# Minisymposium "Geomathematics" 2011

### during the annual meeting of the German Mathematical Union in Cologne

The importance of mathematics in geoscience, in particular geophysics and geodesy, has permanently grown for the last decades. Satellite missions for the observation of the Earth deliver huge amounts of data, whose evaluation is accompanied by new mathematical challenges. The modelling of the complex interactions in the system Earth, no matter if e.g. earthquakes or the climate are concerned, yields systems of equations which require new numerical methods for their resolution. These developments are mirrored by the increasing importance of geomathematics. This new field of mathematics includes mathematical problems and techniques from various mathematical disciplines such as: constructive approximation, integral and differential equations, inverse problems, numerics, potential theory, functional analysis, statistics, and differential geometry.

The purpose of the minisymposium is to observe and communicate new trends in geomathematics and to provide a platform for discussions on open problems. The list of speakers includes mathematicians as well as geoscientists. The following experts will give invited talks (see also the preliminary titles and abstracts) at the minisymposium.

The preliminary programme is now available.

Invited Speakers:

- Jörn Behrens (University of Hamburg, Germany): A Practical Application of Uncertainty Propagation for Tsunami Early Warning
- Katrin Bentel (Norwegian University of Life Sciences, Ås): Point Grid Positions for Radial Base Functions and Their Effect in Regional Gravity Field Representations
- Doreen Fischer (University of Siegen, Germany): Sparse Regularization of an Inversion of Gravitational Data and Normal Mode Anomalies
- Willi Freeden (University of Kaiserslautern, Germany): Spherical Discrepancies
- Christian Gerhards (University of Kaiserslautern, Germany): Multiscale Methods in Geomagnetic Modeling
- Kamil S. Kazimierski (University of Bremen, Germany): Efficiency of Iterative Regularization Methods Using Banach Space Norms
- Zdenek Martinec (Dublin Institute for Advanced Studies, Ireland): The Adjoint Sensitivity Method of Global Electromagnetic Induction for CHAMP Magnetic Data
- Zuhair Nashed (University of Central Florida, Orlando, USA): Moment Problems in Reproducing Kernel Hilbert Spaces
- Isabel Ostermann (Fraunhofer ITWM Kaiserslautern, Germany): Modeling Heat Transport in Deep Geothermal Systems by Radial Basis Functions
- Sergei Pereverzev (RICAM, Linz, Austria): Multiparameter Regularization in Geodetic Data Processing
- Robert Plato (University of Siegen, Germany): The Regularizing Properties of Some Quadrature Methods for Linear Weakly Singular Volterra Integral Equations of the First Kind
- Roland Potthast (Deutscher Wetterdienst, Offenbach, Germany): Convergence Criteria on Ensembles for Local Ensemble Filters and Their Use for Ensemble Control
- Roelof Rietbroek (University of Bonn, Germany): The Use of GRACE Gravimetry and Altimetry to Separate Sea Level Contributions
- Roger Telschow (University of Siegen, Germany): Nonlinear Approximation of Spherical Functions with Dictionaries
- Johannes Wicht (Max Planck Institute for Solar System Research, Katlenburg-Lindau, Germany): Towards Realistic Planetary Dynamo Simulations

The minisymposium will probably take place at the afternoons of Wednesday, 21 September 2011 and Thursday, 22 September 2011 in the auditorium XIa of the University of Cologne. A video projector and a laptop (with Windows operating system) as well as a blackboard will be available. If you need an overhead projector, please send an email to the address above.

Abstracts:

- Jörn Behrens: A Practical Application of Uncertainty Propagation for Tsunami Early Warning
- The challenge of near-field tsunami early warning is to assess the
situation precisely within a few minutes, limited by few and uncertain
measurements of key indicators. In other words, in a short timeframe
only limited information is available, but this information has to be
interpreted in such a way, that false warnings are minimized. This
challenge had not been addressed in existing tsunami early warning
systems until recently and still leads to a large number of false
positive (to be on the save side) tsunami warning messages world wide.In the course of development of the German Indonesian Tsunami Early Warning System (GITEWS), operational since 2008 in Jakarta, Indonesia, a new method to assess the situation has been developed. This method utilizes a simple, yet effective uncertainty propagation model, which leads to more robust and accurate situation assessments under the large uncertainty of the first few minutes after an earthquake event. The system is designed as an analog forecasting system, based on pre-computed scenarios. This allows for a forecast within seconds after measurements are available.In this presentation the basic design of the system is introduced. Examples for the high sensitivity and uncertainty of the forecasting problem are given and an analysis with a simple uncertainty propagation model is given. Based on the analysis, a new method that decreases uncertainties in a robust way, is derived. Finally, examples of successful application of the new method are given.
- Katrin Bentel: Point Grid Positions for Radial Base Functions and Their Effect in Regional Gravity Field Representations
- Global gravity fields are most common represented in spherical harmonic base functions. However, the main drawback of this representation is that regional signals are not necessarily represented in an optimal way. Spherical harmonics have global support, thus, the gravity models are globally optimized best-fit solutions. That means, it is difficult to represent small spatial details, they can even be masked in the solutions.

To represent a gravity signal in a specified region on a sphere appropriately, we use localizing radial base functions for regional gravity field modeling. The distribution of these individual base functions follows a predefined point grid. The type of grid, number of points, area boundaries, point density, and other parameters play a very important role in the representation of a signal. Depending on the type of grid and its characteristics, artificial structures occur in the estimation of gravity field parameters. In this study we present some of these typical structures and investigate in detail various effects of different point grid parameters in the representation of a regional gravity field.

- Doreen Fischer: Sparse Regularization of an Inversion of Gravitational Data and Normal Mode Anomalies
- To recover the density of the Earth we invert Newton's gravitational potential. It is a well-known fact that this problem is ill-posed. Thus, we need to develop a regularization method to solve it appropriately.

We apply the idea of a Matching Pursuit (see Mallat and Zhang 1993) to recover a solution stepwise. At step n+1, the expansion function d_{n+1}and the weight \alpha_{n+1 }are selected to best match the data structure.

One big advantage of this method is that all kinds of different functions may be taken into account to improve the solution stepwise and, thus, the sparsity of the solution may be controlled directly. Moreover, this new approach generates models with a resolution that is adapted to the data density as well as the detail density of the solution.

For the area of South America, we present an extensive case study to investigate the performance and behavior of the new algorithm. Furthermore, we research the mass transport in the area of the Amazon where the proposed method shows great potential for further ecological studies, i.e. to reconstruct the mass loss of Greenland or Antarctica.

However, from gravitational data alone it is only possible to recover the harmonic part of the density. To get information about the anharmonic part as well, we need to be able to include other data types, e.g. seismic data in the form of normal mode anomalies. We present a new model of the density distribution of the whole Earth as the result of such an inversion.

- Willi Freeden: Spherical Discrepancies
- Of practical importance in geomathematics is the problem of generating equidistributed point sets on the sphere. In this respect, the concept of spherical discrepancy, which involves the Laplace-Beltrami operator to give a quantifying criterion for equidistributed point sets, is of great interest. In this lecture, an explicit formula in terms of elementary functions is developed for the spherical discrepancy. Several promising ways are considered to generate point sets on the sphere such that the discrepancy becomes small.

- Christian Gerhards: Multiscale Methods in Geomagnetic Modeling
- With the upcome of high quality satellite magnetic field data from past missions like MAGSAT, Ørsted, CHAMP and future missions like Swarm, it becomes of more and more interest to have adequate mathematical tools at hand for geomagnetic modeling. We present multiscale methods for the modeling of different aspects of the Earth's magnetic field. A special focus is set to the construction of locally supported wavelets for the treated problems, e.g., the reconstruction of radial current densities and the separation of the magnetic field with respect to the sources. Furthermore, some applications to real data sets are presented.

- Kamil S. Kazimierski: Efficiency of Iterative Regularization Methods Using Banach Space Norms
- The main aim of regularization is to provide reconstructions, which exhibit certain, desired properties. In many applications such desired property is a sparse structure. For example in geophysical applications, and in particular when prospecting the soil, sparse structure is given by the layered structure of the ground. One is therefore interested in regularization methods, which enforce sparsity.

A popular method is the Tikhonov regularization with Banach space norms. These methods are well-studied and many results considering necessary parameter-choices, source conditions and convergence rates are known. However, from the practitioners point of view these methods are only feasible if there exists a global, exact minimization scheme for the related Tikhonov functional. In general no such schemes are available. Most minimization schemes (like e.g. steepest descent) output only an approximate minimizer (after finite number of iterations). Finally we remark, that every change of the regularization parameter enforces a rerun of the minimization scheme, which makes the regularization even more computationally expensive.

In contrast, iterative regularization methods, like Landweber-method or conjugate gradient, are exact, i.e. the exact regularizing element is obtained after finite time resp. finite number of iteration steps. Further, since the regularization parameter is the stopping index, one only needs to carry out the iteration scheme once, resp. the iterates of a single run of the iteration scheme may be reused for different regularization parameters.

In this talk we want do discuss several, recently developed methods for iterative regularization. In particular:- Landweber-regularization of linear and non-linear operators (with sparsity constraints);
- conjugate gradient like regularization of linear operators (with sparsity constraints).

The presented results are joint work with Torsten Hein, Matheon, Berlin.

- Zdenek Martinec: The Adjoint Sensitivity Method of Global Electromagnetic Induction for CHAMP Magnetic Data
- Martinec and McCreadie (2004) developed a time-domain spectral-finite element approach for the forward modelling of electromagnetic induction vector data as measured by the CHAMP satellite. Here, we present a new method of computing the sensitivity of the CHAMP electromagnetic induction data on the Earth's mantle electrical conductivity, which we term the adjoint sensitivity method. The forward and adjoint initial boundary-value problems, both solved in the time domain, are identical, except for the specification of prescribed boundary conditions. The respective boundary-value data at the satellite's altitude are the X magnetic component measured by the CHAMP vector magnetometer along satellite tracks for the forward method and the difference between the measured and predicted Z magnetic component for the adjoint method.

The squares of these differences summed up over all CHAMP tracks determine the misfit. The sensitivity of the CHAMP data, that is the partial derivatives of the misfit function with respect to mantle conductivity parameters, are then determined by the scalar product of the forward and adjoint solutions, multiplied by the gradient of the conductivity and integrated over all CHAMP tracks. Such exactly determined sensitivities are checked against numerical differentiation of the misfit, and very good agreement is obtained. The attractiveness of the adjoint method lies in the fact that the adjoint sensitivities are calculated for little cost, regardless of the number of conductivity parameters. However, since the adjoint solution proceeds backwards in time, the forward solution must be stored at each time step, leading to memory requirements that are linear with respect to the number of steps undertaken.

Having determined the sensitivities, we apply the conjugate gradient method to infer 1-D and 2-D conductivity structures of the Earth based on the CHAMP residual time series (after the subtraction of static field and secular variations as described by the CHAOS model) for the year 2001. We show that this time series is capable of resolving both 1-D and 2-D structures in the upper mantle and the upper part of the lower mantle, while it not sufficiently long to reliably resolve the conductivity structure in the lower part of the lower mantle. - Zuhair Nashed: Moment Problems in Reproducing Kernel Hilbert Spaces
The problem of recovery or estimation of a function from its moments arises in several areas of the geosciences. Seminal contributions in this context have been made by Backus and Gilbert, Sabatier, and others. In this talk, I will provide new perspectives on generalizations of the Backus-Gilbert method for the numerical moment problem in the presence of the recently introduced model of weakly bounded noise. The incorporation of a priori information about the signal and the sense of approximation of the delta function by various delta sequences and in different Sobolev spaces of negative norm give rise to interesting optimization and numerical analysis problems. We explore some of these issues in the presence of weakly bounded noise for signals belonging to reproducing kernel Hilbert spaces.

- Isabel Ostermann: Modeling Heat Transport in Deep Geothermal Systems by Radial Basis Functions
- Geothermal power uses the intrinsic heat which is stored in the accessible part of the Earth's crust. Its importance among the renewable energy resources originates from the almost unlimited energy supply of the Earth and its independence from external influences such as seasonal or even daily climatic variability. Nevertheless, there are risks which have to be assessed. In particular, local depletion poses a significant risk during the industrial utilization of deep geothermal reservoirs. In order to reduce this risk, reliable techniques to predict the heat transport and the production temperature are required. To this end, a 3D-model to simulate the heat transport in hydrothermal systems is developed which is based on a transient advection-diffusion-equation for a 2-phase porous medium.

The existence, uniqueness, and continuity of the weak solution of the resulting initial boundary value problem is verified. For the numerical realization, a linear Galerkin scheme is introduced on the basis of scalar kernels. Exemplary applications of this method are investigated for the biharmonic kernel as well as appropriate geometric representations of a hydrothermal reservoir. Moreover, numerical integration methods on geoscientifically relevant bounded regions in 3D are introduced and tested for the considered geometries.

- Sergei Pereverzev: Multiparameter Regularization in Geodetic Data Processing
- We are going to discuss recent developments in multiparameter regularization. The need in this approach becomes apparent when several model uncertainties affect data processing. Focusing on the context of satellite geodesy we discuss theoretical and computational aspects of some multiparameter regularization schemes. Numerical illustrations with synthetic data will be also presented.

- Robert Plato: The Regularizing Properties of Some Quadrature Methods for Linear Weakly Singular Volterra Integral Equations of the First Kind
- The subject of this talk is the stable quadrature of a class of linear weakly singular Volterra integral equations of the first kind. Problems of this kind arise, e.g., in the inversion of seismic flat-earth travel times.

The quadrature methods under consideration are the composite trapezoidal scheme and the composite midpoint rule. In the present talk we consider their regularizing properties, i.e., we discuss appropriate choices of the step size as a function of the noise level for the right-hand side of the considered equation. Different smoothness assumptions on the involved functions are taken into account. Finally some numerical results are presented.

- Roland Potthast: Convergence Criteria on Ensembles for Local Ensemble Filters and Their Use for Ensemble Control
- The goal of data assimilation is to construct the state of some dynamical system from in-situ or remote measurements. It is used for example for numerical weather prediction. Ensemble data assimilation systems are very popular for many applications. They provide a flexible alternative to large-scale variational approaches. Many different versions of ensemble filters have been suggested and tested over the last years, including local ensemble transform Kalman filters (LETKF) and sequential importance resampling (SIR). However, one key question of current research is the setup and control of the ensembles which are used for assimilation and prediction. Here, we will provide some mathematical analysis for the local convergence of such filters and derive mathematical criteria on the ensemble which have the potential to be used for ensemble setup and control.

- Roelof Rietbroek: The Use of GRACE Gravimetry and Altimetry to Separate Sea Level Contributions
- In order to fully understand present and future sea level rise a separation of different sea level contributors is a necessity. Major ice sheets and smaller glaciers contribute to sea level rise, while steric expansion due to thermal and salinity changes play an equally important role. On top of that, the ongoing visco-elastic adjustment of the Earth to former ice loads, may not be neglected. Ocean modeling provides valuable information on the ocean response to melting. While on the other hand, absolute quantification and monitoring of sea level changes require actual observations.

In this study, we take complementary data from GRACE gravimetry data and Jason-1 altimetry and estimate time varying scales associated with predefined sea level patterns. The patterns represent non-uniform gravitationally sea level responses to melting and hydrological loading. Additionally, the steric sea level patterns are obtained from the Finite element Sea-Ice model. We discuss accuracy and separability of the estimation method and provide results in the spatial and time domain.

- Roger Telschow: Nonlinear Approximation of Spherical Functions with Dictionaries
- A nonlinear method using a dictionary to approximate functions on the sphere is derived. The elements of the dictionary added to the approximation are chosen with a matching pusuit algorithm while the dictionary consists of spherical harmonics of low degrees, to approximate global structures, and several radial basis functions such as the Abel-Poisson kernel to be added in areas with more details. The method, therefore, yields a representation of the function to be approximated which is not only sparse but also adapts to the solution, i.e. chooses more basis functions in areas where the function is structured more heavily, which also provides the possibility to easily add more finely resolved data of certain areas to the approximation. Numerical results are presented for both benchmark functions as well as real satellite data.

- Johannes Wicht: Towards Realistic Planetary Dynamo Simulations
- The last years have witnessed an impressive growth in the number and quality of numerical dynamo simulations. These models successfully describe many aspects of the geomagnetic and other planetary magnetic fields. The success is somewhat surprising since numerical limitation force dynamo modelers to run their computations at unrealistic parameters. In particular the Ekman number, a measure for the relative importance of viscous diffusion, is many orders of magnitude too large. We discuss the fundamental dynamo regimes and address the question how well the modern models reproduce the geomagnetic field. First-level properties like the dipole dominance, realistic magnetic field strength, convective flow vigor, and an Earth-like reversal behavior are already captured by larger Ekman number simulations. However, low Ekman numbers are required for successfully modeling features like the low latitude field and torsional oscillations which are thought to be an important part of the decadal geomagnetic field variations. Only low Ekman number models also retain the huge dipole dominance of the geomagnetic field in combination with magnetic field reversals.