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Estimation ff the Global Root Mean Square Error of Geoid Height Calculated by Integral Transforms
Belinger, Jiří ; Pitoňák, Martin ; Trnka, Petr ; Novák, Pavel ; Šprlák, Michal
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 paper, relations defining the estimation of the global root mean square errors of geoid heights 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.
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A New Software For the Calculation of Far–Zone Effects For Spherical Integral
Trnka, Petr ; Pitoňák, Martin ; Belinger, Jiří ; Novák, Pavel ; Šprlák, Michal
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. 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 is being created in the MATLAB environment to calculate far-zone effects in integral transformations for gravitational potential gradients up to the third order.
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Far Zone Effects for Integral Transformations: Theory and Implementation
Trnka, Petr ; Belinger, Jiří ; Šprlák, Michal ; Pitoňák, Martin ; Novák, Pavel
Integral transformations are a useful mathematical apparatus for modelling the gravitational field and require the formulation of integral estimates including error propagation. For classical integral transformations, this issue has already been studied, but the formulation for all available gravitational observables has not been studied yet. The assumption of integral transformations is global data coverage. In practice, however, data availability is limited, so we divide the 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. The theory is implemented in the form of a precise software. In this paper, we present the basic theory for the evaluation of the far zones. We also investigate properties of integral kernels and truncation error coefficients. In the numerical experiments, we compare calculation of the far zones by numerical integration with truncated spherical harmonic series. One of the outputs of this contribution is a software library for computation of the far zones for integral transformations mutually relating all quantities up to the third derivatives of the gravitational potential.
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