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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).

** Next:** Examples of matrices
** Up:** SAMPLING IN VELOCITY DOMAIN
** Previous:** SAMPLING IN VELOCITY DOMAIN
Stanford Exploration Project

1/13/1998