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The modeling equation is similar to Thorson's equation for hyperbolic
transforms. It expresses the NMO-corrected data *d*(*t*,*h*) as a function
of the field *U* in the *t*_{0}-*p* domain:
| |
(3) |

whereas, for Thorson's equation, the term has to be replaced by
. Similarly, it can be compared to the
slant-stack modeling:
Equation (3) will transform a spike in the *t*_{0}-*p* domain
into a parabola (or straight line if *p*=0) in the time-offset domain.
I symbolize equation (3) as: *d*=*L*.*U*. Equation (3)
can be expressed in the frequency domain. For the transform,
Kostov (1989) showed that:

For the parabolic transform, we get:
| |
(4) |

So, for a fixed , an element of the matrix *L* is:
Stacking along parabolas actually corresponds to computing
*U*=*L*^{T}.*d* (as Thorson pointed out in the case of hyperbolic stacking). We
will transform the data *d*(*t*,*h*) into the *t*_{0}-*p* domain by using the
least-squares inverse of *L*, i.e. (*L*^{T}*L*)^{-1}*L*^{T} in the overdetermined
case (or *L*^{T}(*LL*^{T})^{-1} in the underdetermined case):

| |
(5) |

** Next:** Inverse transformation
** Up:** EXPRESSIONS OF THE PARABOLIC
** Previous:** EXPRESSIONS OF THE PARABOLIC
Stanford Exploration Project

1/13/1998