Magnetic potential inversion

Let us consider a problem of inversion of the magnetic potential. According to formula (6.37), we have 

Let us assume that we have observed some magnetic potential Ur C) on the surface of the Earth. The problem is to determine the function /( ) of complex magnetization distribution. We introduce a complex Hilbert space D of data (magnetic potential) with the metric given by formula (7.2), and a complex Hilbert space M of models (functions /(C)) with the metric 

The only difference between the inverse problem formulation for gravity and magnetic fields is that now we use a complex Hilbert space of magnetization models /(C)- 

The magnetic potential inverse problem can be solved by minimization of the misfit functional  

We apply the same technique to solve the minimization problem (7.58) that we used in gravity field inversion. 

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Thus, as in the gravity case, the calculation of the each iteration in the conjugate gradient method for magnetic potential inversion can be based on the migration transformation. 

Note that a similar approach can be outlined for magnetic field inversion as well. The corresponding formulae are obtained from the basic formula for magnetic potential inversion by simple differentiation according to expression (6.36). I leave this derivation as an exercise for interested readers. 

Strakhov, V. N., 1970a, Some aspects of the plane inverse problem of magnetic potential (in Russian) Izvestia AN SSSR, Fizika Zemli, No. 9, 31-41. 

In the incommensurate phase T < 7] the magnetic structure forms a helix along the c-axis. For the theoretical analysis of the magnetic properties of copper metaborate based on a phenomenological thermodynamic potential, it is essential that the crystal symmetry has no center of inversion. The inversion operation enters only in combination with rotational displacement around the c-axis by 90° A and 433. Therefore, in the thermodynamic... 

NMR magnetic field shift in ppm, potential in electrochemistry, distance in crystallography, etc., and frequency its inverse. 

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