Information retrieval from remotely sensed data and a method to remove parameter estimator ambiguity

Several recent developments in nonlinear estimators (e.g. more efficient training algorithms for neural networks) have made it possible to apply them to remote sensing and its associated highly nonlinear mappings between the observation space (Rⁿ) and the parameter space (R^m). One common problem estimators encountering nonlinear applications is when there is not a one-to-one mapping between these two quantities, thus the problem is said to be ill-posed. As a consequence, for a single observation, there may be one or more corresponding sets of parameters which may be responsible for the observation. Ambiguity in estimates can also be attributed to functions which become insensitive to a parameter of interest. To address this problem, a method to determine the region(s) of efficient retrieval in multidimensional space is presented. This is particularly important in a MSE (minimum mean square error) estimator like the neural network which many describe as a ´black-box´ in which an input is applied and the corresponding estimated parameters are presented at the output. In such a configuration, the estimator or the user has no way to detect and/ or avoid this many-to-one mapping problem. Techniques such as the maximum likelihood Cramer-Rao bound will be used to create a multidimensional sensitivity `map´ as a function of the parameters to be estimated. This is useful to: a) determine the areas in parameter space which yield unambiguous estimates of the unknowns, b) determine which combination(s) of measurements will yield the lowest variance on the estimate and c) provide a measure on the confidence of the retrieved parameters. In this article, the theoretical and practical aspects of the removal of estimator ambiguity are discussed in a simple application