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## Imaging in Random Media

### Handbook of Mathematical Methods in Imaging (2020-01-01): 1-54 , January 01, 2020

We give a self-contained presentation of coherent array imaging in random media, which are mathematical models of media with uncertain small-scale features (inhomogeneities). We describe the challenges of imaging in random media and discuss the coherent interferometric (CINT) imaging approach. It is designed to image with partially coherent waves, so it works at distances that do not exceed a transport mean-free path. The waves are incoherent when they travel longer distances, due to strong cumulative scattering by the inhomogeneities, and coherent imaging becomes impossible. In this article we base the presentation of coherent imaging on a simple geometrical optics model of wave propagation with randomly perturbed travel time. The model captures the canonical form of the second statistical moments of the wave field, which describe the loss of coherence and decorrelation of the waves due to scattering in random media. We use it to give an explicit resolution analysis of CINT which includes the assessment of statistical stability of the images.

## On High Reynolds Number Aerodynamics: Separated Flows

### Handbook of Geomathematics (2020-01-01): 1-36 , January 01, 2020

This treatise deals with the occurrence of locally separated, three-dimensional, unsteady high Reynolds number flows. As it is well established, such flows are governed by a triple-deck structure where the wall shear stress in the viscous sublayer of the (in general inviscid) boundary layer is utilized to describe the phenomenon of localized separation bubbles. It is then proved that the Cauchy problem for the local wall shear stress is, in general, ill-posed. Thus, regularization methods need to be applied to numerically compute the time evolution. The numerical scheme comprises a novel technique using rational Chebyshev polynomials. Finally, the breakdown of the triple-deck structure in the sense of a finite time blow-up scenario is shown.

## Forest Fire Spreading

### Handbook of Geomathematics (2020-01-01): 1-34 , January 01, 2020

Due to climate changes, more and more woodlands will be endangered by forest fires in the future. Because of this observation, it is very important to obtain information about how forest fires expand. In this contribution, we are interested in the interacting factors which influence forest fires. Our particular interest is the use of physical models, which consider heat and mass transfer mechanisms. As a result, we are led to a convection-diffusion-reaction problem which is nonstationary and nonlinear. Furthermore, we have a look at the different parameters, which are involved in these equations, especially at the meteorological and fuel data. Finally, we discuss some approaches to solve fire expansion numerically. Moreover, we give some simulations of forest fire spreading.

## Gauss’ and Weber’s “Atlas of Geomagnetism” (1840) Was not the first: the History of the Geomagnetic Atlases

### Handbook of Geomathematics (2020-01-01): 1-32 , January 01, 2020

In the beginning there were geomagnetic charts which were interesting mainly for seafaring nations. The first geomagnetic atlas was printed in London in 1776; its author was the mathematician, cartographer, and astronomer Samuel Dunn, whose aim had been to ameliorate the navigation especially to support the trading of England with the East Indies. The American John Churchman, however, was mainly surveyor; his magnetic atlas was published in four editions, in 1790, 1794, 1800, and 1804. Churchman was in contact with George Washington and with Thomas Jefferson, as far as his geomagnetic charts were concerned; he also became a member of the Academy of Sciences in St. Petersburg. Churchman was convinced that the magnetic pole in the north could be found in northern Canada. The Norwegian astronomer and physicist Christopher Hansteen was convinced that there were two magnetic poles in the north and two in the south; his atlas was published in 1819. One of the magnetic poles in the north should be in Siberia. Hansteen found support by the king of Sweden and Norway so that he undertook an expedition to Siberia (1828–1830). Carl Friedrich Gauss and Wilhelm Weber began to study geomagnetism in 1831: They believed that there were only two magnetic poles, one in the north and one in the south. They were able to calculate their positions by means of Gauss’ new theory of geomagnetism (1839); as sailors found out, their coordinates turned out to be nearly correct. Gauss’ and Weber’s Atlas is without doubt the most famous; it was published in Leipzig in 1840, including 18 geomagnetic charts. On two of these charts, equipotential lines were presented for the first time in history.

## Oblique Stochastic Boundary-Value Problem

### Handbook of Geomathematics (2020-01-01): 1-26 , January 01, 2020

The aim of this chapter is to report the current state of the analysis for weak solutions to oblique boundary problems for the Poisson equation. In this chapter, deterministic as well as stochastic inhomogeneities are treated and existence and uniqueness results for corresponding weak solutions are presented. We consider the problem for inner bounded and outer unbounded domains in $$\mathbb{R}^{n}$$ . The main tools for the deterministic inner problem are a Poincaré inequality and some analysis for Sobolev spaces on submanifolds, in order to use the Lax-Milgram lemma. The Kelvin transformation enables us to translate the outer problem to a corresponding inner problem. Thus, we can define a solution operator by using the solution operator of the inner problem. The extension to stochastic inhomogeneities is done with the help of tensor product spaces of a probability space with the Sobolev spaces from the deterministic problems. We can prove a regularization result, which shows that the weak solution fulfills the classical formulation for smooth data. A Ritz-Galerkin approximation method for numerical computations is available. Finally, we show that the results are applicable to geomathematical problems.

## It’s All About Statistics: Global Gravity Field Modeling from GOCE and Complementary Data

### Handbook of Geomathematics (2020-01-01): 1-24 , January 01, 2020

Since October 2009, ESA’s dedicated satellite gravity mission GOCE (Gravity Field and Steady-State Ocean Circulation Explorer) observes the global gravity field of the Earth. The estimation of the model parameters from the original GOCE observations requires the application of tailored tools of geomathematics and statistics. One of the main constraints is to compute pure GOCE models, which are independent of any other external gravity field information. Up to now, four releases of global GOCE gravity field models have been computed and released. Their continuously increasing accuracy is validated by external gravity field information. A key prerequisite for achieving high-quality results is the correct stochastic modeling of all input data types in the frame of a least-squares adjustment procedure based on the rigorous solution of full normal equation systems. Together with the global gravity field models, parameterized as coefficients of a spherical harmonic series expansion, also the related error variance-covariance matrix is generated, which turns out to describe the true errors of the solutions very accurately. The fourth release achieves global geoid height accuracies of 3.5 cm and gravity anomaly accuracies below 1 mGal at a spatial wavelength of 100 km. Further improvements are expected, also because of the GOCE satellite’s orbit lowering in its final mission phase, which will further improve the spatial resolution. In addition to these pure GOCE-only models, in the frame of the GOCO initiative consistent combined gravity field models are processed by including GRACE and SLR data (improving the long wavelengths), as well as terrestrial gravity information and satellite altimetry (improving the high-frequency component). Also for the computation of these optimum combinations, the tools developed for the GOCE processing can largely be applied. Numerous fields of application in geodesy, oceanography, and geophysics can benefit already now from the new GOCE models. As an example, the derivation of global ocean transport processes from a combination of satellite altimetry and global gravity information demonstrates that GOCE can contribute significantly to an improved understanding of processes in system Earth.

## Electrical Impedance Tomography

### Handbook of Mathematical Methods in Imaging (2020-01-01): 1-57 , January 01, 2020

This chapter reviews the state of the art and the current open problems in electrical impedance tomography (EIT), which seeks to recover the conductivity (or conductivity and permittivity) of the interior of a body from knowledge of electrical stimulation and measurements on its surface. This problem is also known as the inverse conductivity problem and its mathematical formulation is due to A. P. Calderón, who wrote in 1980, the first mathematical formulation of the problem, “On an inverse boundary value problem.” EIT has interesting applications in fields such as medical imaging (to detect air and fluid flows in the heart and lungs and imaging of the breast and brain) and geophysics (detection of conductive mineral ores and the presence of ground water). It is well known that this problem is severely ill-posed, and thus this chapter is devoted to the study of the uniqueness, stability, and reconstruction of the conductivity from boundary measurements. A detailed distinction between the isotropic and anisotropic case is made, pointing out the major difficulties with the anisotropic case. The issues of global and local measurements are studied, noting that local measurements are more appropriate for practical applications such as screening for breast cancer.

## Regularization Methods for Ill-Posed Problems

### Handbook of Mathematical Methods in Imaging (2020-01-01): 1-31 , January 01, 2020

In this chapter are outlined some aspects of the mathematical theory for direct regularization methods aimed at the stable approximate solution of nonlinear ill-posed inverse problems. The focus is on Tikhonov type variational regularization applied to nonlinear ill-posed operator equations formulated in Hilbert and Banach spaces. The chapter begins with the consideration of the classical approach in the Hilbert space setting with quadratic misfit and penalty terms, followed by extensions of the theory to Banach spaces and present assertions on convergence and rates concerning the variational regularization with general convex penalty terms. Recent results refer to the interplay between solution smoothness and nonlinearity conditions expressed by variational inequalities. Six examples of parameter identification problems in integral and differential equations are given in order to show how to apply the theory of this chapter to specific inverse and ill-posed problems.

## Quantitative Remote Sensing Inversion in Earth Science: Theory and Numerical Treatment

### Handbook of Geomathematics (2020-01-01): 1-28 , January 01, 2020

Quantitative remote sensing is an appropriate way to estimate structural parameters and spectral component signatures of Earth surface cover type. Since the real physical system that couples the atmosphere, water, and the land surface is very complicated and should be a continuous process, sometimes it requires a comprehensive set of parameters to describe such a system, so any practical physical model can only be approximated by a mathematical model which includes only a limited number of the most important parameters that capture the major variation of the real system. The pivot problem for quantitative remote sensing is the inversion. Inverse problems are typically ill-posed. The ill-posed nature is characterized by (*C*1) the solution may not exist, (*C*2) the dimension of the solution space may be infinite, and (*C*3) the solution is not continuous with variations of the observed signals. These issues exist nearly for all inverse problems in geoscience and quantitative remote sensing. For example, when the observation system is band-limited or sampling is poor, i.e., there are too few observations, or directions are poor located, the inversion process would be underdetermined, which leads to the large condition number of the normalized system and the significant noise propagation. Hence (*C*2) and (*C*3) would be the highlight difficulties for quantitative remote sensing inversion. This chapter will address the theory and methods from the viewpoint that the quantitative remote sensing inverse problems can be represented by kernel-based operator equations and solved by coupling regularization and optimization methods.

## Potential Methods and Geoinformation Systems

### Handbook of Geomathematics (2020-01-01): 1-21 , January 01, 2020

Geophysical methods “gravity” and “magnetic” belong to potential methods together with geoelectrics. This chapter focuses on gravity and magnetic methods. Their fields can be described by Laplace and Poisson differential equations – if the observation is taken outside or inside the masses. Potential fields were defined to attribute vector fields to scalar fields, because the mathematical treatment of scalar fields is numerically easier. Gravity and magnetic exploration can help to locate faults, mineral or petroleum resources, and groundwater reservoirs. The interpretation of gravity and magnetic fields and their respective anomalies is not unique, and boundary conditions are always required. Geoinformation systems can help to overcome the ambiguity of potential methods and support integrated modeling of potential fields by allocation of boundary conditions, data/information fusion, and advanced visualization at different scales. These systems should help to facilitate 3D interpretation (even 4D), which bases on data from multiple sources. 3D potential field forward modeling and inversion, visualization, and metadata handling facilitate interdisciplinary interpretation crossing the field of geophysics and geoinformatics.