IV Hotine-Marussi Symposium on Mathematical Geodesy

Inhaltsverzeichnis

1. Frontmatter

2. Report on the Symposium

   Riccardo Barzaghi

   Abstract

   The IV Hotine-Marussi Symposium on Mathematical Geodesy was held in Trento from September 14th to September 17th, 1998. It continues the long-standing tradition of symposia that was initiated by Martin Hotine and Antonio Marussi in Venice in 1959. It is the 12th symposium of this series and the fourth one associated with the names of Hotine and Marussi. The arguments treated were connected with the theoretical and methodological aspects of geodesy. Infact, these kind of symposia are usually devoted to the development of the founding aspects of geodesy. The principal themes that were discussed during this last meeting have been related to the boundary value problems, to the satellite geodesy and to the stochastic methods in geodesy. The boundary value problems were tackled both from the theoretical and the numerical points of view. New advancements were presented in the Molodensky scalar boundary value problem as well as in the application of the Slepian theory for the sphere. Furthermore, a theoretical scheme for handling white noise stochastic boundary value problem was illustrated, the importance of ellipsoidal effects were investigated in inverse Stokes problem and the solution of the spheroidal Stokes problem was presented. Methods for computing the solutions of the boundary value problems were also illustrated. The boundary element formulations were introduced and numerically tested in geodetic boundary value problems; the use of Galerkins method has been proposed as a tool for solving geodetic boundary value problems as well as iterative solution applied to the scalar boundary value problem.

3. The Molodensky Scalar Boundary Value Problem in Spherical Coordinates: a New Result

   Jesús Otero, Fernando Sansò

   Abstract

   The theory and analysis of free boundary value problems has attained a certain popularity in Geodesy in the last 20 years, being considered as a sort of reference theory providing a sound scientific background to the problem of determining the figure of the earth and its gravity field. In this framework an effort has been done to come to a solution of the most relevant problem, namely the so-called scalar geodetic boundary value problem, under the most general conditions of regularity of the boundary values. In fact the basic theorem proven here guarantees a solution for the (modified) Molodensky scalar boundary value problem under the conditions that (on the boundary)the gravity belongs to H _α and the gravitational potential to H _1+α.

   One of the basic tools to get this result has been the intermediate Schauder estimates for oblique derivative problems obtained by G.M.Lieberman. The drawback of what we have proved is as always the use of existence theorems in the small, which do not say much about how ”close” should be the data to the reference field in order to guarantee existence and uniqueness of solution. Furthermore, as in many other works in this field, the result is obtained here in the so-called spherical approximation for the reference field and one could think that a better result could be achieved by working with an ellipsoidal approximation for the reference field; this could be a line of research worth some effort in future.

4. The Slepian Problem on the Sphere

   A. Albertella, F. Sansò, N. Sneeuw

   Abstract

   The Slepian problem consists in determining a sequence of functions that constitute an orthonormal basis of a subset of ℝ, (or ℝ²) concentrating the maximum information in the subspace of square integrable function with bandfinite spectrum. The same problem can be stated and solved on the sphere.

   The relation between the new basis and the ordinary spherical harmonic basis can beexplicitely written and numerically studied.

   What turns out is that this tool is a natural solution to capture the maximum amount of information from a non-polar gradiometric mission.

5. White Noise Stochastic BVP’s and Cimmino’s Theory

   F. Sansò, G. Venuti

   Abstract

   The analysis of the Molodensky problem in spherical approximation can be reduced to the simple Dirichlet problem. If the boundary data are noisy (white noise), it is requested to explain what is the meaning of the solution of a B.V.P. with data of this kind. This can be correctly done in the framework of generalized random field theory and an equivalent principle can be stated such that the solution of the stochastic problem exists and is unique iff the analogous deterministic problem has a unique solution with L ² (S) boundary data. This result can be achieved by Cimmino’s theory which is also reviewed here for the ease of the reader.

6. Simulation of the Goce Gravity Field Mission

   Nico Sneeuw, Raul Dorobantu, Christian Gerlach, Jürgen Müller, Helmut Oberndorfer, Reiner Rummel, Radboud Koop, Pieter Visser, Peter Hoyng, Avri Selig, Martijn Smit

   Abstract

   GOCE (Gravity Field and Steady-State Ocean Circulation Explorer) is one of the four selected ESA Earth Explorer Missions (Phase A has started in summer 1998). The main objective of GOCE is the determination of the Earth’s gravity field with high spatial resolution and with high homogeneous accuracy. For this purpose, two observation concepts will be realized. Satellite-to-Satellite Tracking (SST) in high-low mode will be used for orbit determination and for retrieval of the long-wavelength part of the gravity field. Satellite Gravity Gradiometry (SGG) will be employed for the recovery of the medium/short- wavelength parts of the gravity field. The measurement principles and the relation between the various instruments are explained by means of a Flow Chart.

   For its realization, a GPS receiver, a 3-axis gradiometer and further instruments are needed; e.g. star trackers to control the orientation of the spacecraft or thrusters for attitude and drag-free control. Each instrument shows its own error behavior which affects the measurements and the final products in a specific way. Here, the corresponding error Power Spectral Densities (PSD), due to several error sources, are shown.

   Error PSD’s represent the stochastic model in the spectral domain. In connection with frequency-wise modeling of the observables (through the lumped coefficient approach) leastsquares error simulation can be performed. Thus the PSD’s are propagated to spherical harmonic error spectra, geoid heights and gravity anomaly accuracies to assess effects of instrument and measurement errors on the gravity field determination.

7. Quality Improvement of Global Gravity Field Models by Combining Satellite Gradiometry and Airborne Gravimetry

   Johannes Bouman, Radboud Koop

   Abstract

   The expected high resolution and precision of a global gravity field model derived from satellite gradiometric observations is unprecedented compared to nowadays satellite-only models. However, a dedicated gravity field mission will most certainly fly in a non-polar (sun-synchronous) orbit, such that small polar regions will not be covered with observations. The resulting inhomogeneous global data coverage, together with the downward continuation problem, leads to unstable global solutions and regularization is necessary. Regularization gives rise to a bias in the solution, mainly in the polar areas although in other regions as well.

   Undoubtedly, the combination with gravity related measurements in the polar areas, like airborne gravimetry, will improve the stability of the solution. Consequently, the bias is reduced and the quality is likely to be better. Open questions are, for example, how accurate gravity anomalies must be, what spatial sampling is required, and how large the area with observations should be. Moreover, it is unknown whether measurements in one polar area only (e.g. North Pole) is sufficient.

   In order to answer these questions, a gravity field solution from gradiometry-only will be compared with a solution from gradiometry combined with several airborne gravimetric scenarios. Special attention is given to the quality improvement and bias reduction relative to the gradiometry-only solution. The coefficients of a spherical harmonic series are the unknowns and their errors are propagated to, for example, geoid heights.

8. On the Determination of Geopotential Differences from Satellite-to-Satellite Tracking

   Christopher Jekeli

   Abstract

   The Earth’s gravitational potential field can be obtained directly from low-low satellite-to-satellite tracking using range-rates and/or velocity vector differences. There are some possible advantages to measuring potential instead of acceleration or gradients of acceleration, as alternatively proposed. For example, direct geoid modeling in local areas without recourse to Stokes’s integral (but still Poisson’s integral to account for downward continuation). The usual measurement model relates the in situ geopotential difference to the range-rate between two satellites. This model neglects the effect of Earth’s rotation, which is on the order of 0.1kgal*m for polar-orbiting satellites. In this paper an analytic expression is derived for that effect. It is shown that the potential rotation effect can be determined in situ only from velocity and position vector measurements, which is possible using GPS baseline measurements. Applications to two upcoming satellite mission, GRACE and COSMIC are discussed.

9. On the Topographic Effects of Helmert’s Method of Condensation

   Lars E. Sjöberg

   Abstract

   Assuming a constant or laterally variable topographic density the direct and indirect topographic effects on the geoidal and quasigeoidal heights are presented as strict surface integrals with respect to topographic elevation (H) on a spherical approximation of sea level. By Taylor expanding the integrals with respect to H we derive the power series of the effects to arbitrary orders. The study is primarily limited to terms of second order of H, and we demonstrate that current planar approximations of the formulas lead to significant biases, which may range to several decimetres. Adding the direct and indirect geoid effects yields a simple combined effect, while the corresponding combined effect of the quasi-geoid vanishes. Thus we conclude that only the effect of downward continuation of gravity anomaly to sea level under Stokes integral remains as a major computational burden among the topographic effects.

10. Distance Measurement with Electromagnetic Wave Dispersion

    Michele Caputo

    Abstract

    An atmosphere above a half space considered with the dispersion represented by a relation between the electric field and the induction which contains derivatives of rational order and is similar to the empirical formula of Cole and (1941), commonly used in experimental physics, and to the formula used by (1991) in studying the dispersion of energy in electric networks. The dispersion of a monochromatic wave is modelled considering the index of refraction n as a rational function of a rational power of the imaginary frequency if, as usually in geophysics, and is a polymorphic function of f; this function, for each frequency, gives a set of different velocity fields, whose number depends on the rational exponent of if. Each electromagnetic wave leaving the source, with given f and direction, is split in a number of waves with different velocities; if n is a function of position, the paths of the waves are different and reach a given elevation at different points and times. If n is independent of the position, the paths of the waves coincide although the waves have different velocities. The length of a path and the travel time of electromagnetic waves in the atmosphere of a flat Earth model are computed. It is found that the difference between the arc length of the ray and the chord is nil to the second order of refractivity. It is also seen that a change of water content is layers of the atmosphere, leaving the average velocity profile to a given elevation unchanged, may change the length of the ray paths to the elevation. It is found that the separation of the rays with the same frequency and direction at the source, causes small uncertainties in electromagnetic distance measurements which increase with the frequency. In the (1985) atmospheric model we considered frequencies in the range 1 GHz to 2 GHz and found that the arrival of the phases of the rays, with the same frequency in this range, with a zenithal angle smaller that 27π/5 and that a distance of about 10⁴ km, are spread in less than 0.01 ns or 0.3 cm; which does not influence the accuracy presently achieved in distance measurements with electromagnetic waves. The dissipation of energy of the rays in the atmospheric model used, for zenithal angles smaller than 27π/5, is negligible for any length of the path. Formulae are given for the retrieval of a spherical model of the atmosphere of the Earth from a set of differences of the times of arrival, at two observing stations, of the waves emitted from satellites of known orbits.

11. A Global Topographic-Isostatic Model Based on a Loading Theory

    Wenke Sun, Lars E. Sjöberg

    Abstract

    This a preliminary report of our ongoing research on a global topographic-isostatic model. The model originates from a completely new idea—the geoid undulation is the response of an elastic earth to topographic mass load. Assuming the topography as a condensed surface mass load, we derive expressions for calculating the vertical displacement, potential and equipotential surface changes, based on the load theory proposed in (1996). We also discuss the mass conservation problem and some calculating techniques. The modeled geoid is composed of three parts: loading potential, surface displacement and mass redistribution. The surface displacement and mass redistribution of the earth compensate to some extent the topography. In practical calculations we adopt the Getech’s Global Digital Terrain Model with 5x5 minute bloc averages (DTM5). Using the load Love numbers and Green’s functions obtained from the 1066A earth model, we calculate and discuss the vertical displacements and equipotential surface changes for depths: earth’s surface, d = 36 km and the core-mantle boundary. Numerical results show that the displacements at depth 36 km and the earth surface have the same distribution pattern and magnitude, while the vertical movement of the core-mantle boundary appears much smoother and smaller. The contribution of the mass redistribution to the equipotential surfaces is rather small and smooth. The modeled geoid undulations at the earth’s surface caused by the topographic mass load vary between-352 and +555 m. Comparing the modeled and observed geoid undulations shows that there are strong positive correlations between them, but a compensation only by elastic deformations is not sufficient to explain the observed undulations because of the big difference in magnitude between the two geoids. More geodynamic effects should be considered to better explain the long-wavelength geoid features.

12. Stochastic Modelling of Non-stationary Smooth Phenomena

    V. Tornatore, F. Migliaccio

    Abstract

    We propose a new method to model a non-stationary and slowly variable phenomenon. The case of a one-dimensional stochastic process is presented. Particular care is taken in order to prove that the condition of positive definiteness of the covariance function is satisfied. The estimate of the covariance function is afterwards obtained by means of local estimates. The method has been tested on a set of simulated data.

13. Deformation Detection According to a Bayesian Approach

    B. Betti, F. Sansò, M. Crespi

    Abstract

    Sometimes, when we deal with deformations monitoring, the estimated displacements in repeated surveys are small with respect to the measurement precisions; therefore, they are not significant according to the classical testing procedure. Nevertheless, even at a first glance, these data seem to show a correlation pattern: for instance, we can have all sinking benchmarks along a control leveling line. In these cases it seems reasonable to make the hypothesis that the displacements are due to a real slow deformation.

    Therefore, our main goal was to understand how it is possible to use this correlation pattern in order to increase the power of the test. The proposed solution consists in formulating the testing procedure in the frame of the Bayesian theory, which allows to incorporate the correlation pattern into the prior distribution of the parameters.

    We derived the Bayesian test statistics both for the cases with known and unknown variance of unit weight. Moreover we applied the proposed approach to an elementary through realistic example obtaining encouraging results.

14. Block Elimination and Weight Matrices

    Kai Borre

    Abstract

    We describe the technique of block elimination. It is used for explaining Schreiber’s device. We use the technique to derive the equations for sequential least-squares problems. In so doing we derive the basic equations for the Kaiman filter. Finally we study the influence of changing the weight of observations. We bring a list of matrix identities as several steps rely on these.

15. Construction of An-isotropic Covariance-Functions Using Sums of Riesz-Representers

    C. C. Tscherning

    Abstract

    We regard a reproducing kernel Hilbert space (RKHS) of functions harmonic in the set outside a sphere with radius R0, having a reproducing kernel K ₀ (P,Q). (P, Q, and later P _n being points in the set of harmonicity). The degree-variances of this kernel will be denoted σo_n.

    The set of Riesz-representers associated with the evaluation functionals (or gravity functionals) related to distinct points P _n ,n = 1,..., N, on a 2D-surface surrounding the bounding sphere will be linear independent. These functions are used to define a new N-dimensional RKHS with. kernel {fy1|90-1}

    If the points all are located on a concentric sphere with radius R ¹ > Po, and form an є-net covering the sphere, and a _n are suitable area-elements (depending on N) then this kernel will converge towards an isotropic kernel with degree-variances σ_n \( \mathop \sigma \nolimits_n^2 = (2n + 1)*\sigma _{0n}^2 *\left( {\frac{{R_0 }} {{R_1 }}} \right)\left( {2n + 2} \right)*(cons\tan t{\text{)}} \)

    Consequently, if we want K _N (P,Q) to represent an isotropic covariance function of the Earth’s gravity potential, COV(P,Q), we can select σo_n so that σ_n becomes equal to the empirical degree-variances.

    If the points are chosen at varying radial distances R _n > R _o then we have constructed an anisotropic kernel, or equivalent covariance function representation.

    If the points are located in a bounded region, the kernel may be used to modify the original kernel, CON _N (P,Q)=CON(P,Q)+K _N (P,Q)

    Values of an-isotropic covariance functions constructed based on these ideas have been calculated, and some first ideas are presented on how to select the points P _n .

16. New Covariance Models for Local Applications of Collocation

    R. Barzaghi, A. Borghi, G. Sona

    Abstract

    The least-squares collocation method, used to predict or filter a signal, is based on the estimation of the empirical covariance function and the fitting of the empirical values with a proper model function. Generally, with the standard methods on the sphere, we reach a good fitting only up to the first zero of the empirical function. In this work we have investigated how much the collocation filtering is affected by a poor fitting of the empirical covariance.

    Numerical tests have been done both on 1D observed and simulated data to quantify the combined impact on filtering of non stationarity and covariance fitting.

    Furthermore, a new model function on the sphere has been developed which is able to fit in an optimal way the empirical values.

    Simulations have been also carried out on the sphere to test the effectiveness of the collocation filtering using the new covariance model.

17. Approximation of Harmonic Covariance Functions on the Sphere by non Harmonic Locally Supported Ones

    G. Moreaux, C. C. Tscherning, F. Sansò

    Abstract

    Harmonic splines have been used in many branches of geodesy to interpolate and to predict data discretly given on the sphere. Spline technique permits the use of data of different kinds and with different noise-characteristics but has the drawback that one needs to solve a positive definite system as many equations as the number of data. The matrices of such systems are full matrices because the covariance functions even for very long distances are different from zero due to the harmonicity of the covariance function. However, there is an appropriate angle of separation beyond which the covariance function values are negligible small and thus permits the use of finite instead of full covariance function. The computational savings which might be gained if finite covariance functions could be used would be large. If we used a grid with N x N values, the full matrices would contain N ² x (N ²—l)/2 elements. If finite covariance functions are used and if the observations are ordered in a reasonable manner, the matrices would contain non-zero elements of the order N ³. This fact made us propose three techniques to approximate harmonic covariance functions by finite supported positive definite functions. Because all covariance function related to the anomalous potential of the earth can be seen as the spherical convolution of a so-called original function with itself and because the convolution of a finite supported function with itself gives a finite positive definite function, we approximate the covariance function by the self spherical convolution of a finite approximation of the original function.

    In this talk the theoretical background of our three methods is given and comparison between the so-called finite covariance function of Sansò and Schuh (1987) and our three techniques have been carried out for three types of covariance functions associated with the determination of the anomalous gravity potential from gravity anomalies. After that we compare the solution of the full linear system with the sparse one associated with our first method for the third covariance function.

18. Integration of a Priori Gravity Field Models in Boundary Element Formulations to Geodetic Boundary Value Problems

    Roland Klees, Rüdiger Lehmann

    Abstract

    Current high resolution geopotential models of the Earth are based on a combination of satellite and terrestrial data. Satellite data are well-suited to recover the long-wavelength features of the geopotential up to some degree N, whereas terrestrial gravity and height data fix the medium and short wavelengths. Usually, the recovering of the medium and short-wavelengths from terrestrial data is formulated as a boundary value problem (BVP) for the difference between the Earth’s geopotential and the long-wavelength geopotential model as derived from satellite data commonly referred to as the disturbance potential. Since a number of geopotential coefficients of the satellite model cannot be improved by terrestrial data, we should fix them when solving the BVP. Then we are faced with a constrained (overdetermined) BVP for the Laplace equation. This has implications for the representation formula and/or the choice of the trial & test space in Galerkin boundary element methods.

    We consider multipole representation, modified kernel functions, and modified trial spaces. The latter are the best choice for theoretical and numerical reasons. We propose a general method to construct a system of base functions that fix an a priori given set of geopotential coefficients. In addition, we address the problem of compression rates and stability, which implies the use of multiscale base functions. Various implementations are tested and compared for the altimetry-gravimetry II BVP.