Decon in the log domain with variable gain

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Having data , having chosen gain , and having a starting log filter, say , let us see how to update to find a gained output with better hyperbolicity. Our forward modeling operation with model parameters acting upon data (in the Fourier domain where produces deconvolved data (the residual).

 (9) (10) (11)

This follows because shifts the data by units which shifts the residual the same amount. Output formerly at time moves to time . This is not the familiar result that the derivative of an output with respect to a filter coefficient at lag is the shifted input . Here we have the output . This difference leads to remarkable consequences below.

It is the gained residual that we are trying to sparsify. So we need its derivative by the model parameters .

 (12) (13)

Recall and hence . To find the update direction at nonzero lags take the derivative of the hyperbolic penalty function by .
 (14) (15) (16)

This says to crosscorrelate the physical residual with the statistical residual . Notice in reflection seismology the physical residual generally decreases with time while the gain generally increases to keep the statistical variable roughly constant, so grows in time(!)

In the frequency domain the crosscorrelation (16) is:

 (17)

Equation (17) is wrong at . It should be brought into the time domain and have set to zero. More simply, the mean can be removed in the Fourier domain.

Causal least squares theory in a stationary world says the signal output is white (Claerbout, 2009); the autocorrelation of the signal output is a delta function. Noncausal sparseness theory (other penalty functions) in a world of echoes (nonstationary gain) says the crosscorrelation of the signal output with its gained softclip is also a delta function (equation (16), upon convergence).

 Decon in the log domain with variable gain

Next: TAKING THE STEP Up: Claerbout et al.: Log Previous: MINIMUM PHASE EXTENSION

2012-05-10