2007 Special IssueModel inversion by parameter fit using NN emulating the forward model — Evaluation of indirect measurements
Introduction
The problem of the evaluation of indirect measurements is widespread in (but not restricted to) remote sensing. Examples are:
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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;
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flood discharge to be derived from water levels and surface velocities.
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 and that changes in are caused by changes of some underlying variables (cause). The model describing the relation might be physically based but could also be an empirical one. In most relevant cases there will be a certain region in the space where the inverse function exists. (This is the necessary condition which must be fulfilled for whatever retrieval procedure is to be used.)
The model 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 . But if the components of are correlated, the appropriate quantity to minimize3 is where is the covariance matrix of the measurements .
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:
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improve the quality of the retrieved parameters minimizing the error using the Levenberg–Marquardt algorithm;
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regard the full covariance matrix of the measurements and discuss the resulting covariance matrix of the retrieved parameters.
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