Wave-equation migration velocity analysis by residual moveout fitting |

In wave-equation migration, as for example reverse-time migration, the image is computed from the back-propagated receiver wavefield, , and the forward-propagated source wavefield, , where is the recording time, is the model-coordinate vector, is the source position at the surface, and is the slowness model.

The prestack image, , is computed as the zero lag of the temporal cross-correlation between the spatially-shifted back-propagated receiver wavefield and forward-propagated source wavefield as (Rickett and Sava, 2002):

where
is the *half subsurface offset*,
which in this paper I will assume to be horizontal,
but it does not need to be in the general case
(Biondi and Symes, 2004).

The prestack image as a function of subsurface offset can be transformed to an image as a function of reflection aperture angle, by using a linear operator (Sava and Fomel, 2003). In matrix notation, if is a matrix and is a matrix, the image transformation from subsurface offset into the angle domain can be expressed as:

I introduce an objective function that maximizes the flatness of the angle-domain image along the aperture-angle axis at all spatial locations . This objective function aims at maximizing image flatness not directly as a function of the slowness, but indirectly through the application of an angle-domain moveout operator , which depends on the column vector of moveout parameters .

I define the application of the moveout operators to a prestack image computed by equations 1 and 2 with a background slowness , as

(A-3) |

where are the moveout shifts, assumed here to be simple depth shifts. The operator is linear with respect to the input image, but it is nonlinear with respect to the vector of moveout parameters . In matrix notation, the application of the moveout operator to the background image can be expressed as .

I further define the stacking operator that sums the image along the aperture-angle axis . I can now introduce the objective function that measures the flatness of the image as:

where is the slowness vector. This objective function is not a direct function of , but it depends on it indirectly through the moveout parameters . The dependency of the moveout parameters from the slowness function is not defined explicitly; these parameters are defined as the solutions of independent fitting problems, one for each spatial location in the image.

The fitting problems maximize the zero lag of the cross-correlation between the prestack image computed for a realization of the slowness vector and the moved-out image computed with the background slowness . For the sake of keeping the notation as compact as possible, I combine the independent fitting problems into one by defining the following objective function:

where with the notation I indicate the ensemble of inner products between the image vectors and which spans only the aperture-angle axis ; the result of these inner products is a vector of length . The stacking operator sums the elements of this vector to combine the objective functions into one.

The vector of moveout parameters is therefore the solutions of the following maximization problem:

For velocity estimation in the angle domain, an effective parametrization of the moveout is the "curvature" , that defines the following moveout equation

Notice that when the slowness is equal to the background slowness , the corresponding best-fitting moveout parameters are obviously the ones corresponding to no moveout; that is, , and consequently .

- Gradient of the objective function
- Derivatives of objective function ( ) with respect to moveout parameters ( )
- Derivatives of moveout parameters ( ) with respect to slowness ( )

- Computational cost

Wave-equation migration velocity analysis by residual moveout fitting |

2010-11-26