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We define dislocation Love numbers [h•,• ii q , ß • l,•,•, k,•,• _,•,• and Green's functions to describe the elastic deformation of the Earth caused by a point dislocation and study the coseismic displacements caused in a radially heterogeneous spherical Earth model. We derive spherical harmonic expressions for the shear and tensile dislocations, which can be expressed by four independent solutions: a vertical strike slip, a vertical dip slip, a tensile opening in a horizontal plane, and a tensile opening in a vertical plane. We carry out calculations with a radially heterogeneous Earth model (1066A). The results indicate that the dominating deformations appear in the near field and attenuate rapidly as the epicentral distance increases. The shallower the point source, the larger the displacements. Both the Earth's curvature and vertical layering have considerable effects on the deformation fields. Especially the vertical layering can cause a 10% difference at the epicentral distance of 0.1 ø. As an illustration, we calculate the theoretical displacements caused by the 1964 Alaska earthquake (rnw = 9.2) and compare the results with the observed vertical displacements at 10 stations. The results of the near field show that the vertical displacement can reach some meters. The far-field displacements are also significant. For example, the horizontal displacements (•e) can be as large as i cm at the epicentral distance of 30 ø, 0.5 cm a*. about 40 ø, magnitudes detectable by modern instrument, such as satellite laser ranging very long baseline interferometry (VLBI) or Global Positioning System (GPS). Globally, the displacement (•) caused by the earthquake is larger than 0.25 ram. the formulations for a more realistic Earth model have also been advanced through numerous studies [Ben-Menahem and Singh, 1968; Ben-Menahern and Israel, 1970; Srn!fiie and Mansinha, 1971]. These studies re-Paper number 95JB03536. 0148-0227 / 96 / 95 JB-03536505.00 vealed that the effect of the Earth's curvature is neõliõible for shallow events at an epicentral distance less than 200 , while vertical layering may have considerable effects on the deformation fields. 5aito [1967] presented a theory to calculate amplitudes of free oscillations caused by a point source in a spherically symmetric Earth model. He expressed his results in terms of normal mode solutions and source functions. gan [1987a, b] õave the source functions of elementary sources in a õeneral form for both static and dynamic displacements. Okubo [1991, 1992] studied the problem of potential and acceleration chanões caused by point dislocations and by faultinõ on a finite plane in a homoõeneous hMf-space. He derived all expressions in closed All of the above studies, except Saito's [1967], assumed a homogeneous half-space or a homoõeneous nonõravitatinõ sphere. For a more realistic Earth model, Rutrile [1982] studied viscoelastic õravitational defor-8561 8562 SUN ET AL.: COSEISMIC DISPLACEMENT mation by a rectangular thrust fault in a layered Earth. Pollitz [1992] solved the p...

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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...

The main problem of the rigorous definition of the orthometric height is the evaluation of the mean value of the Earth's gravity acceleration along the plumbline within the topography. To find the exact relation between rigorous orthometric and [Molodensky] normal heights, the mean gravity is decomposed into: the mean normal gravity, the mean values of gravity generated by topographical and atmospheric masses, and the mean gravity disturbance generated by the masses contained within geoid. The mean normal gravity is evaluated according to Somigliana-Pizzeti's theory of the normal gravity field generated by the ellipsoid of revolution. Using Bruns's formula, the mean values of gravity along the plumbline generated by topographical and atmospheric masses can be computed as the mean linear potential gradient between the Earth's surface and geoid. Since the gravity disturbance generated by masses inside the geoid (multiplied by the geocentric radius) is harmonic above the geoid (after removal of the topographic and atmospheric masses), its mean value along the plumbline between the Earth's surface and the geoid is obtained by solving the inverse Dirichlet boundary value problem. Numerical results for a test area in the Canadian Rocky Mountains show that the difference between the rigorously defined orthometric height and the Molodensky normal height reaches ~0.5 m.

S U M M A R YThis paper presents a number of new concepts concerning the gravity anomaly. First, it identifies a distinct difference between a surface (2-D) gravity anomaly (the difference between actual gravity on one surface and normal gravity on another surface) and a solid (3-D) gravity anomaly defined in the fundamental gravimetric equation. Second, it introduces the 'no topography' gravity anomaly (which turns out to be the complete spherical Bouguer anomaly) as a means to generate a quantity that is smooth, thus suitable for gridding, and harmonic, thus suitable for downward continuation. It is understood that the possibility of downward continuing a smooth gravity anomaly would simplify the task of computing an accurate geoid. It is also shown that the planar Bouguer anomaly is not harmonic, and thus cannot be downward continued.

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