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

1
Investigation of the Topographic Effect by Using
High Degree Spherical Harmonic Expansion

• Yan Ming Wang
• National Geodetic Survey, USA
• IAG Scientific Meeting
• Buenos Aires 8/31-9/4 2009

2
Overview

• 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 (360ltn 2160)
• Conclusions and discussions

3
Potential of the topography in spherical harmonic
series

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

4
Potential of the condensed surface layer

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

5
Direct, indirect effect the topographic bias in
spherical harmonic series

• By using the coefficients of Hk, k1,2,3, 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

6
Data Used

• 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

7
Statistics of elevation in 1'x1' mean block
values, units are in meters
No 233280000
Mean 376
RMS 933
Min -416
Max 8550
8
Method of the expansion and Parameters of the
ellipsoid used

• Numerical quadrature to degree and order 2700
• R6371000 m
• Parameters of the ellipsoid
• - Semi-major axis (a) 6,378,136.3 m
• - Semi-minor axis (b) 6,356,751.55863 m

9
Cumulative geoid power (n0,2700) of Hk in
meters
RMS value
H 341.949
H2 0.177
H3 0.007
10
Square root of degree variances and cumulative
power of the Hk (k1,2,3)



11
Cumulative power of potentials in geoid, units
are in m, degree 0, 1 included
Potential n2,360 n361,2700 n2,2700
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
12
Indirect Effect (n2,2700)



13
Comparison between the potentials of the
topography and the EGM08

• Legitimacy of comparisons Gravity field at
  wavelength shorter than 100kms are due to the
  topography
• Contribution to geoid at a bandwidth 360ltn2160
• EGM08 12.9 cm Topography 10.9 cm
• Contribution to geoid at a bandwidth 700ltn 2160
• EGM08 4.3 cm Topography 5.5 cm

14
Square root of degree variances and cumulative
power of the topo. and EGM08



15
Conclusions and discussions

• 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.

16
Conclusions and discussions (cont.)

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