### Refine

#### Year of publication

#### Document Type

- Preprint (28)
- Report (3)
- Periodical (1)

#### Language

- English (32) (remove)

#### Has Fulltext

- no (32) (remove)

#### Keywords

- Wavelet (2)
- harmonic density (2)
- wavelets (2)
- Abstract linear systems theory (1)
- Brownian motion (1)
- CHAMP (1)
- CHAMP <Satellitenmission> (1)
- Geomathematik (1)
- Geothermal Flow (1)
- Geothermal Systems (1)

#### Faculty / Organisational entity

- Fachbereich Mathematik (32) (remove)

This report gives an insight into basics of stress field simulations for geothermal reservoirs.
The quasistatic equations of poroelasticity are deduced from constitutive equations, balance
of mass and balance of momentum. Existence and uniqueness of a weak solution is shown.
In order of to find an approximate solution numerically, usage of the so–called method of
fundamental solutions is a promising way. The idea of this method as well as a sketch of
how convergence may be proven are given.

Due to the increasing demand of renewable energy production facilities, modeling geothermal reservoirs is a central issue in today's engineering practice. After over 40 years of study, many models have been proposed and applied to hundreds of sites worldwide. Nevertheless, with increasing computational capabilities new efficient methods are becoming available. The aim of this paper is to present recent progress on seismic processing as well as fluid and thermal flow simulations for porous and fractured subsurface systems. The commonly used methods in industrial energy exploration and production such as forward modeling, seismic migration, and inversion methods together with continuum and discrete flow models for reservoir monitoring and management are reviewed. Furthermore, for two specific features numerical examples are presented. Finally, future fields of studies are described.

Algebraic Systems Theory
(2004)

Control systems are usually described by differential equations, but their properties of interest are most naturally expressed in terms of the system trajectories, i.e., the set of all solutions to the equations. This is the central idea behind the so-called "behavioral approach" to systems and control theory. On the other hand, the manipulation of linear systems of differential equations can be formalized using algebra, more precisely, module theory and homological methods ("algebraic analysis"). The relationship between modules and systems is very rich, in fact, it is a categorical duality in many cases of practical interest. This leads to algebraic characterizations of structural systems properties such as autonomy, controllability, and observability. The aim of these lecture notes is to investigate this module-system correspondence. Particular emphasis is put on the application areas of one-dimensional rational systems (linear ODE with rational coefficients), and multi-dimensional constant systems (linear PDE with constant coefficients).

The inverse problem of recovering the Earth's density distribution from data of the first or second derivative of the gravitational potential at satellite orbit height is discussed for a ball-shaped Earth. This problem is exponentially ill-posed. In this paper a multiscale regularization technique using scaling functions and wavelets constructed for the corresponding integro-differential equations is introduced and its numerical applications are discussed. In the numerical part the second radial derivative of the gravitational potential at 200 km orbitheight is calculated on a point grid out of the NASA/GSFC/NIMA Earth Geopotential Model (EGM96). Those simulated derived data out of SGG (satellite gravity gradiometry) satellite measurements are taken for convolutions with the introduced scaling functions yielding a multiresolution analysis of harmonic density variations in the Earth's crust. Moreover, the noise sensitivity of the regularization technique is analyzed numerically.

The following three papers present recent developments in multiscale gravitational field modeling by the use of CHAMP or CHAMP-related data. Part A - The Model SWITCH-03: Observed orbit perturbations of the near-Earth orbiting satellite CHAMP are analyzed to recover the long-wavelength features of the Earth's gravitational potential. More precisely, by tracking the low-flying satellite CHAMP by the high-flying satellites of the Global Positioning System (GPS) a kinematic orbit of CHAMP is obtainable from GPS tracking observations, i.e. the ephemeris in cartesian coordinates in an Earth-fixed coordinate frame (WGS84) becomes available. In this study we are concerned with two tasks: First we present new methods for preprocessing, modelling and analyzing the emerging tracking data. Then, in a first step we demonstrate the strength of our approach by applying it to simulated CHAMP orbit data. In a second step we present results obtained by operating on a data set derived from real CHAMP data. The modelling is mainly based on a connection between non-bandlimited spherical splines and least square adjustment techniques to take into account the non-sphericity of the trajectory. Furthermore, harmonic regularization wavelets for solving the underlying Satellite-to-Satellite Tracking (SST) problem are used within the framework of multiscale recovery of the Earth's gravitational potential leading to SWITCH-03 (Spline and Wavelet Inverse Tikhonov regularized CHamp data). Further it is shown how regularization parameters can be adapted adequately to a specific region improving a globally resolved model. Finally we give a comparison of the developed model to the EGM96 model, the model UCPH2002_02_0.5 from the University of Copenhagen and the GFZ models EIGEN-1s and EIGEN-2. Part B - Multiscale Solutions from CHAMP: CHAMP orbits and accelerometer data are used to recover the long- to medium- wavelength features of the Earth's gravitational potential. In this study we are concerned with analyzing preprocessed data in a framework of multiscale recovery of the Earth's gravitational potential, allowing both global and regional solutions. The energy conservation approach has been used to convert orbits and accelerometer data into in-situ potential. Our modelling is spacewise, based on (1) non-bandlimited least square adjustment splines to take into account the true (non-spherical) shape of the trajectory (2) harmonic regularization wavelets for solving the underlying inverse problem of downward continuation. Furthermore we can show that by adapting regularization parameters to specific regions local solutions can improve considerably on global ones. We apply this concept to kinematic CHAMP orbits, and, for test purposes, to dynamic orbits. Finally we compare our recovered model to the EGM96 model, and the GFZ models EIGEN-2 and EIGEN-GRACE01s. Part C - Multiscale Modeling from EIGEN-1S, EIGEN-2, EIGEN-GRACE01S, UCPH2002_0.5, EGM96: Spherical wavelets have been developed by the Geomathematics Group Kaiserslautern for several years and have been successfully applied to georelevant problems. Wavelets can be considered as consecutive band-pass filters and allow local approximations. The wavelet transform can also be applied to spherical harmonic models of the Earth's gravitational field like the most up-to-date EIGEN-1S, EIGEN-2, EIGEN-GRACE01S, UCPH2002_0.5, and the well-known EGM96. Thereby, wavelet coefficients arise and these shall be made available to other interested groups. These wavelet coefficients allow the reconstruction of the wavelet approximations. Different types of wavelets are considered: bandlimited wavelets (here: Shannon and Cubic Polynomial (CP)) as well as non-bandlimited ones (in our case: Abel-Poisson). For these types wavelet coefficients are computed and wavelet variances are given. The data format of the wavelet coefficients is also included.

The inverse problem of recovering the Earth's density distribution from satellite data of the first or second derivative of the gravitational potential at orbit height is discussed. This problem is exponentially ill-posed. In this paper a multiscale regularization technique using scaling functions and wavelets constructed for the corresponding integro-differential equations is introduced and its numerical applications are discussed. In the numerical part the second radial derivative of the gravitational potential at 200 km orbit height is calculated on a point grid out of the NASA/GSFC/NIMA Earth Geopotential Model (EGM96). Those simulated derived data out of SGG satellite measurements are taken for convolutions with the introduced scaling functions yielding a multiresolution analysis of harmonic density variations in the Earth's crust.

Integral equations on the half of line are commonly approximated by the finite-section approximation, in which the infinite upper limit is replaced by apositie number called finite-section parameter. In this paper we consider the finite-section approximation for first kind intgral equations which are typically ill-posed and call for regularization. For some classes of such equations corresponding to inverse problems from optics and astronomy we indicate the finite-section parameters that allows to apply standard regularization techniques. Two discretization schemes for the finite-section equations ar also proposed and their efficiency is studied.

Power-ordered sets are not always lattices. In the case of distributive lattices we give a description by disjoint of chains. Finite power-ordered sets have a polarity. We introduct the leveled lattices and show examples with trivial tolerance. Finally we give a list of Hasse diagrams of power-ordered sets.

We show that the intersection local times \(\mu_p\) on the intersection of \(p\) independent planar Brownian paths have an average density of order three with respect to the gauge function \(r^2\pi\cdot (log(1/r)/\pi)^p\), more precisely, almost surely, \[ \lim\limits_{\varepsilon\downarrow 0} \frac{1}{log |log\ \varepsilon|} \int_\varepsilon^{1/e} \frac{\mu_p(B(x,r))}{r^2\pi\cdot (log(1/r)/\pi)^p} \frac{dr}{r\ log (1/r)} = 2^p \mbox{ at $\mu_p$-almost every $x$.} \] We also show that the lacunarity distributions of \(\mu_p\), at \(\mu_p\)-almost every point, is given as the distribution of the product of \(p\) independent gamma(2)-distributed random variables. The main tools of the proof are a Palm distribution associated with the intersection local time and an approximation theorem of Le Gall.

Wavelets on closed surfaces in Euclidean space R3 are introduced starting from a scale discrete wavelet transform for potentials harmonic down to a spherical boundary. Essential tools for approximation are integration formulas relating an integral over the sphere to suitable linear combinations of functional values (resp. normal derivatives) on the closed surface under consideration. A scale discrete version of multiresolution is described for potential functions harmonic outside the closed surface and regular at infinity. Furthermore, an exact fully discrete wavelet approximation is developed in case of band-limited wavelets. Finally, the role of wavelets is discussed in three problems, namely (i) the representation of a function on a closed surface from discretely given data, (ii) the (discrete) solution of the exterior Dirichlet problem, and (iii) the (discrete) solution of the exterior Neumann problem.

For the determination of the earth" s gravity field many types of observations are available nowadays, e.g., terrestrial gravimetry, airborne gravimetry, satellite-to-satellite tracking, satellite gradiometry etc. The mathematical connection between these observables on the one hand and gravity field and shape of the earth on the other hand, is called the integrated concept of physical geodesy. In this paper harmonic wavelets are introduced by which the gravitational part of the gravity field can be approximated progressively better and better, reflecting an increasing flow of observations. An integrated concept of physical geodesy in terms of harmonic wavelets is presented. Essential tools for approximation are integration formulas relating an integral over an internal sphere to suitable linear combinations of observation functionals, i.e., linear functionals representing the geodetic observables. A scale discrete version of multiresolution is described for approximating the gravitational potential outside and on the earth" s surface. Furthermore, an exact fully discrete wavelet approximation is developed for the case of band-limited wavelets. A method for combined global outer harmonic and local harmonic wavelet modelling is proposed corresponding to realistic earth" s models. As examples, the role of wavelets is discussed for the classical Stokes problem, the oblique derivative problem, satellite-to-satellite tracking, satellite gravity gradiometry, and combined satellite-to-satellite tracking and gradiometry.

Wavelet transform originated in 1980's for the analysis of seismic signals has seen an explosion of applications in geophysics. However, almost all of the material is based on wavelets over Euclidean spaces. This paper deals with the generalization of the theory and algorithmic aspects of wavelets to a spherical earth's model and geophysically relevant vector fields such as the gravitational, magnetic, elastic field of the earth.A scale discrete wavelet approach is considered on the sphere thereby avoiding any type of tensor-valued 'basis (kernel) function'. The generators of the vector wavelets used for the fast evaluation are assumed to have compact supports. Thus the scale and detail spaces are finite-dimensional. As an important consequence, detail information of the vector field under consideration can be obtained only by a finite number of wavelet coefficients for each scale. Using integration formulas that are exact up to a prescribed polynomial degree, wavelet decomposition and reconstruction are investigated for bandlimited vector fields. A pyramid scheme for the recursive computation of the wavelet coefficients from level to level is described in detail. Finally, data compression is discussed for the EGM96 model of the earth's gravitational field.

Many problems arising in (geo)physics and technology can be formulated as compact operator equations of the first kind \(A F = G\). Due to the ill-posedness of the equation a variety of regularization methods are in discussion for an approximate solution, where particular emphasize must be put on balancing the data and the approximation error. In doing so one is interested in optimal parameter choice strategies. In this paper our interest lies in an efficient algorithmic realization of a special class of regularization methods. More precisely, we implement regularization methods based on filtered singular value decomposition as a wavelet analysis. This enables us to perform, e.g., Tikhonov-Philips regularization as multiresolution. In other words, we are able to pass over from one regularized solution to another one by adding or subtracting so-called detail information in terms of wavelets. It is shown that regularization wavelets as proposed here are efficiently applicable to a future problem in satellite geodesy, viz. satellite gravity gradiometry.

Metaharmonic wavelets are introduced for constructing the solution of theHelmholtz equation (reduced wave equation) corresponding to Dirichlet's orNeumann's boundary values on a closed surface approach leading to exactreconstruction formulas is considered in more detail. A scale discrete version ofmultiresolution is described for potential functions metaharmonic outside theclosed surface and satisfying the radiation condition at infinity. Moreover, wediscuss fully discrete wavelet representations of band-limited metaharmonicpotentials. Finally, a decomposition and reconstruction (pyramid) scheme foreconomical numerical implementation is presented for Runge-Walsh waveletapproximation.

In modern approximation methods linear combinations in terms of (space localizing) radial basis functions play an essential role. Areas of application are numerical integration formulas on the uni sphere omega corresponding to prescribed nodes, spherical spline interpolation, and spherical wavelet approximation. the evaluation of such a linear combination is a time consuming task, since a certain number of summations, multiplications and the calculation of scalar products are required. This paper presents a generalization of the panel clustering method in a spherical setup. The economy and efficiency of panel clustering is demonstrated for three fields of interest, namely upward continuation of the earth's gravitational potential, geoid computation by spherical splines and wavelet reconstruction of the gravitational potential.

Annual Report
(1997)