Constrained Least Square Inversion

In many geophysical problems it is possible to generate a set of completely different solutions that adequately explain the experimental data, especially where measurement errors are present. Ultimately, one solution has to be selected as the 'best' or most feasible answer to the problem. To do this we have to add to the problem some information not contained in the original equation d=Gm. This extra information is referred to as a priori information and serves to constrain our solutions so as to satisfy any of our quantified expectations of the model parameters. A priori information can take several forms. It may represent previously obtained geophysical, borehole or geological data or may simply be dictated by the physics of the problem. Consequently, constrained inversion takes many forms.

Inversion with prior information 

We can incorporate previously obtained information about the sought model parameters in our problem formulation. This external information could be in the form of results from previous experiments or quantified expectations dictated by the physics of the problem. Generally, these external data help to single out unique solution from among all equivalent ones. The solution process is said to be constrained. The procedure is simple. The constraining equations (data) are arranged to form an expression of the form$$Dm=h$$where, D is a matrix (with all the off-diagonal elements equal to zero) that operates on the model parameters m to yield or preserve the the a priori values of m that are contained in the vector h. The equation \(Dm = h\) means that we are employing linear equality constraints that are to be satisfied exactly. The mathematical development is straightforward. We wish to bias \(m_j\) towards \(h_j\).

We simply minimize,