimizes the peak-frequencyshift between an observed and a predicted trace. For an ac-curate background Qs model, the predicted and the observedarrivals will have the same peak frequency. So, we definef1 = ∆ f to be the frequency shift associated with the actualbackground Qs model. If ∆ f = 0, it indicates that the cor-rect background Qs model has been found and the transmis-sion surface-wave arrivals in the predicted and observed traceshave the same peak frequencies. The derivative ofΦ f1 with re-spect tof1 should be zero at the frequency-shift valuef1 = ∆ f ,i.e.,

Φ∆ f (∆ f ,η(x)) = [∂Φ f1(g, t;s)

∂ f1] f1=∆ f

=

∫

D f (g, t;s)D f−∆ f (g, t;s)obsdt = 0, (8)

whereD f−∆ f (g, t;s)obs = [∂D f− f1(g, t;s)obs/∂ f1] f1=∆ f . Note

that∆ f = 0 if the predicted Qs model is the actual Qs model.Equation 8 is the connective function (Luo and Schuster, 1991)that connects the skeletonized data, i.e., the frequency-shiftresiduals of the Rayleigh-wave arrivals, with the particle-velocityseismograms. Such a connective function is required becausethere is no wave equation that relates the skeletonized data toa single type of model parameter (Dutta and Schuster, 2016).

Using the implicit function theorem and the connective func-tion in equation 8, the Frechet derivative with respect to therelaxation ratioη(x) can be expressed as

∂∆ f∂η(x)

=−

∂ Φ∂η(x)

/∂ Φ∂∆ f

, (9)

where the numerator on the right-hand side is given by

∂ Φ∆ f

∂η(x)=

∫ ∂D f (g, t;s)∂η(x)

D f−∆ f (g, t;s)obsdt, (10)

and the denominator by

∂ Φ∆ f

∂∆ f=

∫

D f (g, t;s) D f−∆ f (g, t;s)obsdt = K2. (11)

Using equation 9, the gradient in equation 5 can be written as

γ(x) =∂ε

∂η(x)=−

∑

s

∑

g

∂ Φ∂ η(x)

∂ Φ∂ ∆ f

∆ f (g,s). (12)

Then, we use the adjoint-state method to derive the Frechetderivative. Combining equations 10-12, the gradient for WQs

20 40 60 80 100 120 140

5

10

15

20

25

30650

700

750

800

850

900

De

pth

(m

)

Distance (m)

(m/s)True Vs Model

20 40 60 80 100 120 140

5

10

15

20

25

30650

700

750

800

850

900

De

pth

(m

)

Distance (m)

(m/s)

(a)

Input Vs Model Used for WQs(c) Starting Qs Model(d)

True Qs Model(b)

20 40 60 80 100 120 140

5

10

15

20

25

30

De

pth

(m

)

Distance (m)

20 40 60 80 100 120 140

5

10

15

20

25

30

De

pth

(m

)

Distance (m)

Qs

Qs

200

150

100

50

10

200

150

100

50

10

De

pth

(m

)

WQs Tomogram(e)

20 40 60 80 100 120 140

5

10

15

20

25

30

Distance (m)

Qs

200

150

100

50

10

De

pth

(m

)

Q-FWI Tomogram(f)

20 40 60 80 100 120 140

5

10

15

20

25

30

Distance (m)

Qs

200

150

100

50

10

Figure 2: (a) True velocity and (b) Qs models used for generat-ing the observed data, (c) background Vs model used for WQsinversion, (d) starting Qs model, (e) the Qs tomogram com-puted by WQs, and (f) the Qs tomogram computed by Q-FWI.

can be written as

γ(x) =−

∑

s

∑

g

1K2

∂ Φ∂η(x)

∆ f ,

=−

∑

s

∫

(2µ∂w∂ z

σxx +2µ∂u∂x

σzz − (µ∂u∂ z

+µ∂w∂x

)σxz

−2µE∂w∂ z

rxx −2µE∂u∂x

rzz +µE(∂w∂x

+∂u∂ z

)rxz) dt.

(13)

Here (u, w, σxx, σzz, σxz, rxx, rzz, rxz) are the adjoint-state vari-ables of(u,w,σxx,σzz,σxz,rxx,rzz,rxz), E = −1

τσandK2 is de-

fined in equation 11. It can be shown that the residual trace,fw, that is backpropagated at every iteration is given by:

fw =1

K2D(g, t;s)obs∆ f (g,s). (14)

NUMERICAL TESTS

We now compare the performance of WQs against that of Q-FWI for the near-surface Vs and Qs models shown in Fig-ure 2. A smooth version of the true S-wave velocity model,shown in Figure 2c, is used as the background velocity modelfor WQs and Q-FWI. The grid spacing and time sampling in-tervals for the 2D viscoelastic finite-difference algorithm are1 m and 0.025 ms, respectively, and the center frequency ofthe source wavelet is 35 Hz. The observed data are generatedby 40 shots evenly distributed on the surface and the data arerecorded by 70 receivers every 2 m on the surface. The initialQs model is a homogeneous half space (Figure 2d).

The Qs tomograms from the WQs and Q-FWI methods after 21iterations are shown in Figures 2e and 2f, respectively. Figure

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WQs Tomography

5 10 15 20 25 30

10

20

30

40

50

60 32

34

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5 10 15 20 25 30

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34

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52

5 10 15 20 25 30

10

20

30

40

50

60 −4

−3

−2

−1

0

1

2

3

4

(a) Peak Frequencies in Observed Data

Re

ce

ive

r (#

)

Source (#)

Hz (b) Peak Frequencies after WQs Hz

(c) Difference between (a) and (b)

Re

ce

ive

r (#

)

Source (#)

Hz

Re

ce

ive

r (#

)

Source (#)

Figure 3: The peak-frequencies for different source-receiverpairs in (a) the observed data, (b) the predicted data after WQs,and (c) their differences.

400

450

500

550

600

650

700

750

800

10

20

30

401600

1800

2000

2200

2400

2600

10

20

30

40

(b) S-velocity Tomogram (m/s)

Depth

(m

)

(a) P-velocity Tomogram (m/s)

Depth

(m

)

Distance (m)

100 200 300 400 500

Distance (m)

100 200 300 400 500

Figure 4: (a) P-wave tomogram with ray tracing tomography,and (b) S-wave velocity tomogram by wave equation disper-sion inversion (Li et al., 2017a).

3 reveals that the predicted peak frequencies from WQs arevery close to the actual ones for many source-receiver pairs.

The WQs method is now tested on a near-surface field data setrecorded over the Qademah fault. There are 60 shots and 60 re-ceivers on the surface placed at 10 m intervals. The first-arrivaltraveltimes are picked in all the CSGs and inverted using ray-based traveltime tomography to obtain the P-wave velocitymodel shown in Figure 4a. The S-wave tomogram is shownin Figure 4b, which is obtained using the wave-equation sur-face wave dispersion method (Li et al., 2017b). These velocitymodels are used as the background velocity models for WQs.For WQs, the starting Qs model is taken to be homogeneouswith Qs = 1000 and the inverted Qs tomogram is shown inFigure 5a. There is reasonable geological agreement betweenthe S-wave velocity model in Figure 4b and the Qs tomogramin Figure 5a. The high attenuation regions in the Qs tomogram(low Qs values) correspond to the low S-wave velocity regions.Previous work by Zhang et al. (2015) demonstrated that areaswith high Vp/Vs ratios tend to have low Q values (or high at-tenuation). We calculate the Vp/Vs ratio using the tomogramsin Figure 4 and the ratio is shown in Figure 5b. It can be seenfrom this figure that areas with high Vp/Vs ratio have low Qs

values.

As a final sanity check, the inverted Qs tomogram is compared

Q

1.6

1.8

2.0

2.2

2.4

2.6

260

210

160

110

60

10

Depth

(m

)

10

20

30

40

Depth

(m

)

10

20

30

40

a) Qs Tomogram

b) Vp/Vs Ratio

Distance (m)

Distance (m)

100 200 300 400 500

100 200 300 400 500

Tim

e (

s)

0.1

0.2

0.3

0.4

c) Common Offset Gather (Offset=50 m)

Distance (m)

100 200 300 400 500

Figure 5: (a) WQs tomogram, (b) ratio (Vp/Vs) computed froma) and b), and (c) common offset gather (offset=50 m) profileafter data processing.

to a common offset gather (COG) profile (Hanafy et al., 2015).Figure 5c shows a COG profile using the processed data for anoffset of 50 m. The black dashed line in this figure showsthe location of the fault which is between 250-300 m. Thelocations of the Qs anomalies in the WQs tomogram and thelow-velocity area in the S-wave tomogram are consistent withthe location of the fault, as indicated in the COG profile.

CONCLUSIONS

We presented a skeletonized surface-wave wave-equation Qs

inversion method, where a Qs model is found that minimizesthe sum of the squared difference squared differences in thepeak-frequencies of the observed and the predicted surface-wave arrivals. The gradient for WQs is obtained by a zero-lagcross-correlation between the forward propagated viscoelas-tic source wavefield and the weighted backprojected residualsthat are obtained by weighting the observed particle velocityseismograms with the residual frequency shifts. This methoddoes not require a layered-medium assumption used in conven-tional Qs estimation techniques and also does not suffer fromthe limitations of ray-tracing based Q tomography methods orQ-FWI.

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EDITED REFERENCES

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SEG Technical Program Expanded Abstracts have been copyedited so that references provided with the online

metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web.

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