A Study on Precise Local Gravity Field Modelling

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of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology in Bratislava 2

2 A Study on Precise Local Gravity Field Modelling R. Čunderlík 1, J. Papčo 2, P. Záhorec 3 cunderli@svf.stuba.sk, juraj.papco@stuba.sk, zahorec@savbb.sk 1 Dept. of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology in Bratislava 2 Dept. of Theoretical Geodesy, Faculty of Civil Engineering, Slovak University of Technology in Bratislava 3 The Earth Science Institute of the Slovak Academy of Sciences, Geophysical division, Banská Bystrica

Realization of Vertical Reference Systems Global approach globally homogenous approach based on precise gravity field modelling W P = U P (h GNSS ) + T P c P = - ( W P - W 0 ) GRACE/GOCE-based

3 Realization of Vertical Reference Systems Global approach globally homogenous approach based on precise gravity field modelling W P = U P (h GNSS ) + T P c P = - ( W P - W 0 ) GRACE/GOCE-based satellite-only GGMs fully independent from LVDs low-frequency part obtained very precisely, however overall accuracy affected by the truncation error Continental approach (e.g. EVRF2007) NAP regional approach based on spirit levelling and potential of the height reference surface W 0i c P = c Pi + W 0 W 0i c Pi W 0i W P P 0i gdh

4 A concept of the realization of IHRS From presentation of Sánchez at al (IAG-IASPEI-2018, KOBE, JAPAN):

resolution of about 100 km affected significantly by stripping noise due to omission

5 Geopotential at points on the Earth s surface GOCE-based satellite-only GGMs low-frequency part obtained very precisely (goal of GOCE) : accuracy of 1 to 2 cm and a spatial resolution of about 100 km affected significantly by stripping noise due to omission errors!!! Geopotential on DTU13 mean sea surface evaluated from GO_CONS_GCF_2_DIR_R5 (SH up to d/o 300)

6 Geopotential at points on the Earth s surface GOCE-based satellite-only GGMs low-frequency part obtained very precisely (goal of GOCE) : accuracy of 1 to 2 cm and a spatial resolution of about 100 km affected significantly by stripping noise due to omission errors!!! necessary to model the high-frequency part combined GGMs (including RTM) national (quasi)geoid models!!! terrestrial gravimetric measurements!!!

7 GNSS-Levelling test of combined GGMs in Slovakia EGM-2008 EIGEN-6C4 GGMPlus (SH up to d/o 2160) (SH up to d/o 2160) (EIGEN-6C4 + residual terrain model) STD: 4.0 cm STD: 3.2 cm STD: 2.4 cm at 336 benchmarks provided by GKÚ Bratislava (10 outliers removed) TOPO

8 Terrestrial gravimetric mapping in Slovakia more than measurements (collected during the last decades) Digital terrain model TOPO

9 Campaign: Tatry 2004

10 Original gravimetric data GGMPlus!!! Local extremes exceed 10 mgal!!!

x Direct BEM formulation Boundary Integral Equation: 1 T 2 G n T n p T q p, q

11 Local gravity field modelling using BEM Linearized fixed gravimetric BVP T (x) = 0 T(x), s(x) = - g(x) T(x) = O ( x -1 ) x ext. x Direct BEM formulation Boundary Integral Equation: 1 T 2 G n T n p T q p, q d q q G p, q d q where G p, q 1 4 p q fundamental solution of the Laplace equation (as a weighted function)

12 Triangulation of the Earth s surface An example of a local refinement of the global triangulation

Local refinement in Slovakia Global resolution: 0.

13 Local refinement in Slovakia Global resolution: local refinement 1: local refinement 2: local refinement 3: local refinement 4: local refinement 5: ( 260 m)

14 Reconstruction of GGMPlus input data Input gravity disturbances GGMPlus TOPO

15 Reconstruction of GGMPlus results GGMPlus BEM - GGMplus GGMPlus BEM STD: 2.44 cm STD: 2.42 cm

16 Terrestrial gravimetric measurements GGMplus nodes replaced by original measurements: if (dist < 120 m) about 55% of all nodes in Slovakia replaced modified triangulation

17 Contribution of gravimetric measurements

18 Contribution of gravimetric measurements

19 Input gravity disturbances generated from CBA Remained GGMplus nodes replaced by generated gravity disturbances about 45% of all nodes in Slovakia CBA2G_SK software Complete Bouguer Anomalies (CBA) (Marušiak et al. 2015) (Pašteka et al. 2014) Geological structures

20 Contribution of terrestrial gravimetric data [m 2 s -2 ]

21 Correlation with Bouguer anomalies? [m 2 s -2 ] Complete Bouguer Anomalies DG (orig GGMplus)

22 GNSS-Levelling test Reconstruced GGMPlus + original gravity data + generatied from CBA STD: 2.42 cm STD: 2.40 cm STD: 2.75 cm

Conclusions precise local gravity field modelling by BEM refinement of the discretization is very straightforward requires very dense distribution of

points on the Earth s surface (many problems vanish!

23 Conclusions precise local gravity field modelling by BEM refinement of the discretization is very straightforward requires very dense distribution of terrestrial gravimetric measurements they can directly represent computational nodes considering their 3D positions BEM solution is obtained directly at points on the Earth s surface (many problems vanish!!!) GNSS-Levelling indicates cm-level accuracy of local BEM solutions improvement in comparison with combined GGMs (EGM-2008, EIGEN-6C4, GGMPlus) contribution of original gravimetric measurements can reach several cm more analyses required for reliable interpretations

24 Thank you For Your attention

Numerical solution of the fixed gravimetric BVP on the Earth s surface its possible contribution to the realization of IHRS.

Numerical solution of the fixed gravimetric BVP on the Earth s surface its possible contribution to the realization of IHRS. Róbert Čunderlík cunderli@svf.stuba.sk Dept. of Mathematics and Descriptive