The data resolution function

Backus and Gilbert (1967) have developed a solution of the linear inverse problem that estimates the resolution of the model parameters. In this section, we will outline the basic ideas of this approach and will show how the regularization can be introduced in the Backus-Gilbert method. 

Suppose that we have observed the data in a fixed number of observation 

In the majority of geophysical problems we know the data kernels 1. It is shown in Appendix C that the solution of the inverse problem which has the minimum nonii in M, can bo presented in the form  

Alternatively, data d can be expressed through the true model by equation (3.109). Thus, substituting (3.109) into (3.117) we find the relationship between the estimated model and true model 

The linear operator R is called the resolution operator. Using the definition of the scalar product (3.110), it can be written 

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