MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA DFHRS (Digital FEM Height Reference Surface) – A Rigorous Approach for the Integrated Adjustment of Fitted Height Reference Surfaces (HRS) - Reiner Jäger Hochschule Karlsruhe Technik und Wirtschaft - University of Applied Sciences Faculty of Geomatics Studiengang Vermessung und Geomatik & International Programme Geomatics (MSc) Institut für Angewandte Forschung (IAF) Moltkestrasse 30, D Karlsruhe ;

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA Height Systems - Definitions and Terrestrial Realisiation H Geoid = Ideal Height Reference Surface (HRS) h N A Gepotential Numbers C Levelling and Gravity Observations Terrestrial Height Determination H=const. Small Area ~ „constant Waterlevel“

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA NewEuropeanNormal-HeightNetwork EUREF/UELN ‚95/98 AdjustmentbyGeopotential Numbers (Differences Datumspoint W 0 bzw. C 0 Amsterdam„NAP“DHHN‘92 ( Diff.: 1 cm) AccuracySituation

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA H = h - N GNSS-Heightung H „H from h-GNSS“ HRS-Model N h h N H „HRS“ Geoid Quasigeoid

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA Gravity Potential W of the Earth at Point P(r,λ, ) Gravitational Potential of the Earth Masses HRS = Geoid / Quasigeoid – Physical Definition & Realisation Centrifgal - Potential due to Earth-Rotation Q-Geoid Geoid (W=W o =const) Spherical Harmonics Coefficients as Parameters HBF

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA Original Observation: Laserranges Derived of it: Orbit Disturbances Derived of it: Spherical Harmonics (Müller/Kaula, 1960,1962) Satellite Geodetic Methods of Gravity Field Determination Satellite Laser Ranging (SLR) „LEOS“ (Low Earth Orbit Satellites) LAGEOS 1, 2 STELLA, etc.

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA Observation Quantities 1.) Abs. Positionierung via GPS (Black-Jack Receiver/NASA, JPL): x(t) 2.) Relative Orbitmovements by Inter-Satellite Observations: 3.) Precise Accelerometers a i STAR (Onera/Frankreich): 4.) Gradiometers (e.g. GOCE) Derived of these: 1.) Orbit Determinations 2.) Gravitational Accelerations g i (x,t) and Gradiometer-Tensor Present Missions Hybride Sensors Aktice Measurements Spherical Harmonics Coefficients Derived from that: Spherical Harmonics Coefficients CHAMP Satellite Geodetic Methods of Gravity Field Determination

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA EIGEN ( = EIG EN EIGEN ( = European Improved Gravity Model of the Earth by New Techniques) [cm] HHJHHJ M6 Earth Geopotential Model 96 (NASA) HHJHHJ Satellite Geodetic Methods of Gravity Field Determination

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA EIGEN ( = EIG EN EIGEN ( = European Improved Gravity Model of the Earth by New Techniques) [cm] HHJHHJ M6 Earth Geopotential Model 96 (NASA) HHJHHJ N(p) Latitude(degree) Longitude(dgree) DFHRS_DB Namibia and Tanzania: 1.) Identical Points (N=h-H) & 2.) EGM96 bzw. EIGEN-GL04C Satellite Geodetic Methods of Gravity Field Determination

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA Bilance for Satellite Geodetic GPM(EGM2008; EIGEN) N n,m N Not sufficient for GNSS-Heighting N H = h – N !!! Further Information for further HRS N Improvement of the HRS and N necessary! Hybrid and redundant Models! Terrestrial Gravity Measurements Identical Points (H, h) N=h-H “GeoidFitting“

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA Short-Waved: ~ (1 – 3) cm Medium-/Long Waved m ! 1.) EGG97 - European Gravimetric QGeoid 1997 (IfE, Hannover) (h, H) Hybride Modeling in HRS-Computation Weg 1: Stokes-based RepresentationGRID

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA (h, H) + + Dgl.-based GPM 98 2.) Prof. Dr.-Ing. H.-G. Wenzel – Fitted Geopotenzialmodell (GPM 98 ) n = m = 1800 Accuracy of N ~ ( ) cm Parametrice Represenation (or Grid) Hybride Modeling in HRS-Computation

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA 3.) „BKG-Model – Parametrisation of the Potentials by additional Mass Points (h, H) Massen-Points as essential additional Parameters Representation of BKG HRS: Grid Hybride Modeling in HRS-Computation

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA H = h GPS (B,L) - DFHRS(B,L,h) korr H H = h GPS (B,L) - (NFEM(p|B,L,h) + ∆m·h) DFHRS-DB Direct - No - NoIdentical Points - - Online - Online - Postpro- cessed DFHRS-Concept

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA FEM representation of height reference surfaces 3D difference at any point P(x,y) along the border SA_SE of the meshes m and n has to vanish

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA Continuous Finite Element Representation of HRS

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA FEM representation of height reference surfaces Result: Continuous Height Reference Surface: NFEM(p)

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA Digital FEM Height Reference Surface (DFHRS)- Concept h GNSS + v = H + NFEM(p) - h GPS ·  m H + v = H N G ‘ j + v j = NFEM(p) +  N G (d j )  j + v = - F B / M(B)  p +   (d ,  ) j  j + v = - F L /(N(B)  cos(B))  p +   (d ,  ) j Complete New Computation of continuous HRS (p and  m)! DFHRS – Adjustment Approach < 2005 State of the Art < 2005  g·S(  )dσ + v = NFEM(p) NFEM(p) N(pj)N(pj) (Global, regional, local) <= Sets of Deflections from Vertical (Zenith Cameras or Geoidmodels) <= GPS/Levelling Fitting Points <= Any number Geoidmodels <= „Gravity“ by correlated Geoidmodels In the sense of an 2 step adjustment

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA EGG97 European Gravimetric Geoid 1997 Mean- up to lang-waved Errors 0.1 – 1.5 m ! => New Concepts, more „precise“ or better: (H,h)-fitted solutions Weak-Shapes of Classical Gravimetric „Geoid“models

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA TallinnExample Pure Geoid- Approach EGG97-Geoid without Datum  N(d) +/- 3.5 cm „Tilt“ Standard GNSS/GPS-heighting Approaches using identical Points

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA TallinnExample Pure Geoidapproach EGG97-Geoid With Datum  N(d) +/- 4 mm HBF:= N G +  N G (d) Standard GNSS/GPS-heighting Approaches using identical Points

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA DFHRS Software Identical Identical„Fitting“Points(B,L,h;H) Meshes Meshes Patches Patches

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA Every geoid- and/or vertical deflection model N can be „patched“ DFHRS Software  N G (d j ) 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

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA DFHRS_DB Design Parameters <_3_cm DFHRS_DB Windhuk, Namibia EGM96 Fitting Point Density (< 10 mm points, EGG97) 50 points per (100 km x 100 km): <_1_cm DFHRS_DB 50 points per (100 km x 100 km): <_1_cm DFHRS_DB 10 points per (100 km x 100 km): < 3_cm DFHRS_DB 10 points per (100 km x 100 km): < 3_cm DFHRS_DB 3-4 points per (100 km x100 km): < 5-10_cm DFHRS_DB 3-4 points per (100 km x100 km): < 5-10_cm DFHRS_DB Meshsize (p=3) km : HRS approximation error < (5-10) cm km : HRS approximation error < (5-10) cm 10 km: HRS approximation error <1 cm10 km: HRS approximation error <1 cm 5 km: HRS approximation error < 0.5 cm 5 km: HRS approximation error < 0.5 cm

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA DFHRS_DB Design Parameters Design Studies < _cm DFHRS Germany Patch-Size (EGG97) km for a < 1_cm DFHRS_DB km for a < 1_cm DFHRS_DB 50 – 60 km for a < 3_cm DFHRS_DB 50 – 60 km for a < 3_cm DFHRS_DB 300 km for a < 10_cm DFHRS_DB 300 km for a < 10_cm DFHRS_DB (3-5) points per patch

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA < 10cm DFHRS Europe – „Fittingpoint-Design“ ETRS89/EVRS „GPS-/Levelling- Points of EVN“ Fitting Points NFEM(p) =: h - H Used for the 1st Version < 10_cm DFHBFS Europe

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA AustriaGermanyEstoniaLatviaLithuania Switzerla nd Number of unused control points RMS [cm] <_10_cm DFHRS_DB - Indepent Quality Control (Present Version, 35 km meshes, 34 Patches) <1_dm EVRF2004

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA Overview about European DFHRS_DB 30 km Mesh Size 10 km 5 km < 1 cm < 3 cm < 10 cm 2005 New

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA DFHRS_DB - Product as CD-Installation of State Land Services in Germany CopyProtection inclusive

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA < 5 cm DFHRS_DB Florida (… Masterthesis )

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA 5 km FEM Meshes European DFHRS_DB… including < 1 cm DFHRS_DB Ungary Test - Area (50 x 90 ) km 1-2_cm DFHRS_DB Budapest DFHRS 4.0 DFHRS 4.0 Software

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA DFHRS in Practice

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA

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

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA DFHRS - Extension to Gravity Observations Sensor-Observation

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA „<1cm-Resolution („2mm“) of HRS“ Instead of classical global spherical harmonics (n=m=7200) Spherical Cap Harmonics (SCHA) Cap P Extension of the DFHRS-Concept to Gravity Observations

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA  0 =90°  0 = arbitrary  0 =90° l = 6, l = 2, l = 4 l(2) = l(6) = l(4) =  0 arbitrary Global Sperical- Harmonics Spherical Cap Harmonics (SCHA)

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA DFHRS - Extension to Gravity Observations SCHA „Handling“ SCHA „Resolution“ („2mm Geoid“, n=7200 u = ) α=α=1° (=110 km area) => k-SCHA = 80 and u = ° (=330 km area) => k-SCHA = 250 and u= α=α= α=α=25° (Europe) => k-SCHA = 2000 and u =

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA DFHRS - Extension to Gravity Observations Sensor-Observation at Position P(x,y,z) Treatment of Gravity Observations

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA Treatment of GPM - EGM96 / EGM99, EIGEN, etc

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA h GNSS + v = H + f T  p - h GPS ·  m H + v = H N G ‘ j + v j = f T  p +  N G (d j )  j + v = - f B T / M(B)  p +   (d ,  ) j  j + v = - f L T /(N(B)  cos(B))  p +   (d ,  ) j  g·S(  )dσ + v = NFEM(p)= f T  p NFEM(p) N(pk)N(pk) Extension of the DFHRS-Concept to gravity observations

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA Example Saarland: 825 Gravity Observations

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA Example Saarland (Jan. 06) DFHRS-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

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA „1_cm DFHRS of Baden-Württemberg“ (EIGEN-GPM, Gravity Values, Fitting Points)

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA „1_cm DFHRS of Baden-Württemberg“ (EIGEN-GPM, Gravity Values, Fitting Points)

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA „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

MOLDPOS – Initial Meeting, Chisinau, HRS-Computation and DFHRS-Concept Prof. Dr. Reiner Jäger, HSKA Solution Concept for HRS-Computation and GNSS-Heighting *DFHRS-Software - 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.q Message used in GNSS-Services, NTRIP etc. - High Capacities for International Cooperations and for EVRS