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Inverse problem

The inverse problem is the task that often occurs in many branches of science and mathematics where the values of some model parameter(s) must be obtained via manipulation of observed data.

The inverse problem can be formulated as follows:

Data→ Model parameters      Eq. 1

The transformation from data to model parameters is a result of the interraction of a physical system, e.g. the Earth, the atmosphere, gravity etc.

Inverse modelling is a term applied to describe the group of methods used to gain information about a physical system based on observations of that system. In other words, it is an attempt to solve the inverse problem.

Linear inverse problems

A linear inverse problem can be described by:
d=Gm     Eq. 2
where G is the matrix operator, or data kernel, which represents the explicit relationship between data and model parameters and is a representation of the `physical system' in Equation 1 above.

Non-linear inverse problems

The other, considerably more complex, set of inverse problems is the class collectively referred to as non-linear problems.

Non-linear inverse problems have a more complex relationship between data and model, represented by the equation:

d=G(m)     Eq. 3

Here g is a non-linear operator and cannot be algebraically separated from the model parameters that form m.

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