Inverse problem and the pseudoempirical orthogonal function method of solution 1: Theory

In situations where a large library of observed distributions of a function, such as temperature vs height, is available, these distributions may be used to form a set of empirical orthogonal functions. When sufficient observed distributions are not available, but when the general mathematical form of the distributions is known, a library may be constructed from the set of mathematical functions. A set of pseudoempirical orthogonal functions may then be constructed from this mathematical library. It is assumed that any distribution of the function may then be constructed from a linear sum of this pseudoempirical orthogonal set. This idea is employed to develop an inversion method using pseudoempirical orthogonal functions when a sufficient library of observations is not available. The technique employs a smoothing constraint as well as a positivity constraint, when warranted by the physical nature of the unknown.