Gaussian character of the distribution of magnetotelluric results working in log-space

Fournier, Hugo G.; Febrer, J.

doi:10.1016/0031-9201(76)90031-5

Cited by 4 publications

(4 citation statements)

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“…On the basis of empirical analyses, Bentley (1973) and Fournier & Febrer (1976) claimed that apparent resistivity is log normally distributed, and this result has been widely cited. It is not difficult to understand this conclusion if it were derived from ordinary least-squares MT response function estimates, as would be standard practice in the 1970s.…”

Section: Discussionmentioning

confidence: 99%

The statistical distribution of magnetotelluric apparent resistivity and phase

Chave¹,

Lezaeta²

2007

Geophysical Journal International

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S U M M A R YThe 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 GaussMarkov 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.

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Section: Discussionmentioning

confidence: 99%

The statistical distribution of magnetotelluric apparent resistivity and phase

Chave¹,

Lezaeta²

2007

Geophysical Journal International

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“…Here our research intends on focusing a sharp boundary, following general Tikhonov regularization inversion within the framework of L 2 norm, which is justified by general Gaussian distribution of EM data (Fournier and Febrer 1976;Weaver et al 2000). In this paper, we propose a new stabilizing functional called the MSG functional, which adds stable constraints to help focusing a sharp boundary, and apply it to a unified form of the objective functional in the regularized inversion.…”

Section: Introductionmentioning

confidence: 99%

Regularized magnetotelluric inversion based on a minimum support gradient stabilizing functional

Xiang

Zhang

et al. 2017

Earth Planets Space

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Regularization is used to solve the ill-posed problem of magnetotelluric inversion usually by adding a stabilizing functional to the objective functional that allows us to obtain a stable solution. Among a number of possible stabilizing functionals, smoothing constraints are most commonly used, which produce spatially smooth inversion results. However, in some cases, the focused imaging of a sharp electrical boundary is necessary. Although past works have proposed functionals that may be suitable for the imaging of a sharp boundary, such as minimum support and minimum gradient support (MGS) functionals, they involve some difficulties and limitations in practice. In this paper, we propose a minimum support gradient (MSG) stabilizing functional as another possible choice of focusing stabilizer. In this approach, we calculate the gradient of the model stabilizing functional of the minimum support, which affects both the stability and the sharp boundary focus of the inversion. We then apply the discrete weighted matrix form of each stabilizing functional to build a unified form of the objective functional, allowing us to perform a regularized inversion with variety of stabilizing functionals in the same framework. By comparing the one-dimensional and twodimensional synthetic inversion results obtained using the MSG stabilizing functional and those obtained using other stabilizing functionals, we demonstrate that the MSG results are not only capable of clearly imaging a sharp geoelectrical interface but also quite stable and robust. Overall good performance in terms of both data fitting and model recovery suggests that this stabilizing functional is effective and useful in practical applications.

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“…The abstract of the paper by Fournier & Febrer [1976], referring to magnetotelluric soundings, states that "This distribution is Gaussian only if the logarithm of the resistivity values is used.…”

Section: The Statistical Distribution Of the Ground Resistivitymentioning

confidence: 99%

“…"On the basis of empirical analyses, Bentley (1973) and Fournier & Febrer (1976) claimed that apparent resistivity is log normally distributed, and this result has been widely cited." The authors conclude that -"However, the correct distribution for the apparent resistivity based on statistical theory is non-central χ² with 2 degrees of freedom, which is always shorter tailed than log-normal, especially as the non-centrality parameter (or the squared response function)…”

Section: The Statistical Distribution Of the Ground Resistivitymentioning

confidence: 99%

HVDC grounding electrode of Rio Madeira - Bipole 1

Freire¹

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My sincere thanks to my supervisor, Prof. Dr. Sueli Yoshinaga Pereira (UNICAMP), who accepted the challenge of guiding me along such a different path from my original formation;  to the Institute of Geosciences of the University of Campinas (IG-Unicamp), for the opportunity of developing this thesis, and to the lecturers who introduced me to the world of the Geosciences, particularly to Prof. Dr. Elson Paiva Oliveira, who was my first lecturer at IG-Unicamp;  to Prof. Antonio Padilha (INPE), also my lecturer, for the support to this research;  to Prof. Dr. José Antonio Jardini (USP), who gave to me the opportunity of developing a Research Project with FDTE (USP) and Eletronorte, concerning the application of geosciences knowledge for the project of HVDC grounding electrodes (2016-2017);  to Eng. Paulo Cesar Esmeraldo (XRTE) and Eng. John Grahan (State Grid), who respectively, indicated my name to develop electrode designs for the Ethiopia-Kenya (Africa) and Belo Monte (Brazil) HVDC projects;  to Eng. Daniel Kovarsky, who, back in 1985, introduced me to HVDC grounding electrode design, for the Itaipu HVDC project, and to Eng. Per Granstrom (ABB), who, in 2010, brought me back to this kind of project in the Rio Madeira, bipole 1, HVDC system;  to my parents Maria de Jesus B. da F. Freire and Paulo Edmundo Lisboa Freire (in memoriam) who gave to me the best education;  to my wife Solange Maria Trainotti, for the invaluable support along all the years devoted to this research, without which I would not have been able to complete this thesis. to my daughter Clara Werneck Freire and son Theo Trainotti Freire, to whom I hope this thesis is an example of how dreams can be achieved with commitment."Scientific knowledge is a body of statements of varying degrees of certaintysome most unsure, some nearly sure, but none absolutely certain."

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Gaussian character of the distribution of magnetotelluric results working in log-space

Cited by 4 publications

References 4 publications

The statistical distribution of magnetotelluric apparent resistivity and phase

The statistical distribution of magnetotelluric apparent resistivity and phase

Regularized magnetotelluric inversion based on a minimum support gradient stabilizing functional

HVDC grounding electrode of Rio Madeira - Bipole 1

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