An invitation to geomathematics
Geomathematics is a subject not see often enough in modern mathematics as many mathematicians are often not enthusiastic about observational concepts. Yet, at a professional level, the role of mathematics in geophysics has led to an increase in the relevance of mathematical intuition as a leading framwork for resolving complex geophysical problems, rather than merely acting a service function. An invitation is more than an introduction. Henceforth, appealing discussions are distributed throughout the text in ways that show how geomathematics can be very career enhancing, just as biologically and epidemiologically driven mathematics has been growing since the 1960s. Besides, the notes will be useful to geoscientists new to this subject, making them aware of new techniques and giving them a new perspective of how mathematics can explain observations. The second chapter starts with several important tools that are used to study terrestrial problems. Classical orthogonal spherical harmonics that use polynomials to the sphere in a Hilbert space are introduced. The reader is then delved further into the power and advantage of multiscale harmonic spherical wavelets that have the capability of breaking up the complicated geophysical structures with simple pieces defined at different scales and positions. The key novelty occurs in Chapter 3, where the authors present the mollifier procedure directly on two geoexploration applications:\begin{itemize}\item[1)] inverse gravimetry; and \item[2)] reflection seismology.\end{itemize} The mathematical analysis for gravimetry starts with standard Newtonian geopotential principles. In this way gravimetric observations are immediately transferred into the language of geomathematics using approximate integration inside a subarea of the Earth. Noting Hadamard's mathematical well-posedness notion on existence, uniqueness and stability of solutions, the authors discuss three multiscale mollifier regularisation inversion methods. Mollifiers, also called Friedrichs mollifiers, are regularization functions used in non-smooth optimization for smoothing sharp features and irregular domains. The chosen solution resolution techniques are: \begin{itemize}\item[1)] Mollified numercial integration (where the negative derivative of the mollified regularisation is modelled by means of Haar wavelet kernel functions); \item[2)] spline interpolation and smoothing (a technique that substitutes the monopole mollified regularisaion by a Gaussian sum representation); and \item[3)] mollified spline interpolation and smoothing (where the density distribution is determined using a Gram Matrix of linearly independent functions expressed in terms of the mollified regularisation substitution with Gaussian sum representation of Method 2).\end{itemize} In the context of seismic reflection inversion for subsurface acoustic wave velocity parameters, the inversion technique utilises: 1) the sparse solution of a linear system (3.143) involving Helmholtz wavelets and scattered wavefield volume potentials; and 2) Helmholtz derivatives \(\{K_{\tau_j}\}\) that reduce to Haar sequences \(\{H_{\tau_j}\}\) (see 3.133, 3.135). An illustration of Helmholtz wavelets \(\{\Psi_{\tau_j}\}\) is given in Fig. 3.21. Helmholtz wavelets (3.129) are computed from multiscale mollified regularisors to the Green's function (see smoothing function (3.124)). Scattered wavefield volume potentials are a central point in the mollified inversion technique used in the determination of the refractive index. The key point is the linear relation between the scattered acoustic wave and the velocity perturbation within the subsurface medium (see 3.122). The linearisation is achieved by a small perturbation procedure for weak scattering used extensively in seismological studies that allows single scattering between source and receiver, the first order Born approximation (3.120, 3.101).
inhomogenous Helmholtz equation inversion
scientific article; zbMATH DE number 7029535
scientific article