DFHRS (Digital FEM Height Reference Surface)

MOLDPOS – Initial Meeting, Chisinau, 10-05-10 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 JägerHochschule 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-76133 Karlsruhe

www.dfhbf.de ; www.geozilla.de; www.dfhbf.de ; www.geozilla.de;

www.galileo-bw.de; www.moldpos.euwww.galileo-bw.de; www.moldpos.eu


Height Systems - Definitions and Terrestrial Realisiation

HH

Geoid = IdealGeoid = IdealHeight Reference Surface Height Reference Surface

(HRS)(HRS)

hh

NN

22

2),,,,,(

),,(),,( pzdydxd

zyxzyxl

zyxzyxWW

ErdeP

APoP WWC i

E

AiAP ngCC

P

PorthP

P

PNormalP g

CHbzw

CH ,, .

GepotentialNumbers C

Levelling and Gravity Observations

Terrestrial Height Determination

H=const. H=const. Small AreaSmall Area ~ „constant~ „constantWaterlevel“Waterlevel“


NewNewEuropeanEuropean

Normal-HeightNormal-HeightNetworkNetwork

EUREF/UELN ‚95/98EUREF/UELN ‚95/98

AdjustmentAdjustmentbyby

GeopotentialGeopotentialNumbers (DifferencesNumbers (Differences

DatumspointDatumspointWW00 bzw. C bzw. C00

AmsterdamAmsterdam„„NAP“NAP“

DHHN‘92DHHN‘92( Diff.: 1 cm)( Diff.: 1 cm)

Accuracy Accuracy SituationSituation


H = h - N

GNSS-Heightung

„HH from h-GNSS“

HRS-Model N

hh

hh NN

HH„„HRS“HRS“GeoidGeoid

QuasigeoidQuasigeoid


)(cosP)msinSmcosC(r

a1(

r

MGV nmnmnm

n

2n

n

0m

Gravity Potential W of the Earth at Point P(r,λ, )

ZVW

22

2sinr

2Z

Gravitational Potential of the Earth Masses

HRS = Geoid / Quasigeoid – Physical Definition & Realisation

Centrifgal - Potential due to Earth-Rotation

Q

80GRSOG

),,r(U),,r(V),,r(N

Hg

NN QGG

Q-Geoid

Geoid (W=Wo=const)

Spherical Harmonics Coefficients as Parameters

HBFHBF


Original Observation: LaserrangesDerived of it: Orbit Disturbances Derived of it: Spherical Harmonics (Müller/Kaula, 1960,1962Spherical Harmonics (Müller/Kaula, 1960,1962))

Satellite Geodetic Methods of Gravity Field Determination

• Satellite Laser Ranging (SLR) „„LEOS“ LEOS“ (L(Low ow EEartharth O Orbitrbit SatellitesSatellites))

LAGEOS 1, 2LAGEOS 1, 2STELLA, etc.


Observation Quantities1.) Abs. Positionierung via GPS (Black-Jack Receiver/NASA, JPL): x(t)2.) Relative Orbitmovements by Inter-Satellite Observations:3.) Precise Accelerometers ai STAR (Onera/Frankreich):4.) Gradiometers (e.g. GOCE)

Derived of these: 1.) Orbit Determinations 2.) Gravitational Accelerations gi(x,t) and Gradiometer-Tensor

(t)xd (t)xd

• Present Missions Hybride Sensors Aktice Measurements

iiii )( axgx

Derived from that: Spherical Harmonics CoefficientsSpherical Harmonics Coefficients

CHAMP



EIGEN ( = EEIGEN ( = European IImproved GGravity Model of the EEarth by NNew Techniques)

[cm]

H

H

J

Earth Geopotential MModel 966 (NASA)

H

H

J



EIGEN ( = EEIGEN ( = European IImproved GGravity Model of the EEarth by NNew Techniques)

[cm]

H

H

J

Earth Geopotential MModel 966 (NASA)

H

H

J -28

-27

-26

-25

-24

-23

-22

-21

-20

-19

-18

-17

N(p)

Latitude(degree)Longitude(dgree)

DFHRS_DB Namibia and Tanzania: 1.) Identical Points (N=h-H) & 2.) EGM96 bzw. EIGEN-GL04C



Bilance for Satellite Geodetic GPM(EGM2008; EIGEN)

)(cosP)msinSmcosC(r

a1(

r

MGV nmnmnm

n

2n

n

0m

n,m < 500 => N < 0.3 m ! =>N < 0.3 m ! => Not sufficient for GNSS-HeightingH = h – NN !!!

Further Information for further Improvement of the HRS HRS and N N

necessary! Hybrid and redundant Models!

TerrestrialGravity

Measurements IdenticalPoints (H, h)

N=h-H

““GeoidGeoidFitting“Fitting“


Short-Waved: ~ (1 – 3) cmMedium-/Long Waved

0.1-1.5 m !

2

096EGM

0

96EGM

dd)(S)gg(4

a

NN

0

1.) EGG97 - EuropeanGravimetric QGeoid 1997

(IfE, Hannover)

(h, H)(h, H)

Hybride Modeling in HRS-Computation

Weg 1: Stokes-based

RepresentationRepresentationGRIDGRID


r

)S,C(W

S,C(W

cosr

1

S,C(W

r

1

g

g

g

nmnm

)nmnm

)nmnm

U

W

N

)S,C(VU)Hh( nmnmref,80GRSQ

Q

PTNHh

Z)S,C(VW nmnm96EGM

(h, H)(h, H)

++

++

Dgl.-based

2.) Prof. Dr.-Ing. H.-G. Wenzel – Fitted Geopotenzialmodell (GPM 98GPM 98)

n = m = 1800 Accuracy of N ~ (10 - 30) cm

Parametrice Represenation (or Grid)



))z,y,x(M und )SC((T1

vN ifRenmmnQ

fRe

))z,y,x(M;)S,C((gg ifRenmmn

3.) „BKG-Model – Parametrisation of the Potentials by additional Mass Points

(h, H)(h, H)

))z,y,x(M und )SC((T1

v)hH( ifRenmmnQ

Massen-Points as essential additional ParametersMassen-Points as essential additional Parameters

Representation of BKG HRS: Grid



H = hGPS

(B,L) - DFHRS(B,L,h) korr

HH = hGPS(B,L) - (NFEM(p|B,L,h) + ∆m·h)

DFHRS-DB

Direct Direct - No - No IdenticalIdenticalPoints -Points -

- Online- Online -

Postpro-cessed

DFHRS-ConceptDFHRS-Concept


FEM representation of height reference surfaces

l

i

il

j

ji

mijnijmnnm

nm

nm

xyaaxyNxyNxyN

y

x

0 0,,,

,

,

0

0

,,,,

0

0

, pp

3D difference at any point P(x,y) along the border SA_SE of the meshes m and n has to vanish


! 0tNxxtxyytyaa n,mj

sasesai

sasesal

0i

il

0jm,ijn,ij

l

0i

il

0j

jim,ijn,ijmn,

xyaa

0

0

N

0

0

! 00

2210,

l

k

kk

kknm tctctctcctN

l,0k ,0x,y;x,y;;c sesesasank ppm

l

i

il

j

ji

mijnijmnnm

nm

nm

xyaaxyNxyNxyN

y

x

0 0,,,

,

,

0

0

,,,,

0

0

, pp

Continuous Finite Element Representation of HRSContinuous Finite Element Representation of HRS


FEM representation of height reference surfaces

Result: Continuous Height Reference Surface: NFEM(p)

nm2,1,0

l

0i

il

0jk

Tjik,ijk

FEM

;C

xyaNN

pp

pfp


DDigital igital FFEM EM HHeight eight RReference eference SSurface (DFHRS)- Concepturface (DFHRS)- Concept

hGNSS+ v = H + NFEM(p) - hGPS· m

H + v = H

NG‘j + v j = NFEM(p) + NG(d j)

j + v = - FB / M(B) p + (d,) j

j + v = - FL/(N(B)cos(B)) p + (d,) j

Complete New Computation of continuous HRS (p and m)!

DFHRS – Adjustment ApproachState of the Art < 2005< 2005

)(4 B

a g·S()dσ + v = NFEM(p)

NFEM(p)

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


EGG97European

GravimetricGeoid 1997

Mean- up to lang-wavedErrors

0.1 – 1.5 m !

=> New Concepts,more „precise“ or better: (H,h)-fitted

solutions

Weak-Shapes of Classical

Gravimetric

„Geoid“models

2

0Re

0

Re

)()(4

0

ddSgga

NN

f

f


Tallinn Tallinn ExampleExample

Pure Geoid-Approach

EGG97-Geoid

without DatumN(d)

+/- 3.5 cm „Tilt“

15km

15mm

Ne t

Re sid ua ls

Standard GNSS/GPS-heighting Approaches using identical Points


Tallinn Tallinn ExampleExample

Pure Geoidapproach

EGG97-Geoid

With DatumN(d)

+/- 4 mm 15kmNe t

20mmRe sid ua ls

HBF:= NG + NG(d)

Standard GNSS/GPS-heighting Approaches using identical Points


DFHRS SoftwareDFHRS Software

• Identical Identical „„Fitting“Fitting“PointsPoints

(B,L,h;H)(B,L,h;H)

• MeshesMeshes

• PatchesPatches


Every geoid- and/or vertical deflection model N can be „patched“

DFHRS SoftwareDFHRS Software

NG(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


DFHRS_DB Design Parameters

<_3_cm DFHRS_DBWindhuk, NamibiaEGM96

Fitting Point Density Fitting Point Density (< 10 mm points, EGG97)(< 10 mm points, EGG97) • 50 points per (100 km x 100 km): <_1_cm DFHRS_DB50 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_DB3-4 points per (100 km x100 km): < 5-10_cm DFHRS_DB

Meshsize Meshsize (p=3)(p=3)• 20-30 km : HRS approximation error < (5-10) cm20-30 km : HRS approximation error < (5-10) cm

•10 km: HRS approximation error <1 cm 10 km: HRS approximation error <1 cm • 5 km: HRS approximation error < 0.5 cm5 km: HRS approximation error < 0.5 cm


DFHRS_DB Design Parameters

Design Studies < 5 - 10_cm DFHRS Germany

Patch-Size Patch-Size (EGG97)(EGG97)

• 30 - 40 km for a < 1_cm DFHRS_DB30 - 40 km for a < 1_cm DFHRS_DB• 50 – 60 km for a < 3_cm DFHRS_DB50 – 60 km for a < 3_cm DFHRS_DB

• 300 km for a < 10_cm DFHRS_DB300 km for a < 10_cm DFHRS_DB

(3-5) points per patch(3-5) points per patch


< 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


Austria Germany Estonia Latvia Lithuania Switzerland

Number of unused control points

9

95

21

25

46

13

RMS [cm]

7.5

4.2

8.8

9.2

6.8

7.0

<_10_cm DFHRS_DB - Indepent Quality Control

(Present Version, 35 km meshes, 34 Patches)<1_dm EVRF2004


Overview aboutEuropean DFHRS_DB

30 kmMesh Size

10 km

5 km < 1 cm

< 3 cm

< 10 cm

2005

New 2006

2003


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


< 5 cm DFHRS_DB Florida (… Masterthesis )< 5 cm DFHRS_DB Florida (… Masterthesis )


5 km FEM Meshes

European DFHRS_DB… European DFHRS_DB… includingincluding

< 1 cm DFHRS_DB Ungary< 1 cm DFHRS_DB Ungary

Test - AreaTest - Area

(50 x 90 ) km (50 x 90 ) km 1-2_cm DFHRS_DB Budapest1-2_cm DFHRS_DB Budapest

DFHRS 4.0 DFHRS 4.0 SoftwareSoftware


DFHRS in PracticeDFHRS in Practice




DFRHS – Extension to Gravity ObservationsDFRHS – Extension to Gravity ObservationsGeodetic Network Optimization - 1st/2nd/3rd Order Design: A,P =>CA,P =>Cpp


DFHRS - Extension to Gravity ObservationsDFHRS - Extension to Gravity Observations

LAVg

g

0

0

Sensor-Observation


„<1cm-Resolution („2mm“) of HRS“

Instead of classical global spherical harmonics (n=m=7200)

Spherical Cap Harmonics(SCHA)

CapPP

Extension of the DFHRS-Concept to Gravity ObservationsExtension of the DFHRS-Concept to Gravity Observations

))'(cosP)'msin'S 'mcos'C(r

a

a

MG)',',r(V m)),k(nm),k(nm),k(n

maxk

0k

k

0m

1)k(n


0=90°

0 =arbitrary

0 =90°

l = 6,

l = 2,

l = 4

l(2) = 109.2714

l(6) = 290.9488 l(4) =200.4828

900P ml );(cos, 20P mkl );(cos),(

)(cos)( 0

P 90,mkl

0)(cos'P 2,m)k(l

0 arbitrary

GlobalSperical-Harmonics Spherical Cap Harmonics (SCHA)



80GRS;refmaxk

0k

k

0mm),k(n'm),k(nm),k(n

1)k(n

V- ) )'(cos'P)'msin'S'mcos'C(r

a()',',r(T

5.0)5.0,k(90

)k(nn SCHASHA

))'(cos'P)'msin'S 'mcos'C(r

a

a

MG)',',r(V m)),k(nm),k(nm),k(n

maxk

0k

k

0m

1)k(n

maxk

0k

k

0mm),k(n'm),k(nm),k(n

1)k(n

)'(cos'P)'msin'S'mcos'C(r

a)',',r(V

SCHA „Handling“

SCHA „Resolution“ („2mm Geoid“, n=7200 u =50.000.000)

α= 1° (=110 km area) => k-SCHA = 80 and u = 6.400

3° (=330 km area) => k-SCHA = 250 and u= 62.500α=

α= 25° (Europe) => k-SCHA = 2000 and u =4.000.000



LAVg

g

0

0

Sensor-Observation at Position P(x,y,z)

TreatmentofGravityObservations

LAV

ECFLAV

ECF

z

y

x

g

0

0

),,L,B(R

g

g

g

TLGVgrav ]

r

V,

V

'sinr

1,

'

V

r

1[

g

ECFlcentrifuga

ECF

z

y

xECF

gravz

y

x

g

g

g

g

'g

'g

'g

ECFgrav

LGVECF

LGV

gravr

E

N

'g),(R

g

g

g

)(dg )'(cosP)'msin'S'mcos'C(r

)1)k(n(

r

avg

0k

k

0mm),k(nm),k(nm)),k(n

1)k(n

rLGVgrav d


Treatment of GPM - EGM96 / EGM99, EIGEN, etc Treatment of GPM - EGM96 / EGM99, EIGEN, etc

0k

k

0mm),k(nm),k(nm),k(n

1)k(n

Q

jm),k(nm),k(n

jGPM

)'(cosP)'msin'S'mcos'C(r

a(

1

)(N)S,C(NvN d

)V 80GRS,ref

)(N jd

)V 80GRS,ref


hGNSS+ v = H + fT p - hGPS· m

H + v = H

NG‘j + v j = fT p + NG(d j)

j + v = - fBT

/ M(B) p + (d,) j

j + v = - fLT/(N(B)cos(B)) p + (d,)

j

)(4 Ba g·S()dσ + v = NFEM(p)= fT p

)(dg )'(cosP)'msin'S'mcos'C(r

)1)k(n(

r

avg

0k

k

0mm),k(nm),k(nm)),k(n

1)k(n

rLGVgrav d

NFEM(p)

N(pk)

Extension of the DFHRS-Concept to gravity observationsExtension of the DFHRS-Concept to gravity observations

0k

k

0m

jm),k(nm),k(nm),k(n

1)k(n

Q

jm),k(nm),k(n

jGPM

)(N)Vref)'(cosP)'msin'S'mcos'C(r

a(

1

)(N)S,C(NvN

d

d

)hm()'S,'C(Nv0 Tm),k(nm),k(nN pf


Example Saarland: 825 Gravity ObservationsExample Saarland: 825 Gravity Observations


Example Saarland (Jan. 06)Example Saarland (Jan. 06) DFHRS-Software DFHRS-Software with Computation with Computation DesignDesign

Version 1 ( 1cm Reference)Version 1 ( 1cm Reference)EGG97 („Gravity indirect“)EGG97 („Gravity indirect“)+ Identical Points+ Identical Points

Version 2 („Gravity indirect“)Version 2 („Gravity indirect“)Gravity Disturbances dgGravity Disturbances dgGPM98GPM98Identical PointsIdentical Points1 CAP 1 CAP

Result: 1_cm Coincidence of the HRS between Version 1 and Version 2Result: 1_cm Coincidence of the HRS between Version 1 and Version 2


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


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


„1_cm DFHRS of Baden-Württemberg“ (EIGEN-GPM, gravity values)

Number B[°] L[°] dg[mgal] v[mgal]--------------------------------------------------------------------6221803000 49.744402 9.325176 17.27052630 0.0166221803100 49.754800 9.320541 18.29553958 0.0046221810000 49.731079 9.322965 48.72880615 -0.0026221810100 49.767690 9.323488 16.90448271 -0.0096221810200 49.739588 9.295846 16.90117365 -0.005 : : : : :6222803100 49.787317 9.482121 25.19440159 -0.0066222803800 49.730625 9.411164 35.48699735 0.0126222803900 49.748882 9.486150 46.45989551 -0.0196222804000 49.735388 9.450240 45.58427649 -0.0226222804104 49.717601 9.439384 46.62347815 -0.0186222804204 49.702843 9.456634 55.15177619 -0.0296222810000 49.699664 9.417770 47.55625221 -0.003 : : : : :

Fig. 8DFHRS-software report for the gravity disturbances dg with the list of corrections v


Solution Concept for Solution Concept for HRS-Computation andHRS-Computation andGNSS-HeightingGNSS-Heighting

- Strict mathematical base for continuous FEM_HRS & **DFHRS-SoftwareDFHRS-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

Documents

DFHRS (Digital FEM Height Reference Surface)