Tomographic full waveform inversion: Practical and computationally feasible approach

Introduction

Seismic velocity-analysis methods can be divided into two major groups. First, there are techniques that aim at minimizing the misfit in the data domain, such as full-waveform inversion (Pratt, 1999; Luo and Schuster, 1991; Tarantola, 1984). Second, there are other techniques that aim at improving the quality in the image domain, such as migration velocity analysis (MVA) (Biondi and Sava, 1999; Symes and Carazzone, 1991; Shen, 2004; Zhang et al., 2012). These techniques try to measure the quality of the image in several ways and then invert the estimated image perturbation using a linearized wave-equation operator.

There are several advantages to minimizing the residual in the image-space, such as global convergence, increasing the signal-to-noise ratio, and decreasing the complexity of the data (Tang et al., 2008). However, a common drawback in doing velocity analysis in the image domain is that only the transmission effect of the velocity are used. This results in a loss of vertical resolution in the estimated model updates. On the other hand, Full-Waveform Inversion (FWI) does not have that problem, since it utilizes the information from both the forward-scattered and back-scattered wavefields. This results in higher resolution in the model estimates. Moreover, the data misfit is computed in the data spaces directly without the need to go to another domain or to separate the data into several components. This direct computation of the errors results in a relatively simple relationship between the data residuals and the model updates. However, FWI has the disadvantage that its objective function is far from being smooth and convex; it requires the starting model to be very close to the true model to avoid converging to local minima.

The conventional solution is to invert first for the velocity model using MVA techniques and then use the output as the initial model for FWI. However, this practice might not work if the results of MVA are not accurate enough for FWI to start. This could be a result of the larger null space that forward-scattered wavefields do not constrain. Moreover, the convergence rate of the MVA techniques is going to be sub-optimal, since they do not use all of the information in the data.

In a companion paper (Biondi and Almomin, 2012), we present a generalize framework called Tomographic Full Waveform Inversion (TFWI) that combines both FWI and WEMVA techniques. This generalized approach utilizes all the components of seismic data to invert for the medium parameters without the cycle-skipping problem. This is achieved in two steps: first by extending the wave equation and adding an offset axis to the velocity model, and second by adding a regularization term that drives the solution towards the zero subsurface offset (Symes, 2008). However, this velocity extension makes the propagation considerably more expensive because each multiplication by velocity becomes a convolution over the subsurface offset axis.

In this paper, we present an approximation that significantly reduces the computational cost of TFWI while maintaining its desirable characteristic of enabling the simultaneous inversion of all wavelengths of the model. First, we use the Born approximation to break the extended velocity model into two components: a background component affecting transmission and a perturbation component affecting reflections. Second, we reduce the background component to zero subsurface offset while keeping the perturbation component at all offsets. This simplification of the background component is the key in reducing the computation cost since the convolution with velocity in propagation becomes a multiplication instead of convolution. However, breaking the model into two components hinders the simultaneous inversion of different wavelengths of the model. Moreover, the scale separation by Born approximation is not perfect since the reflection component of a certain frequency or angle can affect the transmission component of another frequency or angle. Therefore, we add a final step where we mix the gradients of each component and then separate them in Fourier domain.

Another potential advantage of the method presented in this paper compared to the complete method presented in Biondi and Almomin (2012) is the ability to handle variable density without having to adapt a more complicated form of the wave-equation. In the complete method, the inversion tries to fit the data using a velocity model only. Variations in the velocity model will cause both a transmission and a reflection effect. This becomes an issue in the presence of density variations that only change the reflections of the data without affecting the transmission. In the efficient method, this issue can be avoided since the reflectivity is separated from the background. Therefore, the reflectivity part of the model can absorb any reflection effects that do not fit the background part of the model.

Tomographic full waveform inversion: Practical and computationally feasible approach