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You can reject two dips with the operator
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(1) |

Finding the two values *p*_{1} and *p*_{2} is a nonlinear problem
that is easy.
Let *u* be the input signal and *v* be the output signal.
Consider
| |
(2) |

Now recognize that (2)
(which is a separate equation at each point in the (*t*,*x*) space of *v*(*t*,*x*))
is an overdetermined set of linear equations
for the two unknowns *a* and *b*.
It is easy to find *a* and *b* which by comparison
with equation (1)
gives *p*_{1} and *p*_{2} by the nonlinear but easy equations
*a*=*p*_{1}+*p*_{2} and *b*=*p*_{1}p_{2}.
The minimum signal required is three seismograms
(on which to express ).
To recapitulate,
first a simple procedure gives us the required coefficients
for a filter that fits two waves to the dataset *u*(*t*,*x*).
Second, the same dip filter coefficients can be applied
on a mesh in which *t* and *x* are interleaved,
thus introducing new data locations on which we need data.
Third, we interpolate *t* by any method.
Fourth, we find missing traces by minimizing the power in *v*(*t*,*x*).

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Stanford Exploration Project

1/13/1998