Numerical analysis of the azimuth moveout operator for vertically inhomogeneous media

The azimuth-moveout (AMO) operator, unlike the dip-moveout (DMO) operator, has a 3D structure in homogeneous isotropic media, with an out-of-plane (crossline) component. In general, this component is concave downward, giving the operator an overall skewed-saddle shape. The AMO operator, necessary for azimuth correction only, is typically smaller in size than conventional DMO operators, which corresponds to purely offset correction to zero offset. When velocity varies vertically, the operator shape changes depending on the degree of velocity variation. The general shape of the operator, however, remains saddlelike. In fact, for smooth velocity increases with depth, similar to those found in the Gulf of Mexico, the v(z) AMO operator does not differ much from its homogeneous counterpart. In this case, the residual AMO operator, constructed by cascading a forward homogeneous AMO operator with an inverse v(z) one, is extremely small, which suggests that the impact of such v(z) variations on the AMO operator is generally small. Complex vertical velocity variations, on the other hand, result in more complicated AMO operators that include, among other things, triplications at moderate angles. Regardless of the complexity of the model, the v(z) operator has the same first-order behavior as its homogeneous counterpart. As a result, for small dip angles the homogeneous AMO, as a tool for partial stacking, often enhances the image. Moderate to steep dips in complex v(z) media requires the application of an algorithm that honors such velocity variations.