Non Stationary Magnetotelluric Data Processing

Abstract

Studies have proven that the desired signal for Magnetotellurics (MT) in the electromagnetic (EM) field can be regarded as 'quasi stationary' (i.e. sufficiently stationary to apply a windowed Fourier transform). However, measured time series often contain environmental noise. Hence, they may not fulfill the stationarity requirement for the application of the Fourier Transform (FT) and therefore may lead to false or unreliable results under methods that rely on the FT. In light of paucity of algorithms of MT data processing in the presence of non stationary noise, it is the goal of this thesis to elaborate a robust, non stationary algorithm, which can compete with sophisticated, state-of-the-art algorithms in terms of accuracy and precision. In addition, I proof mathematically the algorithm's viability and validate its superiority to other codes processing non stationary, synthetic and real MT data.

Non stationary EM data may affect the computation of Fourier spectra in unforeseeable manners and consequently, the traditional estimation of the MT transfer functions (TF). The TF estimation scheme developed in this work is based on an emerging nonlinear, non stationary time series analysis tool, called Empirical Mode Decomposition (EMD). EMD decomposes time series into Intrinsic Mode Functions (IMF) in the time-frequency domain, which can be represented by the instantaneous parameters amplitude, phase and frequency. In the first part of my thesis, I show that time slices of well defined IMFs equal time slices of Fourier Series, where the instantaneous parameters of the IMF define amplitude and phase of the Fourier Series parameters. Based on these findings I formulate the theorem that non stationary convolution of an IMF with a general time domain response function translates into a multiplication of the IMF with the respective spectral domain response function, which is explicitly permitted to vary over time. Further, I employ real world MT data to illustrate that a de-trended signal's IMFs can be convolved independently and then be used for further time-frequency analysis as done for MT processing.

In the second part of my thesis, I apply the newly formulated theorem to the MT method. The MT method analyses the correlation between the electric and magnetic field due to the conductivity structure of the subsurface. For sufficiently low frequencies (i.e. when the EM field interacts diffusively), the conductive body of the Earth acts as an inductive system response, which convolves with magnetic field variations and results in electric field variations. The frequency representation of this system response is commonly referred to as MT TF and its estimation from measured electric and magnetic time series is summarized as MT processing. The main contribution in this thesis is the design of the MT TF estimation algorithm based on EMD. In contrast to previous works that employ EMD for MT data processing, I (i) point out the advantages of a multivariate decomposition, (ii) highlight the possibility to use instantaneous parameters, and (iii) define the homogenization of frequency discrepancies between data channels. In addition, my algorithm estimates the transfer functions using robust statistical methods such as (i) robust principal component analysis and (ii) iteratively re-weighted least squares regression with a Huber weight function. Finally, TF uncertainties are estimated by iterating the complete robust regression, including the robust weight computation, by means of a bootstrap routine.

The proposed methodology is applied to synthetic and real data with and without non stationary character and the results are compared with other processing techniques. I conclude that non stationary noise can heavily affect Fourier based MT data processing but the presented non stationary approach is nonetheless able to extract the impedances correctly even when the other methods fail.