# Dictionary:Inverse problem

Solving for the spatial distribution that could produce a given set of measurements. Where a distribution of ‘‘causes’’ produces a distribution of measurement values that depend on a system of parameters. The **forward** (or **direct**) **problem**, if linear, is expressible as the matrix equation

**M**=

**P V**,

where **M** is a vector of the measurements *m*_{i}, **P** is a matrix of the parameters *p*_{ij}, and **V** is a vector of the values *v*_{j}. (The problem may also be nonlinear.) This equation expresses the **model**. Solving the equation for *v*_{j} is the **inverse problem** and solving for *p*_{ij} is the **parameter-estimation problem**. Usually *v*_{j} depends on the measurement system. For gravity, *v*_{i} might be the distribution of mass and *m*_{j} measurements of the acceleration of gravity, for well logging *v*_{j} might be the distribution of lithology and porosity and *m*_{i} the values measured by the logs, etc.