Special Issue "Numerical Methods of Geophysical Fields Inversion"
Deadline for manuscript submissions: closed (16 November 2018)
Prof. Vladimir Cheverda
The overwhelming volume of modern knowledge regarding the Earth’s interior became available due to the results of geophysical observations, on or near the surface. Inversion of geophysical fields within the framework of the corresponding mathematical model provides the most complete knowledge about subsurface distributions of desired parameters. However, it is necessary to stress, that we could never describe a real geological medium using such an abstract object as a system of partial differential equations. Any mathematical model would leave unaccounted a series of processes, phenomena and relationships between parameters, no matter how complex it is. On the one hand, neglecting some of them can significantly distort the important physical properties of the studied fields, while the desire to take into account the widest possible their features leads to excessive complication of mathematical models and, as a result, to a sharp increase in the cost of data processing. Hence, the proper mathematical model is necessary in providing reliable results of geophysical inversion.
It is worth mentioning that the emergence and development of such a direction of modern mathematics as inverse and ill-posed problems originates in geophysics. As early as 1907, Gustav Herglotz published the paper Über das Benndorfsche Problem Fortpfianzungsgeschwindigkeit der Erdbebenstrahlen (Zeitschr. fiir Geophys. 1907, 8, 145- 147) devoted to the inverse kinematic problem for the radially-symmetric Earth. The method was developed further by Emil Wiechert in Bestimmung des Weges der Erdbebenwell~n im Erdinnern. l. Theoretisches, (Phys. Z. 1910, 11, 294-304) and forms the basis for the development of modern computer tomography.
The key position in modern theory and numerical methods of inverse and ill-posed problems takes nonlinear least squares (Levenberg K. 1944. A method for the solution of certain nonlinear problems in least squares Quart. Appl. Math., 2, 1944, 164 – 168; Guy Chavent. Nonlinear least squares for inverse problems, Springer, 2009) and various regularization techniques (Tikhonov A. Solution of incorrectly formulated problems and the regularization method. Soviet Mathematics. 1963, 4: 1035 - 1038). It is these two components form the basis of modern methods of geophysical fields inversion (Albert Tarantola: Inverse Problem Theory and methods for model parameter estimation), which is dedicated to this issue.
Prof. Vladimir Cheverda
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- numerical linear algebra
- mathematical modelling
- finite difference simulation
- optimization techniques
- nonlinear least squares