W009
Application of VTI Waveform Inversion with
Regularization and Preconditioning to Real 3D
Data
C. Wang* (ION Geophysical), D. Yingst (ION Geophysical), R. Bloor (ION Geophysical) & J. Leveille (ION Geophysical)
SUMMARY
We describe the methodology and various practical strategies for acoustic anisotropic (VTI) waveform inversion with total variation (TV) regularization and preconditioning. Our time domain approach adopts a joint inversion for both velocity model and source delay time. TV
or edge preserving regularization aims at preserving both the blocky and the smooth characteristics of the velocity model. The preconditioning strategies (spectral shaping etc.) are useful in order to increase the convergence rate. Finally there is an example for a 3D VSO OBC data set.
Introduction
Waveform inversion (WFI) is a seismic method for deriving high-fidelity velocity models for imaging by minimizing some misfit function of the difference between the field and modeled data. It was first intro-duced by Lailly, Tarantola, and Mora using a gradient-based iterative algorithm (Lailly, 1983; Tarantola, 1984, 1987; Mora, 1986). Since these pioneering efforts many researchers have attempted to use various strategies and computational schemes to make WFI implemented either in the time or the frequency domain a processing tool for real data sets (e.g. Sirgue and Pratt, 2004; Shin and Min, 2006; Vigh and Starr, 2006; Operto et al., 2007).
Our WFI has been implemented using two different approaches. One is a time domain approach and the other is a single frequency approach. They were both successfully applied to real 3D data sets (Yingst et al., 2011b). Time domain implementations are straightforward and relatively fast. Since the gradient is quite sensitive to source time shifts, an accurate source signature delay time is necessary to match the observed and predicted data. In this paper, we focus on the time domain approach and develop a joint inversion for recovering both velocity model and source delay time. The single frequency approach was introduced by Sirgue (Sirgue, 2003). The idea is to propagate the wave in time and accumulate the Fourier transform for one or a small number of frequencies. We also use a complex projection technique which allows for both amplitude and phase matching (Yingst et al., 2011b).
WFI is a highly nonlinear, ill-posed problem. As such regularization techniques from optimization theory are beneficial for its solution. Regularization involves introducing additional information in order to solve ill-posed inverse problems. This information is usually in the form of a weighted penalty function, such as restriction of smoothness or sharpness of model parameters. The geophysics of the problem leads to appropriate constraints. In this paper, total variation (TV) regularization is applied. It is a popular method based on the`1-norm of the model derivatives. The TV method provides a more useful recovery of velocity profiles from seismic data including a better delineation of discontinuities by avoiding excessive smoothing (Vogel, 2002).
Another concern of WFI is its high computational cost. An iterative scheme based on a preconditioned nonlinear conjugate gradient method is employed in this paper. Preconditioners are useful in iterative methods since the rate of convergence can often be dramatically decreased as a result of appropriate preconditioning. We apply muting, offset-weighting, and bandpass filter to precondition the data which lets a user focus the inversion on the most suitable parts of the data. We also apply a spectral filter which helps shaping the spectrum of the seismic data and the source wavelet (Lazaratos et al., 2011). This shaping is designed such that the weight of the low-frequency part of the spectrum can be increased. In order to precondition the gradient, a commonly used technique is layer-stripping and/or gradient masking. Other important practical strategies for WFI have also been discussed (Yingst et al., 2011a). For instance, WFI suffers from cycle skipping when the bandwidth is too broad. This can be mitigated using multi-scale inversion, gradually increasing the frequency bandwidth (Bunks et al., 1995).
This paper first presents the time domain implementation with regularization and preconditioning. Then it presents time domain acoustic anisotropic (VTI) WFI results for a 3D marine data set.
Method and Theory
We modify the traditional misfit function (Tarantola, 1987) slightly and add a TV as a penalty term: min
m J[m] = kT(d0−ρd)k
2
2+λR, (1)
whered0=d0(xr,t;xs)is the observed seismic data set andd=d(xr,t;xs)is the predicted data set for
the velocity modelmat source and receiver locations,xs andxr. Tis a data preconditioner andρ is a
normalization scalar (ρ=hd,d0i/kdk2). Note that the preconditionerTcan depend on frequency band-width, spectral filter, offset, and/or time windows.λis a weighting scalar andRis the TV regularization
constraints defined as
R(m) =
Z q
|∇(m−m0)|2+β2dx, (2) whereβ is a small value to avoid singularity of the gradient ofRandm0 is the initial velocity model, usually generated from tomographic inversion.
The misfit function is minimized iteratively by computing the gradient (Tarantola, 1984) at successive velocity models. It uses the gradient at an initial point for an initial direction estimate and updates that direction using the Polak-Ribière implementation of the nonlinear conjugate gradient (CG) method. The line search along the directions uses the BB formula (Barzilai and Borwein, 1988) for an initial estimate of the steplength. BB formula does not require extra forward modeling to evaluate the misfit function and it provides effective initial guess of the steplength for TV regularized problems (Li, 2011). It then applies the backtracking linesearch method to update the steplength.
In this case, the gradient at thek-th iteration is given by gk(x) = − 2 m3k(x)
∑
xs∑
t ∂2p ∂t2(x,t;xs)r +(x,t;xs) − ∇· p ∇(mk(x)−m0(x)) |∇(mk(x)−m0(x))|2+β2 ! , (3)wherem0is the initial velocity model,mkis the inverted velocity model afterk-th iteration,pis the
wave-field modeled frommk, andr+is the wavefield obtained by back-propagating the residualT(d0−ρd).
Convergence can be accelerated using gradient preconditioning. The current gradient preconditioning normalizes by the amplitude of the forward propagated wave:
gkp(x) =p gk(x)
∑xs∑tp(x,t;xs)
2+κ2, (4)
whereκis a whitening factor to avoid zeros in the forward modeled data. We also apply layer-stripping and/or masking as another gradient preconditioning tool to accelerate the convergence.
Notice that an accurate estimation of source signature delay time is critical to match the phase between the observed and predicted data. Since the gradient is quite sensitive to source time shifts, we adopt another step to estimate a source delay time before each iteration in order to overcome this problem. This approach is to find an optimal source time delayτ that maximize the cross-correlation value of the observed and predicted data, i.e.
min τ Js1[τ] = kd0τ−dk22 (5) ⇔ max τ Js2[τ] = d0?d, (6)
whered0τ is the observed datad0shifted by a source time delayτ. Examples
We present an application of 3D acoustic VTI time domain WFI with TV regularization and precondi-tioning to 3D marine data. This deep water VSO OBC survey is located in the Green Canyon area of the Gulf of Mexico. The acquisition area was 160 km2. Offsets range from 0 to 6400m. The lowest frequency in the observed data is about 5 Hz. The initial velocity model for WFI, Figure 1(a), came from an anisotropic VTI tomographic inversion.
We applied layer-stripping, muting, and offset-weighting techniques in order to obtain the optimal up-date. The inversion proceeded with spectral shaping applied to the field data and the source wavelet for
(a) Initial velocity (b) Inverted velocity
Figure 13D velocity model
(a) Migrated gather with initial velocity (b) Migrated gather with inverted velocity
Figure 23D migrated gathers
accelerating convergence. Source delay time and offset weights were automatically computed. We also applied TV regularization in order to preserve the edge of the salt body or any other discontinuity. The inverted velocity model 1(b) after 5 iterations shows a reasonable shallow update at a depth of about 2000mincluding some detailed structures above the salt. The location of the salt body interface is well preserved by TV regularization. The deeper portion of the velocity model was not significantly updated because of limitations imposed by the limited maximum offset in the seismic data. By improving the velocity, the stack gathers 2(b) after the WFI iterations show improvement in flatness compared to the gathers using the initial velocity model 2(a). The structures above the salt were significantly improved by comparing the initial stack 3(a) with the updated stack 3(b), also by comparing the initial depth slice at 1950m4(a) with the updated one 4(b).
Conclusions
In this paper, we presented the methodology, strategies, and 3D data examples for acoustic VTI WFI with TV regularization and preconditioning. Our time domain approach adopts a joint inversion for both velocity model and source delay time. From the results, we showed that TV or edge preserving regularization aimed at preserving both the blocky and the smooth characteristics of the velocity model. The preconditioning strategies (spectral shaping etc.) are useful in order to increase the convergence rate.
Acknowledgements
(a) Migrated stack with initial velocity (b) Migrated stack with inverted velocity
Figure 33D stacks
(a) Depth slice with initial velocity (b) Depth slice with inverted velocity
Figure 4Depth slices at 1950 m
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