Magnetotelluric inversion

The inverse problem in this case is formulated as recovery of the unknown coefficient 7 of the elliptic operator from the known values of the field p(r, u>) in some domain or in the boundary of observations. In a number of brilliant mathematical papers the corresponding uniqueness theorems for this mathematical inverse problem have been formulated and proved The key result is that the unknown coefficient 7 (r) of an elliptic differential operator can be determined uniquely from the boundary measurements of the field, if 7 (r) is a real-analytical function, or a piecewise real-analytical function. In other words, from the physical point of view we assume that 7 (r) is a smooth function in an entire domain, or a piecewise smooth function. Note that this result corresponds well to Wcidelt s and Gusarov s uniqueness theorems for the magnetotelluric inverse problem. I would refer the readers for more details to the papers by Calderon (1980), Kohn and Vogelius (1984, 1985), Sylvester and Uhlmann, (1987), and Isakov (1993). 

DQA approximation in magnetotelluric inverse problem The DQA approximation is particularly suitable for constructing massive 3-D magnetotelluric inversion schemes, because of the low cost and simplicity of the expressions for forward modeling. In this section I discuss the implementation of the DQA approximation in MT inversion, following the paper by Hursan and Zhdanov, 2001. The main advantage of the QA method over the iterative Born method is that now... 

We have introduced above the formulation of a magnetotelluric inverse problem with respect to the anomalous conductivity vector, denoted by <7. However, during the minimization we may obtain anomalous conductivity values such that the total conductivity becomes negative. Thus, we have to transform the anomalous conductivity into a new space of model parameters with the property that the total conductivity always remains positive. We have already introduced above the conventional way of solving this problem, which uses the logarithm of the total conductivity as a model parameter ... 

Newman, G. A., and D. L. Alumbaugh, 2000, Three-dimensional magnetotelluric inversion using non-linear conjugate gradients Geophys. J. Int., 140, 410-424. 

Gusarov, A. L., 1981, On uniqueness of solution of inverse magnetotelluric problem for two-dimensional media (in Russian) Mathematical Models in Geophysics, Moscow State University, 31-61. 

Madden, T. R,., and R. L. Mackie, 1989, Three-dimensional magnetotelluric modeling and inversion Proc. IEEE, 77, 318-333. 

The sections on quasi-analytical (Q. l) inversion and magnetotelluric QA inversion were prepared in collaboration with G. Hursan. 

This model is excited by plane EM waves with four different frequencies 1, 10, 100 and 1000 Hz. Following the traditional approach used in practical MT observations, we calculate the synthetic observed apparent resistivities and phases based on two off-diagonal elements of the magnetotelluric tensor at each observation point. The quantities Py (apparent resistivity) and y are assigned to the nominal TM mode. Note, that this 2-D nomenclature is artificial and approximate in nature for 3-D structures. However, it is widely used in practical MT observations, and often only the quantities Py, y, p y and (t> y are available for inversion. That is why we use the same approach in the model study. The apparent resistivity and phases of impedance were calculated at 195 receivers placed on the nodes of a square grid on the surface. The distance between the observation points is 100 m in the x and y directions. 

Hursan, G., and M. S. Zhdanov, 2001, Rapid 3-D magnetotelluric and CSAMT inversion 71st SEG Annual International Meeting, San Antonio, Texas, 1493-1496. 

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