How Many Parameters Can Be Maximally Estimated From a Set of Measurements?

Remote sensing algorithms often invert multiple measurements simultaneously to retrieve a group of geophysical parameters. In order to create a robust retrieval algorithm, it is necessary to ensure that there are more unique measurements than parameters to be retrieved. If this is not the case, the inversion might have multiple solutions and be sensitive to noise. In this letter, we introduce a methodology to calculate the number of (possibly fractional) “degrees of information” in a set of measurements, representing the number of parameters that can be retrieved robustly from that set. Since different measurements may not be mutually independent, the amount of duplicate information is calculated using the information-theoretic concept of total correlation (a generalization of mutual information). The total correlation is sensitive to the full distribution of each measurement and therefore accounts for duplicate information even if multiple measurements are related only partially and nonlinearly. The method is illustrated using several examples, and applications to a variety of sensor types are discussed.