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S  0 ``  >*  0$ `   @*  0ȝ `   @*H  0޽h ? ̙33 Default Designq- !-- ,(    C `AHC:\Chesanswer\pshow\MC900435987.WMF"  0  4 Let s look at seismic inversion  (using full screen of course). Because the down wave is normally a lot longer than the space between reflectors, primary reflection events mix together, creating the classic  tuning problem. Visible lobes are accidents of this mix, and often show false lineups, especially in complex stratigraphy. The challenge here is to substitute a single signed spike for each reflection generator , thus removing the effect of the long down wave. In other words, the mathematical goal is to solve backwards for these reflection coefficients. This is the definition of inversion. Given this basic need, the only question is the method we use. The currently accepted approach relies on a fixed set of mathematical algorithms. We speak of as  linear . The other major category is spoken of as  non-linear because it employs a series of data-driven logical steps to achieve statistical optimization.. The common problems: We don t know the wavelet shape, the reflector distribution is seldom random and all sorts of things happen from the original recording to our inversion input. To make matters really bad, more than one basic wave shape often exists, due to various types of noise. At best, any system is going to have to average shape information across multiple traces and extended time zones. In linear techniques this averaging is hidden, but it is there. It is important to realize that the problems we face in non-linear approaches affect linear techniques equally. The difference is that they are hidden within the rigid process.V 2" Z laA1y@  0  We need to face the fact that serious error is inevitable! The trick is to make the error manageable. This is where nonlinear techniques shine.4 2; U d  C <A$C:\WEB1\books.gif0`    0   ]The linear approach assumes there exists a closed mathematical set of equations that has the power to transform seismic information from the time domain into the domain where shape and event position is described by frequency and phase. Once there, it is mathematically feasible to compute filter parameters to accomplish the desired spectral modifications, eventually coming back to the time domain for display. This is a beautiful concept that captured the love of seismic researchers. The transform itself may be looked at as a modeling process. If we started with a highly ringing, continuous, reflection train, only one frequency would be required in the model. If we start to dampen the train, other modeling frequencies are needed. When we get to the job of describing complex side lobes we need a bunch. The distribution of vital information now comes into play. As more and more frequencies get involved, the shape information is spread very thinly, and noise becomes a big problem.. \(2 < : |RB  s *Do  LB   c $D  r   C JA2C:\Chesanswer\MEwave.bmp` ~   C VA>C:\Chesanswer\doityourself.gif     0 `@ <`Getting from their best to the spike requires a more intelligent way off attacking the noise problem, and that is where statistical optimization comes in. The fact that our system can start with  pre-whitened input surprises even me. Inversion is not enough! In nature, while beds hopefully can be followed for miles, individual reflecting interfaces often are short lived. The better our inversion got, the more complaints we got about the unfamiliar displays. Early on we saw the need for the integration we discuss below. While we did not invent the non-linear wheel, we sure go back a long way in its development. Sparse spike technique  pitches take us back 10 years to our early 3D efforts. We have come a long way from those first attempts. Optimization is the best way to look at our procedures. We make some initial intelligent guesses and then go into layered iteration loops, building all our hard earned seismic knowledge into logic that drives the process to the best answer possible, under the noise circumstances that exist. In other words we have rearranged the error barriers so that they don t stop us part way. Error still exists in our answers, but it exists in the form of statistical chatter and it doesn t affect the safety of our results. Essentially we merge the development of the best spike guesses with the development of the best wavelet shape, stopping when we do no more good.. We have said that linear techniques can t spike and If they can t spike, they can t integrate. Since our final integration is a large part of the remarkable success on NLI, that should put us at a real competitive advantage. To explain: individual reflection coefficients are the first differences of a sonic log. We have our own log image producer that integrates these computed coefficients for our own use. This process always produces believable curves that match well with other log information. Taking this concept to the logical end, we integrate our spike guesses. We know that imperfect inversion makes integration almost impossible. Extra lobes screw up the whole thing. So, we use the quality of the integration as a major driving test, converging on that wavelet that gave the cleanest output as our best guess. This pushes the system in the right direction. When we show you our well matches we ask that you compare before and after examples so that you can see that removing redundant energy greatly improves the match. So, pay attention to thickness! (2  - )t^   ?fJ6  65jJ0 `  The ideal spectrum is (wavelet influence removed). 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