Division of Geodesy and Geospatial Science, School of Earth Sciences, Ohio State University 1 Christopher Jekeli Data Requirements for a 1-cm Accurate

Division of Geodesy and Geospatial Science, School of Earth Sciences, Ohio State University1

Christopher Jekeli

Data Requirements for a 1-cm Accurate Geoid

10 September 2008

Seminar

National Geodetic SurveySilver Spring, MD

e-mail: [email protected]

Division of Geodesy and Geospatial ScienceSchool of Earth Sciences

Ohio State University125 South Oval Mall

Columbus, OH 43210


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

– vertical accuracy of GPS is approaching 1 cm

H h N

– operative equation:

Nglobal geoid, W0

WGS84 ellipsoid

Earth’s surface

Hlocal

local geoid,

best ellipsoid, U0

local0W

local0 0

0

W W

h

• Consideration: at 1 cm accuracy, one should opt for dynamic heights instead


• 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

• 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:

– these errors are independent

Essential Errors in Geoid Models

– assume all other modeling errors can be reduced significantly below these errors

5/2 = 3.5 cm1/2 = 0.7 cm


TN

• Geoid undulation:

• Approximations

– 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)

Geoid Undulation from Disturbing Potential

= normal gravity on ellipsoid

– 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

– linear approximation: usually insignificant (mm-level)


Disturbing Potential from Gravimetric Data

1

2

( , , ) ( , )n n

nm nm

n m n

GM aT r C Y

a r

, , ', ' ,

4

RT r g S r d

1, ( , )

4 1nm nmC g Y dn

correct for spherical approximationapply topographic correction

on sphere of radius, g R

equiv.

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

Stokes’s formula global spherical harmonic (SH) model

• Errors in other model corrections are not considered


Degree Variances / Power Spectral Density

• Per-degree-variance of geoid undulation: 2 2 2n

N nmnm n

R C

[m2]

• Equivalent (isotropic) power spectral density: 2

22N n N n

Rf

n

– frequency: 2nf n R

[m2/(cy/m)2]

[cy/m]

• Omission error variance: max

2 2

omission1

N N nn n

max

2

omission

1

2N N

f

f f df

• Kaula’s Rule: Kaula2 10 24

2 110 , 6371000 mN n

nR R

n

Kaula

omissionmax

64 [m]N n


1 108

1 107

1 106

1 105

1 104

1 105

1 106

1 107

1 108

1 109

1 1010

1 1011

1 1012

1 1013

1 1014

1 1015

1 1016

1 1017

EGM96EGM08Kaula's Rule

EGM96EGM08Kaula's Rule

frequency [cy/m]

geoi

d un

dula

tion

psd

[m^2

/(cy

/m)^

2]

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, 6 53 10 cy/m 3 10 cy/mf (harmonic degrees 120 – 1200)


• Power Law approximation of geoid undulation psd at very high degrees:

12 3.898 2 24.044 10 [m /(cy/m) ], has units [cy/m]N f bf f f

• Omission error standard deviation: model

0.949omissionmax

60.016 [m]N n

1 108

1 107

1 106

1 105

1 104

1 103

0.11

10100

1 1031 104

1 1051 106

1 1071 1081 109

1 10101 1011

1 10121 1013

1 10141 1015

1 10161 1017

EGM96EGM08power-law model

EGM96EGM08power-law model

frequency [cy/m]

geoi

d un

dula

tion

psd

[m^2

/(cy

/m)^

2]

frequency [cy/m]

New Power Law Model

EGM08 omission error variance model


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

• These are global values; required resolution may be smaller/larger for a specific region.

• Values for nmax > 1200 may be optimistic if power law attenuation model is wrong.

– 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).

• Spatial resolution = 180/nmax = (180/nmax)(111.2 km/ )


Gravity From Topographic Data

• Isostatic gravity anomaly:

Airy isostatic model

free-air anomaly

topographic removal

isostatic adjustment

• If topography is perfectly compensated isostatically, then gI = 0

g C A • Hence:

• For computational efficiency, approximate topography and isostatic compensation as equivalent density layers (Helmert condensation)

• Then, in spectral domain:

Ig g C A

3222 1 xDg G H e e F F


Statistical Analysis of Omission Error Depends on “Randomness” of Field

mmm

EGM96 model

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

mgal


Topographic Elevation Data

• Statistics for latitude, 39, h > 0:

∙ mean = 242 m

∙ st.dev. = 235 m

∙ max = 1543 m

• SRTM 3 data are also available


Gravity Anomaly Simulated from Topography

• Negative elevations (bathymetry) were set to zero

• Estimated from ETOPO2

• Statistics for latitude, 39, h > 0:

∙ mean = 3.59 mgal

∙ st.dev. = 19.32 mgal

∙ max = 150.56 mgal

∙ min = -41.62 mgal

• PSDs computed for local areas:

“smooth” anomaly area

“rough” anomaly area


Power Spectral Densities of Simulated Gravity Anomaly

• periodogram method, with removal of mean and linear trend

• averaged over frequency directions to yield isotropic psd

• 2'2' grids

1 106

1 105

1 104

1 103

1 105

1 106

1 107

1 108

1 109

1 1010

1 1011

1 1012

1 1013

rough area smooth area rough area smooth area

frequency [cy/m]

psd

[m

gal2 /

(cy/

m)2 ]

• straightforward relationship to PSD of geoid


1 106

1 105

1 104

1 103

0.1

1

10

100

1 103

1 104

1 105

1 106

1 107

1 108

1 109

1 1010

1 1011

1 1012

rough areasmooth areaEGM08power-law modelrough area modelsmooth area model

rough areasmooth areaEGM08power-law modelrough area modelsmooth area model

frequency [cy/m]

psd

[m

2 /(c

y/m

)2 ]Power Spectral Densities of Geoid

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.


22

2204 1nm

gC

n

max max22 2 2 2commission 22

02 2

2 1

4 1nm

n nng

C

n m n n

nR R

n

Commission Error vs. Observational Data Noise

• Rapp (1969)1 derived:

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), 108-118.

2g – 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)


Gravity Anomaly Observation Errors

required resolution [arcmin]

nmax allowable comm. error [cm]

g

[mgal]

= = 4.2 2560 3.5 3.3

= = 0.8 13740 0.7 3.3

• 50% commission error does not put stringent requirements on gravity data accuracy.

• Increased data accuracy may relax resolution requirement.

0 0.5 1 1.5 2 2.5 3 3.5 46

7

8

9

10

11

12

g [mgal]

N = 5 cm

reso

luti

on [

km]

0 0.5 1 1.5 2 2.5 3 3.5 41.21.31.41.51.61.71.81.9

22.12.2

g [mgal]

N = 1 cm

reso

luti

on [

km]


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.

Documents

Division of Geodesy and Geospatial Science, School of Earth Sciences, Ohio State University 1 Christopher Jekeli Data Requirements for a 1-cm Accurate