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Integral transformations represent a useful mathematical tool for gravitational field modelling. Basic assumption of integral transformations is the global data coverage. However, availability of high-resolution and accurate gravitational data is restricted. For this reason, we decompose the global integration into two effects: 1) the effect of the near zone calculated by numerical integration of data within a spherical cap, and 2) the effect of the far zone due to data beyond the spherical cap synthesised by external spherical harmonic series and EGM coefficients. Theoretical and numerical aspects of this decomposition have frequently been studied for isotropic integral transforms used in geodesy, such as Poisson’s, Stokes’s, and Hotine’s. In this contribution, we present a complete and unified theory for the far-zone effects of integral transformations mutually relating quantities from the gravitational potential up to the third-order gravitational tensor components. These transformations include both isotropic and non-isotropic integral kernel functions. To understand mathematical aspects of the resulting external spherical harmonic series, we investigate behaviour of truncation error coefficients in the spectral domain in connection with the properties of the corresponding integral kernels in the spatial domain. We systematically quantify magnitudes of the far-zone effects with respect to the signal powers of the corresponding quantities. Preliminary numerical experiments indicate that formulas for evaluation of the far-zone effects are stable and reach less than a hundred percent of the total signal for smoothing integral formulas. For de-smoothing integral transformations, the magnitude of the far-zone effects may be several orders larger than the total signal., The 28th IUGG General Assembly (IUGG2023) (Berlin 2023)

In this contribution, we will investigate a problem of regional gravitational field modelling by the spectral combination of spaceborne first-, second- and third-order radial derivatives of the gravitational potential at the mean altitude of 250 km. Some of the mentioned measurements are not yet observable so we will synthesize them from a global geopotential model (GGM) over a test area. The main goal of a numerical experiment is to compute the gravitational field in a regional area from spatially restricted higher-order radial derivatives of the gravitational potential, and two problems arise. The first is how to estimate local spectral properties, i.e., degree-order variances of local data, and the second is the effect of the omitted distant zone data, i.e., (spatial) truncation error. The spherical harmonic power spectrum from local data will be recovered by 2D-DFT, and the truncation error will be calculated from a GGM. For this purpose, a mathematical model based on the spectral combination method of heterogeneous gravity field data and a GGM is developed. The correctness of derived spectral weights suitable for spectral downward continuation will be verified by a closed-loop test and by direct comparison with the global solution. Moreover, different sizes of integration radii will be tested to minimize the truncation error and various truncation degrees of a GGM.&nbsp, The 28th IUGG General Assembly (IUGG2023) (Berlin 2023)

Integral transformations are a useful mathematical apparatus for gravitational field modelling and form the main mathematical basis for the formulation of integral estimates including error propagation. Theoretical and practical aspects of classical integral transformations, such as Poisson’s, Stokes’s, or Hotine’s, have already been studied. On the other hand, theoretical and practical concepts are not known for many other integral formulas including non-isotropic ones.The basic assumption of integral transformations is global data coverage. In practice, however, data availability is limited. Therefore, we divide global integration into the effects of the near and far zones. The computation of distant zones is a non-negligible systematic effect requiring an accurate calculation.In this paper, we introduce a new software library for the evaluation of the far zones. The software library allows the computation of the far zones for integral transformations mutually relating all quantities up to the third derivatives of the gravitational potential. The library includes scripts for the calculation of the integral kernels, error kernels, and truncation error coefficients for a selected set of input parameters. The implementation and subsequent functionality of the library are tested in a closed-loop simulation by using synthetic data., The 28th IUGG General Assembly (IUGG2023) (Berlin 2023)

Physical geodesy applies the potential theory to study the Earth's gravity field. Among other tools this theory offers, geodetic boundary-value problems are particularly popular to transform and continue harmonic functions. In most cases, the focus is on formulating a functional model relating two functions in mass-free space. Values of an unknown function are unobservable, but they can be recovered from values of another function which can be measured, albeit with errors. Such mathematical models are denoted as deterministic. However, as they include input data burdened with observation errors, respective stochastic models must be derived to provide uncertainties of estimated parameters. Boundary-value problems formulated and solved in the 19th century have been gradually extended for new data types as measured in geodesy. This contribution discusses problems formulated for gradients of the Earth’s gravity potential. Their solutions in a form of surface integrals are given, respective estimators applicable for software implementation are presented, and their stochastic models are formulated. The deterministic and stochastic models represent complete solutions to the problems providing estimates of unknown functions and their variances. As the knowledge of an observation noise is usually incomplete, solutions provided by stochastic models must carefully be considered. Available gravity field data and their noise characteristics are discussed with an attempt to assess their impact on stochastic modelling and its results., The 28th IUGG General Assembly (IUGG2023) (Berlin 2023)