The intent of this book is to provide a rather broad overview of inverse theory as it might be applied to petroleum reservoir engineering and specifically to what has, in the past, been called history matching. It has been strongly influenced by the geophysicists' approach to inverse problems as opposed to that of mathematicians. In particular, we emphasize that measurements have errors, that the quantity of data are always limited, and that the dimension of the model space is usually infinite, so inverse problems are always underdetermined. The approach that we take to inverse theory is governed by the following philosophy.

1. All inverse problems are characterized by large numbers of parameters (conceptually infinite). We only limit the number of parameters in order to solve the forward problem.

2. The number of data is always finite, and the data always contain measurement errors.

3. It is impossible to correctly estimate all the parameters of a model from inaccurate, insufficient, and inconsistent data, but reducing the number of parameters in order to get low levels of uncertainty is misleading.

4. On the other hand, we almost always have some prior information about the plausibility of models. This information might include positivity constraints (for density, permeability, and temperature), bounds (porosity between 0 and 1), or smoothness.

5. […]