A classification of some inverse problems in geochemical modelling

A number of geochemical inversion problems are analysed with the aim of classifying the inherent degree of difficulty and numerical instability involved in the process of obtaining numerical solutions. The approach is to consider relatively abstract representations of the problems so as to concentrate attention on those aspects that cause numerical difficulties. The various geochemical inversion problems are related to numerical differentiation with the order of differentiation indicating the instability of the problem. The less stable problems place more stringent demands on the observational data that are being interpreted, and the results are also more sensitive to any limitations in the formulation of the inversion problem. The examples considered are the determination of CO₂ source strengths, the deconvolution of trace gas concentrations from polar ice and interpretation of CCl₃F observations. The calibration of global carbon cycle models is described schematically as a sequence of poorly-conditioned problems.