Wave-equation migration velocity analysis by inversion of differential image perturbations

Wave-equation migration velocity analysis is based on the linear relation that can be established between a perturbation in the migrated image and the corresponding perturbation in the slowness function. Our method formulates an objective function in the image space, in contrast with other wave-equation tomography techniques which formulate objective functions in the data space. We iteratively update the slowness function to account for improvements in the focusing quality of the migrated image. Wave-equation migration velocity analysis (WEMVA) produces wrong results if it starts from an image perturbation which is not compliant with the assumed Born approximation. Other attempts to correct this problem lead to either unreliable or hard to implement solutions. We overcome the major limitation of the Born approximation by creating image perturbations consistent with this approximation. Our image perturbation operator is computed as a derivative of prestack Stolt residual migration, thus our method directly exploits the power of prestack residual migration in migration velocity analysis. INTRODUCTION Migration velocity analysis based on downward continuation methods, also known as waveequation migration velocity analysis, is a technique designed as a companion to wave-equation migration (Biondi and Sava, 1999; Sava and Fomel, 2002). The main idea of WEMVA is to use downward continuation operators for migration velocity analysis (MVA), as well as for migration. This is in contrast with other techniques which use downward continuation for migration, but traveltime-based techniques for migration velocity analysis (Clapp, 2001; Liu et al., 2001; Mosher et al., 2001). WEMVA is closer to conventional MVA than other wave-equation tomography methods (Noble et al., 1991; Bunks et al., 1995; Forgues et al., 1998) because it tries to maximize the quality of the migrated image instead of trying to match the recorded data. In this respect, our method is related to Differential Semblance Optimization (Symes and Carazzone, 1991) and Multiple Migration Fitting (Chavent and Jacewitz, 1995). However, with respect to these two methods, our method has the advantage of exploiting the power of residual prestack migration to speed up the convergence, and it also gives us the ability to guide the inversion by geologic 1email: [email protected],[email protected] 299 300 Sava and Biondi SEP–113 interpretation. WEMVA benefits from the same advantages over traveltime estimation methods as waveequation migration benefits over Kirchhoff migration. The most important among them are the accurate handling of complex wavefields which are characterized by multipathing, and the band-limited nature of the imaging process, which can handle sharp velocity variations much better than traveltime-based methods (Woodward, 1992). Complex geology, therefore, is where WEMVA is expected to provide the largest benefits. WEMVA is based on a linearization of the downward-continuation operator using the Born approximation. This approximation leads to severe limitations on the magnitude and size of the anomalies that can be handled. It, therefore, cannot operate successfully exactly in the regions of highest complexity. Other methods of linearization are possible (Sava and Fomel, 2002), but neither one allows for arbitrarily large anomalies. In our early tests (Biondi and Sava, 1999), we construct the image perturbation using Prestack Stolt Residual Migration (PSRM) (Stolt, 1996; Sava, 2000). In summary, this residual migration method provides updated images for new velocity maps that correspond to a fixed ratio (ρ) of the new velocity with respect to the original velocity map. Residual migration is run for various ratio parameters, and finally we pick the best image by selecting the flattest gathers at every point. The main disadvantage of building the image perturbation using PSRM is that, for large velocity ratio parameters (ρ), the background and improved images can get more than π/4 out of phase. Therefore, the image perturbation computed by the forward operator and the one computed by residual migration are fundamentally different, and can have contradictory behaviors when using the Born-linearized WEMVA operator for inversion. Alternative methods can be used to create image perturbations for WEMVA, in compliance with the Born approximation and computed directly from the background image (Sava and Biondi, 2001). We use an analytic differential procedure starting from the background image and leading to image perturbations similar to the ones created using the forward WEMVA operator. WAVEFIELD SCATTERING In migration by downward continuation, the wavefield at depth z +1z is obtained by phaseshift from the wavefield at depth z: W (z+1z) =W (z)ez . (1) Using a Taylor series expansion, the depth wavenumber (kz) depends linearly on its value in the reference medium (kzo) and the laterally varying slowness s (x , y, z) in the depth interval under consideration kz ≈ kzo + dkz ds ∣