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## Multiple attenuation

First, I consider that any seismic data is the sum of signal and noise as follows:
 (1)
where are the seismic data, the signal we want to preserve and the noise we wish to attenuate. In the multiple elimination problem, the noise is the multiples and the signal the primaries.

Now, assuming that we know the multidimensional PEFs and for the noise and signal components, respectively, we have
 (2)
by definition of the PEFs. Equations (1) and (2) can be combined to solve a constrained problem to separate signal from spatially uncorrelated noise as follows:
 (3)
We can easily eliminate in the last equation of the fitting goal (3) by convolving with . Doing so, we end up with the following fitting goals:
 (4)
For some field data, it might be useful to add a masking operator on the data and signal residual and in order to perform the noise attenuation. It happens for example when the noise appears after a certain time or offset. I call this masking operator and I weight the fitting goals in equation (5) as follows:
 (5)
Solving for in a least-squares sense lead to the objective function
 (6)
where is a constant to be chosen a-priori that relates to the signal/noise ratio. The least-squares inverse for becomes
 (7)
where (') stands for the adjoint. Note that since is a diagonal operator of zeros and ones, we have . It is interesting to note that is the inverse spectrum of the noise and is the inverse spectrum of the signal where we perform the attenuation. Soubaras (1994) uses a very similar approach for random noise and more recently for coherent noise attenuation Soubaras (2001) with F-X PEFs. Because the size of the data space can be quite large, we estimate iteratively with a conjugate-gradient method. In the next section, I describe the PEFs estimation method I use to compute and needed in equation (7).

Next: Filter estimation Up: Theory of multiple attenuation Previous: Theory of multiple attenuation
Stanford Exploration Project
7/8/2003