G1.1 | Recent Developments in Geodetic Theory
EDI
Recent Developments in Geodetic Theory
Convener: Michal Sprlak | Co-conveners: Zuzana Minarechová, Kevin Gobron, Georgios Panou, Petr Holota
Orals
| Tue, 16 Apr, 16:15–18:00 (CEST)
 
Room -2.91
Posters on site
| Attendance Mon, 15 Apr, 10:45–12:30 (CEST) | Display Mon, 15 Apr, 08:30–12:30
 
Hall X2
Orals |
Tue, 16:15
Mon, 10:45
Remarkable advances over recent years give evidence that geodesy today develops under a broad spectrum of interactions, including theory, science, engineering, technology, observation, and practice-oriented services. Geodetic science accumulates significant results in studies towards classical geodetic problems and also problems that only emerged or gained new interest, in many cases as a consequence of synergistic activities in geodesy and tremendous
advances in the instrumentations and computational facilities. In-depth studies progressed in parallel with investigations that mean a broadening of the traditional core of geodesy. The scope of the session is conceived with a certain degree of freedom, though the session is primarily intended to provide a forum for all investigations and results of a theoretical and methodological nature.

Within this concept, we seek contributions concerning problems of reference frames, gravity field studies, dynamics and rotation of the Earth, and positioning, but also presentations, which surpass the frontiers of these topics. We invite presentations illustrating the use of mathematical and numerical methods in solving geodetic problems, showing advances in mathematical modelling, estimating parameters, simulating relations and systems, using high-performance computations, and discussing methods that enable the exploitation of data essentially associated with new and existing satellite missions. Presentations showing mathematical and physical research directly motivated by geodetic need, practice and ties to other disciplines are welcome. In parallel to theory-oriented results also examples illustrating the use of new methods on real data in various branches of geodetic science and practice are very much solicited in this session.

Session assets

Orals: Tue, 16 Apr | Room -2.91

Chairpersons: Michal Sprlak, Zuzana Minarechová, Georgios Panou
16:15–16:20
Advances in gravity field modelling
16:20–16:40
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EGU24-3550
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solicited
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On-site presentation
Peiliang Xu

The numerical integration method has been routinely used by major institutions worldwide (for example, NASA Goddard Space Flight Center and GFZ) to produce global gravitational models from satellite tracking measurements. Such Earth’s gravitational products have found widest possible multidisciplinary applications. The method is essentially implemented by solving the differential equations of the partial derivatives of the orbit of a satellite with respect to the unknown force parameters under the zero initial conditions. From the statistical point of view, satellite gravimetry from satellite tracking is essentially to estimate the unknown parameters in the Newton’s nonlinear differential equations from satellite tracking measurements --- the mathematical foundation for satellite gravimetry from tracking. From this perspective, it is rather trivial to prove that the numerical integration method, originating from Gronwall on Ann Math almost 100 years ago and currently implemented and used in mathematics/statistics, chemistry/physics, and satellite gravimetry, is groundless, even though, up to this moment, many researchers in the geoscientific community still have problems in understanding this side point of my research. In this talk, we focus on presenting three different methods to derive local solutions to the Newton’s nonlinear differential equations of motion of satellites, given unknown initial values and unknown force parameters. They are mathematically correct and can be used to estimate unknown differential equation parameters, with applications in gravitational modelling from satellite tracking measurements as a typical example in geodesy. These solution methods are generally applicable to any differential equations with unknown parameters. More precisely, we develop the measurement-based perturbation theory and construct global uniformly convergent solutions to the Newton’s nonlinear differential equations of motion of satellites, given unknown initial values and unknown force parameters. From the physical point of view, the global uniform convergence of the solutions implies that they are able to exploit the complete/full advantages of unprecedented high accuracy and continuity of satellite orbits of arbitrary length and thus will automatically guarantee theoretically the production of a high-precision high-resolution global standard gravitational models from satellite tracking measurements of any types. Finally, we develop an alternative method by reformulating the problem of estimating unknown differential equation parameters, or the mixed initial-boundary value problem of satellite gravimetry with unknown initial values and unknown force parameters as a standard condition adjustment model with unknown parameters.
Xu P (2018) Measurement-based perturbation theory and differential equation parameter estimation with applications to satellite gravimetry. Commun Nonlinear Sci Numer Simulat, 59, 515-543. DOI 10.1016/j.cnsns.2017.11.021
Xu P (2008) Position and velocity perturbations for the determination of geopotential from space geodetic measurements. Celest Mech Dyn Astr, 100, 231–249.
Xu P (2009) Zero initial partial derivatives of satellite orbits with respect to force parameters violate the physics of motion of celestial bodies. Sci China Ser D, 52, 562–566.
Xu P (2012) Mathematical challenges arising from earth-space observation: mixed integer linear models, measurement-based perturbation theory and data assimilation for ill-posed problems. Invited talk, joint mathematical meeting of American mathematical society, Boston, January 4–7.

How to cite: Xu, P.: Statistical estimation of differential equation parameters: the mathematical foundation for satellite gravimetry from tracking, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-3550, https://doi.org/10.5194/egusphere-egu24-3550, 2024.

16:40–16:50
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EGU24-1903
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On-site presentation
Volker Michel, Naomi Schneider, and Nico Sneeuw

A multitude of basis functions is available for modelling the gravitational field based on satellite data. The Regularized Functional Matching Pursuit (RFMP) algorithm, which has been developed by the Geomathematics Group Siegen, proved to be able to combine different sets of such trial functions. For this purpose, a dictionary is built as a redundant union of different established basis systems (such as spherical harmonics, radial basis functions and Slepian functions). In an iterative scheme, a best basis is selected by minimizing a Tikhonov-Phillips functional. In a recent add-on (the LRFMP), the dictionary does not have to be discretely predefined but can be learned as part of the algorithm. This is implemented as a non-linear optimization problem. The LRFMP has several benefits, which will be demonstrated in the presentation, where we show numerical tests regarding the inversion of noisy gravity data on real satellite orbits.

References:

N. Schneider, V. Michel and N. Sneeuw, High-dimensional experiments for the downward continuation using the LRFMP algorithm, preprint available at http://arxiv.org/abs/2308.04167, 2023.

N. Schneider, V. Michel: A dictionary learning add-on for spherical downward continuation, Journal of Geodesy, 96 (2022), article 21 (22pp). 

Source Code:

N. Schneider, (L)IPMP source code for gravitational field modelling, v2-dc-2023. Zenodo. https://doi.org/10.5281/zenodo.8223771, 2023. 

 

How to cite: Michel, V., Schneider, N., and Sneeuw, N.: Dictionary learning for downward continuation of gravity data, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-1903, https://doi.org/10.5194/egusphere-egu24-1903, 2024.

16:50–17:00
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EGU24-5992
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ECS
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On-site presentation
Xiaole Deng

Studying the gravitational effects of the Earth's topography and crustal layers is a fundamental topic in gravity field modeling in geodesy and geophysics. The introduction of gravitational curvatures (GC), which are the third-order derivatives of the gravitational potential (GP), has recently broadened theoretical research on gravitational effects. Using tensor analysis, this paper comes up with a general formula for the physical parts of the third-order tensor of the potential in cylindrical coordinates. Then, the expressions for the GC of a vertical cylindrical prism are accordingly derived in cylindrical coordinates. Based on the relation among the vertical cylindrical prism, cylindrical shell, and cylinder, the analytical expressions for gravitational effects up to the GC of a vertical cylindrical shell and a cylinder are derived when the computation point is located on the Z-axis no matter whether it is situated below, inside, or above the cylindrical shell and cylinder. Laplace's equation has been adopted to confirm the correctness of the newly derived formulas of the GC. In addition, a benchmark of a cylindrical shell discretized into cylindrical prisms is proposed to reveal the numerical properties of derived GC formulas with the computation point located on the Z-axis. Numerical results reveal that when the computation point's height increases, the relative and absolute errors of the GP, gravitational vector (GV), gravitational gradient tensor (GGT), and GC decrease, in which the relative errors in log10 scale of the nonzero GP, GV, GGT, and GC components are approximately less than -2 when the computation is located below, inside, and above the cylindrical shell. These newly derived formulas lay the theoretical foundation for the GC in cylindrical coordinates and help to investigate the potential applications of the GC in geodesy and geophysics. This new benchmark can become the standard for testing the correctness of the gravitational effects of the cylindrical prism using different numerical algorithms in cylindrical coordinates in practical applications.

How to cite: Deng, X.: A benchmark for gravitational potential up to its third-order derivatives of a vertical cylindrical shell discretized into cylindrical prisms, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-5992, https://doi.org/10.5194/egusphere-egu24-5992, 2024.

17:00–17:10
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EGU24-20053
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ECS
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On-site presentation
Polynomial families of isotropic windows and filters for geophysical signal analysis on the sphere
(withdrawn)
Dimitrios Piretzidis, Christopher Kotsakis, Stelios Mertikas, and Michael Sideris
17:10–17:20
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EGU24-7800
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ECS
|
On-site presentation
david fuseau, Lucia Seoane, Guillaume Ramillien, José Darrozes, Bastien Plazolles, Didier Rouxel, Thierry Schmitt, and Corinne Salaün

Tesseroid and radial columns decomposition of the undersea relief strategies have been considered to recover the seafloor topography by Kalman Filter (KF) inversion of gravity data in the case of the Great Meteor seamount located in the North Atlantic ocean. These both modeling approaches are shown to be equivalent at high grid sampling rate (<1'). Different types of gravity data functionals for geoid height anomaly, vertical gravity component and gravity gradient (or tensor) are analyzed by spectral decomposition and combined to retrieve most detailed 3-D seafloor topography solutions, as gravity gradient data provide short-wavelength information to have access to high-resolution details. Besides only the vertical gravity tensor Vzz is usually inverted in previous field-related studies, considering up to six components of the gravity gradient is tested for improving the accuracy of the KF solution. The iterative KF scheme has been optimized and parallelized using C++ Armadillo software to accelerate the determination of a very large number of juxtaposed topographic heights.

How to cite: fuseau, D., Seoane, L., Ramillien, G., Darrozes, J., Plazolles, B., Rouxel, D., Schmitt, T., and Salaün, C.: Seafloor topography recovery improved by combination of different gravity data functionals, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-7800, https://doi.org/10.5194/egusphere-egu24-7800, 2024.

Advances in GNSS, geodetic transformations, and refraction models
17:20–17:30
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EGU24-19367
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ECS
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On-site presentation
Lotfi Massarweh and Peter Teunissen

Current theory of integer inference, which is key to many carrier-phase driven observing systems, consists of a rich variety of different estimation principles, each with their own optimality and statistical properties. The various estimators can be classified into different classes of estimators, of which the integer estimation class is the smallest and the integer equivariant class the largest. Although the estimation theory for mixed integer models has matured significantly, there are still some important identifiable open unsolved problems. One of those concerns the way in which in practice the integer-equivariant baseline maximizer of the carrier-phase integer ambiguity function is resolved. Most of the methods employed in practice use rather ad hoc, brute-force grid search techniques, whereby proper considerations of the intrinsic properties of the objective function are lacking. As a result none of the available techniques have a demonstrated proven guarantee of global convergence. In this contribution we will present a novel algorithm for the numerical maximization of the multivariate carrier-phase integer ambiguity function. Our proposed method, which has finite termination with a guaranteed user-defined tolerance, is developed from combining the branch-and-bound principle with the projected-gradient-descent methodology, for which a special continuous differentiable convex-relaxation of the critical elements of the ambiguity objective function is constructed. The methodology of these three constituents is described in an integrated manner and numerical results are provided to illustrate the theory.

How to cite: Massarweh, L. and Teunissen, P.: A novel globally convergent maximizer for the multivariate carrier-phase integer ambiguity function, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-19367, https://doi.org/10.5194/egusphere-egu24-19367, 2024.

17:30–17:40
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EGU24-5214
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On-site presentation
Dimitrios Ampatzidis, Kyriakos Balidakis, and Alexandros Tsimerikas

It is widely known that the geodetic networks (both terrestrial and space) suffer from the so-called rank deficiency. This is the algebraic expression of the weakness of the observations to sense all the necessary information for the reference system definition (in terms of origin scale and orientation). For example, the SLR technique is sensitive to the origin and scale, while the orientation can be only externally defined. On the other hand, the traditional quasar-based VLBI does not sense either origin and orientation but only scale.

In general, the rank deficiency is remedied by the use of the so-called constraints. The constraints can be divided into two major categories: a. The minimum constraints, where they just treat the rank deficiency problem (as the word minimum dictates) and do not interfere with the shape of the network, and b. the over-constraints, which do not only solve the rank deficiency but alter the shape of the geodetic network.

While the minimum constraint solutions are widely discussed in the geodetic literature, regarding their nature, the over-constraints' physical meaning is not so clear (if not vague). The present study aims to provide a physical meaning of the over-constraints solution, under the prism of its stochastic interpretation.

How to cite: Ampatzidis, D., Balidakis, K., and Tsimerikas, A.: On the physical meaning of geodetic networks’ over-constraints solutions, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-5214, https://doi.org/10.5194/egusphere-egu24-5214, 2024.

17:40–17:50
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EGU24-13902
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On-site presentation
Wenxi Zhan, Xing Fang, and Wenxian Zeng

Although centralized coordinates are applied in geodetic coordinate transformations implicitly or explicitly, the centering strategy has not been comprehensively investigated from the theoretical perspective. We rigorously model and extend the empirically used three center strategies based on different models:

  • Original model: Based on the partition representations of the solution, we propose a modified iteration policy, which reduces the parameter number and improves numerical stability during iteration. Also, its simplified version is analyzed when the cofactor matrix has the Kronecker product structures. It can be regarded as the extension of the work of Teunissen, since we essentially follow the same idea of partitioning the transformation parameters and the translation parameters, but more general covariance matrix structures are investigated in our consideration.
  • Shifting model: With the partitioned solution forms, we prove the estimated transformation matrix and the residual vector are translational invariant. For iteration, with the classical iteration policy, the shifts should be chosen properly; with the modified iteration policy, there is no restriction since it is numerically equivalent to the original model. In addition, this model shows the feasibility of conducting the adjustment with the centralized coordinates and the original stochastic model.
  • Translation elimination model: By multiplying the transformation relation with a specific matrix from both sides, we formulate the translation elimination model with the coordinates centralized and the translation parameters eliminated. With this model reduction, the covariance matrix has also been transformed since the observation equations are comprised of coordinate combinations. In addition, Leick’s model reduction strategy is a special case of this model, which is conducted by subtracting one particular observation equation from the remaining equations. 

Test computations with different weight structures show the validity of these strategies.

How to cite: Zhan, W., Fang, X., and Zeng, W.: Center strategies for universal geodetic transformations: modified iteration policy and two alternative models, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-13902, https://doi.org/10.5194/egusphere-egu24-13902, 2024.

17:50–18:00
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EGU24-6872
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ECS
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On-site presentation
Mansoor Sabzali, Gilbert Ferhat, Lloyd Pilgrim, Mehdi Khaki, and Jean-Philippe Malet

Atmospheric refraction is the main source of deviations for laser-based sensors. Having a profound understanding of refraction, and the knowledge of the geometry of the line of sight, assists in identifying an accurate model to correct this error. The height of the point, similar to two other planar coordinates, is also impacted as a result of the refracted beam line. The height can be obtained through numerous geodetic measurement approaches such as spirit levelling or trigonometric levelling. An empirical refraction model was proposed in 1984 to better quantify the observations of trigonometric levelling. In this research, we propose a developed empirical model for the La Valette Landslide (Southeast French Alps) to determine the height of the target benchmarks in a landslide zone. The landslide is located in the Ubaye Valley, where the thrust fault of clay-shale sediments at the bottom and sandstone and limestone competent rocks at the top, control the occurrence of landsliding in this region. The deformation is attributed to the low resistance of the slope material and the increase in pore-fluid pressure resulting from the different hydraulic conductivities of the two geological units. The landslide has been monitored over many years, with several remote sensing techniques, and the task is undertaken as a part of the French Landslide Observation Service - OMIV. Since September 2019, an automated total station Long-Range Trimble S9 has been monitoring 54 reflectors’ positions every 1 to 3 hours with respect to three reference control points. The targets have been uniformly distributed over the landslides at distance from 350m to 2300m from the monitoring station, and at elevations varying from 1300m to 2100m. The research determined the point heights using the empirical trigonometric levelling model with the addition of an improved refraction model incorporating the development refraction correction for the observed angles of the control points.

How to cite: Sabzali, M., Ferhat, G., Pilgrim, L., Khaki, M., and Malet, J.-P.: Developed empirical refraction model for precise trigonometric levelling of the La Valette Landslide, France, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-6872, https://doi.org/10.5194/egusphere-egu24-6872, 2024.

Posters on site: Mon, 15 Apr, 10:45–12:30 | Hall X2

Display time: Mon, 15 Apr, 08:30–Mon, 15 Apr, 12:30
Chairpersons: Michal Sprlak, Zuzana Minarechová, Kevin Gobron
Boundary value problems - analytical and numerical solutions
X2.1
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EGU24-7346
Zuzana Minarechová, Marek Macák, Róbert Čunderlík, and Karol Mikula

The numerical approach for solving the nonlinear geodetic boundary value problem based on the finite element method with mapped infinite elements and itterative procedure is developed and implemented. In this approach, the 3D semi-infinite domain outside the Earth is bounded only by the triangular discretization of the whole Earth's surface and extends to infinity. Then the BVP consists of the Laplace equation for unknown disturbing potential which holds in the domain, the nonlinear boundary condition given directly at computational nodes on the Earth's surface, and regularity of the disturbing potential at infinity. In experiments, a convergence of the proposed numerical scheme to the exact solution is tested and then the numerical study is focused on a reconstruction of the harmonic function above the Earth's topography.

How to cite: Minarechová, Z., Macák, M., Čunderlík, R., and Mikula, K.: On solving the nonlinear geodetic boundary value problem using mapped infinite elements, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-7346, https://doi.org/10.5194/egusphere-egu24-7346, 2024.

X2.2
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EGU24-5272
Marek Macak, Michal Šprlák, and Zuzana Minarechová

Gravity field modelling of irregularly shaped bodies such as the Earth's Moon is a challenging task that can reveal both the strong and weak points of each modelling technique. In our approach we will develop a numerical approach based on the finite element method (FEM) and compare the obtained solutions with the solution by the spectral approach that relies on spherical harmonics. In this way, we aim to study whether the numerical methods such as FEM can overcome the limitations of the spherical-harmonic-based approaches, namely their divergence in the vicinity of the gravitating body. We hope that the presented developed approach could form a valuable alternative to the spherical harmonics.

How to cite: Macak, M., Šprlák, M., and Minarechová, Z.: Modelling gravity field of irregularly shaped bodies by numerical methods, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-5272, https://doi.org/10.5194/egusphere-egu24-5272, 2024.

X2.3
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EGU24-18996
Viktor Popadyev and Roman Sermiagin

The problem of determining the anomalous potential T on the earth's surface can be solved on the basis of various initial available data: gravity anomalies Δg and gravity disturbances δg, their vertical derivatives ∂(Δg)/∂H, ∂(δg)/∂H, gravity gradient anomalies Δ(∂g/∂H) etc. Existing methods of such BVP solution use the integral kernels, elaborated for the sphere and ellipsoid. The attempts to determine the real geoid are closely related to the direct problems of the potential theory, when the mass distribution is assumed to be approximately known (in Molodensky's theory the earth's crust density is used in topographic reductions only for better anomalies interpolation).

Using of the two tipes of related gravity data could be considered as a control, e.g., the anomalous potential T from the gravity anomalies Δg can be used to obtain the gravity disturbances δg, from which we must also get the same anomalous potential T. For the real Earth's surface more flexible is the method of integral equations.

 

(The prime sign indicates a point on the telluroid.)

Molodensky's integral equation for the simple layer density (distributed on the Earth's surface) using the gravity disturbances (1) and gravity anomalies (2) is known, but is usually solved indirectly with an introduction of the small parameter (the Molodensky's parameter k or/and ellipsoid eccentricity e), that lead to the series solution with the well-known integrals. Being the Fredholm equation, the Molodensky's integral equation itself can be solved directly by successive approximations in the ellipsoidal coordinate systems as well as in the spherical one. The integration procedure is probably longer, but any step is of the same type. Then the anomalous potential can be calculated by integration in the form (3).

Figure 1. Simple layers distributed with densities φ on the Earth’s surface S (green). Auxiliary simple layer density χ is distributed on the mean Earth’s sphere Ω  with radius R. In general case, the normal n to the surface, inclination angle α and the radius-vector ρ are slightly different in the two cases. E - reference ellipsoid (blue), the telluroid Σ (red), g - plumb-line.

Some real estimates are possible on the surface and gravity field models. In this study we use the Earth's model in the form of mascons for the surface and gravity field, see Fig. 2. We know all the elements of the anomalous field, the precise coordinates of the points with data and so we can estimate the real theoretical accuracy of the formulas and the number of iterations.

Figure 2. The scheme of the mass forming the anomalous field

In case of gravity anomalies the integration procedure can be considered as an integration over the successively refined  boundary surface. It is enough to find the density distribution of a simple layer on a smoothed surface constructed from the heights of points in the form of the sum of the normal height (from leveling) and the height anomaly from the Stokes approximation.

How to cite: Popadyev, V. and Sermiagin, R.: Control of the accuracy of the Molodensky's integral equation for the gravity anomalies and disturbances on the Earth's models, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-18996, https://doi.org/10.5194/egusphere-egu24-18996, 2024.

Integral transformations - practical and statistical aspects
X2.4
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EGU24-4098
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ECS
Jiri Belinger, Martin Pitonak, Petr Trnka, Pavel Novak, and Michal Sprlak

Integral transformations of the gravitational field gradients are defined over the entire solid angle on the surface of the sphere. Despite the indisputable progress in satellite gravimetry and gradiometry, gravity field focused satellite missions allow accurate determination of the gravity field with a spatial resolution of 100 km, i.e. only in its long-wavelength part. However, there is also a need for high-resolution gravity field models at regional, national or continental scales, especially concerning the determination of the quasi-geoid or geoid. On the other hand, potential weakness of ground-based data is the long-wavelength gravity field accuracy and limited availability due to several constraints (e.g. deserts, lakes and large rivers, forests, or lack of goodwill between neighboring countries to share sensitive data). The ideal scenario combines ground and satellite data that complement each other.

In this contribution, relations defining the estimation of the global root mean square errors of selected gravitational field functionals using integral transformations will be derived and presented. For practical calculation, knowledge about the accuracy of measured terrestrial data and formal errors of global satellite models of the Earth's gravity field will be utilized.

How to cite: Belinger, J., Pitonak, M., Trnka, P., Novak, P., and Sprlak, M.: Estimation of the Global Root Mean Square Error of Selected Gravitational Field Functionals Calculated by Integral Transforms, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-4098, https://doi.org/10.5194/egusphere-egu24-4098, 2024.

X2.5
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EGU24-15596
Pavel Novak, Mehdi Eshagh, and Martin Pitoňák

The geoid model over dry land areas is determined from data observed at or above the Earth’s surface. Observable quantities include various functional of the disturbing potential defined as the difference between the real and model (normal) gravity potential. Transformation of the measured data into the sought-after, but directly unobservable gravity potential is often carried out using mathematical tools of the potential theory. An example of such a transformation is the well-known Hotine integral transform that transforms disturbing gravity, i.e., the first-order vertical gradient of the disturbing potential (Stokes formula is applied to anomalous gravity which is still often encountered in geodesy). Higher-order gradients of the Earth's gravitational potential have been collected by sensors on board aircraft or low-orbiting satellites. This advancement in data availability has resulted in the formulation of new tools based on Green's integral transforms and equations. Associated deterministic models have been well developed, tested, and successfully implemented. However, stochastic models for estimating uncertainties in sought values have only been partially developed. These uncertainties should reflect the inevitable implementation and approximation errors, the propagation of formal errors, as well as external accuracy estimation if relevant independent reference values are available. This contribution discusses mathematical models that can be used to estimate various types of uncertainties related to integral-based transformations of the measured potential gradients into the disturbing potential.

How to cite: Novak, P., Eshagh, M., and Pitoňák, M.: Uncertainties associated with integral-based transforms of measured potential gradients, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-15596, https://doi.org/10.5194/egusphere-egu24-15596, 2024.

X2.6
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EGU24-2824
Martin Pitonak, Petr Trnka, Jiri Belinger, Pavel Novák, and Michal Sprlak

Integral transformations are a useful mathematical apparatus for modelling the gravitational field. They represent the mathematical basis for the formulation of integral estimators of gravity field values, including error propagation. The theoretical and practical aspects of integral transformations traditionally used for the calculation of geoid/quasi-geoid heights in geodesy, such as Stokes’ and Hotine’s integral transformations, have already been studied. However, theoretical and practical concepts regarding other integral transformations, including non-isotropic (azimuth-dependent) transformations, have not yet been explored. One of the basic assumptions of integral transformations is global data coverage. However, the availability of ground measurements is frequently limited. In practice, the global integral is divided into two complementary regions, namely the near and far zones. Non-negligible systematic effects of data in the far zone require accurate evaluation. For this purpose, a new software library entitled FarZone4IT is being created in the MATLAB environment to calculate far-zone effects in integral transformations for gravitational potential gradients up to the third order. The library contains scripts for the calculation of integral kernels, error kernels, truncation error coefficients, and far zone effects for a selected set of input parameters. This contribution concerns the implementation of theoretical equations defining far zone effects and the subsequent numerical testing of the library functionality. Closed-loop tests were carried out using gravitational potential functionals generated from a synthetic model of the Earth's gravity field.

How to cite: Pitonak, M., Trnka, P., Belinger, J., Novák, P., and Sprlak, M.: FarZone4IT: A new software for the calculation of far–zone effects for spherical integral, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-2824, https://doi.org/10.5194/egusphere-egu24-2824, 2024.

Gravity field studies - data combination, data filtering, and seafloor topography
X2.7
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EGU24-5215
Robert Cunderlik, Pavol Zahorec, and Juraj Papčo

The study presents along-track nonlinear diffusion filtering of the airborne gravity data. At first, the provided airborne gravity disturbances from the GRAV-D campaign are transformed into the airborne complete Bouguer disturbances (CBD). This aims to reduce a correlation of the filtered data with the topography. Then the nonlinear diffusion filtering in 1D based on the Perona-Malik model is applied. In this model, a diffusivity coefficient depends on the edge detector, which allows reducing noise while preserving important gradients in the filtered data. As a numerical method we use the finite volume method (FVM). The derived semi-implicit numerical scheme leads to a three-diagonal system matrix that is solved in every iterative step. Here the diffusivity coefficients are updated in every step by new values of the edge detector recomputed from the previous solution.

The numerical experiment presents the along-track nonlinear filtering of the airborne CBD in high mountainous area of the ‘Colorado geoid experiment’. Afterwards, the along-track filtered data are gridded into a 2D map of the airborne CBD. The obtained results show that an appropriate choice of a sensitivity parameter of the diffusivity coefficient can better detect significant structures in the airborne CBD, especially their edges that are usually smoothed by the Gaussian filtering. Finally, the filtered and gridded airborne CBD are backward transformed into the airborne gravity disturbances.

How to cite: Cunderlik, R., Zahorec, P., and Papčo, J.: Along-track nonlinear filtering of airborne gravity data from the GRAV-D campaign: case study for the ‘Colorado geoid experiment’, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-5215, https://doi.org/10.5194/egusphere-egu24-5215, 2024.

X2.8
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EGU24-9318
Lucia Seoane, David Fuseau, Guillaume Ramillien, José Darrozes, Bastien Plazolles, Didier Rouxel, Corinne Salaün, and Thierry Schmitt

During the last decades, several inversion approaches have been proposed to derive sea floor topography from satellite-based gravity data. Unfortunately, the most accurate non linear ones are based on iterative schemes that remain very time-consuming, especially if the number of topographic heights to be fitted is very important, e.g. when the oceanic domain is large and/or the gravity data is geographically dense and thus the maximum grid resolution to be accessible is high. Our strategy of computation is to decompose the total area into geographical cells that are overlapped to cancel the edge effects. The reference ocean depth given by GEBCO and the elastic thickness for regional compensation in function of the square root of the age of the oceanic crust are assumed to be constant in each cell. The initial inversion code has been translated into C++ and optimized using Armadillo software and LAPACK library to obtain a gain of speed of 1000 for a large region such as the complete North Atlantic Ocean (-54,-26,18,37). Post-fit and absolute errors are typically less than 200 m and 50 m r.m.s. respectively. These new detailed maps of bathymetry represent a precious source of information for geophysical applications. 

How to cite: Seoane, L., Fuseau, D., Ramillien, G., Darrozes, J., Plazolles, B., Rouxel, D., Salaün, C., and Schmitt, T.: Fast seafloor topography mapping of large oceanic provinces by optimization/parallelization, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-9318, https://doi.org/10.5194/egusphere-egu24-9318, 2024.

Applications of statistical methods
X2.9
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EGU24-4313
Georgios Panou and Urs Marti

The need for the determination of the parameters of an equipotential rotational ellipsoid of revolution that approximates closer and closer to the Earth requires analysis of the most currently available data. The main objective of this study is to examine such new data and to perform an adjustment technique to estimate the parameters and their standard deviations of the mean Earth ellipsoid. The parameters considered are the geocentric gravitational constant, the angular velocity, the geoidal potential, the dynamical form factor, and the major and minor semi-axes. A-priori estimates of these quantities, which may be determined independently, are treated as “observations” and their adjusted values from a weighted least-squares procedure are presented. Since a level ellipsoid of revolution and its gravity field are completely determined by four constants, we use two non-linear condition equations to relate the six parameters for performing the adjustment. Among the products of the adjustment is the correlation matrix of the adjusted values of parameters, which helps us in the selection of a consistent set of four parameters for the definition of a new Geodetic Reference System (GRS). After the selection of the four parameters, we compute by Newton’s method the two derived parameters and estimate their standard deviations by applying the law of propagation of variances. Furthermore, all the derived geometric and physical constants are computed from the defining constants of a GRS, by means of closed formulas. Finally, in order to yield numerical results of high precision, all computations are executed in computer algebra system software using variable precision floating point numbers.

How to cite: Panou, G. and Marti, U.: Current adjustment of the mean Earth ellipsoid parameters, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-4313, https://doi.org/10.5194/egusphere-egu24-4313, 2024.

X2.10
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EGU24-8993
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ECS
Bernd Uebbing, Jan Höckendorff, Caroline Jungheim, Anne Driemel, Christian Sohler, and Jürgen Kusche

The Earth’s system is warming due to natural and human driven climate change. Observing, analyzing and understanding the associated geophysical processes is important in order to improve prediction of future changes and mitigate impacts on society and infrastructure. Investigating individual climate processes, such as sea level change, often requires partitioning of the total signal for identifying sub-signals and drivers; in the sea level example these could be trend and seasonal signals or impacts from the El Niño Southern Oscillation (ENSO).

A commonly applied method is the (real) Principal Component Analysis (PCA), which factorizes a given input dataset into time-invariant Empirical Orthogonal Functions (EOF), i.e. spatial patterns, and time-variable Principal Components (PC) based on the most dominant eigenvalues. However, this real-EOF analysis assumes more or less static patterns over time and, thus, lacks the ability to capture temporal variations in the patterns. This can be circumvented by the application of complex or Hermitian EOF analysis, which also enables capturing phase shifts or in other words allows for time-varying spatial patterns.

Here, we present first results from a novel approach utilizing dynamic time warping (DTW) for extracting dominant modes in the form of spatially distributed amplitudes and lags with respect to a ‘base curve’. While classic PCA methods are sensitive to outlier influence on the partitioning, our approach represents a robust alternative. Furthermore, base curves are computed that represent spatial modes via traversal matrices, which act as extensions of the base curves to capture individual lag. We introduce our new approach, compare to complex/Hermitian EOF, explain the numerical scheme, and present some first results based on gridded sea level change data.

How to cite: Uebbing, B., Höckendorff, J., Jungheim, C., Driemel, A., Sohler, C., and Kusche, J.: An alternative to PCA utilizing Dynamic Time Warping, EGU General Assembly 2024, Vienna, Austria, 14–19 Apr 2024, EGU24-8993, https://doi.org/10.5194/egusphere-egu24-8993, 2024.