Numerical aspects of the omission errors due to limited grid size in geoid computations Yan Ming Wang National Geodetic Survey, USA VII Hotine-Marussi.

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Numerical aspects of the omission errors due to limited grid size in geoid computations Yan Ming Wang National 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 integration Numerically 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

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 sizeMeanRMS valueMin/Max 1’ /531 2’ /749 5’ /1498

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

Statistics of the geoid omission errors (cm) Grid sizeMeanRMS valueMin/Max 1’ (~1.9 km) /0.9 2’(~3.7 km) /8.6 5’(~9.3km) /11.4

. Results (RMS values) based on Kaula’s rule (Jekeli 2008) This WorkKaula’s Rule. For 1’ grid size:0.1 cm 0.5 cm (Kaula). For 5’ grid size: 1.1 cm3.0 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 cm for 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?

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

Statistics of gravity of residual terrain (mGal) Grid sizeMeanRMS valueMin/Max 1’ /23.6 2’ /40.8 5’ /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 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.)