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## PDE to reject two dips

You can reject two dips with the operator
 (1)
Finding the two values p1 and p2 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 p1 and p2 by the nonlinear but easy equations a=p1+p2 and b=p1p2. 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).

Next: Does it work? Up: INTERPOLATION WITH P.D.E. DIP Previous: Spatial interpolation with
Stanford Exploration Project
1/13/1998