The linear discrete inverse problem

Let us consider a general inverse geophysical problem, described by the operator equation  

In a case where A is a linear operator, equation (3.1) can be rewritten in matrix notation  

Expression (3.2) describes a system of N linear equations with respect to L unknown parameters, mi, m2, m3. mi,  

solving the inverse problem (3.1) means solving the system of linear equations (3.3) with respect to parameters mi, m2, m3,. ..,rriL- 

The system (3.3) is called underdetermined if L. The system (3.3) is called overdetermined if L. Very often in geophysical applications, we work with an overdetermined system wherein the number of observations exceeds the number of model parameters. At the same time, in many situations it may be necessary to work with an underdetermined system. We will examine both types of linear equation systems below. 

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