Functional representation of geophysical data and an inverse problem

3 Functional representation of geophysical data and an inverse problem 

Assume that we have geophysical measurements in a fixed number of observation points dj, j = 1,2,. ..n dj G Ey These measurements depend on parameters of the corresponding geophysical models and therefore can be treated as the functionals 

According to the Riesz representation theorem there exist vectors (elements of the space M) such that 

Suppose that we know the data kernels. The problem is to determine the model m which fits the observed data. In other words we have to find the solution of the system of equations (C.9). 

To solve this problem we assume that F j = l,2.n is a system of linear independent vectors, which forms the subspace L C M. If the dimension of M is greater than L, the element m is not unequally defined by (C.9). So we can find the solution of (C.9) which possesses the additional properties, for example the smallest norm. 

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Functional representation of geophysical data and an inverse problem Thus we have the following solution for m ... 

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