Non Stationary Magnetotelluric Data Processing

[eng] 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

On the Utility of Horizontal-to-Vertical Spectral Ratios of Ambient Noise in Joint Inversion with Rayleigh Wave Dispersion Curves for the Large-N Maupasacq Experiment

Horizontal-to-Vertical Spectral Ratios (HVSR) and Rayleigh group velocity dispersion curves (DC) can be used to estimate the shallow S-wave velocity (VS) structure. Knowing the VS structure is important for geophysical data interpretation either in order to better constrain data inversions for P-wave velocity (VP) structures such as travel time tomography or full waveform inversions or to directly study the VS structure for geo-engineering purposes (e.g., ground motion prediction). The joint inversion of HVSR and dispersion data for 1D VS structure allows characterising the uppermost crust and near surface, where the HVSR data (0.03 to 10s) are most sensitive while the dispersion data (1 to 30s) constrain the deeper model which would, otherwise, add complexity to the HVSR data inversion and adversely affect its convergence. During a large-scale experiment, 197 three-component short-period stations, 41 broad band instruments and 190 geophones were continuously operated for 6 months (April to October 2017) covering an area of approximately 1500km2 with a site spacing of approximately 1 to 3km. Joint inversion of HVSR and DC allowed estimating VS and, to some extent density, down to depths of around 1000m. Broadband and short period instruments performed statistically better than geophone nodes due to the latter’s gap in sensitivity between HVSR and DC. It may be possible to use HVSR data in a joint inversion with DC, increasing resolution for the shallower layers and/or alleviating the absence of short period DC data, which may be harder to obtain. By including HVSR to DC inversions, confidence improvements of two to three times for layers above 300m were achieved. Furthermore, HVSR/DC joint inversion may be useful to generate initial models for 3D tomographic inversions in large scale deployments. Lastly, the joint inversion of HVSR and DC data can be sensitive to density but this sensitivity is situational and depends strongly on the other inversion parameters, namely VS and VP. Density estimates from a HVSR/DC joint inversion should be treated with care, while some subsurface structures may be sensitive, others are clearly not. Inclusion of gravity inversion to HVSR/DC joint inversion may be possible and prove useful

Horizontal-to-Vertical Spectral Ratio of Ambient Vibration Obtained with Hilbert–Huang Transform

The Horizontal-to-Vertical Spectral Ratio (HVSR) of ambient vibration measurements is a common tool to explore near surface shear wave velocity (Vs) structure. HVSR is often applied for earthquake risk assessments and civil engineering projects. Ambient vibration signal originates from the combination of a multitude of natural and man-made sources. Ambient vibration sources can be any ground motion inducing phenomena, e.g., ocean waves, wind, industrial activity or road traffic, where each source does not need to be strictly stationary even during short times. Typically, the Fast Fourier Transform (FFT) is applied to obtain spectral information from the measured time series in order to estimate the HVSR, even though possible non-stationarity may bias the spectra and HVSR estimates. This problem can be alleviated by employing the Hilbert–Huang Transform (HHT) instead of FFT. Comparing 1D inversion results for FFT and HHT-based HVSR estimates from data measured at a well studied, urban, permanent station, we find that HHT-based inversion models may yield a lower data misfit χ2 by up to a factor of 25, a more appropriate Vs model according to available well-log lithology, and higher confidence in the achieved mode

On the effect of non stationary (synthetic) sources in the magnetotelluric method

European Geosciences Union General Assembly 2013, 7-12 April, Vienna, AustriaA new, non stationary scheme for the statistical magnetotelluric (MT) transfer function estimation is used to assess the effects of non stationary noise in MT data processing. The scheme uses Empirical Mode Decomposition (EMD) to process spectral data and is referred to as Empirical-mode-decomposition-based Magneto-Telluric processing (EMT). We compare EMT with BIRRP by Chave and Thomson [2004], a traditional and efficient processing code based on the Fourier Transform. Two tests are performed, first, synthetic, non stationary data is constructed from two non stationary sources to demonstrates the inability of a Fourier based method to deal with non stationary sources. Then, secondly, these sources are used as a noise source. The non stationary noise is added only to the electric fields and leaves the magnetic and remote channels completely unaffected. Therefore, we can show that the computation of the spectra via Fourier Transform fails, because uncorrelated stationary noise in the spectra should be cleaned by the remote referencing technique, which is applied in the test. Since any uncorrelated (random) non stationary noise acts as any random stationary noise and does not affect the measurements other than decreasing the confidence in the results (larger error bars), this test shows that the mere fact that the added noise is non stationary affects the estimated results by a Fourier Transform based method or even makes it impossible to extract reasonable transfer functions, whereas the EMT algorithm is able to deal with the non stationarity and allows a more precise estimation to a lower signal-to-noise ratio. In summary, we show that non stationary sources can heavily impact on traditional MT processing routines which rely on the Fourier Transform but that this effect can be diminished by relying on a purely non stationary analysis. The non stationary source is specifically designed to disturb the Fourier Transform and to break its assumptions, however the results provide an insight in how bad real non stationary noise can affect MT measurements and encourage to verify these findings on a real world problem with data that is suspected to contain, in particular, non stationary noise, e.g. data that is acquired close to train lines, mining shafts and elevators or electric fencesPeer Reviewe

EMT - Empirical-mode-decomposition-based Magneto-Telluric Processing

European Geosciences Union General Assembly 22-27 April 2012, Vienna, Austria.-- 1 pageWe present a new Magneto-Telluric (MT) data processing scheme based on an emerging non linear, non stationary time series analysis tool, called the Empirical Mode Decomposition (EMD) or Hilbert-Huang Transform (HHT), to transform data into a non-stationary frequency domain and a robust principal component regression to estimate the most likely MT transfer functions from the data with the 2- confidence intervals computed by a bootstrap algorithm. Optionally, data quality can be controlled by a physical coherence and a signal power filter. MT sources are assumed to be quasi stationary and therefore a (windowed) Fourier Transform is often ap- plied to transform the time series into the frequency domain in which Transfer Functions (TF) are defined between the electromagnetic field components. This assumption can break down in the presence of noise or when the sources are non stationary, and then TF estimates can become unreliable when obtained through a stationary transform like the Fourier transform. Our TF estimation scheme naturally deals with non stationarity without introducing artifacts and, therefore, potentially can distinguish quasi-stationary sources and non-stationary noise. In contrast to previous works on using HHT for MT processing, we argue the necessity of a multivariate EMD to model the MT problem physically correctly and highlight the resulting possibility to use instantaneous parameters as independent and identically distributed variables. Furthermore, we define a homogenization between data channels of frequency discrepancies due to non stationarity and noise. The TF estimation in the frequency domain bases on a robust principal component analysis in order to find two source polarizations. These two principal components are used as predictor to regress robustly the data channels within a bootstrap algorithm to estimate the Earth¿s Transfer function with 2-¿ confidence interval supplied by the measured data.The scheme can be used with and without aid by any number of remote reference stations. The performance of this scheme will be demonstrated on MT data and compared with BIRRP, a widely used MT processing software by Alan ChavePeer Reviewe

Frequency Shift in the Convolution of Non Stationary Time Series

International Workshop in Recent Advances in Time Series Analysis (RATS), 9-12 June 2012, Protaras, CyprusThe Hilbert-Huang Transform (HHT, [1]) is a novel tool to analyze non stationary time series and describe them with their instantaneous, spectral information. HHT decomposes a time series into a number of zeromean, oscillatory modes (Intrinsic Mode Functions, IMF) in order to ensure existence of an interpretable analytic signal of each IMF. Then, it is possible to express the analytic signal in terms of time series of the instantaneous parameters: amplitude, phase and frequency. Therefore, each IMF resides in the timefrequency domain and is described by amplitude and phase as a function of time. In [2] we show that the convolution of IMFs with any spectral response function translates into a complex multiplication of the IMF with that response function. Since the convolution of a (non) stationary time series with a spectral response function in the time domain can be transformed into a basic algebraic formulation, in this work we focus on the repercussions of a convolution of non stationary signals by analyzing the instantaneous parameters and their time derivations of the resulting convolved signal. Most notably, we find that there can be a frequency shift in the resulting signal with respect to the original signal depending on the degree of non stationarity. This finding may be important for non stationary time series, which are filtered by a system response for technical reasons, as it is often the case for physical measurements. However, the work is in a preliminary state and only carried out in the theoryPeer Reviewe

Inverting Capacitive Resistivity (Line Electrode) Measurements with Direct Current Inversion Programs

11 pages, 11 figuresThe capacitive resistivity (CR) method is a time- and labor-saving alternative to traditional direct current (DC) resistivity methods. The line electrode variant of CR suffers from the absence of data inversion programs as available for the DC resistivity method. Direct current inversion programs were applied to determine the resistivity distribution from CR measurements using an approximately equivalent four-point dipole-dipole configuration. We optimized configurations to minimize the systematic error applying DC inversion programs to CR, using data based on the comparison of the two-dimensional sensitivities of the proposed DC approximations. The optimal four-point dipole-dipole geometry has a dipole length of 80%. © Soil Science Society of AmericaPeer Reviewe