Kirchhoff migration often fails in areas of complex geology, such as sub-salt, because the wavefield is severely distorted by lateral velocity variations leading to complex multipathing. As the shortcomings of Kirchhoff migration have become apparent (68), there has been renewed interest in wave-equation migration and computationally efficient 3D prestack depth migration methods have been developed (6; 62; 9). However, no corresponding progress has been made in the development of Migration Velocity Analysis (MVA) methods based on the wave-equation. My goal is to fill this gap through a method that, at least in principle, can be used in conjunction with any downward-continuation migration method. In particular, this methodology can be applied to downward continuation based on the Double Square Root (119; 23; 71) or common-azimuth (9) equations.
As for migration, wave-equation MVA (WEMVA) is intrinsically more robust than ray-based MVA because it avoids the well-known instability problems that rays encounter when the velocity model is complex and has sharp boundaries. The transmission component of finite-frequency wave propagation is mostly sensitive to the smooth variations in the velocity model. Consequently, WEMVA produces smooth, stable velocity updates. In most cases, no smoothing constraints are needed to assure stability in the inversion. In contrast, ray-based methods require strong smoothing constraints to avoid divergence. These smoothing constraints often reduce the resolution of the inversion that would be otherwise possible given the characteristics of the data (e.g. geometry, frequency content, signal-to-noise ratio, etc.). Eliminating, or substantially reducing, the amount of smoothing increases the resolution of the final velocity model.
Wave-equation MVA belongs to a much larger family of methods using wavefield-based tomography or inversion techniques for velocity estimation, ultrasonic data or for surface and crosswell seismic data (103; 107; 108; 114; 115; 116; 25; 26; 29; 30; 31; 44; 45; 59; 60; 63; 72; 73; 74). The main reason for the large interest in wavefield-base tomography methods is related to the potential for robustness and high resolution that all such methods proclaim.
A well-known limitation of wave-equation tomography or MVA is represented by the linearization of the wave equation based on the truncation of the Born scattering series to the first order term. This linearization is hereafter referred to as the Born approximation. If the phase differences between the modeled and recorded wavefields are larger than a fraction of the wavelet, then the assumptions made under the Born approximation are violated and the velocity inversion methods diverge (114; 28; 52; 72). Overcoming these limitations is crucial for a practical MVA tool. This goal is easier to accomplish with methods that optimize an objective function that is defined in the image space than with methods that optimize an objective function that is defined in the data space.
Wave-equation MVA also employs the Born approximation to linearize the relationship between the velocity model and the image. However, I ``manipulate'' the image perturbations to assure that they are consistent with the Born approximation, and replace the image perturbations with their linearized counterparts. I compute image perturbations by analytically linearizing the image-enhancement operator (e.g prestack residual migration presented in storm) and applying this linearized operator to the background image. Therefore, the linearized image perturbations are approximations to the non-linear image perturbations that are caused by arbitrary changes of the velocity model. Since I linearize both operators (migration and residual migration) with respect to the amplitude of the images, the resulting linear operators are consistent with each other. Therefore, the inverse problem converges for a wider range of velocity anomalies than the one implied by the Born approximation.
This method is more similar to conventional MVA than other proposed wave-equation methods for estimating the background velocity model (18; 37; 66) because it maximizes the migrated image quality instead of matching the recorded data directly. I define the quality of the migrated image by flatness of the migrated angle-domain common image gathers (adcig) along the aperture angle axis. In this respect, this method is related to Differential Semblance Optimization (DSO) (105; 97) and Multiple Migration Fitting (20). With respect to DSO, this method has the advantage that at each iteration it optimizes an objective function that rewards flatness in the ADCIGs globally (for all the angles at the same time), and not just locally as DSO does (minimizing the discrepancies between the image at each angle and the image at the adjacent angles). This characteristic should speed-up the convergence, although I have no formal proof for this assertion.
This section describes the theoretical foundations of wave-equation MVA with simple examples illustrating the main concepts and techniques. In example I present applications of wave-equation MVA to the challenging problem of velocity estimation under salt. Here, I begin by discussing wavefield scattering in the context of one-way wavefield extrapolation methods. Then, I introduce the objective function for optimization and finally address the limitations introduced by the Born approximation. Appendix C details the wave-equation scattering operator and the computation of linearized image perturbations.