# Inverse Problems

## General

In a physical sense, it can be useful to define inverse problems as those in which the direction of mathematical inference is opposite to the direction of real-world causality. Since many real-world processes are dissipative, leading to loss of information, inversions that attempt to recover this attenutated information are subject to what is termed ill-conditioning, i.e. extreme sensitivity to errors in the models and data. Quantifying the sensitivity to model error can be particularly difficult since there are few ways to identify model error and inclusion of model error often requires non-linear estimation techniques.
Mathematical studies of inverse problems often take the ill-conditioning as the defining characteristic of inverse problems.

Inverse problems occur widely in fields such as seismology, satellite-based remote sensing, medical imaging and oceanography.

## Trace gas inversions

The trace gas inversion problem is that of using measurements of concentrations of trace atmospheric gases such as carbon dioxide in order to deduce the sources and sinks of these gases. My recent book, Inverse Problems in Atmospheric Constituent Transport, by I.G. Enting (CUP, 2002) reviews the field of global-scale trace gas inversions. A bibliography of papers on CO2 inversions, including all references from my book, is on this site (143kb pdf) .

The study of inversions of CO2 data, designed to enhance understanding of the global carbon cycle, have been greatly assisted by the TransCom  activity. The TransCom intercomparison aims to investigate differences between transport models in order to help quantify the model error in trace gas inversions. (In the earth system sciences community, the term "intercomparison" has been adopted to denote comparison between models, often at greater detail that that for which observational data are available).

## MASCOS projects on inversions

#### Statistical diagnostics for inversions

This project is undertaken within the TransCom project, jointly with Anna Michalak of U. Michigan and other members of the TransCom group.

The aim is to develop and apply statistical diagnostics for synthesis inversion of CO2 and similar trace gases.