Instability of the inverse problem solution

Another critical problem of inversion theory is instability. This problem reflects the practical fact that two observed data sets could differ only within the noise level, while the corresponding model parameter distributions could be completely different. 

We will discuss below a typical example of instability in solving a downward analytical continuation problem for the gravity field potential. 

According to equation (1.7), the gravity potential outside masses satisfies the equation  

Let us introduce the Cartesian system of coordinates x,y,z) with the vertical axis directed downwards. We assume that we observe the gravity potential at some level z = 0  

We can write an operator equation which connects the potential at a level z = h with the potential at the observation line, 2 = 0, 

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