Maximum likelihood estimation of magnitude-squared multiple and ordinary coherence

Walden, A. T.

doi:10.1016/0165-1684(90)90008-m

Cited by 7 publications

(6 citation statements)

References 10 publications

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“…The distribution functions for the coherence square estimator of two Gaussian processes are complicated expressions involving gamma functions and generalized hypergeometric functions. Some approximations are given by Walden [1990b]. The validity of these expressions has been verified experimentally in Monte Carlo experiments [Guarnieri and Prati, 1997].…”

Section: B4 Statistics Of Multiwindow Coherence Estimatesmentioning

confidence: 99%

Spatiospectral localization of isostatic coherence anisotropy in Australia and its relation to seismic anisotropy: Implications for lithospheric deformation

2003

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[1] We investigate the two-dimensional (2-D) nature of the coherence between Bouguer gravity anomalies and topography on the Australian continent. The coherence function or isostatic response is commonly assumed to be isotropic. However, the fossilized strain field recorded by gravity anomalies and their relation to topography is manifest in a degree of isostatic compensation or coherence which does depend on direction. We have developed a method that enables a robust and unbiased estimation of spatially, directionally, and wavelength-dependent coherence functions between two 2-D fields in a computationally efficient way. Our new multispectrogram method uses orthonormalized Hermite functions as data tapers, which are optimal for spectral localization of nonstationary, spatially dependent processes, and do not require solving an eigenvalue problem. We discuss the properties and advantages of this method with respect to other techniques. We identify regions on the continent marked by preferential directions of isostatic compensation in two wavelength regimes. With few exceptions, the short-wavelength coherence anisotropy is nearly perpendicular to the major trends of the suture zones between stable continental domains, supporting the geological observation that such zones are mechanically weak. Mechanical anisotropy reflects lithospheric strain accumulation, and its presence must be related to the deformational processes affecting the lithosphere integrated over time. Threedimensional models of seismic anisotropy obtained from surface wave inversions provide an independent estimate of the lithospheric fossil strain field, and simple models have been proposed to relate seismic anisotropy to continental deformation. We compare our measurements of mechanical anisotropy with our own model of the azimuthally anisotropic seismic wave speed structure of the Australian lithosphere. The correlation of isostatic anisotropy with directions of fast wave propagation gleaned from the azimuthal anisotropy of surface waves decays with depth. This may support claims that above $200 km, internally coherent deformation of the entire lithosphere is responsible for the anisotropy present in surface wave speeds or split shear waves.

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Section: B4 Statistics Of Multiwindow Coherence Estimatesmentioning

confidence: 99%

Spatiospectral localization of isostatic coherence anisotropy in Australia and its relation to seismic anisotropy: Implications for lithospheric deformation

2003

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“…The appropriateness of using least squares is not something that can be taken for granted but rather needs to be carefully assessed, as was pointed out early on in this context by Dorman & Lewis (1972), Banks et al (1977), Stephenson & Beaumont (1980) and Ribe (1982), which, however, also focused on other issues that have since received more attention. Admittance and coherence are "statistics": functions of the data with non-Gaussian distributions even if the data themselves are Gaussian (Munk & Cartwright 1966;Carter et al 1973;Walden 1990; Thomson & Chave 1991;Touzi & Lopes 1996;Touzi et al 1999). Estimators for flexural rigidity based on any given method have their own distributions, though not necessarily ones with a tractable form.…”

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confidence: 99%

Maximum-likelihood estimation of lithospheric flexural rigidity, initial-loading fraction and load correlation, under isotropy

Simons

Olhede

2013

Geophysical Journal International

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Topography and gravity are geophysical fields whose joint statistical structure derives from interface-loading processes modulated by the underlying mechanics of isostatic and flexural compensation in the shallow lithosphere. Under this dual statistical-mechanistic viewpoint an estimation problem can be formulated where the knowns are topography and gravity and the principal unknown the elastic flexural rigidity of the lithosphere. In the guise of an equivalent "effective elastic thickness", this important, geographically varying, structural parameter has been the subject of many interpretative studies, but precisely how well it is known or how best it can be found from the data, abundant nonetheless, has remained contentious and unresolved throughout the last few decades of dedicated study. The popular methods whereby admittance or coherence, both spectral measures of the relation between gravity and topography, are inverted for the flexural rigidity, have revealed themselves to have insufficient power to independently constrain both it and the additional unknown initial-loading fraction and load-correlation factors, respectively. Solving this extremely ill-posed inversion problem leads to non-uniqueness and is further complicated by practical considerations such as the choice of regularizing data tapers to render the analysis sufficiently selective both in the spatial and spectral domains. Here, we rewrite the problem in a form amenable to maximum-likelihood estimation theory, which we show yields unbiased, minimum-variance estimates of flexural rigidity, initial-loading fraction and load correlation, each of those separably resolved with little a posteriori correlation between their estimates. We are also able to separately characterize the isotropic spectral shape of the initial-loading processes. Our procedure is well-posed and computationally tractable for the two-interface case. The resulting algorithm is validated by extensive simulations whose behavior is well matched by an analytical theory with numerous tests for its applicability to real-world data examples.

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“…The coherence estimate is asymptotically Gaussian (Carter et al, 1973;Touzi and Lopes, 1996;Walden, 1990). The grand average is the constant denoted γ 2 .…”

Section: Anisotropy Tests For Coherence and T Ementioning

confidence: 99%

On the robustness of estimates of mechanical anisotropy in the continental lithosphere: A North American case study and global reanalysis

Kalnins

Simons

Kirby

et al. 2015

Earth and Planetary Science Letters

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Publisher's copyright statement: NOTICE: this is the author's version of a work that was accepted for publication in Earth and Planetary Science Letters. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reected in this document. Changes may have been made to this work since it was submitted for publication. Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. AbstractLithospheric strength variations both influence and are influenced by many tectonic processes, including orogenesis and rifting cycles. The long, complex, and highly anisotropic histories of the continental lithosphere might lead to a natural expectation of widespread mechanical anisotropy. Anisotropy in the coherence between topography and gravity anomalies is indeed often observed, but whether it corresponds to an elastic thickness that is anisotropic remains in question. If coherence is used to estimate flexural strength of the lithosphere, the null-hypothesis of elastic isotropy can only be rejected when there is significant anisotropy in both the coherence and the elastic strengths derived from it, and if interference from anisotropy in the data themselves can be plausibly excluded. We consider coherence estimates made using multitaper and wavelet methods, from which estimates of effective elastic thickness are derived. We develop a series of statistical and geophysical tests for anisotropy, and specifically evaluate the potential for spurious results with synthetically generated data. Our primary case study, the North American continent, does not exhibit meaningful anisotropy in its mechanical strength. Similarly, a global reanalysis of continental gravity and topography using multitaper methods produces only scant evidence for lithospheric flexural anisotropy.

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Maximum likelihood estimation of magnitude-squared multiple and ordinary coherence

Cited by 7 publications

References 10 publications

Spatiospectral localization of isostatic coherence anisotropy in Australia and its relation to seismic anisotropy: Implications for lithospheric deformation

Spatiospectral localization of isostatic coherence anisotropy in Australia and its relation to seismic anisotropy: Implications for lithospheric deformation

Maximum-likelihood estimation of lithospheric flexural rigidity, initial-loading fraction and load correlation, under isotropy

On the robustness of estimates of mechanical anisotropy in the continental lithosphere: A North American case study and global reanalysis

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