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## Physical nonlinearity

When standard methods of physics relate theoretical data to model parameters ,they often use a nonlinear relation, say .The power-series approach then leads to representing theoretical data as
 (87)
where is the matrix of partial derivatives of data values by model parameters, say ,evaluated at .The theoretical data minus the observed data is the residual we minimize.
 (88) (89)
It is worth noticing that the residual updating (89) in a nonlinear problem is the same as that in a linear problem (44). If you make a large step , however, the new residual will be different from that expected by (89). Thus you should always re-evaluate the residual vector at the new location, and if you are reasonably cautious, you should be sure the residual norm has actually decreased before you accept a large step.

The pathway of inversion with physical nonlinearity is well developed in the academic literature and Bill Symes at Rice University has a particularly active group.

Next: Statistical nonlinearity Up: THE WORLD OF CONJUGATE Previous: THE WORLD OF CONJUGATE
Stanford Exploration Project
4/27/2004