investigation of the topographic effect by using high degree spherical harmonic expansion
DESCRIPTION
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. Overview. - PowerPoint PPT PresentationTRANSCRIPT
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.)