DFHRS (Digital FEM Height Reference Surface) A Rigorous Approach for the Integrated Adjustment of Fitted Height Reference Surfaces (HRS) -

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1 DFHRS (Digital FEM Height Reference Surface) DFHRS (Digital FEM Height Reference Surface) A Rigorous Approach for the Integrated Adjustment of Fitted Height Reference Surfaces (HRS) - Reiner Jäger and Sascha Schneid Hochschule Karlsruhe Technik und Wirtschaft - University of Applied Sciences Faculty of Geoinformatics Studiengang Vermessung und Geomatik & International Programme Geomatics (MSc) Institut für Angewandte Forschung (IAF) Moltkestrasse 3, D Karlsruhe

2 Direct - No Identical Points - DFHRS-Concept - Online - Postprocessed DFHRS-DB H = h GPS (B,L) - DFHRS(B,L,h) korr H = h GPS (B,L) - (NFEM(p B,L,h) + m h)

3 IAG IAG Reference Reference Frame Frame Sub Sub-Commission Commission for for Europe (EUREF) Symposium 26, Riga, 14 Europe (EUREF) Symposium 26, Riga, June FEM representation of height reference surfaces ( ) ( ) ( ) ( ) = = = = l i i l j j i m ij n ij m n n m n m n m x y a a x y N x y N x y N y x,,,,,,,,,, p p 3D difference at any point P(x,y) along the border SA_SE of the meshes m and n has to vanish

4 Continuous Finite Element Representation of HRS x y N m, n N n, m l l i i= j= m, n m, n l l i ( ) ( ) ( ) ( ) i j y, x N p, y, x N p, y, x a a y x = ( a a ) y + t ( y y ) ij,n N = l l i i= j= ij,m n i j ( a a ) y x ij,n i j ( ) ( x + t ( x x )) = N ( t)! sa ij,m se sa l m, n 1 2 k k k= sa se sa m, n 2 k k ( t) = c + c t + c t + L+ c t = c t! c m ( p m ; p ;y,x ;y,x ) =, k, l k n sa sa se se = = i= j= ij, n ij, m

5 FEM representation of height reference surfaces Result: Continuous Height Reference Surface: NFEM(p) N FEM N ( p ) = l i a k = i= j= l C,1,2 ij,k y i x j = T p ( p ) m; pn f k

6 Digital FEM Height Reference Surface (DFHRS)- Concept Complete New Computation of continuous HRS (p and m)! DFHRS Adjustment Approach State of the Art < 25 h GNSS + v = H + NFEM(p) - h GPS m H + v = H N G j + v j = NFEM(p) + N G (d j ) NFEM(p) N(p j ) <= GPS/Levelling Fitting Points <= Any number Geoidmodels (Global, regional, local) ξ j + v = - F B / M(B) p + ξ (d ξ,η ) j <= Sets of Deflections from Vertical η j + v = - F L /(N(B) cos(b)) p + η(d ξ,η ) j (Zenith Cameras or Geoidmodels) a 4 π γ ( B) σ g S(ψ)dσ + v = NFEM(p) <= Gravity by correlated Geoidmodels In the sense of an 2 step adjustment

7 N = N a 4πγ ψ Weak-Shapes of Classical Gravimetric Geoid models Re f 2π + Ψ= α = ( g g Re f EGG97 European Gravimetric Geoid 1997 ) S( ψ ) dψ dα Mean- up to lang-waved Errors m! => New Concepts, more precise or better: (H,h)-fitted solutions

8 Standard GNSS/GPS-heighting Approaches using identical Points Tallinn Example Pure Geoid- Approach EGG97-Geoid without Datum N(d) +/- 3.5 cm Tilt Residuals 15mm Net 15km

9 Standard GNSS/GPS-heighting Approaches using identical Points Tallinn Example HBF:= N G + N G (d) Pure Geoidapproach EGG97-Geoid With Datum N(d) Net +/- 4 mm 15km Residuals 2mm

10 DFHRS Software Identical Fitting Points (B,L,h;H) Meshes Patches

11 DFHRS Software Every geoidand/or vertical deflection model N can be patched Hochschule für Technik und Wirtschaft - University of Applied Sciences LVA Baden-Württemberg LVA Hessen LVA Rheinland-Pfalz LVA Riga, Latvia University of Federal Forces Munich University Darmstadt N G (d j )

12 DFHRS_DB Design Parameters <_3_cm DFHRS_DB Windhuk, Namibia EGM96 Meshsize (p=3) 2-3 km : HRS approximation error < (5-1) cm 1 km: HRS approximation error <1 cm 5 km: HRS approximation error <.5 cm Fitting Point Density (< 1 mm (< 1 mm points,, EGG97) 5 points per (1 km x 1 km): <_1_cm DFHRS_DB 1 points per (1 km x 1 km): < 3_cm DFHRS_DB 3-4 points per (1 km x1 km): < 5-1_cm 5 DFHRS_DB

13 DFHRS_DB Design Parameters Design Studies < 5-1_cm DFHRS Germany Patch-Size (EGG97) 3-4 km for a < 1_cm DFHRS_DB 5 6 km for a < 3_cm DFHRS_DB 3 km for a < 1_cm DFHRS_DB (3-5) points per patch

14 < 1cm DFHRS Europe Fittingpoint-Design ETRS89/EVRS GPS-/Levelling- Points of EVN Fitting Points NFEM(p) =: h - H Used for the 1st Version < 1_cm DFHBFS Europe

15 <_1_cm DFHRS_DB - Indepent Quality Control <1_dm EVRF24 (Present Version, 35 km meshes, 34 Patches) Austria Germany Estonia Latvia Lithuania Switzerland Number of unused control points RMS [cm]

16 Overview about European DFHRS_DB 23 5 km < 1 cm 1 km < 3 cm 25 3 km Mesh Size < 1 cm New 26

17 DFHRS_DB - Product as CD-Installation of State Land Services in Germany CopyProtection inclusive

18 < 5 cm DFHRS_DB Florida ( Masterthesis ) New 25

19 European DFHRS_DB including New 25 5 km FEM Meshes < 1 cm DFHRS_DB Ungary Test - Area (5 x 9 ) km 1-2_cm DFHRS_DB Budapest DFHRS 4. Software

20 DFHRS in Practice

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22 DFHRS in Practice TOPSurv many other independent GNSS- Controller Software and GIS Packages See

23 DFRHS Extension to Gravity Observations Geodetic Network Optimization - 1st/2nd/3rd Order Design: A,P =>C p

24 DFHRS - Extension to Gravity Observations LAV g = g Sensor-Observation

25 Extension of the DFHRS-Concept to Gravity Observations <1cm-Resolution ( 2mm ) of HRS Instead of classical global spherical harmonics (n=m=72) Cap P Spherical Cap Harmonics (SCHA) V(r, λ', θ' ) = G M a k max a r n(k) + 1 k k = m= (C' n(k),m cos mλ' + S' n(k),m sin mλ' ) P n(k)), m (cos θ' ))

26 Global θ =9 Sperical- Harmonics SHA Spherical Cap Harmonics (SCHA) θ arbitrary General Case of SHA!!! l = 2, Pl(k),m '(cosψ) ψ= 2 θ = l(2) arbitrary = = ψ Pl ( k),m(cosψ ) ψ = 9 = ψ l = 6, l = 4 l(6) = l(4) = P l, m (cos θ ); < θ < 9 P l ( k ), m (cos θ ); < θ < 2

27 DFHRS - Extension to Gravity Observations SCHA Handling V(r, λ', θ' ) = G M a k max a r n(k) + 1 k k= m= (C' n(k),m cos mλ' + S' n(k),m sin mλ' ) P' n(k)), m (cos θ' )) V(r, λ', θ' ) = k max n(k) + 1 a (C' n(k), m cos mλ ' + S' n(k), m' sin mλ' ) P' n(k), m (cos θ' ) r k = m = k k max n(k) + 1 k a T(r, λ', θ' ) = ( (C' n(k),m cos mλ ' + S' n(k),m' sin mλ' ) P' n(k),m (cos θ' ) r k= m= ) - V ref ;GRS8 SCHA Resolution ( 2mm Geoid, n=72 u =5..) n SHA = n(k) 9 α (k, SCHA +.5).5 α= 1 (=11 km area) => k-scha = 8 and u = 6.4 α= 3 (=33 km area) => k-scha = 25 and u= 62.5 α= 25 (Europe) => k-scha = 2 and u =4..

28 DFHRS - Extension to Gravity Observations Treatment of Gravity Observations g LAV = g Sensor-Observation at Position P(x,y,z) g g g x y z ECF = R(B, L, η, ξ) ECF LAV g LAV g g g x y z ' ' ' ECF grav = g g g x y z ECF ECF [ g ] centrifuga l g g g N E r LGV grav = R( ϕ, λ) LGV ECF g' ECF grav g 1 r V θ' 1 r sin θ' V λ V r LGV T grav = [,, ] g LGV grav r + v = n(k) + 1 a (n(k) + 1) (C' n(k)),m cosmλ ' + S' n(k),m sin mλ') Pn(k),m (cosθ') r r k= m= k + dg( d)

29 Treatment of GPM - EGM96 / EGM99, EIGEN, etc Vref,GRS8 ) N j GPM 1 = γ Q ( + v = N(C n(k),m n(k) + 1 k, S ) + N( d a (C' n(k),m cos mλ ' + S' n(k),m sin mλ') Pn(k),m (cosθ') r k= m= n(k),m j ) V 8 ) ref,grs + N( d j )

30 Extension of the DFHRS-Concept to gravity observations h GNSS + v = H + f T p H + v = H - h GPS m NFEM(p) N G j + v j = f T p + N G (d j ) N(p k ) ξ j + v = - f B T / M(B) p + ξ (d ξ,η ) j η j + v = - f L T /(N(B) cos(b)) p + η(d ξ,η ) j a 4 π γ ( B) σ g LGV grav r + v = g S(ψ)dσ + v = NFEM(p)= f T p n(k) + 1 a (n(k) + 1) (C' n(k)),m cos mλ ' + S' n(k),m sin mλ') Pn(k),m (cosθ') r r k= m= k + dg( d) N j GPM 1 = γ Q + v = N(C n(k),m n(k) + 1 k, S k= m= n(k),m j ) + N( d ) a j ( (C' n(k),m cos mλ ' + S' n(k),m sin mλ') Pn(k),m (cosθ') Vref ) + N( d ) r + v = N(C', S' ) ( f p + m h) N n(k),m n(k),m T

31 Example Saarland: 825 Gravity Observations

32 Example Saarland (Jan. 6) DFHRS-Software Software with Computation Design Version 1 ( 1cm Reference) EGG97 ( Gravity indirect ) + Identical Points Version 2 ( Gravity indirect ) Gravity Disturbances dg GPM98 Identical Points 1 CAP Result: : 1_cm Coincidence of the HRS between Version 1 and Version 2

33 1_cm DFHRS of Baden-Württemberg (EIGEN-GPM, Gravity Values, Fitting Points)

34 1_cm DFHRS of Baden-Württemberg (EIGEN-GPM, Gravity Values, Fitting Points)

35 1_cm DFHRS of Baden-Württemberg (EIGEN-GPM, gravity values) Number B[ ] L[ ] dg[mgal] v[mgal] : : : : : : : : : : Fig. 8 DFHRS-software report for the gravity disturbances dg with the list of corrections v

36 Solution Concept for HRS-Computation and GNSS-Heighting - Strict mathematical base for continuous FEM_HRS & *DFHRS-Software* - New concept for an overdetermined BVP =>parametric HRS determination - Mesh and patch-design => Any accuracy and any! area size (<= FEM) - Open for all geometrical & physical (e.g. gravity) observations! - DFHRS = (Leading) Geoidfitting Concept - Ready for 1 cm EVRS using existing data +EPN densification fitting-points! - High practical relevance for GNSS services all over the world - Industrial Standard in GNSS-Equipment and GIS - DFHRS_DB => RTCM 3. Message used in GNSS-Services, NTRIP etc. - High Capacities for International Co-operations and for EVRS