Investigation of the Topographic Effect by Using High Degree Spherical Harmonic Expansion

Yan Ming WangNational Geodetic Survey, USA

IAG Scientific MeetingBuenos Aires 8/31-9/4 2009

• Potentials of the topography and its condensed surface layer are expanded into spherical harmonic series to degree and order 2700 using numerical quadrature

• Topographic effects on the geoid (direct, indirect effect of Helmert 2nd condensation, topographic bias) are derived from above the spherical harmonic series

• Comparison between potentials of the topography and EGM08 at the spectral band (360<n ≤ 2160)

• Conclusions and discussions

Overview

where the subscripts “e” and “i” denote the quantities of exterior and interior spaces, respectively.

Potential of the topography in spherical harmonic series

)()(3

1)(

0 0

3

13

12nm

knmnm

n

n

m

knm

n

k

kn

n

P

et SbRaC

r

R

nGRPV

)()(2

1)(

0 11)2(

2nm

knm

n

n

m

nmknm

k

kn

nPit SbRaC

R

r

nGRPV

dS

R

R

H

b

a

nm

nmkknm

knm

)(

4

1

and

Our task is to compute the integrals above for all n and m up to 2700.

Potential of the condensed surface layer

0

1)()(n

Cn

n

PC V

r

RPV

n

m

nmCnmnm

Cnm

Cn SbRaV

0

)(

dS

R

R

H

R

H

R

H

n

GR

b

a

nm

nm

Cnm

Cnm

])(

3

11)[(

)12(

4 22

• By using the coefficients of H**k, k=1,2,3…, the indirect and direct effects of Helmert 2nd condensation can be expressed in terms of spherical harmonics

• The topographic bias can also be put into spherical harmonic series

• The gravity of the topography can be computed and subtracted from surface or airborne gravity anomalies to form the Bouguer anomaly

Direct, indirect effect & the topographic bias in spherical harmonic series

• 1'x1' block means from the SRTM-derived Digital Elevation Model in 30 arc-seconds grid

• All 1'x1'oceanic cells contain a nominal orthometric height value of zero

Data Used

Statistics of elevation in 1'x1' mean block values, units are in meters

No 233280000

Mean 376

RMS 933

Min -416

Max 8550

• Numerical quadrature to degree and order 2700• R=6371000 m

• Parameters of the ellipsoid

- Semi-major axis (a) = 6,378,136.3 m - Semi-minor axis (b) = 6,356,751.55863 m

Method of the expansion and Parameters of the ellipsoid used

Cumulative geoid power (n=0,2700) of H**k in meters

RMS value

H 341.949

H**2 0.177

H**3 0.007

Square root of degree variances and cumulative power of the H**k (k=1,2,3)

0.001

0.01

0.1

1

10

400 800 1200 1600 2000

h1h2h3h1h2h3

Degree

Cumulative power of potentials in geoid, units are in m, *degree 0, 1 included

Potential n=2,360 n=361,2700 n=2,2700

Indi. Effect 0.126 0.015 0.127

Topog. Bias 0.239 0.040 0.242

Topography 76.086 0.110 76.086

Topography* 342.075 0.110 342.075

Indirect Effect (n=2,2700)

• Legitimacy of comparisons: Gravity field at wavelength shorter than 100kms are due to the topography

• Contribution to geoid at a bandwidth 360<n≤2160EGM08: 12.9 cm Topography: 10.9 cm

• Contribution to geoid at a bandwidth 700<n ≤ 2160EGM08: 4.3 cm Topography: 5.5 cm

Comparison between the potentials of the topography and the EGM08

Square root of degree variances and cumulative power of the topo. and EGM08

0.01

0.1

1

10

400 800 1200 1600 2000

TopoEGM08Kaula's RuleTopoEGM08c

m

Degree

• The potentials of the topography and its condensed surface layer are expanded into spherical harmonic series to degree and order 2700 by using the numerical quadrature.

• Topographic effects (e.g. gravity of the topography, direct and indirect effect, topographic bias) can be computed from above developed spherical harmonic series in about 5 minute resolution.

• The topography has larger power than the EGM08 above degree and order 700. Topographic potential may provide more accurate information for gravity field at this frequency bandwidth, if the impact of the density variations of the topography is not significant.

Conclusions and discussions

• The EGM08 has larger power from degree 361 to 700, indicates that the topography may be somewhat isostatic compensated at this spectral bandwidth.

Conclusions and discussions (cont.)