A Problem in the Application of Inverse Methods to Tracer Data [and Discussion]

Attempts to apply inverse methods to the interpretation of tracer data usually seek some least squares solution of the flux divergence equations AC = S, where A is a matrix of transport coefficients, C a vector of concentrations and S a vector of sources/sinks. However, what is often really required is a set of values for the elements of A, which will give a satisfactory prediction of the concentrations. This corresponds to finding a least squares solution of C = A$^{-1}$ S. The two problems are not equivalent. The latter corresponds to an extensively reweighted version of the former, where the weights depend on the solution (the elements of A). The former is linear in the elements of A: the latter is highly nonlinear. In addition, the matrix A is invariably sparse, and required to be so. However, A$^{-1}$ is not, nor is it guaranteed that its inverse will be if its elements are determined freely. It is not clear whether the standard methods of generalized inverse theory are applicable to the more difficult `real' problem nor, if they are not, what other methods might be used. It is, however, possible that solutions of the `real' problem, if they can be found, would be more informative.