Decon in the log domain with variable gain

UNIQUENESS

As the figures show, our results are excellent, amazing even, but we've had a continuing problem with uniqueness. We find the pseudo-code presented here can spike any of the three lobes of the Ricker wavelet defining the sea floor. This is particularly annoying as it amounts to apparent time shifts and polarity changes. For about a year we ascribed this difficulty to nonlinear problems having many solutions, so we concentrated on controlling the descent. Now it looks like the problem is much simpler.

We now ascribe our uniqueness problem to a familiar problem in linear optimization. We believe we have what amounts to a null space. Tiny changes in initialization or other conditions lead to a wide variety of solutions.

For example, we often found by the third iteration we could see the spiking, and we could see the bubble estimation was well underway. By the tenth iteration it was pretty much settled down, and we would begin to be happy. But the computation was quick, so we were tempted to continue iterating. Maybe about the 150th iteration we would notice that spiking on the center of the Ricker wavelet would begin transition to spiking the first or third lobes of the Ricker wavelet (including the accompanying apparant polarity change). To make matters worse, only slight changes in the gain function $g_t$ would determine the selection of which final lobe.

We wasted a lot of time believing nonlinearity was responsible for multiple solutions. Our early primative attempts at regularization had failed. With the pseudocode above you can have results like in this paper in a dozen iterations, however, the theory below explains the missing regularization that should allow you all the iterations you like.

Decon in the log domain with variable gain