11.3.6 GPS Heighting

GRAVIMETRIC (STOKES' INTEGRAL) METHODS

The basic method involves the integration of terrestrial gravity observations using Stokes' Integral:

(11.3-3)

where:

R	is the mean radius of the earth,
	is normal gravity on the sphere,
()	is the Stokes' function (HEISKANEN & MORITZ, 1967),
g	is the gravity anomaly (= observed gravity reduced to the geoid minus normal gravity at the corresponding point on the ellipsoid) for the surface element d, and
	is the angular radius, measured from the geocentre, between the computation point and the point at which the gravity anomaly is located.

Note that, in principle, the integration is carried out over the entire globe. The requirement to have gravity anomalies over the entire globe had limited the use of the gravimetric techniques until the advent of the Space Age, and in particular the availability of global gravity field models such as those afforded by spherical harmonic coefficients. Nowadays the gravimetric technique is actually a combination of Stokes' Integral and spherical harmonic models (SCHWARZ & SIDERIS, 1993), such that:

N = NL + NS

(11.3-4)

where N_L is given by eqn (11.3-2), and N_S is the short wavelength information derived from the integration of surface gravity, but derived from a modification of eqn (11.3-3):

(11.3-5)

in which the integration is carried out only for a limited "cap", centred at the computation point, with radius _o. Usually this is of the order of a degree (110km) or so, and the gravity anomaly is actually a residual quantity:

g' = g + gL

(11.3-6)

The quantity g_L is derived from the spherical harmonic model:

(11.3-7)

There are a number of software packages that use the above formulation to derive geoid height. A description of the package developed at the School of Geomatic Engineering, University of New South Wales, can be found in such references as HOLLOWAY (1988) and KEARSLEY (1988).

The following are some characteristics, advantages and disadvantages of this method:

• Geoid height information is obtained relative to a geocentric ellipsoid.

   

• The basic data is terrestrial gravity observations in the vicinity of the computation point. The computation is essentially a "weighted mean" of gravity anomalies (appropriately scaled by Stokes' Function).

   

• Suitable only in areas with good local gravity coverage (as in most developed countries).

   

• Information is sometimes provided in the form of maps of geoid contours. Otherwise needs to be computed, but requires sophisticated software and detailed (and large) gravity data sets. The software and/or the gravity data may not be generally available.

   

• Variable accuracy, dependent upon:
  • quality and density of gravity coverage, and
  • sophistication of software.

   

• Relative geoid height (between two points up to several 100km apart) accuracy is higher than absolute values. Tests in Australia, U.S. and Europe indicate that relative geoid height can be determined at the few parts per million level.

Summary: Highest accuracy possible, but most inconvenient of the GPS heighting techniques to use by non-experts. Must be precomputed.

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© Chris Rizos, SNAP-UNSW, 1999