Iterated linear inversion theory can often solve nonlinear inverse problems. Also, linear inversion theory provides convenient error estimates and other interpretive measures. But are these interpretive measures valid for nonlinear problems? We address this question in terms of the joint probability density function (pdf) of the estimated parameters. Briefly, linear inversion theory will be valid if the observations are linear functions of the parameters within a reasonable (say, 95%) confidence region about the optimal estimate, if the optimal estimate is unique. We use Bayes' rule to show how prior information can improve the uniqueness of the optimal estimate, while stabilizing the iterative search for this estimate. We also develop quantitative criteria for the relative importance of prior and observational data and for the effects of nonlinearity. Our method can handle any form of pdf for observational data and prior information. The calculations are much easier (about the same as required for the Marquardt method) when both observational and prior data are Gaussian. We present calculations for some simple one and two-parameter nonlinear inverse problems. These examples show that the asymptotic statistics (those based on the linear theory) may in some cases be grossly erroneous. In other cases, accurate observations, prior information, or a combination of the two may effectively linearize an otherwise nonlinear problem.
[1]
R. Wiggins,et al.
The general linear inverse problem - Implication of surface waves and free oscillations for earth structure.
,
1972
.
[2]
A. Tarantola,et al.
Inverse problems = Quest for information
,
1982
.
[3]
G. Backus,et al.
The Resolving Power of Gross Earth Data
,
1968
.
[4]
G. Backus,et al.
Uniqueness in the inversion of inaccurate gross Earth data
,
1970,
Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.
[5]
D. Jackson.
The use of a priori data to resolve non‐uniqueness in linear inversion
,
1979
.
[6]
D. Jackson.
Interpretation of Inaccurate, Insufficient and Inconsistent Data
,
1972
.
[7]
G. Backus,et al.
Inference from Inadequate and Inaccurate Data, III.
,
1970,
Proceedings of the National Academy of Sciences of the United States of America.
[8]
Naoshi Hirata,et al.
Generalized least-squares solutions to quasi-linear inverse problems with a priori information.
,
1982
.
[9]
D. Marquardt.
An Algorithm for Least-Squares Estimation of Nonlinear Parameters
,
1963
.
[10]
A. Tarantola,et al.
Generalized Nonlinear Inverse Problems Solved Using the Least Squares Criterion (Paper 1R1855)
,
1982
.