Data Requirements for a 1-cm Accurate Geoid Christopher Jekeli e-mail: jekeli@osu.edu Division of Geodesy and Geospatial Science School of Earth Sciences Ohio State University 125 South Oval Mall Columbus, OH 43210 10 September 2008 Seminar National Geodetic Survey Silver Spring, MD

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

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

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

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

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:

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)

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:

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).

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

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

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

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. = 19.32 mgal max = 150.56 mgal min = -41.62 mgal PSDs computed for local areas: “rough” anomaly area “smooth” anomaly area

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

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.

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), 108-118.

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]

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.