The maximum entropy approach to inverse problems - spectral analysis of short data records and density structure of the Earth
Article Sidebar
Main Article Content
Abstract
The maximum entropy principle as described in the first, introductory part of the paper is applied to 2 problems: the estimation of the power spectrum from a finite number of values of the autocovariance function, and the determination of the density within the Earth from its mass, radius, and moment of inertia. In both cases the available information is given in terms of known values of linear functionals and the maximum entropy principle is used to derive a probability distribution for the values of the unknown function. The expectation value of the probability distribution for the spectral power is shown to be equal to the well-known maximum entropy power spectrum. The expectation value for the density within the Earth is in ― with respect to the few data used ― good agreement with that of accepted Earth models.
ARK: https://n2t.net/ark:/88439/y025164
Permalink: https://geophysicsjournal.com/article/87
Article Details
Authors who publish with this journal as of Vol. 63 agree to the following terms:
a. Authors share the copyright with this journal in equal parts (50% to the journal, 50% to the lead author), and grant the journal right of first publication, with the work after publication simultaneously licensed under Creative Commons Attribution License CC BY-NC-ND 4.0 that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
b. Authors may enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal, and a reference to this copyright notice.
c. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) during the submission process, as this can lead to productive exchanges and earlier and greater citation of published work and better sales of the copyright.
Author Self-archiving
Authors retain copyright and grant the Journal of Geophysics right of first publication, with the work three years after publication simultaneously licensed under the Creative Commons BY-NC-ND 4.0 License that allows others to share the work (with an acknowledgment of the work's authorship and initial publication in this journal), except for commercial purposes and for creating derivatives.
Authors can enter into separate, additional, but non-commercial contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository, but not publish it in a book), with an acknowledgment of its initial publication in this journal.
Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) before and during the submission process, as that can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
Additional Notes
This journal is one of a handful of scholarly journals that publish original scientific works under CC BY-NC-ND 4.0 - the only Creative Commons license affording the authors' intellectual property absolute worldwide protection.
Journal of Geophysics is published under the scholar-publishers model, meaning authors do not surrender their copyright to us. Instead, and unlike corporate publishers like Elsevier or Springer Nature that resell copyright to third-parties for up to $80,000 (per paper, per transaction!), the Journal of Geophysics authors share copyright equally with this journal.
Therefore, all the proceeds from reselling copyright to third parties get shared to equal parts (50% to the journal, 50% to the lead author). Under the Berne Convention, this protection is an inheritable right that lasts for as long as the rightsholder lives + 50 years.
By submitting to this journal, the lead author, on behalf of all co-authors, grants permission to this journal to represent all co-authors in negotiating copyright sales and collecting proceeds. The lead author should negotiate with his/her co-authors the modalities of distributing the lead author's portion of the proceeds. Usually, this is per pre-agreed percentage of each co-author's contribution to creating the copyrighted work. (more...)
References
Akhiezer, N.I. (1956) Theory of approximation. Ungar, New York
Backus, G., Gilbert, F. (1967) Numerical applications of a formalism for geophysical inverse problems. Geophys. J. 13:247 276
Backus, G., Gilbert, F. (1968) The resolving power of gross Earth data. Geophys. J. 16:169-205
Backus, G., Gilbert, F. (1970) Uniqueness in the inversion of inaccurate gross Earth data. Phil. Trans. Roy. Soc. London, Ser. A 266:123 192
Bullen, K.E. (1975) The Earth's density. Chapman and Hall, London
Cook, A.H. (1971) The dynamical properties and internal constitutions of the Earth, the Moon, and the Planets. Quarterly J. Roy. astron. Soc. 12:154-168
Edward, J.A., Fitelson, M.M. (1973) Notes on maximum entropy processing. IEEE Trans. Inf. Theory (Corresp.). IT-19:232-234
Jaynes, E.T. (1968) Prior probabilities. IEEE Trans. Systems Sci. Cybern. SSC-4:227-241
Jensen, O.G., Ulrych, T.J. (1973) An analysis of the perturbations on Barnard's star. Astron. J. 78:1104·1114
Kanasewich, E.R. (1975) Time sequence analysis in geophysics. 2nd ed. The University of Alberta Press, Edmonton, Alb. Canada
Khinchin, R.T. (1957) Mathematical foundations of information theory. Dover Publications, New York
Papoulis, A. (1973) A new class of Fourier series kernels. IEEE Trans. Circuit Theory CT-20:101-107
Parker, R.L. (1972) Inverse theory with grossly inadequate data. Geophys. J. 29:123-138
Phillips, J.D., Cox, A. (1976) Spectral analysis of geomagnetic reversal time scales. Geophys. J. 45:19-33
Rao, C.R. (1965) Linear statistical inference and its applications. Wiley, New York
Rawlinson, I.S. (1970) Probability, information and entropy. Nature 225:1196-1198
Smylie, D.E., Clarke, G.K.C., Ulrych, T.J. (1973) Analysis of irregularities in the Earth's rotation. Methods Computat. Phys. 13:391-430
Ulrych, T.J. (1972) Maximum entropy power spectrum of long period geomagnetic reversals. Nature 235:218-219