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Topography is a problem in geoid determination by the Stokes formula, a high degree Earth Gravitational Model (EGM), or for a combination thereof. Herein, we consider this problem in analytical/harmonic downward continuation of the external potential at point P to a geocentric spherical sea level approximation in geoid determination as well as to a sphere through the footpoint at the topography of the normal through P. Decomposing the topographic bias into a Bouguer shell component and a terrain component, we derive these components. It is shown that there is no terrain bias outside a spherical dome of base radius equal to the height H P of P above the sphere, and the height of the dome is about 0.4 × H P. In the case of dealing with an EGM, utilizing Molodensky truncation coefficients is one way to cope with the bias.

The topographic bias (TB) is defined as the difference between the disturbing potential on the geoid and the harmonic downward continued one. Sjöberg (J Geoid 81:345–350, 2007, 10.1007/s00190-006-0112-2SearchinGoogleScholar; Geophys J Int 176:14–18, 2009) showed that the TB is that of a Bouguer plate, and the terrain contribution is zero. Out of curiosity, we inspected the TB of a homogeneous right circular cylinder and a cone—the terrain correction is zero for the cylinder, and it reaches the maximum for the cone at the top center points. As expected, the TB of the cylinder is the same as a Bouguer plate of the same height, and it is independent of the base size. The TB of the cone is quite different, it is not equal to that of a Bouguer plate with the same height of the cone; rather, it is a function of the shape of the cone or the ratio of its base radius to the height. The TB becomes smaller with a steeper slope of the slant height of the cone, and it approaches the TB of the Bouguer plate with a gentle slope of the slant height. For a cone with the same height and base radius, the relative error of ignoring the terrain reaches 80%. The TB of the cone disproves that the TB of the terrain is zero. [ABSTRACT FROM AUTHOR]

We consider the topographic bias in gravimetric geoid determination when analytically downward continuing the disturbing potential from the Earth's surface to sea level. The total bias is subdivided into those of the Bouguer shell or plate and the terrain. In this process, the potential of the Bouguer shell always has a downward continuation bias in the process, which increases with the square of the topographic height and typically exceeds 1–2 cm for elevations higher than 1 km. The main conclusion is that the terrain does not provide a potential bias except possibly for masses located inside a dome of height of about 0.4 times the height of the computation point, and base radius equal to the height of the computation point. This result implies that the potential of all terrain masses of arbitrary density located exterior to the Bouguer shell as well as those outside the dome are unbiasedly downward continued to sea level. [ABSTRACT FROM AUTHOR]

The topographic potential bias at geoid level is the error of the analytically continued geopotential from or above the Earth’s surface to the geoid. We show that the topographic potential can be expressed as the sum of two Bouguer shell components, where the density distribution of one is spherical symmetric and the other is harmonic at any point along the normal to a sphere through the computation point. As a harmonic potential does not affect the bias, the resulting topographic bias is that of the first component, i.e. the spherical symmetric Bouguer shell. This implies that the so-called terrain potential is not likely to contribute significantly to the bias. We present three examples of the geoid bias for different topographic density distributions.

It is well known that the success in precise determinations of the gravimetric geoid height (N) and the orthometric height (H) rely on the knowledge of the topographic mass distribution. We show that the residual topographic bias due to an imprecise information on the topographic density is practically the same for N and H, but with opposite signs. This result is demonstrated both for the Helmert orthometric height and for a more precise orthometric height derived by analytical continuation of the external geopotential to the geoid. This result leads to the conclusion that precise gravimetric geoid heights cannot be validated by GNSS-levelling geoid heights in mountainous regions for the errors caused by the incorrect modelling of the topographic mass distribution, because this uncertainty is hidden in the difference between the two geoid estimators.

All operant behaviors have multiple characteristics in addition to those criterial for reinforcement, and variation occurs across all. All such characteristics can also reflect topographic bias due to historic and physiological factors. The revealed operant is constructed so that topographic aspects and variation are measurable. In two experiments humans performed a revealed operant response of 14 or more keystrokes. The first and last were mandated, while the middle 12 or more were allowed to vary. There were significant differences in variability among participants, as well as systematic effects of the experimental designs. Despite not being reinforced, variability among complete sequences was high. Test conditions in Experiment 2 resulted in a much larger increase in variability than did suspension of reinforcement in Experiment 1. There was systematic topographic bias both for and against letter keys in the center of the keyboard. There were also correlations between measures of variability and bias. [ABSTRACT FROM AUTHOR]

In 1962 A. Bjerhammar introduced the method of analytical continuation in physical geodesy, implying that surface gravity anomalies are downward continued into the topographic masses down to an internal sphere (the Bjerhammar sphere). The method also includes analytical upward continuation of the potential to the surface of the Earth to obtain the quasigeoid. One can show that also the common remove-compute-restore technique for geoid determination includes an analytical continuation as long as the complete density distribution of the topography is not known. The analytical continuation implies that the downward continued gravity anomaly and/or potential are/is in error by the so-called topographic bias, which was postulated by a simple formula of L E Sjöberg in 2007. Here we will numerically test the postulated formula by comparing it with the bias obtained by analytical downward continuation of the external potential of a homogeneous ellipsoid to an inner sphere. The result shows that the postulated formula holds: At the equator of the ellipsoid, where the external potential is downward continued 21 km, the computed and postulated topographic biases agree to less than a millimetre (when the potential is scaled to the unit of metre).

Historic aerial images are invaluable sources of aid to archaeological research. Often collected with large-format photogrammetric quality cameras, these images are potential archives of multidimensional data that can be used to recover information about historic landscapes that have been lost to modern development. However, a lack of camera information for many historic images coupled with physical degradation of their media has often made it difficult to compute geometrically rigorous 3D content from such imagery. While advances in photogrammetry and computer vision over the last two decades have made possible the extraction of accurate and detailed 3D topographical data from high-quality digital images emanating from uncalibrated or unknown cameras, the target source material for these algorithms is normally digital content and thus not negatively affected by the passage of time. In this paper, we present refinements to a computer vision-based workflow for the extraction of 3D data from historic aerial imagery, using readily available software, specific image preprocessing techniques and in-field measurement observations to mitigate some shortcomings of archival imagery and improve extraction of historical digital elevation models (hDEMs) for use in landscape archaeological research. We apply the developed method to a series of historic image sets and modern topographic data covering a period of over 70 years in western Sicily (Italy) and evaluate the outcome. The resulting series of hDEMs form a temporal data stack which is compared with modern high-resolution terrain data using a geomorphic change detection approach, providing a quantification of landscape change through time in extent and depth, and the impact of this change on archaeological resources. [ABSTRACT FROM AUTHOR]

Geoid determination below the topographic surface in continental areas using analytical continuation of gravity anomaly and/or an external type of solid spherical harmonics determined by an Earth GravitationalModel (EGM) inevitably leads to a topographic bias, as the true disturbing potential at the geoid is not harmonic in contrast to its estimates. We show that this bias differs for the geoid heights represented by Stokes’ formula, an EGMand for the modified Stokes formula. The differences are due to the fact that the EGM suffers from truncation and divergence errors in addition to the topographic bias in Stokes’ original formula.

This article is a personal review of the development of physical geodesy during 1984-2014. The period is characterized by an intensive advance in both data and theory to meet the growing technical demands in GPS/GNSS applications and scientific needs in geoscience. As a result,many parts of theworld are nowmapped with a 1cmdetailed geoid model, and the global long- to mediumwavelengths of the gravity field and geoid are homogeneously determined to 1 mGal and 1 cm by satellite-only dedicated satellite gravity missions. The future can expect to see even higher demands for accuracy and reliability to satisfy the specifications for a pure geoid model based vertical datum.

Since the continuation of an external gravity field inside topographic masses by a harmonic function results in topographic bias, geoid computation by means of global gravity models (GGMs) in terms of external-type series of spherical harmonics, at locations where the GGMs are evaluated inside the topographic masses, will be biased. Consequently, if the reference ellipsoid is defined based on the geoid, it will also be biased. In this paper, the effects of topographic bias on the geoid and reference ellipsoid of Venus, Mars, and the Moon are studied. Moreover, a thorough error analysis in the geoid and reference ellipsoid computation is presented, and it is shown that the estimated standard deviation (STD) of the geoid potential value, the geoidal heights, and the semimajor and semiminor axes of the reference ellipsoid are independent of the topographic bias. According to the results, the effects of topographic bias on the geoid potential value and the semimajor and semiminor axes of the reference ellipsoid in comparison with their estimated STDs are insignificant for Venus, Mars, and the Moon. Moreover, the effect of topographic bias on the geoidal heights of Venus as compared with the estimated STD of its geoidal heights is insignificant. However, the effects of topographic bias on the geoidal heights of Mars and the Moon can be significant, especially in high mountains such as the Tharsis volcanic region on Mars. [ABSTRACT FROM AUTHOR]

Abstract: Today, the recent global Earth''s gravity model, EGM08, is successfully utilised for different purposes in geosciences. Here, EGM08 is used to compute a geoid model for Fennoscandia and since it is restricted to degree and order 2160, the higher frequencies of the geoid, or the truncation bias, is recovered directly from terrestrial gravity anomalies using a simple formula. The total topographic and atmospheric effects are computed and added to the derived geoid as well. A very simple EGM08-based non-integral geoid estimator is developed and applied for computing the geoid of Fennoscandia. The outcome of the estimator is compared with the Global Positioning System (GPS)/levelling data of Sweden, Denmark, Finland and Norway. Numerical results show the successful performance of the presented estimator as the geoid become closer to GPS/levelling data than the one computed solely with EGM08. This study will show that considering the truncation bias of EGM08 will reduce the root mean square error (RMSE) of the differences between the geoid and GPS/levelling data by about 1.3cm and the additive topographic and atmospheric corrections by 1cm further. It is shown that the correlations among the data have no significant influence on the estimated geoid. [Copyright &y& Elsevier]