Elsevier

Neural Networks

Volume 20, Issue 4, May 2007, Pages 479-483
Neural Networks

2007 Special Issue
Model inversion by parameter fit using NN emulating the forward model — Evaluation of indirect measurements

https://doi.org/10.1016/j.neunet.2007.04.022Get rights and content

Abstract

The usage of inverse models to derive parameters of interest from measurements is widespread in science and technology. The operational usage of many inverse models became feasible just by emulation of the inverse model via a neural net (NN).

This paper shows how NNs can be used to improve inversion accuracy by minimizing the sum of error squares. The procedure is very fast as it takes advantage of the Jacobian which is a byproduct of the NN calculation. An example from remote sensing is shown. It is also possible to take into account a non-diagonal covariance matrix of the measurement to derive the covariance matrix of the retrieved parameters.

Introduction

The problem of the evaluation of indirect measurements is widespread in (but not restricted to) remote sensing. Examples are:

  • a sensor onboard a satellite measures spectral resolved radiances emanating from a certain area of the ocean surface — the parameters of interest are the concentrations of the water constituents;

  • flood discharge to be derived from water levels and surface velocities.

A common method for tackling the problem is first to model the causal dependency of the measured quantities from the parameters of interest: in our former example the parameters of interest, the concentrations of water constituents, cause a certain coloration of the water which is registered by the measurement of spectral resolved radiances by a sensor onboard a satellite. In the second step, to derive the parameters of interest from the measurements, the model must be inverted. Nowadays, in the case of high model complexity, it is common practice to emulate the inverse model by a neural network (NN). In order not to apply the inverse NN outside its scope, a domain check (Schiller & Krasnopolsky, 2001) can be performed by combining the inverse NN with an NN emulating the forward model (forward NN). A comprehensive discussion of problems related to NNs emulating forward/inverse models can be found in Krasnopolsky and Schiller (2003).

The idea of the scope check can be extended: the output of the inverse net is not only fed into the forward net to check if its output complies to the original measurements but it is also used to improve iteratively the accuracy of the parameters of interest (Schiller & Doerffer, 2005). This becomes feasible since the NN emulation of the forward model allows us to calculate the Jacobian of the forward model efficiently and thus the Levenberg–Marquardt optimization scheme can be used to determine the parameters of interest best fitting the measurements.

The paper is organized as follows. In Section 2 the construction of the inverse/forward NN is sketched and then the implementation of the optimization loop is presented. A realization of this procedure is discussed in Section 3. A generalization of this scheme for the case of a non-diagonal covariance matrix of measurement errors is given in Section 4. Conclusions are drawn in Section 5.

Section snippets

Minimization of sum of error squares

We assume that the result of a measurement is a vector of quantities r and that changes in r are caused by changes of some underlying variables c (cause). The model describing the relation r=m(c) might be physically based but could also be an empirical one. In most relevant cases there will be a certain region in the c space where the inverse function c=m1(r) exists. (This is the necessary condition which must be fulfilled for whatever retrieval procedure is to be used.)

The model m is used to

Example from remote sensing

This section presents a remote sensing application of the algorithm described. It is used to derive the concentrations of water constituents from MERIS (the Medium Resolution Imaging Spectrometer). The primary mission of MERIS is the measurement of sea color in the oceans and in coastal areas. The aim is to convert such measurements of the sea color into a measurement of concentrations of chlorophyll pigment, suspended sediment and gelbstoff (dissolved organic material). The measurement of the

Minimization in case of non-diagonal covariance matrix

The sum-of-square error function equation (1) is the quantity to minimize in the case of uncorrelated components of the measurement rM. But if the components of rM are correlated, the appropriate quantity to minimize3 is χ2=(rMm(c))TC1(rMm(c)) where C is the covariance matrix of the measurements rM.

Conclusions

The emulation of an inverse model by an NN allows us to invert very complex models operationally for a large amount of data. To calculate the Jacobian as a byproduct of the emulation of the corresponding forward model in addition allows us to:

  • improve the quality of the retrieved parameters minimizing the error using the Levenberg–Marquardt algorithm;

  • regard the full covariance matrix of the measurements and discuss the resulting covariance matrix of the retrieved parameters.

References (7)

There are more references available in the full text version of this article.

Cited by (7)

View all citing articles on Scopus
View full text