By Renata Dmowska (Eds.)
The seriously acclaimed serialized evaluate magazine for over 50 years, Advances in Geophysics is a hugely revered book within the box of geophysics. on account that 1952, each one quantity has been eagerly awaited, often consulted, and praised by way of researchers and reviewers alike. Now in its 52nd quantity, it includes a lot fabric nonetheless appropriate today--truly a necessary ebook for researchers in all fields of geophysics.
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However, note that from the point of view of the probabilistic inverse theory, the task of estimating the forward modeling operator G is by no means very different from the goal of estimating the model parameters. It simply corresponds to a different use of the a posteriori information. In the traditional inversion, the a posteriori information is used to estimate unknown parameters (indirect measurement). However, it is fully justified to use it also for evaluating the “correlation” between data and parameters—evaluation of the G operator.
Alternatively, inspection of the evidence terms (see Eq. (70)) shown as bars on the right side can be used to select the best theory. 7. Using the probabilistic approach to solve such inverse problems makes sense only if the observational errors have unusual statistics. If they can be described by a Gaussianlike function, the algebraic solution method is the most appropriate approach (Menke, 1989; Parker, 1994). Inversion errors can be reliably estimated by the covariance matrix provided observational errors follow Gaussian-like statistics.
It is also the convenient tool for comparing different theories and/or selecting the best one. Let us discuss some special cases illustrating each of the above ways of using the a posteriori PDF. 1. A Posteriori PDF in the M Â D Space Although Eq. (53) provides the most general formula for the a posteriori PDF distribution, it can be further simplified in most practical applications, as the following conditions are usually met: 1. m and d are intrinsically independent and the homogeneous PDF factorizes μðm; dÞ ¼ μm ðmÞμd ðdÞ; ð56Þ 2.