An evaluation of some systematic error sources affecting terrestrial gravity anomalies

Terrestrial free-air gravity anomalies form a most essential data source in the framework of gravity field determination. Gravity anomalies depend on the datums of the gravity, vertical, and horizontal networks as well as on the definition of a normal gravity field; thus gravity anomaly data are affected in a systematic way by inconsistencies of the local datums with respect to a global datum, by the use of a simplified free-air reduction procedure and of different kinds of height system. These systematic errors in free-air gravity anomaly data cause systematic effects in gravity field related quantities like e.g. absolute and relative geoidal heights or height anomalies calculated from gravity anomaly data. In detail it is shown that the effects of horizontal datum inconsistencies have been underestimated in the past. The corresponding systematic errors in gravity anomalies are maximum in mid-latitudes and can be as large as the errors induced by gravity and vertical datum and height system inconsistencies. As an example the situation in Australia is evaluated in more detail: The deviations between the national Australian horizontal datum and a global datum produce a systematic error in the free-air gravity anomalies of about −0.10 mgal which value is nearly constant over the continent     

Gravimetric and astro-geodetic geoids and mean free-air anomalies in India

A preliminary gravimetric geoid with respect to the International Spheroid and the latest astro-geodetic geoid computed on the Everest and International Spheroids are given in the form of undulation maps over the Indian Sub—continent. 1⁰x1⁰ mean free-air anomalies (modified) on the Geodetic Reference System, 1967 (GRS-67) are also given for the whole country in the form of a chart. For the purpose of computing the gravimetric geoid, 5⁰x5⁰ mean free-air anomalies were used outside the area bounded by latitudes 0⁰ to 40⁰ N and longitudes 60⁰ to 100⁰ E and 1⁰x1⁰ mean free-air anomalies within these limits. The anomalies partly computed by Survey of India and mostly collected from other sources (such as B.G.I.) were utilised for this purpose. The astro-geodetic geoid is based on the astronomical data observed in India up to 1978.     

The geopotential to (14, 14) from a combination of satellite and gravimetric data

A set of2261 5°×5° mean anomalies were used alone and with satellite determined harmonic coefficients of the Smithsonian' Institution to determine the geopotential expansion to various degrees. The basic adjustment was carried out by comparing a terrestrial anomaly to an anomaly determined from an assumed set of coefficients. The (14, 14) solution was found to agree within ±3 m of a detailed geoid in the United States computed using1°×1° anomalies for an inner area and satellite determined anomalies in an outer area. Additional comparisons were made to the input anomaly field to consider the accuracy of various harmonic coefficient solutions. A by-product of this investigation was a new γ_E=978.0463 gals in the Potsdam system or978.0326 gals in an absolute system if −13.7 mgals is taken as the Potsdam correction. Combining this value of γ_E withf=1/298.25, KM=3.9860122·10 ²² cm ³ /sec ² , the consistent equatorial radius was found to be6378143 m.     

Determination of Potential coefficients to degree 52 from 5° Mean Gravity Anomalies

A set of 38406 1°×1° mean free air anomalies were used to derive a set of 1507 5° equal area anomalies that were supplemented by 147 predicted anomalies to form a global coverage of 1654 anomalies. These anomalies were used to derive potential coefficients to degree 52 using the summation formulae. In these computations, a smoothing operator was introduced and found to significantly effect the results at higher degrees. In addition, the effects of the atmosphere, spherical approximation and terrain were studied. It was found that the atmospheric effects and spherical approximation effects were about 0.3% of the actual coefficients. The terrain correction effects amounted to 10 to 25% of the low degree coefficients depending on a specific terrain correction model chosen; however, the correction terms found from the models did not yield solutions that agreed better with current satellite derived potential coefficient determinations. Anomalies were computed from the derived potential coefficients for comparison to the original anomalies. These comparisons showed that the agreement between the two anomalies became significantly better as the degree of expansion increased to the maximum considered. These comparisons shed some doubt on the rule of thumb that a block of size θ^° can be represented by a spherical harmonic expansion to 180^°/θ^°.     

Improvements on existing geopotential coefficients models for precise determination of 1° global geoid and mean gravity anomalies

The concept of an idealised earth having 1° averaged heights over its land surface is introduced as a means to improve upon the existing geopotential coefficient solutions without the use of additional observed data, in order to provide more precise knowledge of the earth’s gravity field in the form of 1° global geoid and 1° mean free-air gravity anomalies especially over the mountainous regions with the visible topography condensed into the actual geoid, first by referring them to the idealised earth and then by reducing the same to the actual earth on applying appropriate corrections for the differences between the two earths.     

Improving Orthometric Heights Using Precise GPS Measurements

The vertical component obtained from the Global Positioning System (GPS) observations is from the ellipsoid (a mathematical surface), and therefore needs to be converted to the orthometric height, which is from the geoid (represented by the mean sea level). The common practice is to use existing bench marks (around the four corners of a project area and interpolate for the rest of the area), but in many areas bench marks may not be available, in which case an existing geoid undulation is used. Present available global geoid undulation values are not generally as detailed as needed, and in many areas they are not known better than ±1 to ±5 m, because of many limitations. This article explains the difficulties encountered in obtaining precise geoid undulation with some example computations, and proposes a technique of applying corrections to the best available global geoid undulations using detailed free-air gravity anomalies (within a 2° × 2° area) to get relative centimeter accuracy. Several test computations have been performed to decide the optimal block sizes and the effective spherical distances to compute the regional and the local effects of gravity anomalies on geoid undulations by using the Stokes integral. In one test computation a 2° × 2° area was subdivided into smaller surface elements. A difference of 37.34 ± 1.6 cm in geoid undulation was obtained over the same 2° × 2° area when 1° × 1° block sizes were replaced by a combination of 5' × 5' and 1' × 1' subdivision integration elements (block sizes).     

Investigations on the accuracy concerning the derivation of absolute gravimetric height anomalies and deflections of the vertical based on the theory of Molodensky

The investigations refer to the compartment method by using mean terrestrial free air anomalies only. Three main error influences of remote areas (distance from the fixed point >9°) on height anomalies and deflections of the vertical are being regarded:

1. The prediction errors of mean terrestrial free air anomalies have the greatest influence and amount to about ±0″.2 in each component for deflections of the vertical and to ±3 m for height anomalies;
2. The error of the compartment method, which originates from converting the integral formulas of Stokes and Vening-Meinesz into summation formulas, can be neglected if the anomalies for points and gravity profiles are compiled to 5°×5° mean values.
3. The influences of the mean gravimetric correction terms of Arnold—estimated for important mountains of the Earth by means of an approximate formula—on height anomalies may amount to 1–2 m and on deflections of the vertical to 0″0.5–0″.1, and, therefore, they have to be taken into account for exact calculations.

The computations of errors are carried out using a global covariance function of point free air anomalies.     

DEM-induced errors in developing a quasi-geoid model for Africa

Errors in digital elevation models (DEMs) will introduce errors in geoid and quasi-geoid models, via their use in interpolating free-air gravity anomalies and (in the case of the quasi-geoid) their use in computing the Molodensky G ₁ term. The effects of these errors and those of datum shifts are assessed using three independent DEMs for a test region in South Africa. It is shown that these effects are significant and that it is important to choose the best-possible DEM for use in geoid and quasi-geoid modelling. Acknowledgments.The land gravity data used for this research were provided by the South African Council for Geoscience. Marine gravity anomalies were provided by the Danish National Survey and Cadastre (Kort & Matrikelstyrelsen). The GLOBE DEM was provided by the US National Geophysical Data Centre, and the CDSM DEM was provided by the South African Chief Directorate for Surveying and Mapping. The constructive comments of the reviewers are gratefully acknowledged.     

Determination of geoidal heights and deflections of the vertical for the hellenic area using heterogeneous data

Mean gravity anomalies, deflections of the vertical, and a geopotential model complete to degree and order180 are combined in order to determine geoidal heights in the area bounded by [34°≦ϕ≤42°, 18°≦λ≦28°]. Moreover, employing point gravity anomalies simultaneously with the above data, an attempt is made to predict deflections of the vertical in the same area. The method used in the computations is least squares collocation. Using empirical covariance functions for the data, the suitable errors for the different sources of observations, and the optimum cap radius around each point of evaluation, an accuracy better than±0.60m for geoidal heights and±1″.5 for deflections of the vertical is obtained taking into account existing systematic effects. This accuracy refers to the comparison between observed and predicted values.     

Investigating the vegetation and agricultural responses to El Nino/Southern Oscillation using AVHRR data

This study explores the possible linkages of El Nino/Southern Oscillation (ENSO) with vegetation and rainfall patterns, vegetation activity and food grain yields, in arid and semi-arid regions of western India. A sequence of 20-year (1981–2000) monthly maximum Normalized Difference Vegetation Index (NDVI) data from the Advanced Very High Resolution Radiometer (AVHRR) and monthly rainfall from 160 stations were examined to study the seasonal patterns and their relation to ENSO activity. In addition, a direct (ENSO-crop yield) linkage and an intermediate (ENSO-NDVI) linkage of agricultural responses to ENSO were also investigated. The results indicate below-normal seasonal NDVI and rainfall associated with El Nino (warm) events, except during 1997, while positive anomalies occur during La Nina (cold) events. Sea surface temperature (SST) anomalies from NINO 3 region (5°N–5°S; 150°W–90°W), as an indicator of ENSO were significantly correlated with NDVI anomalies, rainfall anomalies and yield anomalies but the Southern Oscillation Index (SOI) was significantly related to NDVI anomalies only. NDVI anomaly patterns correspond to rainfall variability including that associated with ENSO activity. The observed strong intermediate linkage between yield anomalies and NDVI anomaly signal (r = 0.609) indicates that NDVI is an ideal index for understanding and analysing agricultural response to ENSO climate teleconnections.     

On the error of analytical continuation in physical geodesy

Analytical continuation of gravity anomalies and height anomalies is compared with Helmert's second condensation method. Assuming that the density of the terrain is constant and known the latter method can be regarded as correct. All solutions are limited to the second power of H/R, where H is the orthometric height of the terrain and R is mean sea-level radius. We conclude that the prediction of free-air anomalies and height anomalies by analytical continuation with Poisson's formula and Stokes's formula goes without error. Applying the same technique for geoid determination yields an error of the order of H², stemming from the failure of analytical continuation inside the masses of the Earth.     

Bathymetric inversion of South China Sea from satellite altimetry data

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Detailed gravimetric geoid for the North Atlantic

A detailed gravimetric geoid in the North Atlantic Ocean, named DGGNA-77, has been computed, based on a satellite and gravimetry derived earth potential model (consisting in spherical harmonic coefficients up to degree and order 30) and mean free air surface gravity anomalies (35180 1°×1° mean values and 245000 4′×4′ mean values). The long wavelength undulations were computed from the spherical harmonics of the reference potential model and the details were obtained by integrating the residual gravity anomalies through the Stokes formula: from 0 to 5° with the 4′×4′ data, and from 5° to 20° with the 1°×1° data. For computer time reasons the final grid was computed with half a degree spacing only. This grid extends from the Gulf of Mexico to the European and African coasts. Comparisons have been made with Geos 3 altimetry derived geoid heights and with the 5′×5′ gravimetric geoid derived byMarsh andChang [8] in the northwestern part of the Atlantic Ocean, which show a good agreement in most places apart from some tilts which porbably come from the satellite orbit recovery.     

Accounting for topography in the calculation of quasigeoidal heights and plumb-line deflections from gravity anomalies

A calculation of quasigeoidal heights and plumb-line deflections according to Molodensky formulae was carried out under elimination of the effect of topography from gravity anomalies. After the masses of topography had been removed a smoothed-out surface passing through astronomical and gravity stations was considered as representing the physical surface of the Earth. Thus it has been practically rendered possible to use the first-approximation formulae of Molodensky, and, in many cases, also the “zero-approximation” formulae analogous to the formulae of Stokes and Vening-Meinesz. The effect of the restored masses of topography was then added to the quantities found; the said effect was expressed as the effect of topography condensed on the normal equipotential surface passing through the point under investigation, plus a correction for condensation. Following some transformations, the resulting formulae (13) and (18) were obtained which formulae differ in their “zero-approximation” (15) and (20) from traditional formulas in that they contain terrait reductions added to free-air anomalies. Moreover, in the calculation of plumb-line deflections directly in mountain regions a correction for differing effects of topography before and after its condensation is to be introduced. A tentative expansion of terrain reduction in terms of spherical harmonics up to the third order is given; it can be seen therefrom that the Stokes series in its usual form is subject to a mean arror about 15–20%. It is also shown that the expansion of free-air anomalies in terms of spherical functions contains a first-order harmonic with a mean values about ±0.3 mgl. The said harmonic practically disappears in the expansion of the sum of free-air anomalies and terrain reductions.     

MAPPING OUT OF CROWN LANDS IN CEYLON

Abstract

The Island of Ceylon has an extent of 25,332 square miles, and a population of nearly seven millions; the range of latitude is from 5° 55′ to 9° 50′ North and of longitude 79° 42′ to 81° 53′ East.     

NOTES ON THE BASE LINES OF THE CEYLON TRIANGULATION

Abstract

The triangulation of Ceylon depends for its scale upon two bases, each about 5½ miles long, situated at Negombo on the West Coast (latitude 7° 10′) and at Batticaloa on the East Coast (latitude 7° 40′). Both bases are in low, flat country; brick towers up to 70 feet high had to be built over the terminals to enable observations to be taken to surrounding points. These lines have recently been re-measured.     

Determination of Indian absolute geodetic datum by least-square solution technique

The Everest spheroid, 1830, in general use in the Survey of India, was finally oriented in an arbitrary manner at the Indian geodetic datum in 1840; while the international spheroid, 1924, in use for scientific purposes; was locally fitted to the Indian geoid in 1927. An attempt is here made to obtain the initial values for the Indian geodetic datum in absolute terms on GRS 67 by least-square solution technique, making use of the available astro-geodetic data in India, and the corresponding generalised gravimetric values at the considered astro-geodetic points, as derived from the mean gravity anomalies over1°×1° squares of latitude and longitude in and around the Indian sub-continent, and over5° equal area blocks covering the rest of the earth’s surface. The values obtained independently by gravimetric method, were also considered before actual finalization of the results of the present determination.     

Regional geopotential model improvement for the Iranian geoid determination

Spherical harmonic expansions of the geopotential are frequently used for modelling the earth’s gravity field. Degree and order of recently available models go up to 360, corresponding to a resolution of about50 km. Thus, the high degree potential coefficients can be verified nowadays even by locally distributed sets of terrestrial gravity anomalies. These verifications are important when combining the short wavelength model impact, e.g. for regional geoid determinations by means of collocation solutions. A method based on integral formulae is presented, enabling the improvement of geopotential models with respect to non-global distributed gravity anomalies. To illustrate the foregoing, geoid computations are carried out for the area of Iran, introducing theGPM2 geopotential model in combination with available regional gravity data. The accuracy of the geoid determination is estimated from a comparison with Doppler and levelling data to ±1.4m.     

Generation and study of satellite gravity over Gujarat,India and their possible correlation with earthquake occurrences

High-resolution satellite gravity data have been generated and utilized to infer subsurface geological structures in the area of devastating earthquake that struck the Bhuj region in Gujarat on 26 January 2001. Latitudinal gravity profiles have been generated in the Bhuj, Anjar and IBF regions across the epicentres (23.5° N, 69.8° E/M_w 7.0 in 2001; 23.2° N, 70° E/M_w 7.0 in 1956; 24.2° N, 69.2° E/M_w 7.8 in 1819). Substantial differences in gravity anomaly patterns as high as 37 mGal could be observed existing near the epicentre regions. These gravitational differences might have caused due to the plate tectonic processes and due to the changes in densities of different lithospheric zones/sedimentary layers. Temporal variations of the satellite-derived gravity and their probable relations with already occurred major earthquakes in this region have been studied. Hence we conclude that drastic changes in gravity anomalies can be considered as a precursor for occurrences of substantially large earthquakes.