1. Data Requirements for a 1-cm Accurate Geoid
Christopher Jekeli
Division of Geodesy and Geospatial Science
School of Earth Sciences
Ohio State University
125 South Oval Mall
Columbus, OH
10 September 2008
Seminar
National Geodetic Survey
Silver Spring, MD

2. Introduction
Geoid: the equipotential (i.e., level) surface that closely approximates mean sea level
global geoid: approximates global mean sea level
local geoid: vertical datum with origin at one point close to mean sea level
Why do we need a 1-cm accurate geoid?
height modernization is facilitated enormously using GPS instead of standard leveling N
global geoid, W0
WGS84 ellipsoid
Earth’s surface
Hlocal
local geoid,
best ellipsoid, U0 h
operative equation:
vertical accuracy of GPS is approaching 1 cm
Consideration: at 1 cm accuracy, one should opt for dynamic heights instead

3. Essential Errors in Geoid Models
Geoid models are subject to two types of data error
commission error - due to observational error in the gravimetric data
omission error - due to lack of resolution in the gravimetric data
these errors are independent
assume all other modeling errors can be reduced significantly below these errors
A particular accuracy goal for a regional geoid model requires corresponding accuracy and resolution in the gravimetric data
these requirements can be estimated:
using statistical analyses of the gravitational field
using gravimetric data simulated from a global model and topography
Accuracy goals for local geoid: 5 cm (st.dev.), 1 cm (st.dev.)
assume equal contribution from each error type:
5/2 = 3.5 cm
1/2 = 0.7 cm

4. Geoid Undulation from Disturbing Potential
g = normal gravity on ellipsoid
Approximations
linear approximation: usually insignificant (mm-level)
neglect topographic effect (evaluate T inside masses): can be significant (cm-dm level)
Assumptions
N refers to global geoid (global geoid is not the same as national vertical datum)
best-fitting ellipsoid (WGS84 ellipsoid is not ellipsoid of T according to SH/Stokes)
zero-tide model (this may not be consistent with other applications of tidal effect)
all these assumption introduce biases in a comparison between gravimetric and GPS/leveling geoid undulations

5. Disturbing Potential from Gravimetric Data
correct for spherical approximation
apply topographic correction
equiv.
Stokes’s formula
global spherical harmonic (SH) model
Errors:
finite, discrete data  limited resolution  finite degree, nmax  omission error
observational noise  propagated error  commission error
Analysis of omission/commission errors in SH model applies equally to Stokes’s formula
Errors in other model corrections are not considered

6. Degree Variances / Power Spectral Density
Per-degree-variance of geoid undulation:
[m2]
Equivalent (isotropic) power spectral density:
frequency:
[m2/(cy/m)2]
[cy/m]
Omission error variance:
Kaula’s Rule:

7. Power Spectral Density of Geoid Undulation
Notes:
Kaula’s rule over-estimates power at low freq. and under-estimates power at high freq.
EGM96 is under-powered at its high frequencies
Gravitational field appears to follow power law (constant fractal dimension) for frequencies,
(harmonic degrees 120 – 1200)

8. New Power Law Model
frequency [cy/m]
EGM08 omission error variance model
Power Law approximation of geoid undulation psd at very high degrees:
Omission error standard deviation:

9. Standard Deviation of Omission Error
Spatial Res.
nmax
New Power Law
Kaula’s Rule
56 km
360
22.5 cm
17.8 cm
9.3 km
2160
4.1 cm
3.0 cm
7.8 km
2560
3.5 cm
2.5 cm
1.5 km
13740
0.7 cm
0.5 cm
Spatial resolution = 180/nmax = (180/nmax)(111.2 km/ )
Values for nmax > 1200 may be optimistic if power law attenuation model is wrong.
These are global values; required resolution may be smaller/larger for a specific region.
For example, in South Korea, the st.dev. of the omission error is likely only 3.6 cm for gravimetric data resolution of 9.3 km (5 arcmin).

10. Gravity From Topographic Data
free-air anomaly
Isostatic gravity anomaly:
isostatic adjustment
topographic removal
If topography is perfectly compensated isostatically, then DgI = 0
Hence:
For computational efficiency, approximate topography and isostatic compensation as equivalent density layers (Helmert condensation)
Then, in spectral domain:
Airy isostatic model

11. Statistical Analysis of Omission Error Depends on “Randomness” of Field
EGM96 model
mgal
Statistics of inner box
mean
st.dev.
min
max
gravity anomaly [mgal]
20.3
15.8
-26.4
67.2
geoid undulation [m]
26.3
3.8
18.0
33.2

12. Topographic Elevation Data
Statistics for latitude, f  39, h > 0:
mean = 242 m
st.dev. = 235 m
max = 1543 m
SRTM 3 data are also available

13. Gravity Anomaly Simulated from Topography
Estimated from ETOPO2
Negative elevations (bathymetry) were set to zero
Statistics for latitude, f  39, h > 0:
mean = 3.59 mgal
st.dev. = mgal
max = mgal
min = mgal
PSDs computed for local areas:
“rough” anomaly area
“smooth” anomaly area

14. Power Spectral Densities of Simulated Gravity Anomaly
frequency [cy/m]
psd [mgal2/(cy/m)2]
2'2' grids
periodogram method, with removal of mean and linear trend
averaged over frequency directions to yield isotropic psd
straightforward relationship to PSD of geoid

15. Power Spectral Densities of Geoid
frequency [cy/m]
psd [m2/(cy/m)2]
power-law model
implied required resolution
global
5 arcmin
local rough area
7 arcmin
local smooth area
10 arcmin
These are tentative (illustrative) results - local simulated gravity anomaly field may be improved using ground data and higher resolution topographic data.
However, it appears possible that better than 5 arcmin resolution would not be required.
Further studies are needed for a final recommendation.

16. Commission Error vs. Observational Data Noise
Rapp (1969)1 derived:
variance in observational noise
angular data resolution 
Assumptions:
observational errors are uncorrelated (pure white noise with no systematic errors)
data are uniformly distributed (i.e., uniform angular resolution)
1 Rapp, R.H. (1969): Analytical and numerical differences between two methods for the combination of gravimetric and satellite data. Boll. di Geofisica Teorica ed Applicata, XI(41-42),

17. Gravity Anomaly Observation Errors
required resolution [arcmin]
nmax
allowable comm. error [cm]
sDg
[mgal]
Dq = Dl = 4.2
2560
3.5
3.3
Dq = Dl = 0.8
13740
0.7
50% commission error does not put stringent requirements on gravity data accuracy.
Increased data accuracy may relax resolution requirement.
sDg [mgal]
sN = 5 cm
resolution [km]
sDg [mgal]
sN = 1 cm
resolution [km]

18. Summary
An analysis of data requirements for a 5-cm (1-cm) accurate geoid must consider both commission and omission errors (and other model and observational errors).
Allowable omission error determines required resolution in the gravimetric data.
Omission error variance can be determined using a stochastic interpretation of the high-frequency gravity field in the form of degree variances or psd.
Global psd models indicate that 4.2 arcmin (7.8 km) resolution is required for 3.5 cm omission error; and, 0.79 arcmin (1.5 km) resolution for 0.7 cm omission error.
Resolution requirements can be refined based on regional characteristics of the gravity field.
Rough/smooth characteristics can be determined in many cases from topographic data under reasonable assumptions.
Considering present-day gravimetric accuracy, the resolution of the data rather than their measurement accuracy (~3 mgal) is the driving requirement for high accuracy geoid computations.