Magnetotelluric data, stable distributions and impropriety: an existential combination

Author Posting. © Author, 2014. This article is posted here by permission of The Royal Astronomical Society for personal use, not for redistribution. The definitive version was published in Geophysical Journal International 198 (2014): 622-636, doi: 10.1093/gji/ggu121.The robust statistical model of a Gaussian core contaminated by outlying data that underlies robust estimation of the magnetotelluric (MT) response function has been re-examined. The residuals from robust estimators are systematically long tailed compared to a distribution based on the Gaussian, and hence are inconsistent with the robust model. Instead, MT data are pervasively described by the alpha stable distribution family whose variance and sometimes mean are undefined. A maximum likelihood estimator (MLE) that exploits the stable nature of MT data is formulated, and its two-stage implementation in which stable parameters are first fit to the data and then the MT responses are solved for is described. The MLE is shown to be inherently robust, but differs from the conventional robust estimator because it is based on a model derived from the data, while robust estimators are ad hoc, being based on the robust model that is inconsistent with actual data. Propriety versus impropriety of the complex MT response was investigated, and a likelihood ratio test for propriety and its null distribution was established. The Cramér-Rao lower bounds for the covariance matrix of proper and improper MT responses were specified. The MLE was applied to exemplar long period and broad-band data sets from South Africa. Both are shown to be significantly stably distributed using the Kolmogorov–Smirnov goodness of fit and Ansari-Bradley non-parametric dispersion tests. Impropriety of the MT responses at both sites is pervasive, hence the improper Cramér-Rao bound was used to estimate the MLE covariance. The MLE is shown to be nearly unbiased and well described by a Gaussian distribution based on bootstrap simulation. The MLE was compared to a conventional robust estimator, establishing that the standard errors of the former are systematically smaller than for the latter and that the standardized differences between them exhibit excursions that are both too frequent and too large to be described by a Gaussian model. This is ascribed to pervasive bias of the robust estimator that is to some degree obscured by their systematically large confidence bounds. Finally, a series of topics for further investigation is proposed.This work was supported by NSF grant EAR0809074

On the statistics of magnetotelluric rotational invariants

Author Posting. © Author, 2013. This article is posted here by permission of Oxford University Press on behalf of The Royal Astronomical Society for personal use, not for redistribution. The definitive version was published in Geophysical Journal International 196 (2014): 111-130, doi:10.1093/gji/ggt366.The statistical properties of the Swift skew, the phase-sensitive skew and the WAL invariants I1−I7 and Q are examined through analytic derivation of their probability density functions and/or simulation based on a Gaussian model for the magnetotelluric response tensor. The WAL invariants I1−I2 are shown to be distributed as a folded Gaussian, and are statistically well behaved in the sense that all of their moments are defined. The probability density functions for Swift skew, phase-sensitive skew and the WAL invariants I3−I4, I7 and Q are derived analytically or by simulation, and are shown to have no moments of order 2 or more. Since their support is semi-infinite or infinite, they cannot be represented trigonometrically, and hence are inconsistent with a Mohr circle interpretation. By contrast, the WAL invariants I5−I6 are supported on [ − 1, 1], and are inferred to have a beta distribution based on analysis and simulation. Estimation of rotational invariants from data is described using two approaches: as the ratio of magnetotelluric responses that are themselves averages, and as averages of section-by-section estimates of the invariant. Confidence intervals on the former utilize either Fieller's theorem, which is preferred because it is capable of yielding semi-infinite or infinite confidence intervals, or the less accurate delta method. Because section-by-section averages of most of the rotational invariants are drawn from distributions with infinite variance, the classical central limit theorem does not pertain. Instead, their averaging is accomplished using the median in place of the mean for location and an order statistic model to bound the confidence interval of the median. An example using real data demonstrates that the ratio of averages approach has serious systematic bias issues that render the result physically inconsistent, while the average of ratios result is a smooth, physically interpretable function of period, and is the preferred approach.Supported by NSF grant EAR101518

A note about Gaussian statistics on a sphere

Author Posting. © The Author, 2015. This article is posted here by permission of The Royal Astronomical Society for personal use, not for redistribution. The definitive version was published in Geophysical Journal International 203 (2015): 893-895, doi:10.1093/gji/ggv324.The statistics of directional data on a sphere can be modelled either using the Fisher distribution that is conditioned on the magnitude being unity, in which case the sample space is confined to the unit sphere, or using the latitude–longitude marginal distribution derived from a trivariate Gaussian model that places no constraint on the magnitude. These two distributions are derived from first principles and compared. The Fisher distribution more closely approximates the uniform distribution on a sphere for a given small value of the concentration parameter, while the latitude–longitude marginal distribution is always slightly larger than the Fisher distribution at small off-axis angles for large values of the concentration parameter. Asymptotic analysis shows that the two distributions only become equivalent in the limit of large concentration parameter and very small off-axis angle

A multitaper spectral estimator for time-series with missing data

Author Posting. © The Authors, 2019. This article is posted here by permission of The Royal Astronomical Society for personal use, not for redistribution. The definitive version was published in Geophysical Journal International 218(3), (2019): 2165-2178, doi: 10.1093/gji/ggz280.A multitaper estimator is proposed that accommodates time-series containing gaps without using any form of interpolation. In contrast with prior missing-data multitaper estimators that force standard Slepian sequences to be zero at gaps, the proposed missing-data Slepian sequences are defined only where data are present. The missing-data Slepian sequences are frequency independent, as are the eigenvalues that define the energy concentration within the resolution bandwidth, when the process bandwidth is [−1/2,1/2) for unit sampling and the sampling scheme comprises integer multiples of unity. As a consequence, one need only compute the ensuing missing-data Slepian sequences for a given sampling scheme once, and then the spectrum at an arbitrary set of frequencies can be computed using them. It is also shown that the resulting missing-data multitaper estimator can incorporate all of the optimality features (i.e. adaptive-weighting, F-test and reshaping) of the standard multitaper estimator, and can be applied to bivariate or multivariate situations in similar ways. Performance of the missing-data multitaper estimator is illustrated using length of day, seafloor pressure and Nile River low stand time-series.The length of day utilized in Section 3 are available from http://hpiers.obspm.fr. The pressure data used in Section 4 are available from https://doi.org/10.1029/2018JC014586. A Matlab function MDmwps.m to compute missing-data power spectra is available from the Mathworks file exchange website. The author thanks Jeff Park and editor F.J. Simons for thorough reviews. This work was supported by an Internal Research and Development award at WHOI, and by the Walter A. and Hope Noyes Smith Chair for Excellence in Oceanography

Applications of time series analysis to geophysical data

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution June, 1980This thesis consists of three papers applying the techniques of time series analysis to geophysical data. Surface wave dispersion along the Walvis Ridge, South Atlantic Ocean, is obtained by bandpass filtering the recorded seismogram in the frequency domain. The group velocity is anomalously low in the period range of 15-50 s, and formal inversion of the data indicates both crustal thickening to 12.5 km and low shear velocity (4.25-4.35 km/s) to depths of 40-50 km. The electromagnetic induction fields at a deep ocean site northeast of Hawaii were used to determine the electrical conductivity of the earth to 400 km depth. Singular value decomposition of the data matrix indicates three degrees of freedom, suggesting source field complications and a two dimensional conductive structure. Inversion of one of the principal terms in the response function shows an abrupt rise in electrical conductivity to 0.05 mho/m near 160 km with no resolvable decrease below this. A model study suggests that moving source fields influence the induction appreciably in the other principal response tunction. A set of piston cores from the northeast Atlantic Ocean are used to construct paleomagnetic time series covering the interval 25-127 kybp. Stratigraphic control is provided by counts of planktonic toraminifera, and empirical orthogonal function analysis shows a significant decrease in sedimentation rate at the interglaciai/glacial transition. The sediments are magnetically stable and reliable relative paleointensity measurements could be obtained. Spectral analysis of the directions reveals a predominant 10 ky periodicity and no dominant looping direction.I was supported for the early parts of this work by a NSF Graduate Fellowship. The Walvis Ridge study was supported by the WHOI Education Office and the Defense Advanced Research Projects Agency. The induction study was funded by the NSF under grants OCE74-12730 and OCE77-8633, and by the WHOI Ocean Industries Program. The paleomagnetic study was supported by NSF contracts OCE77-82255 and ÖCE79-19258

The statistical distribution of magnetotelluric apparent resistivity and phase

Author Posting. © The Authors, 2007. This article is posted here by permission of John Wiley & Sons for personal use, not for redistribution. The definitive version was published in Geophysical Journal International 171 (2007): 127-132, doi:10.1111/j.1365-246X.2007.03523.x.The 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 Gauss–Markov 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.This work was supported by NSF grant EAR0309584

On the physics of frequency domain controlled source electromagnetics in shallow water, 2: transverse anisotropy

Author Posting. © The Authors, 2017. This article is posted here by permission of Oxford University Press for personal use, not for redistribution. The definitive version was published in Geophysical Journal International 211 (2017): 1046–1061, doi:10.1093/gji/ggx360.In recent years, marine controlled source electromagnetics (CSEM) has found increasing use in hydrocarbon exploration due to its ability to detect thin resistive zones beneath the seafloor. It is the purpose of this paper to evaluate the physics of CSEM for an ocean whose electrical thickness is comparable to or much thinner than that of the overburden using the in-line configuration through examination of the elliptically-polarized seafloor electric field, the time-averaged energy flow depicted by the real part of the complex Poynting vector, energy dissipation through Joule heating and the Fréchet derivatives of the seafloor field with respect to the sub-seafloor conductivity that is assumed to be transversely anisotropic, with a vertical-to-horizontal resistivity ratio of 3:1. For an ocean whose electrical thickness is comparable to that of the overburden, the seafloor electromagnetic response for a model containing a resistive reservoir layer has a greater amplitude and reduced phase as a function of offset compared to that for a halfspace, or a stronger and faster response, and displays little to no evidence for the air interaction. For an ocean whose electrical thickness is much smaller than that of the overburden, the electric field displays a greater amplitude and reduced phase at small offsets, shifting to a stronger amplitude and increased phase at intermediate offsets, and a weaker amplitude and enhanced phase at long offsets, or a stronger and faster response that first changes to stronger and slower, and then transitions to weaker and slower. By comparison to the isotropic case with the same horizontal conductivity, transverse anisotropy stretches the Poynting vector and the electric field response from a thin resistive layer to much longer offsets. These phenomena can be understood by visualizing the energy flow throughout the structure caused by the competing influences of the dipole source and guided energy flow in the reservoir layer, and the air interaction caused by coupling of the entire sub-seafloor resistivity structure with the sea surface. The Fréchet derivatives are dominated by preferential sensitivity to the vertical conductivity in the reservoir layer and overburden at short offsets. The horizontal conductivity Fréchet derivatives are weaker than to comparable to the vertical derivatives at long offsets in the substrate. This means that the sensitivity to the horizontal conductivity is present in the shallow parts of the subsurface. In the presence of transverse anisotropy, it is necessary to go to higher frequencies to sense the horizontal conductivity in the overburden as compared to an isotropic model with the same horizontal conductivity. These observations in part explain the success of shallow towed CSEM using only measurements of the in-line component of the electric field.This work was supported at WHOI by an Independent Research and Development award, and by the Walter A. and Hope Noyes Smith Chair for Excellence in Oceanography

Applications of time series analysis to geophysical data

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Earth and Planetary Sciences, 1980.Microfiche copy available in Archives and Science.Vita.Includes bibliographies.by Alan Dana Chave.Ph.D

Correction of shallow-water electromagnetic data for noise induced by instrument motion

Author Posting. © The Authors, 2004. This is the author's version of the work. It is posted here by permission of Society of Exploration Geophysicists for personal use, not for redistribution. The definitive version was published in Geophysics 70 (2005): G127–G133, doi:10.1190/1.2080748.An unexpected noise source was found in magnetic and sometimes electric field data recorded on the bottom of lakes in the Archean Slave craton (NW Canada) during warm seasons. The noise is due to instrument motion and in some instances direct induction by wind-driven surface gravity waves when the lakes are not ice covered. The noise can be reduced or eliminated by pre-filtering the data with an adaptive correlation noise cancelling filter using instrument tilt records, prior to estimation of magnetotelluric (MT) response functions. Similar effects are to be expected in other shallow water environments, and the adaptive correlation canceller is a suitable method to pre-process MT data to reduce motional noise in the magnetic field. This underscores the importance of ancillary tilt measurements in shallow water MT surveys. In coastal or lake bottom surveys, special efforts to reduce hydrodynamic effects on the instrument should also be pursued.This project was funded by NSF grant EAR-9725556 and EAR-0087699. P.L. ac- knowledges the Fundacion Andes for a postdoctoral grant

High-Q spectral peaks and nonstationarity in the deep ocean infragravity wave band: Tidal harmonics and solar normal modes

Author Posting. © American Geophysical Union, 2019. This article is posted here by permission of American Geophysical Union for personal use, not for redistribution. The definitive version was published in Journal of Geophysical Research-Oceans 124(3), (2019):2072-2087, doi:10.1029/2018JC014586.Infragravity waves have received the least study of any class of waves in the deep ocean. This paper analyzes a 389‐day‐long deep ocean pressure record from the Hawaii Ocean Mixing Experiment for the presence of narrowband (≲2 μHz) components and nonstationarity over 400–4,000 μHz using a combination of fitting a mixture noncentral/central χ2 model to spectral estimates, high‐resolution multitaper spectral estimation, and computation of the offset coherence between distinct frequencies for a given data segment. In the frequency band 400–1,000 μHz there is a noncentral fraction of 0.67 ± 0.07 that decreases with increasing frequency. Evidence is presented for the presence of tidal harmonics in the data over the 400‐ to 1,400‐μHz bands. Above ~2,000 μHz the noncentral fraction rises with frequency, comprising about one third of the spectral estimates over 3,000–4,000 μHz. The power spectrum exhibits frequent narrowband peaks at 6–11 standard deviations above the noise level. The widths of the peaks correspond to a Q of at least 1,000, vastly exceeding that of any oceanic or atmospheric process. The offset coherence shows that the spectral peaks have substantial (p = 0.99–0.9999) interfrequency correlation, both locally and between distinct peaks within a given analysis band. Many of the peak frequencies correspond to the known values for solar pressure modes that have previously been observed in solar wind and terrestrial data, while others are the result of nonstationarity that distributes power across frequency. Overall, this paper documents the existence of two previously unrecognized sources of infragravity wave variability in the deep ocean.This work was supported at WHOI by an Independent Research and Development award and the Walter A. and Hope Noyes Smith Chair for Excellence in Oceanography. At the University of Hawaii, Martin Guiles provided a number of consequential data analyses, and work was supported by NSF‐OCE1460022. D. J. T. acknowledges support from Queen's University and NSERC. The data used in this study are available from the supporting information.2019-08-2