# Dictionary:Inverse problem

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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

${\displaystyle \mathbf {M} =\mathbf {PV} }$

where M is a vector of the measurements mi, P is a matrix of the parameters pij, and V is a vector of the values vj. (The problem may also be nonlinear.) This equation expresses the model. Solving the equation for vj is the inverse problem and solving for pij is the parameter-estimation problem. Usually vj depends on the measurement system. For gravity, vi might be the distribution of mass and mj measurements of the acceleration of gravity, for well logging vj might be the distribution of lithology and porosity and mi the values measured by the logs, etc.