Geodetic leveling and its applications

Abstract

The inherent precision of spirit leveling has preserved its utility as a geodetic measurement system for over a century. While various instrumental and procedural modifications designed to enhance this precision have been introduced over the years, the basic measurement system has remained virtually unchanged since the mid‐nineteenth century. Possible systematic error has dictated the majority of the procedural and instrumental requirements associated with geodetic leveling; the physical source(s) of several of these errors remain poorly understood. Statistically independent random errors, which accumulate according to the square root of the survey distance, are generally controlled through redundancy and procedural randomization; they range from 0.5 mm L1/2 for the highest‐order modern leveling to about 6 mm L1/2 for the lowest‐order nineteenth‐century geodetic surveys, where L is the survey distance in kilometers. Height differences are conceptually distinct from observed or measured elevation differences in the sense that the former are uniquely defined, whereas the latter are path dependent, a distinction that arises from the nonparallelism of the equipotential surfaces of the earth’s gravity field. The number of possible height systems is virtually limitless. They include the systems of geopotential numbers and dynamic heights; although neither of these systems is geometrically informative, each provides perfectly valid height characterizations that may be especially useful in the solution of certain physical problems. The most generally used system of heights is the orthometric height system; the resulting heights are true geometric heights above the geoid. Normal height systems are referred to the quasi‐geoid rather than the geoid. Each of the various height systems meets the requirement of uniqueness, and none can be viewed as being conceptually superior. Conversion of the observed elevation differences obtained from leveling into uniquely defined height differences requires the application of a gravity‐dependent correction. Because gravity coverage in North America was generally sparse until recently, an approximation for this correction, which provides for the effects of the poleward covergence of the equipotential surfaces, has usually been used on this continent. Heights have been traditionally referred to mean sea level as a datum, a usage that implies coincidence between mean sea level and the geoid (or quasi‐geoid). Because the determination of mean sea level is dependent on the length of the observation period, because its definition makes no allowance for vertical crustal displacements or changes in eustatic sea level, and because its definition disregards the demonstrable existence of sea surface topography, local mean sea level generally departs from the geoid. This introduces errors in computed heights that probably equal or exceed those due to leveling. Repeated levelings continue to provide the best basis for determining terrestrial vertical displacements. These displacements are necessarily measured with respect to the time‐dependent (instantaneous) geoid. Analysis of changes in the orthometric correction and changes in the geoid as a function of changes in gravity indicates that height determinations are almost insensitive to temporal variations in gravity. The simplest and probably the most accurate procedure for determining vertical displacements is based on direct comparisons between observed elevations derived from repeated levelings along the same line. The estimated errors associated with these displacements are a function of the precision of the utilized levelings and have been shown to be much smaller than those associated with heights. A smoothed representation of the vertical displacement field can be obtained by the fitting of a mathematical surface to the results of segmented relevelings. Surface‐fitting techniques usually tend to subdue short‐wavelength features, but they are especially useful in depicting artificially induced subsidence and broadly defined tectonic uplift or collapse.