On the robust estimation of power spectra, coherences, and transfer functions

Abstract

Robust estimation of power spectra, coherences, and transfer functions is investigated in the context of geophysical data processing. The methods described are frequency-domain extensions of current techniques from the statistical literature and are applicable in cases where section-averaging methods would be used with data that are contaminated by local nonstationarity or isolated outliers. The paper begins with a review of robust estimation theory, emphasizing statistical principles and the maximum likelihood or M-estimators. These are combined with section-averaging spectral techniques to obtain robust estimates of power spectra, coherences, and transfer functions in an automatic, data-adaptive fashion. Because robust methods implicitly identify abnormal data, methods for monitoring the statistical behavior of the estimation process using quantile-quantile plots are also discussed. The results are illustrated using a variety of examples from electromagnetic geophysics. INTRODUCTIONReliable estimation of power spectra for single data sequences or of transfer functions and coherences between multiple time series is of central importance in many areas of geophysics and engineering. While the effects of the underlying Gaussian distributional assumptions on such estimates are generally understood, the ability of a small fraction of non-Gaussian noise or localized nonstationarity to affect them is not. These phenomena can destroy conventional estimates, often in a manner that is difficult to detect.Problems with conventional (i.e., nonrobust) time series procedures arise because they are essentially copies of classical statistical procedures parameterized by frequency. Once Fourier transforms are taken, estimating a spectrum is the same process as computing a variance, and estimating a transfer function is a similar procedure to linear regression. Because these methods are based on the least squares or Gaussian maximum likelihood approaches to statistical inference, their advantages include simplicity and the optimality properties established by the Gauss- Paper number 5B5911 0148-0227/87/005B-5911505.00 residuals are drawn from a multivariate normal probability distribution, then the least squares result is also a maximum likelihood, fully efficient, minimum variance estimate. In practice, the regression model is rarely an accurate description due to departures of the data from the model requirements. Most data contain a small fraction of unusual observations or "outliers" that do not fit the model distribution or share the characteristics of the bulk of the sample. These can often be described by a probability distribution which has a nearly Gaussian shape in the center and tails which are heavier than would be expected for a normal one, or by mixtures of Gaussian distributions with different variances.Two forms of data outliers are common: point defects and local nonstationarity. Point defects are isolated outliers that exist independent of the structure of the process under study. In this paper the principles o...

A study of transient variations in the Earth's electromagnetic field at equatorial electrojet latitudes in western Africa (Mali and the Ivory Coast)

“…Apparent resistivity values were computed with the odiagonal term yx of the impedance tensor estimates obtained using the robust method for magnetotelluric data processing proposed by Chave et al (1987). Both night-time and daytime impedances are estimated with tens of samples from a period ranging from December 1992 to March 1993.…”

New application of wavelets in magnetotelluric data processing: reducing impedance bias

“…Outliers may originate from several issues including non-stationarity, the source effect of MT signals and brief overwhelming noise. Robust statistics were then applied to MT processing to accommodate these outliers (Chave et al 1987;Egbert and Booker 1986). Another approach was the use of wavelet techniques to look for MT sources or remove noise in the time series.…”

“…At each scale for each pair of geomagnetic events, the response function is estimated by using the Huber M-estimator described by Chave et al (1987) in a MT context. From the distribution of response functions at each scale estimated from all available pairs of geomagnetic events, we represent the final estimation by taking the median value (less sensitive to outliers).…”

“…To assess the validity of our method, we have compared our results with the ones obtained with robust processing codes from the work of Alan D. Chave (BIRRP, Bounded Influence Remote Reference Processing, Chave et al 1987;Chave and Thomson 2004) and Gary Egbert (EMTF, Egbert 1997). For simplicity, we consider error bars as they were calculated by the robust processing codes, even if they can be discussed in more detail Chave 2015;Wawrzyniak et al 2015).…”

Electrical conductivity images of Quaternary faults and Tertiary detachments in the California Basin and Range

“…Data were recorded for periods of up to one month per site. The time series for these sites were analyzed for impedance tensors and magnetic transfer functions using robust processing [Chave et al, 1987]. When the long period impedances are compared to the broadband ones, the former are clearly smoother and less noisy at periods longer than 100 s ( Figure A2).…”