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11-13-92In general, the inverse methods can be summarized in three categories. The first one is a data base approach. Whenever a measurement result comes in, it will be compared with all the pre-stored patterns in a library to identify the possible targets. In industry, it is called data interpretation which strongly relies on previous experience. The accumulation of data in the library is a learning process. The advantage of this method is that the unknowns can be identified exactly if such target has been seen before so that the cause and the result are directly connected. The disadvantage is that if the data base is not big enough, the target can not be identified, or if the data base is too big, it will take too much time to search for and match the right pattern. The second method is the iterative approach. A forward model of deducing results from cause has been established before the inversion. When a measurement is obtained, a set of guessed values for the unknowns will be put in the forward model as the possible causes. If the predicted result of the forward model does not match the measurement, an adjusted set of parameters will be used as a new guess. The whole procedure will be repeated until the match is found. During iterations, how to correct the error to reach fastest the convergent value is very important.This procedure is also called optimization or error minimization, which by itself is an active research area. Iterative methods are particularly useful when the solution of an integral equation can not be found in an explicit form. As an example of the iterative methods, the Born approximation approximates the forward model in a linearized form, then iterates towards convergence. The advantage of the iterative approach is that an accurate final solution can be found at a relative fast (compare to data base search) speed. The method is also flexible for different kinds of targets, unlike the data base search where the target has to have been seen before. The disadvantage is that the convergence of iteration is not always guaranteed. There are usually more than one minimum of the cost function.
3The initial guess is sometimes so critical to assure the right convergent result that a Priori information needs to be used. Non-uniqueness is a serious problem in all inversion methods.Sufficient number of measurements are helpful of reducing the degree of non-uniqueness.Formulation of the forward model for the final inversion equation determines how efficient the inversion scheme is, which sometimes also affect the degree of non-uniqueness.A recently developed inversion method referred to as the renormalized Source-TypeIntegral Equation (STIE) approach solves the integral equation derived from the Green's function without the linear approximation. The STIE approach formulates an exact forward model, hence is not restricted to low contrast profiles or weak scattering. The STIE method has been applied to profile inversion problems of the soil moisture and oil formation in bore...