Advances in Geophysics by Renata Dmowska (Eds.)

By Renata Dmowska (Eds.)

The seriously acclaimed serialized assessment magazine for over 50 years, Advances in Geophysics is a hugely revered booklet within the box of geophysics. considering the fact that 1952, every one quantity has been eagerly awaited, usually consulted, and praised through researchers and reviewers alike. Now in its 52nd quantity, it comprises a lot fabric nonetheless suitable today--truly an important e-book for researchers in all fields of geophysics.

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1997; Scales and Snieder, 1997; Bernardo and Smith, 2001; Mohammad-Djafari, 2001a; Sambridge and Mosegaard, 2002). 3. Marginal a Posteriori PDF in the D Space Let us assume again that the m and d parameters are independent so the homogeneous PDF in the J space can be factorized the same way as in Eq. (56). Let us also assume that the a priori PDF also factorizes according to Eq. (57). Then, the marginal a posteriori PDF in the D space reads dpos ðd; dapr ; d obs Þ ¼ dexp ðd; dobs Þ d apr ðd; dapr Þ d ðdÞ Z M th ðm; dÞ m ðmÞ dm: apr ðm dÞ If there is no available a priori information on d, dapr (d) = d (d) and Z th ðm; dÞ m dpos ðd; dobs Þ ¼ dexp ðd; dobs Þ apr ðmÞ dm: ðm dÞ ð67Þ ð68Þ M This formula is the counterpart of Eq.

Example of the most common inverse problem: weakly nonlinear forward problem, approximate theory, and noisy data. 7. M D M�D FIG. 9. Example of a “degenerated” PDF. In spite of the high quality of data (narrow σD(d) distribution) in the D space, the m parameter is poorly resolved due to the existence of the “null space” for the forward operator. In such a situation the a priori information becomes crucial for estimating the “best” model. 7. 45 Probability PROBABILISTIC INVERSE THEORY M D M�D FIG.

7. M D M�D FIG. 7. Example of regularization by the a priori PDF (cf. Fig. 6). Adding the a priori information on m (taking a nontrivial σA(m)) has removed the ambiguity in the joint and thus also in the marginal a posteriori PDFs. 7. DE¸BSKI Probability 44 M D M�D Probability FIG. 8. Example of the most common inverse problem: weakly nonlinear forward problem, approximate theory, and noisy data. 7. M D M�D FIG. 9. Example of a “degenerated” PDF. In spite of the high quality of data (narrow σD(d) distribution) in the D space, the m parameter is poorly resolved due to the existence of the “null space” for the forward operator.

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