By Renata Dmowska (Eds.)
The severely acclaimed serialized evaluate magazine for over 50 years, Advances in Geophysics is a hugely revered ebook within the box of geophysics. given that 1952, every one quantity has been eagerly awaited, usually consulted, and praised by means of researchers and reviewers alike. Now in its 52nd quantity, it includes a lot fabric nonetheless correct today--truly an important booklet for researchers in all fields of geophysics.
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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.
49). The above result highlights the importance of the geometrical properties of the M space for the inverse theory. In fact, the volumetric measure (Eq. , 1983; Sharpe, 1997; Lang, 1999; Nestruev, 2003). If M is a flat space isomorphic with RN, it is possible to introduce the global Cartesian coordinate system in which dV = dx1dx2 ÁÁÁ dxn so the volume measure is constant over M. However, such a system cannot be introduced if M is a manifold with more complex geometry defined, for example, by inequalities as in Eq.
Next, let us assume that there is no observational data so � � ð116Þ dexp d ; dobs ¼ d ðdÞ: In this case we arrive at the following formula for dpos (d): Z th ðm; dÞ dpos ðdÞ ¼ dapr ðd ; dapr Þ m apr ðmÞ ðm dÞ dm: ð117Þ M This formula is the counterpart of Eq. (63) and shows that in the case of missing observational data the a posteriori PDF in the data space is the product of the a priori PDF in D space and the function which describes how the a priori information about m is projected into the D space.