Two-parameters residual-moveout analysis for wave-equation migration velocity analysis

Introduction

The measurement of the residual moveout (RMO) in migrated common image gathers (CIG) is an important component of any migration velocity analysis (MVA) method. The choice of a robust method for measuring RMO is particularly important if the MVA process is automatized to avoid explicit picking of RMO parameters from coherency spectra (Biondi, 2010,2008; Zhang and Biondi, 2011). In Biondi (2011), I illustrate with a simple example some of the challenges that these methods may encounter when a one-parameter RMO analysis is employed in presence of strong lateral velocity variations. To address these concerns, in this report I introduce two possible choices of RMO functions defined by two parameters instead of one. Both choices add a term to the RMO function, in addition to the usual term that is proportional to the square of the tangent of the aperture angle.

The first choice of RMO function adds a term proportional to the fourth power of the tangent of the aperture angle, and thus I will dub it the ``Taylor'' RMO function. The second choice adds a term proportional to the absolute value of the sine of the normalized aperture angle. The angle is normalized to enable the sine to complete a full cycle between zero and the maximum aperture angle used for the analysis. This choice is motivated by the fact that it is theoretically desirable to have the terms of the RMO function to be mutually orthogonal (Siliqi, 2009). The sine function is only approximately orthogonal to the square of the tangent, but it has the advantage of being extremely simple. I will dub this RMO function ``Orthogonal".

I test the efficacy of the proposed RMO functions using two CIGs that were obtained by migrating two different synthetic data sets. The first CIG represents the challenges presented by strong lateral velocity variations. The data were modeled assuming a strong negative velocity anomaly above a flat reflector (Biondi, 2011). The second CIG represents the effects of anisotropy on RMO analysis. The data were modeled assuming a strongly anisotropic VTI medium ( $\epsilon=0.0975$ and $\delta=-0.11$ ) above a flat reflector, and migrated assuming an isotropic velocity (Biondi, 2005).

In the following section I introduce the new RMO functions and apply them to compute two-dimensional spectra measuring the stack power as a function of the moveout parameters. In the subsequent section I analyze the accuracy of the potential search direction that would be computed by evaluating the gradient of the stack power as a function of the RMO parameters, and compare the results with the results of a similar analysis when the conventional one-parameter RMO analysis is applied.

Two-parameters residual-moveout analysis for wave-equation migration velocity analysis