IV Hotine-Marussi Symposium on Mathematical Geodesy

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.

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.

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.

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 L2 (S) boundary data. This result can be achieved by Cimmino’s theory which is also reviewed here for the ease of the reader.

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.

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.

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.

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.

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.

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.

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.

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.

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.

We regard a reproducing kernel Hilbert space (RKHS) of functions harmonic in the set outside a sphere with radius R0, having a reproducing kernel K₀(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 R1 > 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 .

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.

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 N2 x (N2—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 N3. 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.

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.

The inverse Stokes problem deals with the determination of gravity anomalies Δg from given disturbing potential T of geoidal height data, which are based on the processing of satellite altimetry data. By introducing a couple of simplifications —the so-called spherical approximation and the constant radius approximation—a very simple solution is achieved, which can be represented by a spherical integral formula (inverse Stokes integral) in the space domain. This type of approximation has often been applied in the past for the calculation of gravity anomalies from altimetric sea surface heights, and is implicity contained also in Least Sqares Collocation and FFT procedures. A more accurate solution of the inverse Stokes problem is achieved by retaining the first order ”ellipsoidal effects” in the boundary condition relating Δg and T, neglecting terms of order f2 (f= flattening of the reference ellipsoid approximating the earth). These ellipsoidal effects are due to the non-radial direction of the derivative appearing in the boundary condition,terms in the reference potential depending on J ₂ (zonal harmonic of degree 2) and ω (angular velocity of the earth’s rotation), anddeviations of the true boundary surface from a sphere.As a consequence of these ellipsoidal terms the solution of the inverse Stokes problem can no longer be represented by a simple spherial integral formula. Starting from an approximation of the boundary condition of order 0(f) the solution of the inverse Stokes problem is provided in the frequency domain, using spherical harmonics as base functions. Special attention is given to the null space of the boundary operator as well as to small eigenvalues related to the harmonics ot first degree. The results are transformed from the frequency domain into the space domain, resulting in two alternatives for the procedure of evaluation. In the first alternative the altimetry-derived disturbing potential (or geoidal height) data are inserted into the inverse Stokes formula, and a correction term is applied to the resulting gravity anomalies. In contrast, in the second alternative a correction term is added to the disturbing potential data, and this reduced data is inserted in the inverse Stokes formula. A numerical evaluation based on the EGM96 geopotential model results in estimates of the order of magnitude of the correction terms, which is ±3 • 10-6ms-2 and ±2m, respectively; these numbers mirror the approximations existent in the ”spherical” inverse Stokes problem. The correction terms show a low-frequency behaviour, dominated by spectral terms of degree ≤ 20.

The target of the spheroidal Gauss-Listing geoid determination is presented as a solution of the spheroidal fixed-free two-boundary value problem based on a spheroidal Bruns transformation (spheroidal Bruns formula). The nonlinear spheroidal Bruns transform (nonlinear spheroidal Bruns formula), and the spheroidal fixed part and the spheroidal free part of the two-boundary value problem have been derived. Four different spheroidal gravity models are treated, in particular whether they pass the test to fit to the postulate of a level ellipsoidal gravity field, namely of Somigliana-Pizzetti type. The free part of the fixed-free two-boundary value problem in its linearized version agrees with the spheroidal Stokes boundary value problem.

Based on the contribution ”The oblique Mercator projection on the ellipsoid of revolution $$ _{n,m}^{(0)} $$ ” , Journal of Geodesy 70 (1995) 38-50, also known as the Hotine rectified skew orthomorphic projection, an alternative representation of the conformai mapping of Hotine type is presented.An oblique meta-equatorial coordinate system is established, namely meta-longitude and metalatitude, which s tuned to solve the first boundary value problem of the Korn-Lichtenstein equation (pde) which govern conformai mapping of the oblique Mercator. Numerical examples are presented.

The differential equations which generate a general conformai mapping of a two-dimensional Riemann manifold found by Korn and Lichtenstein are reviewed. The Korn-Lichtenstein equations subject to the integrability conditions of type vectorial Laplace-Beltrami equations are solved for the geometry of an ellipsoid of revolution (International Reference Ellipsoid), specifically in the function space of bivariate polynomials in terms of surface normal ellipsoidal longitude and normal ellipsoidal latitude. The related coefficient constraints are collected and the constraints to the general solution of the Korn-Lichtenstein equations which directly generates Gauß-Krüger conformai coordinates as well as the Universal Transverse Mercator Projection (UTM) in one step, avoiding any intermediate isometric coordinate representation, are presented. Namely, the equidistant mapping of a meridian of reference generates the constraints in question. The detailed computation of the solution is given in terms of bivariate polynomials up to degree five with coefficients listed in closed form. In addition, a fresh derivation of the Korn-Lichtenstein equations of conformai mapping for a (pseudo-) Riemann manifold of arbitrary dimension extending initial results for higher-dimensional manifolds of Riemann type is presented.

In the formulation of the scalar free boundary value problem we solve for the gravity potential in the external space outside the earth’s surface and for the vertical position of the boundary surface. After linearization the reduced boundary condition refers to the Telluroid s ∋ p, and the new difference quantity δω, the disturbing potential, is introduced. To represent the unknown disturbing potential in the global basis of spherical harmonics a harmonic analysis has to be applied to the given boundary data. In this context the boundary data have to be (downward) continued from s to a reference surface which shows a rotational symmetry with respect to the earth’s mean rotational axis.In general a sphere K or the surface E of an ellipsoid of revolution is selected.After the analytical continuation of the evaluation operator E _s the boundary condition can be split in two parts. The main component is covered by the isotropic term which corresponds to the Stokes-problem. The second part consists of the ellipsoidal and topographical component which are functionals of δω. Therefore an iterative solution stategy is appropriate. In numerical investigations the different behaviour in convergence—whether K or E is used as reference surface—is object of this contribution.

A generalized formulation of the geodetic boundary value problem and the concept of its weak solution were the starting point of this paper. The construction of a bilinear form connected with the problem in question was discussed and associated functional-analytic aspects reviewed. However, the main emphasis of the paper was on the interpretation of the above mentioned approach in terms of function bases. For this purpose the respective Galerkin’s system was constructed and a considerable attention was given to the computation of individual elements of Galerkin’s matrix and to an estimate of their accuracy. A majority of these problems were demonstrated for base functions given by elementary potentials of Laplace’s equation with a particular view to the influence of the topography of the boundary and the depth of individual mass concentrations.

The paper deals with the altimetry-gravimetry problems, taking into account the different situation with available terrestrial geodetic data on the land and sea part of the Earth surface. A unified formulation is given, from which all relevant problems can be derived as special cases. The related theory is briefly discussed. The primary subject of the paper is the numerical treatment of altimetry-gravimetry problems. For this purpose, a numerical experiment is invented, in which special numerical aspects of these problems can be investigated.

When standard boundary element methods (BEM) are used in order to solve the linearized vector Molodensky problem we are confronted with two problems: (i) the absence of O (|x|-2) terms in the decay condition is not taken into account, since the single layer ansatz, which is commonly used as representation of the disturbing potential, is of the order O (|x|_1) as x → ∞. This implies that the standard theory of Galerkin BEM is applicable since the injectivity of the integral operator fails; (ii) the N x N stiffness matrix is dense, with N typically of the order 105. Without fast algorithms, which provide suitable approximations to the stiffness matrix by a sparse one with O(N ∙ log8N), s ≥ 0, non-zero elements, high-resolution global gravity field recovery is not feasible.We propose solutions to both problems, (i) A proper variational formulation taking the decay condition into account is based on some closed subspace of codimension 3 of L2(T). Instead of imposing the constraints directly on the boundary element trial space, we incorporate them into a variational formulation by penalization with a Lagrange multiplier. The conforming discretization yields an augmented linear system of equations of dimension (N+3) x (N+3). The penalty term guarantees the well-posedness of the problem, and gives precise information about the incompatibility of the data, (ii) Since the upper left submatrix of dimension N x N of the augmented system is the stiffness matrix of the standard BEM, the approach allows to use all techniques to generate sparse approximations to the stiffness matrix such as wavelets, fast multipole methods, panel clustering etc. without any modification. We use a combination of panel clustering and fast multipole method in order to solve the augmented linear system of equations in 0(N) operations. The method is based on an approximation of the kernel function of the integral operator by a degenerate kernel in the far field, which is provided by a multipole expansion of the kernel function.In order to demonstrate the potential of the method we solve a Robin problem on the sphere with a nullspace of dimension 3. For N = 65538 unknowns the matrix assembly takes about 600 s and the solution of the sparse linear system using GMRES without any preconditioning takes about 8 s. 30 iterations are sufficient to make error smaller than the discretization error.

At the First Hotine-Marussi Symposion, this author compared varieous methods to pretict the sate vector” within a Random Effects Model ineluding inhomBLIP (”lesst-squar” collocation”), homI3LIP and horn- BLUP (”robust collocation”); see (1986). The term ”robust” refers to the specific restance of homBLUP against scale errore In the prior infonnation. The case of resistance against multiple scale errors was later discussed in tvo popers by 8. Sehaf-frin/B, Middd tl989; 1991) related to terrestrial gravity field studies.In the meantime, the question was asked whether there ia a continuous transition between homBLUP (which is weakly unbiased) and homBLIP (which is ”more efficient” than homBLUP). Such an intermediate predictor would allow us to give up some rigor in the un-biasednes condition in order to gain more ”efficiency”, i. e., a smaller Mean-Square-Error matrix. First results have been presented for univariate spatial processes in the context of Ordinary Kriging by (1997).In the following we shall introduce the notion of ”Softly Unbiased Prediction” and present the corresponding formulas for the homBLISUP (Best homogeneously Linear Softly Unbiased Predictor) within a Random Effects Model.A similar approach within the GaussMarkov Model led to the BLUSUE (Best Linear Uniformly Softly Unbiased Eatimator) of the parameters; see (1998).

The connection between Brownian motion, the Wiener process and fractal geometry is outlined. Incremental stochastic processes are described by incremental variance-covariance functions, called Kolmogorov structure function or Krige variogram.Geodetic examples (geodetic networks on differential manifolds / curved surfaces) are presented, in particular with respect to spacetime criterion matrices, namely towards stationarity, homogeneity and isotropy.

It is well known that the least squares estimation is equivalent to the Best Linear Unbiased Estimation (BLUE) in the case. that the covariance matrix of the observations is positive definite. If the variance covariance matrix is positive semi-definite the inverse of this matrix (the weight matrix) is in general not unique so that the question arises: which weight matrix has to be used in least squares estimation to get the BLUE? This question is discussed in this paper and an alternative solution by means of the condition adjustment is given. The numerical problems that can occur when the modified covariance matrix is used can be avoided by using the alternative solution. With respect to the theory presented in this paper, it is shown how important a correct definition of weight matrix in the least squares method is.

Earlier a solution of the gravimetric free scalar boundary value (BV) problem was derived with respect to the potential coefficients C _n,m , taking into account the correction terms due to both the earth’s ellipticity and surface topography. In the present paper the behavior of this solution is studied for large values of degree n. It is shown that the absolute value of the ellipsoidal correction to the spherical approximation $$ _{n,m}^{(0)} $$ , having the sign opposite to the latter, becomes of the same order for n ≈ 360, is equal to it at n ≈ 600 and then exceeds it, increasing with growing n. A similar situation is observed for the ellipsoidal correction in{152-2} derived by other authors, in particular by Pellinen, Rapp and Cruz. In the present paper, proceeding from the same BV solution, an alternative iteration procedure for evaluating $$ _{n,m} $$ is elaborated. The initial approximation in it is not $$ _{n,m}^{(0)} $$ but a modified quantity $$ _{n,m,}^{(1)} $$ depending on e2. The new formulas are well-suited for evaluating the potential coefficients of any degree n and order m.

A detailed and statistically rigorous treatment of adjustment problems in combined GPS/levelling/geoid networks is given in this paper. The two main types of ”unknowns” in this kind of multi-data ID networks are the gravimetric geoid accuracy and a 2D spatial field that describes all the systematic distortions among the available height data sets. An accurate knowledge of the latter becomes especially important when we consider employing GPS techniques for levelling purposes with respect to a local vertical datum. Various modeling alternatives for the correction field are presented, namely a pure discrete deterministic model, a hybrid deterministic and stochastic model, and finally a pure stochastic model. Variance component estimation is also introduced as an important tool for assesing the actual geoid noise level, and checking a-priori given geoid error models. In addition, theoretical comparisons are made with some of the already established adjustment models that have been used in practice. The problem of statistical testing of various model components (data noise, deterministic model, stochastic model) in such networks is also discussed. Finally, some conclusions are drawn and a few recommendations for further study are pointed out.

This paper presents some alternative formulas reduction. The alternative formulas are checked against the existing formulas. They are checked with the TC-program written originally by (1984) and with the rigorous formulas of terrain reduction given by Abd-Elmotaal (1993), (1995). The checking process are made into two procedures. In the first procedure, we assumed ideal topography of constant height performing an exact spherical shell, and comparison with the results of the closed formula of the exact spherical shell is carried out. In the second procedure, actual data sets are used and CPU-time comparison with the existed formulas is performed. The alternative formulas give relatively good accuracy in a reasonable computer time.

The main purpose of this article is to present the idea of using Multi-process models as a method of detecting errors in GPS observations.The theory behind Multi-process models, and. double differenced phase observations in GPS is presented shortly.It is shown how to model cycle slips in the Multi-process context by means of a simple simulation. The simulation is used to illustrate how the method works, and it is concluded that the method deserves further investigation.

A satellite’s trajectory around a celestial body, e.g. the Earth, is mainly influenced by the gravitational field of the body. This causes perturbations of the orbit which would be a Keple-rian ellipse in the ideal case. Measuring these perturbations allows an analysis of the gravitational field itself. It is quite obvious that different trajectories are perturbed in different ways depending on the direction, position and velocity of the satellite’s movement in the gravitational field. For analysis purposes it is vital to know which components of the gravitational field have the strongest effect on the satellite’s orbit, with other words, which component of the gravitational field’s description can be obtained from an analysis of the measured orbit perturbations. This task is called satellite orbit sensitivity analysis. It is mainly performed during the planning process for a new geodetic satellite mission to get an idea of the benefits the mission may yield. Today there are two main methods to perform a sensitivity analysis. The classical way is based on the solution of the Lagrange planetary equations for the satellite using Kaula’s representation of the perturbing potential by use of the six Ke-plerian elements. The second method is some how a try-and-error method: A complete orbit integration (orbit synthesis) is performed using a reference gravitational field model in it’s original state first. After that single representation coefficients are varied, the integration performed again and the perturbations caused with respect to the reference orbit are a measure for the sensitivity of the satellite with respect to the varied coefficient. Especially the second method yields high computational time expenses.The idea of the method presented in the following is to gain some sort of pre-information on the coefficients of a gravitational field’s representation a satellite is most sensitive to by means of a Fourier analysis of both the perturbing potential and the satellite’s orbit data. To be more precise the pre-information will, as we will see, consist of information on the so-called lumped coefficients, linear combinations of gravitational coefficients being the Fourier coefficients of the development of the gravitational field. This information will be introduced to a parameter adjustment process to determine the accuracy with which the preselected coefficients are possibly estimated from satellite orbit data. This adjustment procedure will be based solely on an existing gravitational model as used for the reference trajectory before and the approximate orbit positions computed from it. Thus it will yield no improvement of the unknown gravitational coefficients but their variance-covariance-matrix. By using this two-step method for orbit analysis we want to reduce the dimension of the equation systems to be evaluated during the adjustment process and with that the computation time expenses by omitting irrelevant gravitational coefficients introducing pre-computed information from the Fourier representations.