Truncated SVD methods for discrete linear ill-posed problems

SUMMARY Truncated singular value decomposition (TSVD) techniques have been widely used in inversion. The method of truncation determines the quality of a truncated SVD solution, but truncation has often been done arbitrarily. The ¢rst workable criterion for truncation was based on F-statistical testing, but has only rarely been used in geophysical inversion. Recently, the L-curve approach was proposed for the same purpose, and soon found many applications in interdisciplinary inverse problems. Up to the present, very little has been known of the solution quality using these two workable criteria. We have thus proposed in this paper a new quality-based method for truncation. Six TSVD estimators have been investigated for comparison in regularizing discrete unstably ill-posed problems, based on the statistically frequently used F-statistic, the L-curve and our new quality-based mean squared error (MSE) criteria. The three F-statistic-based TSVD estimators tested can indeed marginally improve the least squares (LS) solution to the ill-posed downward continuation problem in the sense of long-term averaging, depending on pre-selected signi¢cance levels.The simulations have shown that estimators of this type can hardly guarantee the improvement of the condition number of the linearized unstably ill-posed system. In other words, F-statistic criterion can frequently lead to incorrect discardings of components.TheTSVD estimatorby means ofan L-curve is thebest in (over)stabilizing ill-posed problems, but results in an over-discarding of components. It is extremely poor in terms of biases and mean squared error (MSE) roots of the solution. The qualitybased TSVD estimator with the basic ridge estimate of x as its initial value has outperformed the other twoTSVD techniques tested in terms of solution stability, bias and MSE. The simulations have clearly shown signi¢cant quality advantages of the new method. The simulations have also indicated that the new method is able to achieve a mean accuracy of 5 mgal for gravity anomalies from satellite gradiometry, if the few largest biases are left out of the computation.

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