Efficient τ-p domain waveform inversion, part 2: sensitivity to p component setting, source spacing and noiseWenyong Pan∗, Kristopher A. Innanen, Gary F. Margrave, CREWES, University of Calgary and Danping Cao, China University ofPetroleum (Huadong)

SUMMARY

Full Waveform Inversion (FWI) has been widely studied in recentyears but challenges remain. Issues include computational cost, slowconvergence rate, cycle skipping problem and so on. Aiming at theseobstacles, we develop the τ-p domain waveform inversion with a as-semblage of strategies and present the inversion results with differentscaling methods in a companion paper (Pan et al., 2014b). Generally,for per iteration in FWI, slant stacking over a set of p values should beperformed to balance the updates. To reduce the computational bur-den further, we illustrate slant update strategy with varied p values inwhich the model updates can be balanced as the iteration proceeds.The phase-encoding method in τ-p domain can reduce the computa-tional cost considerably, but unfortunately, it can also involve seriouscrosstalk artifacts especially for sparsely sampled sources. A furtherexamination of the anti-aliasing rules in the Random transform revealsthat the source spacing has a negative relationship with ray parame-ter spacing. Different ray parameters are responsible to illuminate thesubsurface layers with different dip angles. So, in this paper, we ana-lyze the influences of source spacing and ray parameter range on theτ-p domain FWI. In practical application, the presence of noise can in-crease the ill-posedness of the least-squares inversion problem. Hence,we also analyze the stability of τ-p domain FWI with noise data.

INTRODUCTION

Full Waveform Inversion (FWI) is a very important method to build thevelocity model for high resolution seismic imaging (Tarantola, 1984;Virieux and Operto, 2009; Margrave et al., 2011) by minimizing thedifference between the synthetic data and observed data. It has beenwidely observed that the gradient construction for a least-squares FWIobjective function corresponds to migrating the data residuals(Gaoet al., 2012). Reverse time migration (RTM) and FWI share the samealgorithmic structure (Shin et al., 2001) and the gradient calculation inFWI is formally identical to constructing a RTM image with a cross-correlation imaging condition. Margrave et al. (2011) shows that anyother wave-equation migration technology is also able to calculate anapproximate gradient.

The traditional shot-profile RTM can provide high quality images buttypically at a greater cost (Romero et al., 2000). The plane-wavemigration with densely distributed sources was firstly introduced inseismic imaging to reduce the computational cost (Morton and Ober,1998; Romero et al., 2000; Zhang et al., 2005; Stoffa et al., 2006; Shan,2008; Dai and Schuster, 2013). This strategy forms supergathers bysumming densely distributed individual shots into supergather, whichcan be viewed as a τ-p transform. We can also employ phase-encodingtechnique to construct the gradient or even Hessian approximations inFWI. Generally, slant stacking sufficient ray parameters should be per-formed to balance the amplitudes of the gradient for per iteration inFWI. In this research, to reduce the computational cost further, we in-troduce the slant update strategy (or iteration-dependent p componentsetting strategy), in which the model is updated using the slant gra-dient with one single ray parameter but the p value is changed as theiteration proceeds. Thus, the model updates can be balanced during agroup of iterations (Pan et al., 2014a, 2013).

Considering the τ-p transform as Fourier transform, Zhang et al. (2005)used the sampling rules in discrete Fourier transform to determine thenumber of ray parameters Np and ray parameter spacing ∆p under theassumptions that the shot sampling interval ∆xs is small enough andthe spread length is great enough. While in practical application, es-

pecially for 3D survey, the sources are always sparsely sampled inthe cross-line direction. When the sources are distributed densely andregularly in the whole acquisition geometry, the phase-encoding tech-nique shows a limited amount of noise (Liu et al., 2006). While whenthe shots are sparsely and irregularly sampled, the crosstalk noise ris-ing form the undesired interactions between the unrelated sources andreceivers wavefields becomes a problem. The plane-wave source canbe defined as pseudo plane-wave source or phase encoded sources inthis condition. The τ-p transform is also known as Radon transform,which can be regarded as a transformation of a line in Cartesian co-ordinate space to a point in polar coordinate space. According to theanti-aliasing rules in Radon transform, we find that the ray parameterspacing and source spacing are negatively related. This means thatfor densely distributed sources, sparse p sampling is enough to createhigh-quality image or gradient. While for sparsely sampled sources,large number of p values should be used to build the comparable one.Because different ray parameters are responsible to illuminate or up-date the subsurface layers with different dip angles. Proper ray param-eter range should be designed to balance the updates and guaranteethe convergence rate. In this paper, we analyze the influence of sourcespacing and ray parameter range on τ-p domain FWI. Furthermore,in the presence of noise, the ill-posedness of the least-squares inverseproblem will be increased. Therefore, in this paper, we study the effectof Gaussian noise on our proposed strategies and examine the stabilityof the τ-p domain FWI with noisy data.

This paper is organized as follows. Firstly, we introduce the generalprinciple of the τ-p domain FWI with slant update strategy and thetheories used to determine the ray parameter spacing and source spac-ing. Finally, we apply the proposed strategies on a modified Marmousimodel and discuss the influences of ray parameter range, source spac-ing and Gaussian noise on the τ-p domain FWI.

THEORY AND METHOD

Slant update with varied p valuesIt is known that the gradient construction in FWI is equivalent to amigration process. And the gradient is formally identical to a reversetime migration (RTM) image with cross-correlation imaging condi-tion. Because the traditional shot-profile method for gradient con-struction is extremely expensive. The gradient and Hessian approxi-mations in τ-p domain FWI are constructed using the phase-encodingtechnique, which was firstly introduced in seismic imaging for reduc-ing the computational burden (Morton and Ober, 1998; Romero et al.,2000; Zhang et al., 2005; Dai and Schuster, 2013). The linear phase-

Figure 1: Diagram for slant update with phase-encoding and varied pvalues.

encoding method is implemented by applying linear phase shifts (ortime delays in time domain) to the sources. The phase shift function

τ− p domain waveform inversion

γ(xs, p,ω) = ω p(xs−x0) is controlled by the ray parameter p (or slantparameter) and source’s position xs. So, a common-receiver gather canbe transformed into a single trace from a linear source wavefields byτ-p transform (Zhang et al., 2005):

d̃(rg,rs, p,ω) =∑

rs

d(rg,rs,ω)eiω p(rs−r0), (1)

where rs = (xs,ys = 0,zs = 0) and rg = (xg,yg = 0,zg = 0) mean thelocations of the sources and receivers. So, the slant gradient withslant parameter pg

j in τ-p domain can be obtained by applying a zero-lag cross-correlation between the plane-wave forward modeling wave-fields and backpropagated wavefields:

g̃(r, pgj ,ω) =

∑ω,rs ,r′s

ℜ{

ω2 |A (ω) |2 G(r,rs,ω)

× G∗(r,r′s,ω

)eiω pg

j (rs−r′s)},

(2)

where r = (x,y,z) indicates the subsurface position, j is the index ofthe slant parameter, A (ω) is the real function depending upon theangular frequency ω (Liu et al., 2006; Tang, 2009). Generally, in oneFWI iteration, slant stacking over a set of ray parameters should beperformed to balance the updates and reduce the crosstalk artifacts:

g̃(r,pg,ω) =∑

ω,rs ,r′s

Ngp∑

j=1

ℜ{

ω2 |A (ω) |2 G(r,rs,ω)

× G∗(r,r′s,ω

)eiω pg

j (rs−r′s)},

(3)

where pg means the ray parameter vector and Ngp indicates the maxi-

mum number of the ray parameter for constructing the gradient. Andif Ng

p is large enough, the crosstalk artifacts can be suppressed com-pletely and the phase encoded gradient (equation (3)) is identical tothe shot-profile gradient.

Generally, 2 ∗Ngp simulations should be performed in one FWI itera-

tion with phase-encoding technique. To reduce the computational costfurther, we propose an iteration-dependent ray parameter setting strat-egy, in which the slant gradient with single slant parameter is usedas a substitution of the phase-encoded gradient with slant stacking, asshown in Figure 1. Furthermore, the slant parameter is changed regu-larly in the whole iteration process. Thus, we only need 2 simulationsfor one FWI iteration and the biased slant update in each iteration canbe balanced through a group of iterations.

p component sampling issueOne issue for phase-encoding technique is how to determine the num-ber of ray parameters Np and ray parameter spacing ∆p, which hasbeen investigated by many researchers (Stork and Kapoor, 2004; Et-gen, 2005; Zhang et al., 2005; Gray, 2013). Equation 1 can be con-sidered as a Fourier transform, when replacing the wavenumber k withω p. So, the number of ray parameters Np and ray parameter spac-ing ∆p can be determined using the sampling rules in discrete Fouriertransform based on the assumptions that the shot sampling interval ∆xsis small enough and the spread length is great enough (Zhang et al.,2005).

Np ≥2Ns4xs f sinθmax

v, (4)

4p =2sinθmax

vNp, (5)

where Ns and4xs are the number of encoded sources and source spac-ing, Ns×4xs is the spread length. f and v are the frequency and topsurface velocity. θmax indicates the maximum take-off angle for a sym-metric slant stack. The p component sampling theory by (Zhang et al.,2005) is based on the assumption of infinite source coverage. So, Npand ∆p are independent of source spacing ∆xs.

When the sources are distributed densely and regularly in the wholeacquisition geometry, the phase-encoding technique shows a limited

amount of noise (Liu et al., 2006). While when the shots are sparselyand irregularly sampled, the crosstalk noise rising form the undesiredinteractions between the unrelated sources and receivers wavefieldsbecomes very serious. The τ-p transform is also known as Radontransform. So, according to the anti-aliasing rules in Radon transform:

∆xs ≤1

Np∆p f, (6)

We can notice that to create the artifacts-free image or gradient, thesource spacing ∆xs has a negative relationship with ray parameter spac-ing ∆p. So, for densely distributed sources, sparse p sampling can beused to create high-quality image or gradient. While for sparsely sam-pled source experiments, large number of p values should be used tobuild the comparable one.

NUMERICAL EXPERIMENTS

In the companion paper (Pan et al., 2014b), we compare the scalingstrategies for τ-p domain FWI. And in this paper, firstly, we comparethe effects for fixed ray parameter and varied ray parameter strategies.We also discuss the sensitivities of τ-p domain FWI on ray parameterrange, source spacing and Gaussian noise.

The τ-p domain FWI with proposed assemblage of strategies is prac-ticed and applied on one portion of the Marmousi Model. The modelis further modified by introducing one water layer with a thicknessof 70m and a velocity of 1500m/s. And the model has 180× 767grid cells with the same grid interval of 5m in horizontal and vertical.380 point sources are distributed on the surface with a source inter-val of 10m from 10m to 3800m and 765 receivers are deployed onthe surface with a receiver interval of 5m from 5m to 3825m. Thesource function is a Ricker wavelet with a dominant frequency of30Hz. The ray parameter range used for linear phase-encoding methodis [−0.3s/km,0.3s/km] with a step of 0.1s/km. The lowest frequencyband is [0Hz,5Hz] and the frequency band increases by 5Hz every 10iterations. Figure 2a and Figure 2b show the exact P-wave velocitymodel and initial velocity model.

Figure 2: (a) is true velocity model; (b) is the initial velocity model.

Figure 3a, b and c show the slant gradients when the slant parameter pis 0s/km, −0.2s/km and 0.2s/km respectively. It can be observed thatthe amplitudes of the slant gradients are biased and some artifacts arepresent. Figure 3d shows the phase encoded gradient by slant stack-ing the ray parameters from −0.2s/km to 0.2s/km with a spacing of0.1s/km. We can see that the amplitudes of the gradient are balancedand the artifacts are suppressed.

Figure 4 show the FWI results after 50 iterations with fixed ray pa-rameter and varied ray parameter in each iteration. Figure 4a, b and cshow the inversion results when ray parameter is fixed at p = 0s/km,p =−0.2s/km and p = 0.2s/km respectively for every iteration in τ-pdomain FWI. It can be seen that the updated model is biased with justone fixed ray parameter during iterations. Furthermore, some noise oranomalies are present. Figure 4d shows the inversion result when theray parameter is changed from −0.3s/km to 0.3s/km with a spacingof 0.1s/km regularly during the whole iteration process. We can seethat the model can be updated in balance and the updated model ismuch more smooth. Figure 5 show the relative least-squares errors for

τ− p domain waveform inversion

Figure 3: (a), (b) and (c) show the slant gradient with single slantparameter p = 0s/km, p = −0.2s/km and p = 0.2s/km respectively.(d) shows the phase encoded gradient with ray parameter ranging from−0.3s/km to 0.3s/km with a step of 0.1s/km.

different ray parameter arrangements. Varying ray parameter duringiterations can provide a higher quality inversion result after 50 itera-tions, as indicated by the black-solid line in Figure 5.

Figure 4: FWI results after 50 iterations with single ray parameterin each FWI iteration. (a) ray parameter is fixed at p = 0s/km; (b)ray parameter is fixed at p = −0.2s/km; (c) ray parameter is fixed atp= 0.2s/km; (d) ray parameter varies from−0.3s/km to 0.3s/km witha step of 0.1s/km.

Figure 5: Relative least-squares errors comparison for different rayparameter settings. The gray-dash line, black-dash line and gray-solidline indicate the normalized least-squares errors when the ray param-eter p is fixed at −0.2s/km, 0s/km and 0.2s/km respectively. And theblack-solid line indicates the least-squares errors when varying the rayparameter from −0.3s/km to 0.3s/km with a spacing of 0.1s/km.

Because the ray parameter is controlled by the take-off angle and topsurface velocity, generally different ray parameters are responsible toilluminate the subsurface layers with different dip angles. We com-pare the inversion results after 50 iterations with different ray param-eter ranges as shown in Figure 6. Figure 6a shows the inversion resultwhen the ray parameter ranges from −0.1s/km to 0.1s/km with a rayparameter spacing of 0.05s/km. We can see that because the ray pa-rameter range is too small, the subsurface layers with dip angles cannotbe updated in balance. Figure 6b and c show the inverted results whenthe ray parameter ranges are [−0.5s/km,0.5s/km] with a spacing of0.1s/km and [−0.6s/km,0.6s/km] with a spacing of 0.2s/km. It canbe seen that if the ray parameter range is too large, the convergencerate will be decreased.

Figure 6: Sensitivity to the ray parameter range. (a), (b) and(c) show the inversion results when the ray parameter ranges are[−0.1s/km,0.1s/km], [−0.5s/km,0.5s/km] and [−0.6s/km,0.6s/km]respectively.

When the sources are distributed densely and regularly in the wholeacquisition geometry, the phase-encoding technique shows a limitedamount of noise. While if the source arrangement is sparse and ir-regular, the crosstalk artifacts become very obvious. As indicated by

τ− p domain waveform inversion

equation (6), the source spacing ∆xs has a negative relationship withray parameter spacing ∆p for anti-aliasing rule. We also discuss the

Figure 7: Sensitivity to the source spacing. (a) and (c) show the in-version results when the source spacing ∆xs is 100m and 50m respec-tively; (b) shows the inversion result when the source spacing is also100m but slant stacking ray parameters from −0.2s/km to 0.2s/kmwith a spacing of 0.1s/km in each FWI iteration.

Figure 8: Well data comparison at 0.5km (a) and 3km (b) for differ-ent source spacings. The black-bold-solid and black-thin-solid linesindicate the true velocity and initial velocity respectively. And thegray-bold-dash lines, black-bold-dash lines and gray-bold-solid linesindicate the inversion results when source spacing ∆xs is 100m, 50mand 10m respectively.

influence of source spacing on the τ-p domain FWI with slant updatestrategy. Figure 7a, c and Figure 4d show the inversion results whenthe source spacing ∆xs is 100m, 50m and 10m respectively. For Figure7a, when the sources are sparsely distributed (∆xs = 100m), we cannote that the inversion result is contaminated by the artifacts seriously,especially for the shallow layers. With decreasing the source spac-ing, as shown in Figure 7c and Figure 4d, the crosstalk noise becomesweaker and the convergence rate also increase. Figure 8a and b com-pare the well logs at 0.5km and 3.0km for different source spacings.The gray-bold-dash lines, black-bold-dash lines and gray-bold-solidlines indicate the inversion results when source spacing ∆xs is 100m,50m and 10m. We can recognize that the with decreasing the sourcespacing, the inversion result approaches the true velocity model (thebold-black-solid lines) better, especially for the deep layers. Figure 7bis the inversion result when the source spacing is also 100m. But weapply a slant stacking over ray parameter from −0.2s/km to 0.2s/kmwith a spacing of 0.1s/km in each iteration. Compared to Figure 7a,

Figure 9: The inversion results when SNR = 5 (a), SNR = 4 (b) andSNR = 3 (c) respectively.

the artifacts are suppressed and inversion result is de-blurred to someextent. So, the τ-p domain FWI with slant update strategy is sensi-tive to the source spacing. For sparsely sampled experiments, slantstacking densely sampled ray parameters should be performed in eachiteration to create high quality inversion result but at a greater cost.

In practical application, the seismic data is contaminated by all kindsof noise seriously and the ill-posedness of the least-squares inverseproblem will be increased with noisy data. So, in this paper, we studythe effects of Gaussian noise on our proposed strategies and exam-ine the stability of the τ-p domain FWI. Figure 9a, b and c show theinversion results when the signal to noise ratio (SNR) is 5, 4 and 3 re-spectively. It can be seen that with increasing the strength of Gaussiannoise, the inverted results become worse and the convergence rate isalso decreased. Some thin layers in the shallow parts of the velocitymodel cannot be estimated. But we can still recognize the main geo-logical structures in the deep parts of the velocity model. So, the τ-pdomain FWI is relative stable in the presence of noise.

CONCLUSION

In this paper, we introduce the slant update strategy for τ-p domainFWI and analyze the influences of ray parameter range, source spac-ing and noise. It is concluded that proper ray parameter range shouldbe determined for balancing the updates and guarantee the conver-gence rate. The τ-p domain FWI is sensitive to the source spacing.For sparsely distributed sources, slant stacking densely sampled rayparameters should be performed in each iteration to create high qual-ity inversion result but at a greater cost. When the Gaussian noise ispresent, the τ-p domain FWI is relative stable to recover the geologicalstructures.

ACKNOWLEDGMENTS

This research was supported by the Consortium for Research in ElasticWave Exploration Seismology (CREWES). Thanks also to Sam Ka-plan (Chevron), Sheng Xu (CGG, now Staoil), Danping Cao, WenyuanLiao, Vladimir Zubov, Raul Cova and Babatunde Arenrin for theirvaluable discussions.