Decon in the log domain with variable gain |

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

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 .

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:

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 *auto*correlation of the signal output is a delta function.
Noncausal sparseness theory (other penalty functions) in a world of echoes (nonstationary gain)
says the
*cross*correlation 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 |

2012-05-10