Research On Numerical Algorithm Of A Kind Of Geophysical Inverse Problem

The paper is about numerical algorithm of geophysical inverse problem whose model is ill-posed linear equations.Firstly, we analyze various models by Picard theory, and by the simple model, give the method analyzing the degree of the model contaminated by noise.Then we propose five methods which fall into two classes for handling the ill-posed problems. By revised the strategies for choosing regularization parameter, all these methods can give true solution of the problems. And we can obtain stable T₂ relaxation spectra from nuclear magnetic resonance (NMR) log data. The first class of regularization method is named stationary method which include two methods, Tikhonov regularization and truncated singular value decomposition regularization(TSVDR). For TSVDR,we give a method of choosing truncated parameter which can balance the resolution and covariance of the solution. The second class of regularization method is named iteration regularization method, it includes Landweber iteration regularization, Tikhonov iteration regularization and trust region method. Trust region method is also an iteration regularization which can automatically moderate regularization parameter.The paper also presents the method of choosing the regularization parameter, including the discrepancy principle,GCV, L-curve. And, we analyze the advanges and disadvanges of these methods.At last, in the field of echo-stripping ,firstly, we implement the existing method, including regularization method, which we give three different results, the smallest length result, the slippiest result, the smoothest result, truncated singular value decomposition, improved singular value decomposition, SIRT and so on. Moreover, we present two new methods which aren't be applied in echo-stripping. They are trust regoin method and improved iteration regularization method. And then, we analyze the applicability of the two methods, give the difference with the existing method and first give the method choosing the regularization parameter using L-curve at home and aborad.By analyzing in theory and experimental data and the logging data in the oil field, we prove that all these methods equipped with an appropriate strategy and appropriate stopped criteria can obtain stable T₂ relaxation spectra. At the same time,we research the changes of the result when we apply different logging equipment, different relaxation points, different relaxation time. Finally,we compare our T₂ relaxation spectra with the Schlumberger's.