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.

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.

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 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.

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.

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.

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.

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.

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.

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.