Comparison of stable maximum likelihood estimator with traditional robust estimator in magnetotelluric impedance estimation
Introduction
The magnetotelluric (MT) method has been proved to be an effective geophysical method to image subsurface electrical conductivity. It has been widely used in tectonic studies (Xiang et al., 2018; Guo et al., 2019; DeLucia et al., 2019), geohazard investigation (Foudili et al., 2019; Pazzi et al., 2019), geothermal exploration (Blake et al., 2016; Gao et al., 2018), and mineral exploration (Piña-Varas and Dentith, 2018; Zhang and Li, 2019). Due to the unknown source, the time varying MT data are Fourier transformed to estimate the MT impedance tensor, which is further interpreted to obtain the subsurface structure. Thus, the impedance estimation is considered as an important step for MT exploration (Kelbert et al., 2019).
In early time, the impedance tensor is estimated from Fourier-transformed time series based on the least square method by minimizing the sum of square of residuals (Sims et al., 1971). However, the least square method can be affected by noisy input magnetic data and produce biased estimation. To avoid the bias caused by the magnetic noise, the remote reference method (Goubau et al., 1978; Gamble et al., 1979) is proposed. This method can handle the input noise efficiently if the noise of remote sites is uncorrelated with local one. Even with the remote reference, the least square method can be strongly affected by the outliers in the data. These can be eliminated by robust methods (Egbert and Booker, 1986; Chave et al., 1987), which is now in common use in the MT application. Later on different robust variants are developed to handle different challenges in the MT impedance estimation(Siegel, 1982; Chave and Thomson, 1989; Egbert and Livelybrooks, 1996; Egbert, 1997; Rousseeuw and Leroy, 2003; Chave and Thomson, 2004).
Now, various estimators for the MT impedance tensor are available, which makes the MT practitioner hard to choose a particular estimator for their specific applications. Jones et al. (1989) had a complete comparison of different traditional techniques for MT impedance estimation. In their work, they think the remote reference robust estimator should be always applied in MT impedance estimation. At the same time, they noted that even with the remote reference the estimated impedance could still be biased.
Most recently, the realization of the persistent long-tailed data residual from traditional estimators leads to the development of a stable maximum likelihood estimator (MLE) (Chave, 2014) to obtain the precise MT impedance tensor, which results in data residuals with stable distributed. Based on real data, the work compares the results from both the traditional robust estimator and stable MLE. The conclusion is that the robust estimator is biased with the underestimated error estimation, while the stable MLE solution is unbiased based on statistic tests (Chave, 2017). However, in their work, the real impedance and its errors are unknown, making their conclusions unjustified, which is the primary goal of this paper.
In this paper, we compare the stable MLE with the traditional robust method in processing several MT stations from southeast Tibet Plateau collected under the SinoProbe project (Dong et al., 2013, Dong et al., 2016). We generate synthetic MT data with errors of different distributions (e.g. Gaussian and Cauchy distributions) based on the estimated MT impedance and measured magnetic data (as the input) for one site in this area. Then we investigate the performance of the two methods and their variants on processing the synthetic data with different noise pattern. Finally, they are applied to process two real sites in the same area to further demonstrate their performance.
Section snippets
Stable distributions
The stable distributions, allowing skews and heavy tails, were firstly introduced by Lévy (1925) and later used to describe the error distributions of MT data (Chave, 2014). If a random process X satisfies a stable distribution, its characteristic function which is the inverse Fourier transform of the probability density function (pdf), can be given bywhere ξ represents the expectation, and the four parameterization
Results and analysis
A wide coverage of MT data has been collected from 2010 to 2012 at the support of the China-wide, multi-discipline geophysical deep probing project SinoProbe (Dong et al., 2013, Dong et al., 2016). Among them, we consider several broadband MT stations with good data quality from southeast Tibet Plateau collected by Phoenix MTU-5 systems. The electromagnetic time series are collected in the geomagnetic coordinates (with x and y positively pointing north and east, respectively) and the typical
Conclusion
In this paper, we make a complete comparison of the traditional robust method and stable MLE estimator. For this purpose, we generate synthetic MT data with desired errors distributions to model different real measurement errors, based on the estimated MT impedance and measured magnetic data for one site from southeast Tibet Plateau collected at the support of SinoProbe project. The performance of the two methods is completely tested based on the synthetic data. Then they are applied to process
Declaration of Competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was financially supported by the National Natural Science Foundation of China (41674079, 41674080, 41404061 and 41904158), the Fundamental Research Funds for the Central Universities of Central South University (2019zzts159) and China Postdoctoral Science Foundation funded project (2019M652385). The STABLE software package at www.RobustAnalysis.com is used in this work, and authors thanks their agreement for free use.
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