# A Geometrical Interpretation of Inverse Problems

Chapter

## Abstract

Most data interpretation problems in geophysics have no unique solutions since the information collected about a set of unknown parameters is frequently insufficient or contradictory or even both. Consider for simplicity a linear problemwith unknowns

$$\sum\limits_{k=1}^{K}D_{lk}p_{k}=b_{l}\quad (l=1,\ldots,L)$$

(1)

*p*_{ k },*k*= 1,…,*K*. This has no solution at all if*L*>*K*(overdetermined case) and an infinity of solutions if*L*<*K*(underdetermined case). We assume here for the moment that (1) is not further degenerated, i.e. that*D*_{ lk }has the full rank (either*L*or*K*). Unique solutions for the underdetermined case can be only given if some additional criteria of selection on the infinity of solutions is assumed. One can of course also consider the complete*K*-*L*-dimensional hyperplane defined by (1) as “the solution” of the problem. Solutions for the overdetermined case can only be meaningfully defined if*L*-*K*equations are discarded. We will outfit here the possible solutions to these problems with geometrical interpretations. The content of this paper is a supplement to Wunsch’s outline of inverse problems in this book.## Keywords

Singular Value Decomposition Geometrical Interpretation Canonical Variable Minimum Norm Case View
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Kluwer Academic Publishers 1989