By Renata Dmowska (Eds.)

ISBN-10: 0123749107

ISBN-13: 9780123749109

The seriously acclaimed serialized assessment magazine for over 50 years, Advances in Geophysics is a hugely revered e-book within the box of geophysics. due to the fact that 1952, each one quantity has been eagerly awaited, often consulted, and praised by means of researchers and reviewers alike. Now in its 52nd quantity, it comprises a lot fabric nonetheless proper today--truly a vital e-book for researchers in all fields of geophysics.

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A priori information about m and d is independent so σapr(Á) can be represented as the product m d σ apr ðm; mapr ; d; dapr Þ ¼ σ apr ðm; mapr Þσ apr ðd; dapr Þ; ð57Þ 3. if m and d are independent physical parameters, then the results of measurement of d are statistically independent of m (Tarantola, 1987; Cook, 1994) and σ exp ðm; d; dobs Þ ¼ μm ðmÞσ exp ðd; dobs Þ: ð58Þ After taking these postulates into account, the a posteriori PDF is simplified and reads obs apr apr d pos ðm; dÞ ¼ m apr ðm; m Þapr ðd; d Þexp ðd; d Þth ðm; dÞ ðm dÞd ðdÞ ; ð59Þ where the dependence of pos(m, d) on mapr, dapr, and dobs has been omitted for the sake of clarity.

61) reads Z th ðm; dÞ apr obs m apr d dd: ð62Þ ðm; m ; d Þ ¼  ðm; m Þ exp ðd; dobs Þ m pos apr ðm; dÞ D Traditionally this formula is written as m obs pos ðmÞ ¼ m apr ðmÞLðm; d Þ; ð63Þ where the likelihood function L(m) is defined as follows: Z th ðm; dÞ Lðm; dobs Þ ¼ exp ðd; dobs Þ dd: ðm; dÞ ð64Þ D This time the likelihood function L(m) expresses how well a given model can explain (predict) the observational data taking into account both observational and experimental errors. If the forward problem can be written as d ¼ GðmÞ; ð65Þ the a posteriori PDF reads m apr obs m pos ðmÞ ¼ apr ðm; m Þtot ðd −GðmÞÞ; ð66Þ ⋅tot (Á) stands for the statistics of the sum of observational and modeling errors.

In this section we would like to illustrate the whole process of probabilistic inference described pre­ viously using simple 1D models. Throughout this section the model and data spaces are 42 DE¸BSKI Probability assumed to be 1D—we have a single datum and one parameter to be estimated, so the joint space J Â M Â D is 2D and all PDFs can easily be visualized. All figures in this section consist of three parts. The main, central panel shows the joint J ¼ M Â D space where the model subspace M is represented by one of the axes and the D subspace by the second one.

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Advances in Geophysics by Renata Dmowska (Eds.)

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