IX Hotine-Marussi Symposium on Mathematical Geodesy

Many studies in the past have discussed potential orbit configurations of future satellite gravity missions. Most of those works have targeted orbit optimization of the satellite missions of the next generation in the so-called Bender formation. The studies have investigated the impact of the Keplerian orbital parameters, especially the influence of the repeat orbits and mission altitude of both satellite pairs and the inclination of the second pair in Bender formation on the satellite configurations’ gravity field recovery quality performance.

Obviously, the search space for the orbit optimization in the Bender formation is vast and, therefore, different approaches have been suggested for optimal orbit design. Among approaches, however, different assumptions about input geophysical models as well as the error models into the simulation software play a role. Our paper shows how different assumptions for input models change the orbit optimization results. For this purpose, the genetic algorithm has been utilized for orbit optimization of the Bender formation where different input models were considered. Those input models include (1) the updated ESA geophysical models, and (2) error models for the Ocean Tide (OT error) and Atmosphere-Ocean (AO error). Here, we focus on the impact of the models on relative difference of the longitude of ascending nodes between the two pairs in Bender formation. The results of the paper clearly state that our current and future knowledge about signal and error models can significantly affect the orbit optimization problem.

The study of the Earth’s time-varying gravity field using GRACE data requires the removal of correlated errors using filtering techniques in the spherical harmonic domain. The empirical decorrelation filter is an effective method of decorrelating order-wise series of spherical harmonic coefficients, although its improper implementation can lead to signal attenuation. To reduce geophysical signal over-filtering, decorrelation should be performed only for orders that show evidence of high correlation. In this paper we investigate and compare the behavior of three criteria, i.e., the root mean square ratio, the angle distribution of phase spectrum and the geometric properties of order-wise coefficient series, that can be used for the identification of correlated orders in GRACE data. Our analysis indicates that the root mean square ratio is the most reliable criterion, due to its simple implementation and for providing averaged time series of equivalent water height with smaller root mean square error, based on a simulation.

The computation of second- and third-order derivatives of the Somigliana-Pizzetti reference gravity field (reference gravity gradients and reference gravity field curvature values) is investigated. Closed expressions for these second- and third-order derivatives are derived in spheroidal coordinates. Rigorous equations for the second-order derivatives in a local north-oriented frame are also given. It is shown that on the surface of the reference ellipsoid, these lengthy expressions can be reduced to simple elegant formulas, akin to Somigliana’s formula for the first-order derivative. Numerical results provide insight into the curvature of the reference plumb lines and spheropotential surfaces. It is shown that spheropotential surfaces up to 10,000 m in altitude differ from an oblate ellipsoid of revolution by less than 0.04 m. It is also shown that this fact can be utilised to approximate the reference gravity gradients through simple formulas.

This paper analyzes the arguments in the report “The shape of the quasigeoid” by Robert Kingdon, Petr Vaníček, Marcelo Santos presented in Rome (IX Hotin-Marussi Symposium on Theoretical Geodesy, Italy, Rome, June 18–June 22, 2018), which contains the criticisms of the basic concepts of Molodensky’s theory: normal height and height anomaly of the point on the earth’s surface, plotted on the reference ellipsoid surface and forming the surface of a quasigeoid. Also are presented the main advantages of the system of normal heights. They are closely related to the theory of determination of the external gravitational field and the Earth’s surface, are presented.

Despite the fact that the main core of Molodensky’s theory is the rigorous determining of the anomalous potential on the Earth’s surface, the advantage of the normal heights system can be shown and proved separately. And this can be easily demonstrated by a simple hypothetical example of the spherical non-rotating Earth where the change of marks along the floor of a strictly horizontal tunnel in the spherical mountain massif serves as criterion for the convenience of the system. In this example, the difference in orthometric heights comes up to 3 cm per 1.5 km. It will require the same corrections to the measured elevations what with the effect of the orthometric heights system. Also the knowledge of the inner structure of the rock mass is necessary. In turn, the normal heights are constant along the tunnel and behave as dynamic ones and there is no need to introduce corrections.

Neither the ellipsoid nor the quasi-geoid is a reference surface for normal heights, because until now the heights are referenced to the initial tide gauge. The numerical values of heights are assigned to the physical surface. This is similar to the ideas of prof. L. V. Ogorodova about the excessive emphasis on the concept of quasigeoid itself. According to prof. V. V. Brovar the more general term is the “height anomaly” that exists both for points on the Earth’s surface and at a distance from it and decreases together with an attenuation of the anomalous field.

The aim of the paper is to implement the Green’s function method for the solution of the Linear Gravimetric Boundary Value Problem. The approach is iterative by nature. A transformation of spatial (ellipsoidal) coordinates is used that offers a possibility for an alternative between the boundary complexity and the complexity of the coefficients of Laplace’s partial differential equation governing the solution. The solution domain is carried onto the exterior of an oblate ellipsoid of revolution. Obviously, the structure of Laplace’s operator is more complex after the transformation. It was deduced by means of tensor calculus and in a sense it reflects the geometrical nature of the Earth’s surface. Nevertheless, the construction of the respective Green’s function is simpler for the solution domain transformed. It gives Neumann’s function (Green’s function of the second kind) for the exterior of an oblate ellipsoid of revolution. In combination with successive approximations it enables to meet also Laplace’s partial differential equation expressed in the system of new (i.e. transformed) coordinates.

In global studies, the Earth’s gravitational field is conveniently described in terms of spherical harmonics. Four integral-based solutions to a gravitational curvature boundary-value problem can formally be formulated for the vertical-vertical-vertical, vertical-vertical-horizontal, vertical-horizontal-horizontal and horizontal-horizontal-horizontal components of the third-order gravitational tensor. Each integral equation provides an independent set of spherical harmonic coefficients because each component of the third-order gravitational tensor is sensitive to gravitational changes in the different directions. In this contribution, estimations of spherical harmonic coefficients of the gravitational potential are carried out by combining four solutions of the gravitational curvature boundary-value problem using three methods, namely an arithmetic mean, a weighted mean and a conditional adjustment model. Since the third-order gradients of the gravitational potential are not yet observed by satellite sensors, we synthesise them at the satellite altitude of 250 km from a global gravitational model up to the degree 360 while adding a Gaussian noise with the standard deviation of 6.3 × 10⁻¹⁹ m⁻¹ s⁻². Results of the numerical analysis reveal that the arithmetic mean model provides the best solution in terms of the RMS fit between predicted and reference values. We explain this result by the facts that the conditions only create additional stochastic bindings between estimated parameters and that more complex numerical schemes for the error propagation are unnecessary in the presence of only a random noise.

Our planet Earth is constantly deforming under the effects of geophysical processes that cause linear and nonlinear displacements of the geodetic stations upon which the International Terrestrial Reference Frame (ITRF) is established. The ITRF has traditionally been defined as a secular (linear) frame in which station coordinates are described by piecewise linear functions of time. Nowadays, some particularly demanding applications however require more elaborate reference frame representations that can accommodate non-linear displacements of the reference stations. Two such types of reference frame representations are reviewed: the usual linear frame enhanced with additional parametric functions such as seasonal sine waves, and non-parametric time series of quasi-instantaneous reference frames. After introducing those two reference frame representations, we briefly review the systematic errors present in geodetic station position time series. We finally discuss the practical issues raised by the existence of these systematic errors for the implementation of both types of non-linear reference frames.

LARES, an Italian satellite launched in 2012, and its successor LARES-2 approved by the Italian Space Agency, aim at the precise measurement of frame dragging predicted by General Relativity and other tests of fundamental physics. Both satellites are equipped with Laser retro-reflectors for Satellite Laser Ranging (SLR). Both satellites are also the most dense particles ever placed in an orbit around the Earth thus being nearly undisturbed by nuisance forces as atmospheric drag or solar radiation pressure. They are, therefore, ideally suited to contribute to the terrestrial reference frame (TRF). At GFZ we have implemented a tool to realistically simulate observations of all four space-geodetic techniques and to generate a TRF from that. Here we augment the LAGEOS based SLR simulation by LARES and LARES-2 simulations. It turns out that LARES and LARES-2, alone or in combination, can not deliver TRFs that meet the quality of the LAGEOS based TRF. However, once the LARES are combined with the LAGEOS satellites the formal errors of the estimated ground station coordinates and velocities and the co-estimated Earth Rotation Parameters are considerably reduced. The improvement is beyond what is expected from error propagation due to the increased number of observations. Also importantly, the improvement concerns in particular origin and scale of the TRF of about 25% w.r.t. the LAGEOS-combined TRF. Furthermore, we find that co-estimation of weekly average range biases for all stations does not change the resulting TRFs in this simulation scenario free of systematic errors.

The contribution discusses the optimal design of a Global Navigation Satellite System (GNSS) network compromising between the estimation of the tectonic motion with other geodetic criteria. It considers the case of a pre-existing network to be densified by the addition of new stations. An optimization principle that minimizes the error of the estimated background motion and maximizes the spatial uniformity of the stations is formulated. A means to solve approximately the proposed target function is presented. The proposed procedure is preliminary tested for the case of the densification of the Agenzia Spaziale Italiana (ASI) GNSS network in Italy.

In this contribution, we extend the Gauss-Helmert model (GHM) with t-distributed errors (previously established by K.R. Koch) by including autoregressive (AR) random deviations. This model allows us to take into account unknown forms of colored noise as well as heavy-tailed white noise components within observed time series. We show that this GHM can be adjusted in principle through constrained maximum likelihood (ML) estimation, and also conveniently via an expectation maximization (EM) algorithm. The resulting estimator is self-tuning in the sense that the tuning constant, which occurs here as the degree of freedom of the underlying scaled t-distribution and which controls the thickness of the tails of that distribution’s probability distribution function, is adapted optimally to the actual data characteristics. We use this model and algorithm to adjust 2D measurements of a circle within a closed-loop Monte Carlo simulation and subsequently within an application involving GNSS measurements.

The DIA-method, for the detection, identification and adaptation of modeling errors, has been widely used in a broad range of applications including the quality control of geodetic networks and the integrity monitoring of GNSS models. The DIA-method combines two key statistical inference tools, estimation and testing. Through the former, one seeks estimates of the parameters of interest, whereas through the latter, one validates these estimates and corrects them for biases that may be present. As a result of this intimate link between estimation and testing, the quality of the DIA outcome \(\bar {x}\) must also be driven by the probabilistic characteristics of both estimation and testing. In practice however, the evaluation of the quality of \(\bar {x}\) is never carried out as such. Instead, use is made of the probability density function (PDF) of the estimator under the identified hypothesis, say \(\hat {x}_{i}\), thereby thus neglecting the conditioning process that led to the decision to accept the i ^th hypothesis. In this contribution, we conduct a comparative study of the probabilistic properties of \(\bar {x}\) and \(\hat {x}_{i}\). Our analysis will be carried out in the framework of GNSS-based positioning. We will also elaborate on the circumstances under which the distribution of the estimator \(\hat {x}_{i}\) provides either poor or reasonable approximations to that of the DIA-estimator \(\bar {x}\).

It is well known that the MInimum NOrm LEast-Squares Solution (MINOLESS) minimizes the bias uniformly since it coincides with the BLUMBE (Best Linear Uniformly Minimum Biased Estimate) in a rank-deficient Gauss-Markov Model as typically employed for free geodetic network analyses. Nonetheless, more often than not, the partial-MINOLESS is preferred where a selection matrix \(S_k := \operatorname {\mathrm {Diag}}(1,{\ldots },1,0,{\ldots },0)\) is used to only minimize the first k components of the solution vector, thus resulting in larger biases than frequently desired. As an alternative, the Best LInear Minimum Partially Biased Estimate (BLIMPBE) may be considered, which coincides with the partial-MINOLESS as long as the rank condition \( \operatorname {\mathrm {rk}}(S_k N) = \operatorname {\mathrm {rk}}(N) = \operatorname {\mathrm {rk}}(A) =: q\) holds true, where N and A are the normal equation and observation equation matrices, respectively. Here, we are interested in studying the bias divergence when this rank condition is violated, due to q > k ≥ m − q, with m as the number of all parameters. To the best of our knowledge, this case has not been studied before.

The two layers inverse gravimetric problem is to determine the shape of the two layers in a body B, generating a given gravitational potential in the exterior of B. If the constant density of the two layers is given, the problem is reduced to the determination of the geometry of the interface between the two. The problem is known to be ill posed and therefore it needs a regularization, that for instance could have the form of the optimization of a Tikhonov functional. In this paper it is discussed why neither L ² nor H ^{1, 2} are acceptable choices, the former giving too rough solutions, the latter too smooth. The intermediate Banach space of functions of Bounded Variation is proposed as a good solution space to allow for discontinuities, but not too wild oscillations of the interface. The problem is analyzed by standard variational techniques and existence of the optimal solution is proved.

In this paper, the three kind of solutions of TLS problem, the common solution by singular value decomposition (SVD), the iteration solution and Partial-EIV model are firstly reviewed with respect to their advantages and disadvantages. Then a newly developed Converted Total Least Squares (CTLS) dealing with the errors-in-variables (EIV) model is introduced. The basic idea of CTLS has been proposed by the authors in 2010, which is to take the stochastic design matrix elements as virtual observations, and to transform the TLS problem into a traditional Least Squares problem. This new method has the advantages that it cannot only easily consider the weight of observations and the weight of stochastic design matrix, but also deal with TLS problem without complicated iteration processing, if the suitable approximates of parameters are available, which enriches the TLS algorithm and solves the bottleneck restricting the application of TLS solutions. CTLS method, together with all the three TLS models reviewed here has been successfully integrated in our coordinate transformation programs and verified with the real case study of 6-parameters Affine coordinate transformation. Furthermore, the comparison and connection of this notable CLTS method and estimation of Gauss-Helmert model are also discussed in detail with applications of coordinate transformations.

In this contribution, a robust Bayesian approach to adjusting a nonlinear regression model with t-distributed errors is presented. In this approach the calculation of the posterior model parameters is feasible without linearisation of the functional model. Furthermore, the integration of prior model parameters in the form of any family of prior distributions is demonstrated. Since the posterior density is then generally non-conjugated, Monte Carlo methods are used to solve for the posterior numerically. The desired parameters are approximated by means of Markov chain Monte Carlo using Gibbs samplers and Metropolis-Hastings algorithms. The result of the presented approach is analysed by means of a closed-loop simulation and a real world application involving GNSS observations with synthetic outliers.

The authors conceived the GNSS for Meteorology (G4M) procedure to remote-sense the Precipitable Water Vapor (PWV) content in atmosphere with the aim to detect severe meteorological phenomena. It can be applied over an orographically complex area, exploiting existing networks of Global Navigation Satellite System (GNSS) Permanent Stations (PSs) and spread meteorological sensors, not necessarily co-located. The results of a posteriori analysis of four significant meteorological events are here presented, also in comparison with rain gauge data, to show the effectiveness of the method. The potentiality of G4M to detect and locate in space and time intense rainfall events is highlighted. The upcoming application of G4M in near-real time could provide a valuable support to existing Decision Support System for meteorological alerts.

The Laplace equation represents harmonic (i.e., both source-free and curl-free) fields. Despite the good performance of spherical harmonic series on modeling the gravitational field generated by spheroidal bodies (e.g., the Earth), the series may diverge inside the Brillouin sphere enclosing all field-generating mass. Divergence may realistically occur when determining the gravitational fields of asteroids or comets that have complex shapes, known as the Complex-boundary Value Problem (CBVP). To overcome this weakness, we propose a new spatial-domain numerical method based on the equivalence transformation which is well known in the fluid dynamics community: a potential-flow velocity field and a gravitational force vector field are equivalent in a mathematical sense, both referring to a harmonic vector field. The new method abandons the perturbation theory based on the Laplace equation, and, instead, derives the governing equation and the boundary condition of the potential flow from the conservation laws of mass, momentum and energy. Correspondingly, computational fluid dynamics (CFD) techniques are introduced as a numerical solving scheme. We apply this novel approach to the gravitational field of comet 67P/Churyumov-Gerasimenko with a complex shape. The method is validated in a closed-loop simulation by comparing the result with a direct integration of Newton’s formula. It shows a good consistency between them, with a relative magnitude discrepancy at percentage level and with a maximum directional difference of 5°. Moreover, the numerical scheme adopted in our method is able to overcome the divergence problem and hence has a good potential for solving the CBVPs.

Advancements in the Global Geodetic Observing System (GGOS) have enabled us to investigate the effects of lateral heterogeneities in the internal Earth structure on long-term surface deformations caused by the Glacial Isostatic Adjustment (GIA). Many theories have been developed so far to consider such effects based on analytical and numerical approaches, and 3D viscosity distributions have been inferred. On the other hand, fewer studies have been conducted to assess the effects of lateral heterogeneities on short-term, elastic deformations excited by surface fluids, with 1D laterally homogeneous theories being frequently used. In this paper, we show that a spectral finite-element method is applicable to calculate the elastic deformation of an axisymmetric spherical Earth. We demonstrate the effects of laterally heterogeneous moduli with horizontal scales of several hundred kilometers in the upper mantle on the vertical response to a relatively large-scale surface load. We found that errors due to adopting a 1D Green’s function based on a local structure could amount to 2–3% when estimating the displacement outside the heterogeneity. Moreover, we confirmed that the mode coupling between higher-degree spherical harmonics needs to be considered for simulating smaller-scale heterogeneities, which agreed with results of previous studies.

The increased use of areal measurement techniques in engineering geodesy requires the development of adequate areal analysis strategies. Usually, such analysis strategies include a modelling of the data in order to reduce the amount of data while preserving as much information as possible. Free form surfaces like B-splines have been proven to be an appropriate tool to model point clouds. The complexity of those surfaces is among other model parameters determined by the number of control points. The selection of the appropriate number of control points constitutes a model selection task, which is typically solved under consideration of parsimony by trial-and-error procedures. In Harmening and Neuner (J Appl Geod 10(3):139–157, 2016; 11(1):43–52, 2017) a model selection approach based on structural risk minimization was developed for this specific problem. However, neither this strategy, nor standard model selection methods take correlations into account. For this reason, the performance of the developed model selection approach on correlated data sets is investigated and the respective results are compared to those provided by a standard model selection method, the Bayesian Information Criterion.

In 2008 P. Xu (Celest Mech Dyn Astron, 100:231–249) proposed a strictly kinematic perturbation method for determining the Earth’s gravitational field from continuous satellite tracking. The main idea is to process orbital arcs of arbitrary length, thus minimizing superfluous parameter estimation associated with stitching together short-arc solutions, and at the same time formulating the problem in terms of standard linear parameter estimation. While the original formulation appears mathematically robust, its nested quadruple along-track integrations are computationally challenging. We reduce the formulation to double integrals and show that the method is numerically not feasible as originally envisaged. On the other hand, by abandoning the rigorous Gauss-Markov formalism, we show the numerical feasibility of processing multiple-day orbital arcs. The methodology lends itself to high-low and low-low satellite-to-satellite tracking, or combinations thereof, as for GRACE-like systems.

A stochastic process can be represented and analysed by four different quantities in the time and frequency domain: (1) the process itself, (2) its autocovariance function, (3) the spectral representation of the stochastic process and (4) its spectral distribution or the spectral density function, if it exits. These quantities and their relationships can be clearly represented by the “Magic Square”, where the quantities build the corners of this square and the connecting lines indicate the transformations into each other.

The real-valued, time-discrete, one-dimensional and covariance-stationary autoregressive process of order p (AR(p) process) is a frequently used stochastic process for instance to model highly correlated measurement series with constant sampling rate given by satellite missions. In this contribution, a reformulation of an AR(p) to a moving average process with infinite order is presented. The Magic Square of this reformulated process can be seen as an alternative representation of the four quantities in time and frequency, which are usually given in the literature. The results will be evaluated by discussing an AR(1) process as example.

In this paper, we intend to test whether the random deviations of an observed regression time series with unknown regression coefficients can be described by a covariance-stationary autoregressive (AR) process, or whether an AR process with time-variable (say, linearly changing) coefficients should be set up. To account for possibly present multiple outliers, the white noise components of the AR process are assumed to follow a scaled (Student) t-distribution with unknown scale factor and degree of freedom. As a consequence of this distributional assumption and the nonlinearity of the estimator, the distribution of the test statistic is analytically intractable. To solve this challenging testing problem, we propose a Monte Carlo (MC) bootstrap approach, in which all unknown model parameters and their joint covariance matrix are estimated by an expectation maximization algorithm. We determine and analyze the power function of this bootstrap test via a closed-loop MC simulation. We also demonstrate the application of this test to a real accelerometer dataset within a vibration experiment, where the initial measurement phase is characterized by transient oscillations and modeled by a time-variable AR process.

Many geodetic measurements which are automatically gathered by sensors can be interpreted as a time series. For instance, measurements collected by a satellite platform along the satellite’s track can be seen as a time series along the orbit. Special treatment is required if the time series is contaminated by outliers or non-stationarities, summarized as ‘suspicious data’, stemming from sensor noise variations or changes in environment. Furthermore, the collected measurements are often – for instance due to the sensor design – correlated along the track.

Here, we apply the procedure to gravity gradient observations as collected by the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) satellite mission in the low orbit measurement campaign. The estimated autoregressive process is used as a stochastic model of the gravity gradients in a gradiometer-only gravity field determination following the time-wise approach. The resulting estimates are compared to the counterparts of the official EGM_TIM_RL05 processing. Additionally, with newly processed level 1B GOCE gravity gradients at hand we pursue comparison of the robust and conventional approaches for original and reprocessed data.

Based on the success of the satellite mission GOCE in providing information on the global gravity field with high quality and spectral resolution, the realization of the 1 cm-geoid is at reach, leading to an increased interest in regional geoid modeling. It is therefore necessary to review theoretical and numerical aspects of regional geoid modeling, including availability of adequate data. In this study, we deal with the latter aspect, specifically the representation error implied by the available gravity data.

We use least-squares collocation to derive formal errors of block mean gravity anomalies and geoid heights for given distributions of scattered gravity stations. By comparison with independent error measures, we validate a generalized procedure in which we do not base the solution on an empirical covariance function of a specific test area, but rather use band-pass filtered global functions. This implies that the procedure is applicable beyond our specific test-bed and can be used to give general error measures, e.g., for network design in poorly surveyed regions.

The computations are carried out in a medium size test area along the Norwegian coast, where the national gravity basis network had been densified in recent years. This allows to show the gain in geoid accuracy that can be expected from adding the new gravity data. We show that the signal variance of the regional gravity field corresponds well with the one derived from the global covariance function, thus validating our generalized procedure. In previous studies, the accuracy of gravity anomalies and geoid heights in Norway were estimated to be (on average) around 2 mGal and 3 cm, respectively. We find good agreement of the formal gravity anomaly error with the empirical measure. By adding the new data, the gravity anomaly error can be reduced to almost 1 mGal. The formal geoid error can be reduced from around 1.7 to 1.3 cm (on average). The discrepancy between the formal error and the empirical measure of 3 cm is probably due to contributions from GNSS and leveling errors, which are not considered in our formal estimate. The results presented here show larger errors over ocean areas, because the computations are restricted to land data. Available airborne and marine gravity will be considered in the future.

The paper highlights arguments that, coming from Mathematics, have fostered the advancement of Geodesy, as well as those that, generated by geodetic problems, have contributed to the enhancement of different branches in Mathematics. Furthermore, not only examples of success are examined, but also open questions that can constitute stimulating challenges for geodesists and mathematicians.