Gravity field and levelled heights in Canada
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Abstract
Modern trends in geodesy demand an increased accuracy of relative heights and height changes. Precise spirit levelling is known to be the most accurate method available to meet such requirements. It is also known that unique height determination can be made only by taking into account the convergence and irregularities of the equipotential surfaces of the earth’s actual gravity field. In the context of levelling, this is accomplished by supplementing the spirit levelling with actual gravity values observed along levelling routes.
In Canada, and the U.S.A, because of the lack of actual gravity values (during the period of building up and extending the levelling networks) the normal gravity was used instead to define the heights. The normal gravity values were computed along levelling routes from a simplified mathematical model of the earth.
Two systems of heights – orthometric and dynamic – are used in Canada, both taking into account only the broadest features of the gravity field expressed via the computed normal gravity. This implies the neglect of the effect of local irregularities of the actual gravity field on the defined heights, which results in systematic distortions of the computed heights.
The study contained herein focuses on the investigation of the influence of actual gravity variations (anomalies) on heights currently used in Canada. These influences are referred to here as “GRAVITY CORRECTIONS”, GC’s. The GC’s are to be added to the existing height differences (based on normal gravity) to obtain the corresponding rigorous height difference based on actual gravity.
The GC’s for three systems of heights – Dynamic, Helmert and Vignal – are modelled in terms of practically obtainable quantities: free-air gravity anomalies, observed heights and latitudes of levelling bench marks along the levelling routes. Although the developed formulae for the GC’s can be readily used for the evaluation of these corrections, tables are provided to facilitate field estimation of the GC’s.
Results based on real data indicate that the GC’s can be evaluated with adequate reliability, whether we use observed or predicted gravity anomalies, for all three systems of heights under investigation. This reliability is characterized by the small standard deviations associated with the GC’s compared to the magnitude of the corrections themselves.
The behaviour of GC’s along real levelling lines and loops are investigated and compared to the corresponding standard errors, ׁ[subscript Δh], of precise levelling as specified in the Canadian specification for vertical control [Surveys and Mapping Branch, 1961: Boal, 1971b; Surveys and Mapping Branch, 1973]. The results show that the influence of the GC’s on the derived heights of most of the bench marks along the tested lines and loops is significant.
A computational approach, based on the least-squares surface fitting techniques, is proposed for the prediction of GC’s. The aim of this approach is to treat the problem of GC’s in two-dimensions, so as to enable to one to determine the geographical areas in Canada where actual gravity influence on heights is significant and should be taken into account. Gravity corrections within each 1 ° X 1 ° block, that vary with direction, have been predicted for the entire country, using real gravity data. Supplied by the Earth Physics Branch, Ottawa. Obtained results – compared to a prespecified significance criterion of 0.14 mm/km (10% of the standard deviation of a height difference in the Canadian Precise Level Net, CPLN) reveal the significance of the GC’s, in practically all the Canadian areas, at least in the direction of its maximum value. In many cases, the GC even exceeds the standard error of precise levelling, especially in Helmert system.
Based on the analysis and results of this study, it appears necessary to begin basing the heights in Canada on actual gravity in order to maintain the standard of accuracy required for the CPLN. Such procedure was recommended by the International Association of Geodesy as early as 1950 [IAG, 1950]. This has become feasible since the coverage of the Canadian territory with gravity observations has become sufficiently dense. The information and findings contained in this thesis should thus contribute to the forthcoming new adjustment and analysis of the CPLN, as planned by the geodetic survey of Canada for early 1980’s.