() introduced azimuth moveout correction (AMO) as a single operator to correct for azimuth variations in homogeneous isotropic media. They analytically derived the AMO operator, and used it in a Kirchhoff-type of implementation on multi-azimuth seismic data sets. Though the AMO operator had a 3-D structure, it had an overall small aperture, thus the Kirchhoff implementation of AMO is relatively cheap. Figure shows an AMO operator in homogeneous media. It is clearly 3-D in structure and has a general skewed saddle shape.
Like the DMO operator, the AMO operator is applied after NMO correction. Despite the simplicity of the homogeneous-medium AMO operator and its application, the earth subsurface is rarely homogeneous. Velocity increase with depth is very common in the subsurface, and an important question is how much of an error can be attributed to ignoring such vertical velocity variation. Using the 3-D SEG/EAEG salt-dome model, () shows that the homogeneous AMO operator produces reasonable results in smooth vertical velocity variations. Is this a general conclusion or only holds for the cases he tested?
Through the combined action of gravity and sedimentation, velocity variation with depth represents the most important first-order inhomogeneity in the Earth subsurface. This is one reason why time migration works well in so many places. Therefore, studying the AMO operator for such 1-D models can be useful in many parts of the Earth, and since the AMO operator is generally small, the v(z) AMO operator might be useful even in relatively complex areas.
In this paper, we will numerically construct the AMO operator for vertically inhomogeneous media, as well as observe how the operator shape is influenced by vertical inhomogeneity. Next, we will generate the residual AMO operator constructed by cascading a forward homogeneous-medium AMO operator and an inverse v(z)-medium AMO operator. The size and shape of the residual operator provides us with valuable information regarding the impact of vertical inhomogeneity on AMO. The smaller the residual operator the lesser the impact of vertical velocity gradients on AMO. Examples will include three types of vertical velocity variations: linear increase as a function of depth, a low velocity layer embedded in an overall increase in velocity with depth, and a high velocity layer embedded in an increase in velocity with depth. The last example is similar to what can be observed in the North Sea, as a result of the Austin Chalk layer.