Engineering topics

Rapp, R. H.

Publications and source records attributed to Rapp, R. H..

At least 19 records

NASA NTRS1998

The Development of the Joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) Geopotential Model EGM96

The NASA Goddard Space Flight Center (GSFC), the National Imagery and Mapping Agency (NIMA), and The Ohio State University (OSU) have collaborated to develop an improved spherical harmonic model of the Earth's gravitational potential to degree 360. The new model, Earth Gravitational Model 1996 (EGM96), incorporates improved surface gravity data, altimeter-derived gravity anomalies from ERS-1 and from the GEOSAT Geodetic Mission (GM), extensive satellite tracking data-including new data from Satellite Laser Ranging (SLR), the Global Postioning System (GPS), NASA's Tracking and Data Relay Satellite System (TDRSS), the French DORIS system, and the US Navy TRANET Doppler tracking system-as well as direct altimeter ranges from TOPEX/POSEIDON (T/P), ERS-1, and GEOSAT. The final solution blends a low-degree combination model to degree 70, a block-diagonal solution from degree 71 to 359, and a quadrature solution at degree 360. The model was used to compute geoid undulations accurate to better than one meter (with the exception of areas void of dense and accurate surface gravity data) and realize WGS84 as a true three-dimensional reference system. Additional results from the EGM96 solution include models of the dynamic ocean topography to degree 20 from T/P and ERS-1 together, and GEOSAT separately, and improved orbit determination for Earth-orbiting satellites.

Lemoine, F. G.↗

NASA NTRS1990

The development of an isostatic gravitational model to degree 360 and its use in global gravity modelling

Consideration is given to the possibility of combining low-degree satellite-derived geopotential models with the harmonic coefficients of the topographic-isostatic potential implied by the Airy/Heiskanen isostatic hypothesis. The compilation of a topographic database providing information pertaining to terrain type classification is discussed. The formulation for the determination of harmonic coefficients of the topographic-isostatic potential is extended beyond to cases discussed by Lachapelle (1976) to include various terrain types. This formulation and the series expansion approach of Rummel et al. (1988) are implemented for potential coefficient determinations complete to degree and order 360. The topographic-isostatic coefficients are used with satellite-derived geopotential models to estimate mean gravity anomalies. The results are compared with observations to evaluate the quality of different estimation procedures.

Pavlis, N. K.↗

NASA NTRS1989

The decay of the spectrum of the gravitational potential and the topography for the earth

The spectrum of the earth's gravitational potential and topography, as represented by spherical harmonic expansions to degree 180, have been computed. Modeling the decay in the form of (A x l) exp-Beta, values of A and Beta for several degree (l) ranges were computed. For degree range 5-180, Beta was 2.54 for the potential and 2.16 for equivalent rock topography. The potential decay was somewhat slower than that implied by Kaula's rule. However, at high degree ranges, the Beta values were larger agreeing better with recent determinations from terrestrial gravity data and geoid undulations implied by satellite altimetric data. The values imply that the potential decays faster at higher l values.

Rapp, R. H.↗

NASA NTRS1987

Terrestrial gravity data and comparisons with satellite data

Figures that demonstrate the state of terrestrial gravity coverage, and comparisons between satellite derived gravity field and terrestrial gravity data are presented. It is shown that only a few areas of the world have information accurate enough for geodesy and geophysics. A gravity field mapping space mission is recommended.

Rapp, R. H.↗

NASA NTRS1984

The precise computation of geoid undulation differences with comparison to results obtained from the global positioning system

Ellipsoidal height differences have been determined for 13 station pairs in the central Ohio region using measurements made with the Global Positioning System. This information was used to compute geoid undulation differences based on known orthometric heights. These differences were compared to gravimetrically-computed undulations (using a Stokes integration procedure, and least squares collocation having an internal r.m.s. agreement of plus or minus 1 cm in undulation differences). The two sets of undulation differences have an r.m.s. discrepancy of plus or minus 5 cm while the average station separation is of the order of 14 km. This good agreement suggests that gravimetric data can be used to compute accurate geoid undulation differences that can be used to convert ellipsoidal height differences obtained from GPS to orthometric height differences.

Engelis, T.↗

NASA NTRS1984

A comparison of satellite Doppler and gravimetric geoid undulations considering terrain-corrected gravity data

Geoid undulations derived from Doppler satellite positioning are compared with corresponding values derived from terrestrial and satellite information using three different procedures. Doppler undulations are compared with uncorrected gravimetric undulations, and systematic differences of 1.6 m and 0.4 m are found for western and eastern U.S. stations, respectively. The systematic difference of 1.6 m was reduced to 0.1 m when terrain-related corrections were applied. Undulations computed from a spherical harmonic expansion to degree 180 yielded standard deviations only slightly poorer than the combined data. A more systematic difference in the western than in the eastern stations remained.

Rapp, R. H.↗

NASA NTRS1982

Degree variances of the earth's potential, topography and its isostatic compensation

A spherical harmonic expansion of the earth's gravitational potential and equivalent rock topography to degree and order 180 is described. The potential implied by the topography considered as uncompensated and with isostatic compensation has been computed. Good agreement with the observed potential field is found when the depth of compensation in the Airy theory is assumed to be 50 km. At the higher degrees the correlation coefficient between the potential expansion and the equivalent rock topography is about 0.5. The Lachapelle equations for the topographic isostatic potential were tested using 1 x 1 deg equivalent rock topography. The degree variances agree at the lower degrees but at degree 36 the Lachapelle results using 5 deg data underestimate the potential degree variances by about one-third.

Rapp, R. H.↗

NASA NTRS1981

Ellipsoidal corrections for geoid undulation computations using gravity anomalies in a cap

Ellipsoidal correction terms have been derived for geoid undulation computations when the Stokes equation using gravity anomalies in a cap is combined with potential coefficient information. The correction terms are long wavelength and depend on the cap size in which its gravity anomalies are given. Using the regular Stokes equation, the maximum correction for a cap size of 20 deg is -33 cm, which reduces to -27 cm when the Stokes function is modified by subtracting the value of the Stokes function at the cap radius. Ellipsoidal correction terms were also derived for the well-known Marsh/Chang geoids. When no gravity was used, the correction could reach 101 cm, while for a cap size of 20 deg the maximum correction was -45 cm. Global correction maps are given for a number of different cases. For work requiring accurate geoid computations these correction terms should be applied.

Rapp, R. H.↗

NASA NTRS1981

Ellipsoidal corrections for geoid undulation computations

The computation of accurate geoid undulations is usually done combining potential coefficient information and terrestrial gravity data in a cap surrounding the computation point. In doing this a spherical approximation is made that can cause the errors that were investigated. The equations dealing with ellipsoidal corrections developed by Lelgemann and by Moritz were used to develop a computational procedure considering the ellipsoid as a reference surface. Terms in the resulting expression for the geoid undulation are identified as ellipsoidal correction terms. These equations were developed for the case where the Stokes function is used, and for the case where the modified Stokes function is used. For a cap of 20 deg the correction can reach -33 cm.

Rapp, R. H.↗

NASA NTRS1980

Accuracy of the determination of mean anomalies and mean geoid undulations from a satellite gravity field mapping mission

Improved knowledge of the Earth's gravity field was obtained from new and improved satellite measurements such as satellite to satellite tracking and gradiometry. This improvement was examined by estimating the accuracy of the determination of mean anomalies and mean undulations in various size blocks based on an assumed mission. In this report the accuracy is considered through a commission error due to measurement noise propagation and a truncation error due to unobservable higher degree terms in the geopotential. To do this the spectrum of the measurement was related to the spectrum of the disturbing potential of the Earth's gravity field. Equations were derived for a low-low (radial or horizontal separation) mission and a gradiometer mission. For a low-low mission of six month's duration, at an altitude of 160 km, with a data noise of plus or minus 1 micrometers sec for a four second integration time, we would expect to determine 1 deg x 1 deg mean anomalies to an accuracy of plus or minus 2.3 mgals and 1 deg x 1 deg mean geoid undulations to plus or minus 4.3 cm. A very fast Fortran program is available to study various mission configurations and block sizes.

Jekeli, C.↗

NASA NTRS1980

A comparison of altimeter and gravimetric geoids in the Tonga Trench and Indian Ocean areas

Geoids computed from GEOS-3 altimeter data are compared with gravimetric geoids computed by various techniques for 30 x 30 deg areas in the Tonga Trench and the Indian Ocean. The gravimetric geoids were calculated using the standard Stokes integration with the Molodenskii truncation procedure, the modified Stokes integration suggested by Ostach (1970) and Meissl (1971) with modified Molodenskii truncation functions, and three sets of potential coefficients including one complete to degree 180. It is found that the modified Stokes procedure with a cap size of 10 deg provides better results when used with a combined altimeter terrestrial anomaly field data set. Excellent agreement at the plus or minus 1 m level is obtained between the altimeter and gravimetric geoid using the combined data set, with the modified Stokes procedure having a greater accuracy. Coefficients derived from the 180 x 180 solution are found to be of an accuracy comparable to that of the modified Stokes method, however to require six times less computational effort.

Rapp, R. H.↗

NASA NTRS1979

GEOS 3 data processing for the recovery of geoid undulations and gravity anomalies

The paper discusses the analysis of GEOS 3 altimeter data for the determination of geoid heights and point and mean gravity anomalies. Methods are presented for determining the mean anomalies and mean undulations from the GEOS 3 altimeter data available by the end of September 1977 without having a complete set of precise orbits. The editing of the data is extensive to remove questionable data, although no filtering of the data is carried out. An adjustment process is carried out to eliminate orbit error and altimeter bias. Representative point anomaly values are computed to investigate anomaly behavior across the Bonin Trench and over the Patton seamounts.

Rapp, R. H.↗

NASA NTRS1979

Global anomaly and undulation recovery using GEOS-3 altimeter data

The data were adjusted to remove orbit error and altimeter bias in a primary adjustment and four regional adjustments. The root mean square crossover discrepancy was about + or - 55 cm after the adjustment. The adjusted altimeter data, now considered to give geoid undulations, was used to predict values at 1 deg intersections from which an oceanic geoid map, with predicted accuracies, was prepared at a two meter contour interval. This geoid was compared to the Gemini 9 geoid over very long profiles to examine the long wavelength error in the altimeter geoid. At a wavelength of 13,010 km the root mean squares difference was 57 cm. The altimeter geoid was also compared to altimeter geoids fixed by precise orbits. A root mean square difference was found of about 1 m with a systematic difference that implied the equatorial radius of the earth was 6,378,137 meters. The adjusted altimeter data was also used to determine a total of 29,479 1 deg x 1 deg anomalies (and undulations).

Rapp, R. H.↗

NASA NTRS1979

Gravity anomaly and geoid undulation results in local areas from GEOS-3 altimeter data

The adjusted GEOS-3 altimeter data, taken as averages within a data frame, have been used to construct free air anomaly and geoid undulation profiles and maps in areas of geophysical interest. Profiles were constructed across the Philippine Trench (at a latitude of 6 deg) and across the Bonin Trench (at a latitude of 28 deg). In the latter case an anomaly variation of 443 mgals in 143 km was derived from the altimeter data. These variations agreed reasonably with terrestrial estimates, considering the predicted point accuracy was about + or - 27 mgals. An area over the Patton Sea mounts was also investigated with the altimeter anomaly field agreeing well with the terrestrial data except for the point directly over the top of the sea mount. It is concluded that the GEOS-3 altimeter data is valuable not only for determining 5 deg and 1 deg x 1 deg mean anomalies, but also can be used to describe more local anomaly variations.

Rapp, R. H.↗

NASA NTRS1978

Mean gravity anomalies and sea surface heights derived from GEOS-3 altimeter data

Approximately 2000 GEOS-3 altimeter arcs were analyzed to improve knowledge of the geoid and gravity field. An adjustment procedure was used to fit the sea surface heights (geoid undulations) in an adjustment process that incorporated cross-over constraints. The error model used for the fit was a one or two parameter model which was designed to remove altimeter bias and orbit error. The undulations on the adjusted arcs were used to produce geoid maps in 20 regions. The adjusted data was used to derive 301 5 degree equal area anomalies and 9995 1 x 1 degree anomalies in areas where the altimeter data was most dense, using least squares collocation techniques. Also emphasized was the ability of the altimeter data to imply rapid anomaly changes of up to 240 mgals in adjacent 1 x 1 degree blocks.

Rapp, R. H.↗

NASA NTRS1978

The determination of gravity anomalies from geoid heights using the inverse Stokes' formula, Fourier transforms, and least squares collocation

A numerical method for the determination of gravity anomalies from geoid heights is described using the inverse Stokes formula. This discrete form of the inverse Stokes formula applies a numerical integration over the azimuth and an integration over a cubic interpolatory spline function which approximates the step function obtained from the numerical integration. The main disadvantage of the procedure is the lack of a reliable error measure. The method was applied on geoid heights derived from GEOS-3 altimeter measurements in the calibration area of the GEOS-3 satellite.

Rummel, R.↗

NASA NTRS1978

Comparison of surface and satellite gravity data

Satellite derived potential coefficients (GEM 9) are compared to terrestrial gravity data by degree in terms of coefficient differences and in terms of mean anomaly differences. The root mean square undulation difference (to degree 20) was 9 + or - m and the anomaly difference was + or - 7 mgals with GEM 9 commission errors of + or - 1.7 m and + or - 3.8 mgals. The standard deviations of the GEM 9 implied undulations increased from + or - 4 cm at degree 2 to + or - 53 cm at degree 20. The corresponding values implied by a recent (June 1978) terrestrial 5 deg field were + or - 2.53 m and + or - 0.38 m (at degree 20). Comparisons between Geos-3 altimeter derived anomalies and 1 deg X 1 Deg terrestrial data showed that + or - 6-8 mgals is a reasonable accuracy estimate for the altimeter derived anomalies. Anomalies derived from satellite to satellite tracking data indicate an accuracy of about + or - 6 mgals for the recovery of 5 deg equal area blocks.

Rapp, R. H.↗

NASA NTRS1977

The relationship between mean anomaly block sizes and spherical harmonic representations

The frequently used rule specifying the relationship between a mean gravity anomaly in a block whose side length is theta degrees and a spherical harmonic representation of these data to degree l-bar is examined in light of the smoothing parameter used by Pellinen (1966). It is found that if the smoothing parameter is not considered, mean anomalies computed from potential coefficients can be in error by about 30% of the rms anomaly value. It is suggested that the above mentioned rule should be considered only a crude approximation.

Rapp, R. H.↗