Numerical aspects of the omission errors due to limited grid size in

geoid computations

Yan Ming WangNational Geodetic Survey, USA

VII Hotine-Marussi Symposium Rome 6-10 July 2009

• Objective: to determine the maximum grid size at which the omission error is less than 1-cm

• Omission errors are estimated by using the 3” SRTM elevation data, focus is on the spectral band corresponding to grid size 5’ to 3”

• Results are compared with those based on Kaula’s rule.

• Conclusions and discussions

Overview

1. Spherical harmonic series up to ultra high degree and order Numerical difficulty to expand the potential of the topography into degree and order of 216,000 (3” resolution); Kaula’s rule can be used

2. Newtonian integrationNumerically doable. Computation only need to be extended in 1ºx1º integration area around computation point for both potential and gravity.

Methods of estimation of the omission errors

• Newtonian integral

• Assumption 1: All short wavelengths of the gravity field (< 10km resolution) are due to the residual topography (isostay uncompensated)

• Assumption 2: Constant density of the residual topography

• Assumption 3: Contribution of the wavelengths shorter than 3” (<=90m) are negligible

Method used in this work

• Newtonian integral

where and are the radial distances to the reference surface and the Earth’s surface, respectively.

Potential of the residual topography

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• SRTM3” elevation data for North and central America and the Caribbean (10°≤ ϕ ≤ 60° ; 190° ≤ λ ≤ 308°).

• It contains 13,992,031 gaps assigned an elevation value of –1.

• SRTM30/GTOPO30 30”x30” global DEM is used to fill-in the gaps.

Data Used

1. Compute 5’, 2’ and 1’ bloc-mean values using the SRTM 3” elevation data

2. Using the block-mean values as reference surfaces, compute the potentials of the residual topography

3. Convert these potentials to geoid heights by using Bruns formula

4. Using the block-mean values as reference surfaces, compute the gravity of the residual topography

Computation procedure

Elevation along latitude band 41º24’15”. It peaks at 4157 m in the Rocky Mountains. The smaller peak is in

the Appalachian.

Heights after removing 1’, 2’ and 5’ block-mean values

Statistics of residual heights (meters)

Grid size Mean RMS value Min/Max

1’ -0.6 32.4 -342/531

2’ -0.3 44.8 -488/749

5’ 2.2 79.3 -696/1498

Geoid omission errors at grid size 1’, 2’ and 5’, in cm

Statistics of the geoid omission errors (cm)

Grid size Mean RMS value Min/Max

1’ (~1.9 km) -0.00 0.09 -0.8/0.9

2’(~3.7 km) -0.06 0.54 -1.2/8.6

5’(~9.3km) 0.07 1.08 -4.3/11.4

. Results (RMS values) based on Kaula’s rule (Jekeli 2008)

This Work Kaula’s Rule. For 1’ grid size:0.1 cm 0.5 cm (Kaula)

. For 5’ grid size: 1.1 cm 3.0 cm (Kaula)

. The geoid omission errors by Kaula’s rule are several times larger

. Question: does Kaula’s rule overestimate the power of the gravity field at ultra high frequencies?

. This study provides ranges of geoid omission errors:

for 1’: -0.8/0.9 cmfor 5’: -4.3/11.4 cm

Comparisons with Jekeli’s results

What are the contributions of the residual terrain to gravity at 5’, 2’ and 1’ grid size?

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Gravity of residual terrain at grid size 1’, 2’ and 5’ (mGal)

Statistics of gravity of residual terrain (mGal)

Grid size Mean RMS value Min/Max

1’ 0.4 1.4 -05.4/23.6

2’ 0.7 2.4 -11.1/40.8

5’ 1.6 5.5 -18.2/114.5

. The geoid omission error at 5’ grid size reaches dm in high rough mountains. It is not suitable for cm-geoid.

. The geoid omission error at 1’ grid size is less than 1 cm

everywhere in CONUS, even in very rough areas. It seems 1’ grid size is sufficient for cm-geoid.

. The omission errors are smaller than those obtained by Kaula’s rule. Overestimation by Kaula’s rule at ultra high frequency band?

. Is the assumption that the gravity field in the frequency band

5’ – 3” is purely due to the residual terrain reasonable and accurate?

Conclusions and discussions

. The RMS values of the gravity omission errors are 1.4 and 5.5 mGal for 1’ and 5’ grid size, respectively.

. The extreme values are 23.6 and 114.5 mGal for 1’ and 5’ grid

size, respectively.

. The removal of omission errors from the gravity observations should help improving the quality of block-mean computation and data gridding.

. Constant density assumption should not change the conclusions.

Conclusions and discussions (cont.)