Automatic model for finding the one-dimensional magnetotelluric problem

Summary. A method is described for finding a resistivity model that fits given magnetotelluric data in the one-dimensional case. The procedure is automatic and objective in that no a priori model structure is imposed. Starting with a uniform half space derived directly from the data, the procedure gradually transforms the half space to one with a continuous and smooth resistivity distribution whose response fits the measured data. The method is illustrated by application to two magnetotelluric data sets.     

Finite-difference modelling of magnetotelluric fields in two-dimensional anisotropic media

An algorithm for the numerical modelling of magnetotelluric fields in 2-D generally anisotropic block structures is presented. Electrical properties of the individual homogeneous blocks are described by an arbitrary symmetric and positive-definite conductivity tensor. The problem leads to a coupled system of partial differential equations for the strike-parallel components of the electromagnetic field. E _x, and H _x These equations are numerically approximated by the finite-difference (FD) method, making use of the integro-interpolation approach. As the magnetic component H _x, is constant in the non-conductive air, only equations for the electric mode are approximated within the air layer. The system of linear difference equations, resulting from the FD approximation, can be arranged in such a way that its matrix is symmetric and band-limited, and can be solved, for not too large models, by Gaussian elimination. The algorithm is applied to model situations which demonstrate some non-trivial phenomena caused by electrical anisotropy. In particular, the effect of 2-D anisotropy on the relation between magnetotelluric impedances and induction arrows is studied in detail.     

Calculating the two-dimensional magnetotelluric Jacobian in finite elements using reciprocity

To speed up the calculation of the field Jacobian for 2-D magnetoteliuric inversion using finite elements, the principle of electromagnetic reciprocity is applied. The governing relationship for the Jacobian of the field along strike is obtained by differentiating the Helmholtz equation with respect to the resistivity of each region in the finite-element mesh. The result is a similar Helmholtz equation for the Jacobian, with new sources distributed over all nodes within the parameter medium. However, according to the principle of electromagnetic reciprocity, the roles of sources and receivers are interchangeable. Utilizing reciprocity, the field values obtained from the original forward problem and for new unit sources imposed at the receivers are then utilized in the calculation of the Jacobian by simple multiplication and summation with finite-element terms at each rectangle in the mesh. For the auxiliary (across-strike) fields, the Jacobian terms are obtained by solving source vectors loaded with parabola coefficients used in the approximation to Maxwell's equations. Jacobian terms for the apparent resistivity ( p _a), the impedance phase (φ) and the vertical magnetic field ( K _zy) are then calculated utilizing the parallel- and auxiliary-field Jacobians. Comparison of Jacobian values obtained from reciprocity calculations and by differencing two forward solutions show that the reciprocity method is accurate and can be used to decrease the number of calculations required to obtain sensitivities by one to two orders of magnitude.     

A comparison of the stability of stationary and non-stationary magnetotelluric analysis methods

A Magnetotelluric (MT) sounding was carried out at a site in south-east Queensland, in the Clarence-Moreton Basin. The synoptic recordings were taken over a period of four months at sampling frequencies from 500 Hz to 5 × 10-5 Hz. The resulting data was analysed by the stationary cross-frequency and the Cone kernel time-frequency distribution (TFD) methods of MT analysis. The results were compared as apparent resistivities on a daily basis for frequencies above 1 Hz, as well as over all the available data. The TFD MT apparent-resistivity results were more stable and less noisy on an daily basis than the cross-frequency results. Similarly the TFD analysis gave less noisy results than the cross-frequency analysis when all available data was processed. Application of these new non-stationary analysis techniques to MT processing should decrease the bias error problem of the MT methods and so increase reliability and repeatability of MT soundings.     

The statistical distribution of magnetotelluric apparent resistivity and phase

The marginal distributions for the magnetotelluric (MT) magnitude squared response function (and hence apparent resistivity) and phase are derived from the bivariate complex normal distribution that describes the distribution of response function estimates when the Gauss–Markov theorem is satisfied and the regression random errors are normally distributed. The distribution of the magnitude squared response function is shown to be non-central chi-squared with 2 degrees of freedom, with the non-centrality parameter given by the squared magnitude of the true MT response. The standard estimate for the magnitude squared response function is biased, with the bias proportional to the variance and hence important when the uncertainty is large. The distribution reduces to the exponential when the expected value of the MT response function is zero. The distribution for the phase is also obtained in closed form. It reduces to the uniform distribution when the squared magnitude of the true MT response function is zero or its variance is very large. The phase distribution is symmetric and becomes increasingly concentrated as the variance decreases, although it is shorter-tailed than the Gaussian. The standard estimate for phase is unbiased. Confidence limits are derived from the distributions for magnitude squared response function and phase. Using a data set taken from the 2003 Kaapvaal transect, it is shown that the bias in the apparent resistivity is small and that confidence intervals obtained using the non-parametric delta method are very close to the true values obtained from the distributions. Thus, it appears that the computationally simple delta approximation provides accurate estimates for the confidence intervals, provided that the MT response function is obtained using an estimator that bounds the influence of extreme data.     

Lithology-derived structure classification from the joint interpretation of magnetotelluric and seismic models

Magnetotelluric and seismic methods provide complementary information about the resistivity and velocity structure of the subsurface on similar scales and resolutions. No global relation, however, exists between these parameters, and correlations are often valid for only a limited target area. Independently derived inverse models from these methods can be combined using a classification approach to map geologic structure. The method employed is based solely on the statistical correlation of physical properties in a joint parameter space and is independent of theoretical or empirical relations linking electrical and seismic parameters. Regions of high correlation (classes) between resistivity and velocity can in turn be mapped back and re-examined in depth section. The spatial distribution of these classes, and the boundaries between them, provide structural information not evident in the individual models. This method is applied to a 10 km long profile crossing the Dead Sea Transform in Jordan. Several prominent classes are identified with specific lithologies in accordance with local geology. An abrupt change in lithology across the fault, together with vertical uplift of the basement suggest the fault is sub-vertical within the upper crust.     

A magnetotelluric study of the Alpine Fault, New Zealand

Magnetotelluric soundings have been made at seven locations on a 4  km profile crossing the Alpine Fault in the South Island of New Zealand. The 'distortion' techniques of Groom & Bailey (1989 ) and Lilley (1998a , b ) have been used to derive regional apparent resistivity and phase curves that correspond to electromagnetic induction in orientations parallel and perpendicular to the fault. 2-D inversion of the regional responses reveals that a narrow (<1  km wide) conductive zone is associated with the Alpine Fault. This conductor is most probably related to the heating of deep circulating meteoric water in a region in which enhanced temperatures occur at shallow depth due to the tectonic uplift of the Southern Alps.