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Matrix expression of stacking operators

Stacking along hyperbolas is a linear operator mapping a two-dimensional (t,x)-space into two-dimensional -space. The operator may therefore be expressed as a matrix in a four-dimensional space. This is not convenient for our imagination. The representation of the operator in two dimensions is preferable. This can be achieved by ordering all the columns of (t,x)-space into one vector, and the same for -space. To get an idea of what the matrix looks like, let us write the equation transforming (t,x)-space into -space when the number of samples is six, two offsets and two sloths. The nearest neighbor interpolation was used.

 (14)

Each submatrix in the matrix represents transformation (Claerbout, 1989). The transpose matrix applied on u gives d:

 (15)
Generally, if we denote as the i-th trace of a gather, as the i-th trace of a velocity analysis panel and as an NMO matrix transforming i-th offset with j-th sloth, then the velocity stacking transformation may be expressed by the equation
 (16)
If we want to compute from by least squares, then the matrix
 (17)
should be inverted. This matrix is not even approximately unitary, as it is for (Claerbout, 1989). For our example we have
 (18)
Hence just a transpose operation to stacking cannot be used for inverting the velocity analysis panel.

The rank of the matrix cannot be higher than max(n.nt,k.nt), where nt is the number of samples of a trace. From this it follows that the matrix is singular if the number of traces in the -space is less than the number of traces in (t,x)-space. Generally, this matrix may be expected to be singular even for the opposite case, as we can see from equation (18).

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Stanford Exploration Project
1/13/1998