# Verification of gravimetric geoidal models by a combination of GPS and orthometric heights

## Abstract

Gravimetric geoidal models such as “UNB Dec.’86” and “UNB ‘90” may be verified by a combination of GPS and orthometric heights. Ideally, the following relationship should equal zero: h – H _ N, where h is the height above a reference ellipsoid obtained from GPS, H is its orthometric height, and N is the geoidal undulation obtained from the gravimetric model. In many cases users are interested in relative positioning and the equation becomes: Δ(h – H – N).
This study looks at each aspect of these equations. The geometric heights (or height difference) is defined and the principal sources of error that are encountered in GPS levelling such as tropospheric delay, orbit biases etc. are examined.
The orthometric height (or height difference) is discussed by looking at various systems of height determination and deciding under which system the Canadian vertical network may be categorized, as well as what errors, and of what magnitude, are likely to be encountered. Orthometric heights are measured from the geoid which in practice is difficult to determine. The surface, not in general coincident with the geoid, from which these measurements are actually made, is investigated.
The three campaigns discussed in this study - North West Territories, Manitoba, and Ontario – are in areas where lvelled heights are referenced to the Canadian Geodetic Datum of 1928 in the case of the former two and the International Great Lakes Datum in the case of the latter. These two reference surfaces are discussed in some detail.
The geoidal solutions - “UNB Dec. ‘86” and “UNB ‘90” are described. The models are fairly similar as both use the same modified version of Stoke’s function so as to limit the area of the earth’s surface over which integration has to take place in order to determine the undulation at a point. “UNB ‘90” makes use of an updated gravity data collection. Both solutions makes use of terrestrial data for the high frequency contribution and a satellite reference field for the low frequency contribution. “UNB Sec. ‘86” uses Goddard Earth Model, GEM9, whereas “UNB ‘90” uses GEM-T1. The implications of changes inference field are discussed.
All measurements are prone to error and thus each campaign has associated with it a series of stations characterized by a misclosure obtained from h – H – N. These misclosures may be ordered according to any argument – latitude, ɸ, longitude, λ, orthometric height, H, etc., in order to search for a statistical dependency between the misclosure and its argument, or in other words, a systematic effect.
The autocorrelation function will detect the presence of a systematic “error” and least squares spectral analysis will give more information on the nature of this dependency. Both these tools are described and their validity is demonstrated on a number of simulated data series.
The field data collected during the three campaigns is analysed. The geometric and orthometric heights are combined with geoidal undulations from the “UNB Dec. ‘86” and then from “UNB ‘90” using the misclosure h – H – N. The resulting data series are ordered according to vaious arguments and examined for presence of systematic effect by means of the autocorrection function and spectral analysis. Similar tests are carried out on the data series yield by Δ(h – H – N) ordered according to azimuth and baseline length.
Clear evidence of statistical dependency is detected. Reasons for these dependencies are discussed.