An example of gravimetric geoid computation:

The Iberian Gravimetric Geoid of 2005

The figure of Earth in first approximation:the revolution ellipsoid

normal

Relationship between orthometric (H) and ellipsoidal (h) height:the undulation of the geoid (N)

H = h N

The figure of Earth in second approximation: the geoid undulation (N) measured over the ellipsoid

Ellipsoid

Geoid

Earthsurface

H

hNA

A

B

C

HA = hA NA

HB = hB NB

HBA = hBA NBA

NB

NBA = B – NA 0 HBA hBA

The leveling with GPS computing height differencesrequires the use of a geoid to calculate N

Calculating N for two paths of 100 km, it can be observed that the values of h H are related to the topography

N calculated in plane zone: path AB

N calculated in mountainous zone: path CD

• As conclusion, a geoid model is required to use GPS heights (h) for leveling.

• The geoid models available in the study area, have not enough precision to be used in the major part of the engineering problems.

• For this reason, it is necessary the computation of a new geoid for the Iberian area, which has the major precision possible.

Problem: the computation of an Iberian geoid

NECCESARY DATA

• Gravity data available from several international centers (usually, from internet links).

• Digital terrain models (DTM) that can be combined (and interpolated, if it is necessary) to get a high-resolution Iberian DTM.

• Validation data (GPS/leveling data) that can be supplied by several European centers.

Land and Marine Gravity Data

http://www.ngdc.noaa.gov/seg/

Land and Marine Gravity Data

http://bgi.cnes.fr:8110/bgi_debut_a.html

Land and Marine Gravity Data

http://www.usgs.gov/

The gravity data needed for the computation canbe obtained from the above-mentioned centers

http://www2.jpl.nasa.gov/srtm/

http://www.ngdc.noaa.gov/mgg/gdas/gd_designagrid.html

The elevation data needed for the computation canbe obtained from the above-mentioned centers

The short and long wavelength effects are removed from the gravity data by means of :

Thus, the interpolation of the gravity data randomly distributed over the study area to a regular grid, is facilitated (Corchete et al., 2005)

The term href corresponds to the elevations filtered with a long wavelength filter of 60 arc-minutes

n

0m

nmnmnm

n

2n

n

2GM msenKmcosJ)(cosP)1n(r

a

r

GMg

máx

EIGEN-CG01C Gravity Anomalies (nmax = 360)

The term gGM is calculated using a geopotential model

The short-wavelength contribution must be recovered after the interpolation by means of:

For it, the digital terrain model previously computed will be used (Corchete et al., 2005)

Obtained by interpolation

Obtained by interpolation

Determination of a geoid model (N)

N = NGM + Nb + NI

NGM : contribution of the geopotential model

Nb : contribution of the residual gravity

NI : indirect effect

n

0m

nmnmnm

n

2n

n

GM msenKmcosJ)(cosPr

a

r

GMN

máx

EIGEN-CG01C Geoid (nmax = 360)

NGM : contribution of the geopotential model

http://www.gfz-potsdam.de/pb1/op/grace/results/index_RESULTS.html

Nb : contribution of the residual gravity

g = gfree + c + g ,, g = 0.3086 NI

where:

d)(S g4

RN ),(b

f(x,y) and g are:

c: terrain correction(only considering the masses over the geoid)

]dxdy

)zh()yy()xx(

dz)hz([Kc

h

h23

2p

2p

2p

p

Ap

gh)hf(h2fh2

Kc 22

2/322 )yx(

1)y,x(f

A

dxdy)y,x(fg

Nb : contribution of the residual gravity(obtained after integration with the FFT 1D)

F1 = FFT 1D F1-1 = FFT 1D backward

where:

PQ222

2

1sins ,, s + s )ln6s - (3 - 10s + s 6 - 4-

s

1=)sS(

coscos)(2

1sin)(

2

1sin

2

1sin 22

PQ2

n

1

)S( cos)g(G4

R),(N 11

11 FFF

f(x,y) and g are:

NI : indirect effect

dxdy)yy()xx(

)y,x(h)y,x(h

6

K)y,x(h

KN

A2/3 2

p2

p

pp33

pp2

I

2/322 )yx(

1)y,x(f

A

dxdy)y,x(fg

332I gh

6

Khf

6

Kh

KN

MODEL VALIDATION

• The model validation is performed comparing its values with high-precision data.

• These data are the geoid undulations (N0) obtained by means of GPS/leveling.

• These data can be supplied by European centers for our study area: the Iberian Peninsula.

GPS/leveling process: calculation of the observed geoid undulations (N0)

N0 = hGPS - Hlev GPS satellite receiver

Leveling instrument

          

European Vertical Reference System

(EVRS)

         

http://crs.bkg.bund.de/evrs/

Organisms that can supply validation data

http://crs.bkg.bund.de/evrs/tabelle_neu.html

-10 -8 -6 -4 -2 0 2 4

36

38

40

42

44

Data of the European Vertical Network (EVRS) on Iberia

COMPARISON WITH OTHERS MODELS

• The official geoid used in Spain (Sevilla, 1997): the IBERian GEOid of 1995 (IBERGEO95).

• The European Geoid (Denker and Torge, 1998): the European Gravimetric Geoid 1997 (EGG97).

• The worldwide geoids EGM96 y EIGEN-CG01C.

IBERian GEOid 1995 (IBERGEO95)

European Gravimetric Geoid 1997 (EGG97)

EGM96 geoidEIGEN-CG01C Geoid

Worldwide geoid models

http://cddis.gsfc.nasa.gov/926/egm96/egm96.html

CONCLUSIONS

• The IGG2005 geoid model improves all previous geoid, being much more precise.

• The IGG2005 is a first step towards a centimetric precision geoid for the Iberian area.

• The centimetric precision in the geoid computation will be achieved, if more and more precise gravity data are available for the Iberian area.

REFERENCES

Corchete V., Chourak M. and Khattach D., 2005. The high-resolution gravimetric geoid of Iberia: IGG2005. Geophys. J. Int., 162, 676–684.

Denker, H., and W. Torge. The European Gravimetric Quasigeoid EGG97. International Association of Geodesy Syposia, Vol. 119, Geodesy on the Move. Springer-Verlag, Berlin-Heidelberg-New York, S. 249-254, 1998.

Sevilla, M. J. A new gravimetric geoid in the Iberian Peninsula. BGI Bull. D’Inf. Nº 77 (Toulouse) and IGeS Bull. Nº 4 (Milano), 163-180, 1995.

CONTACT

Prof. Dr. Víctor CorcheteDepartment of Applied Physics

Higher Polytechnic School - CITE II(A)UNIVERSITY OF ALMERIA

04120-ALMERIA. SPAINFAX: + 34 950 015477

e-mail: corchete@ual.es