- Geodetic Science Reports (School of Earth Sciences)

# Geodetic Science Reports (School of Earth Sciences)

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Item The downward continuation to the earth's surface of truncated spherical and ellipsoidal harmonic series of the gravity and height anomalies(Ohio State University. Division of Geodetic Science, 1981-12) Jekeli, Christopher, 1953-Show more The problem of the divergence of the geopotential spherical harmonic series at the earth's surface is investigated from a numerical, rather than a theoretical, approach. It is shown that previous numerical evaluations, based on the expansion of the potential generated by the masses between the point of computation and the bounding sphere, are inconclusive with respect to the magnitude of the downward continuation error that is attributable to series divergence. A more representative model of the earth's potential is devised on the basis of a density layer, which, in the spherical approximation, generates a gravity field whose harmonic constituents decay according to an accepted degree variance model. This field, expanded to degree 300, and a topographic surface specified to a corresponding resolution of 67 km are used to compute the differences between truncated inner and outer series of the gravity and height anomalies at the surface of the earth model. Up to degree 300, these differences attain RMS values from 0.33 μgal to 86 μgal for the gravity anomaly and from 0.32 μm to 410 μm for the height anomaly, in areas ranging respectively from near the equator to the vicinity of the pole. In addition to these values, there is an expected truncation effect, caused by the neglect of higher degree components of the inner series, of about 30 mgal and 36 cm, respectively. The field is then subjected to a Gaussian filter which effectively cuts off information at degree 300 (at the 5% level). The RMS downward continuation error to degree 300 is thereby reduced by factors of 10 to 20, with a concomitant reduction in the truncation effect to about 0.3 mgal and 0.7 cm. Testing the downward continuation of ellipsoidal harmonic series on a similar model yields RMS values of the first 300 degrees of the error of 1x10-4 to 3x10-4 μgal for the (point) gravity anomaly, from pole to equator, respectively. [Some mathematical expressions are not fully represented in the metadata. Full text of abstract available in document.]Show more Item The earth's gravity field to degree and order 180 using SEASAT altimeter data, terrestrial gravity data, and other data(Ohio State University. Division of Geodetic Science, 1981-12) Rapp, Richard H.Show more The spherical harmonic expansion of the earth's gravitational field has been obtained to degree 180 by combining several sources of data. The first data set was an a priori set of potential coefficients to degree 36 based on a number of recent solutions including a substantial number of resonance terms. A second data set was a 1° x 1° anomaly field derived from the Seasat data set, while the third data set was an updated 1° x 1° terrestrial field. The last two fields were combined into one set containing 56761 1° x 1° values. The remaining values were computed from the a priori potential coefficients. A rigorous combination solution was not carried out. Instead all anomalies were weighted in such a way that the normal equations were diagonal. The results of the adjustment were 64800 1° x 1° anomalies that were expanded into spherical harmonics using the optimum quadrature procedures developed by Colombo. Accuracy estimates for each coefficient were obtained considering noise propagation and sampling error caused by the finite block size (1° x 1°) in which the anomalies are given. The percentage error of the solution reaches 100% near degree 120. The coefficients and their accuracy to degree 50 are listed in an appendix and the complete set is available on tape. The coefficients have been compared to other coefficient sets such as GEM10C and GRIM3.Show more Item Mathematical modelling of the behavior of the LaCoste and Romberg "G" gravity meter for use in gravity network adjustments and data analyses(Ohio State University. Division of Geodetic Science, 1981-11) Krieg, Lenny A.Show more This report deals with the modelling of the behavior of the LaCoste Romberg "G" Gravity Meter. This is accomplished by using a model which includes a linear term plus a long wave sinusoidal term to represent the calibration information contained in the Calibration Table 1 supplied with each gravity meter. Additional sinusoidal terms are added to model any periodic screw error effects. Using gravity meter data collected for the United States Gravity Base Station Network with the majority of the data being along the Mid-Continent Calibration Line, various models were tested. The control used were absolute gravity station values determined by the Italians, Marson and Alasia, and by Hammond of AFGL. The results indicate the presents of periodic screw error terms having a period of approximately 70.941 counter units with amplitudes less than 20 μgal and the estimated accuracy of gravity meter observations to be from 20-25 μgal. Due to the inconsistencies of the absolute gravity station determinations, the accuracy of the absolute values is probably closer to 20 μgal than the purported 10 μgal.Show more Item Prediction of earth rotation and polar motion(Ohio State University. Division of Geodetic Science, 1981-09) Zhu, Sheng YuanShow more Based on the analysis of the polar motion behavior, we found the possibility of predicting polar motion up to one year in advance. Comparing these predicted polar coordinates with the observed ones (smoothed), the rms of the differences is about 0."02. The differences of the relative polar motion are much smaller. For any time interval of 20 - 30 days throughout the whole year, the rms of the relative polar motion differences is about 0."01. Compared with the best available VLBI results (from 1977 to 1980), the rms of (pred. - obs.) is 0."013, and the relative rms (for time intervals less than two months) is 0.008 (here the observed data is unsmoothed). It appears that 80 - 90% of the polar motion is composed of the stable, predictable Chandler and annual terms. UT1-UTC has more complicated changes than polar motion making it difficult to find a satisfactory method of long-term prediction. So far the rms prediction error is 0.s0023 for up to 30 days. [Some mathematical expressions are not fully represented in the metadata. Full text of abstract available in document.]Show more Item The Stokes' problem for the ellipsoid using ellipsoidal kernels(Ohio State University. Division of Geodetic Science, 1981-06) Zhu, ZhuowenShow more A brief review of Stokes' problem for the ellipsoid as a reference surface is given. Another solution of the problem using an ellipsoidal kernel, which represents an iterative form of Stokes' integral, is suggested with a relative error of the order of the flattening. On studying of Rapp's method in detail the procedures of improving its convergence are discussed.Show more Item Theories of nutation and polar motion II(Ohio State University. Division of Geodetic Science, 1981-09) Moritz, Helmut, 1933-Show more The report describes and interrelates various theories of nutation and polar motion based on an earth model consisting of an elastic mantle and a liquid core. In order to make basic features transparant and to facilitate interrelation and intercomparison, the liquid core is considered homogeneous and incompressible. A detailed treatment of the theory of Molodensky serves as a basis for describing recent advances by Shen and Mansinha, Smith, Wahr, and others; an elementary treatment of the variational approach by Jeffreys and Vicente and a numerical comparison conclude the report.Show more Item Global geopotential modelling from satellite-to-satellite tracking(Ohio State University. Division of Geodetic Science, 1981-10) Colombo, Oscar L.Show more The error analysis of the global modelling of the geopotential has been carried out up to degree and order 331 of the spherical harmonic expansion, for data from a low-low satellite-to-satellite tracking (SST) mission. The sphericity and the rotation of the Earth have been considered, as well as the discrete nature of the data, assumed to consist of time averages of the measured range-rate sampled at regular intervals. The expansion of the potential has been truncated at degree n = 331, because little information on higher degrees is likely to be present in the data. Two theories have been used: that of least squares adjustment, and that of least squares collocation; above degree n = 200 the accuracies predicted according to collocation are significantly better than those according to least squares. In this report there is also a discussion on how to process SST data to obtain very high resolution models of the gravitational field. Descriptions and listings of computer programs are included. To reduce the computer time and storage needed to set up and to invert the normal matrix, a somewhat simplified orbital geometry and an approximate model of the data have been adopted; no orbit determination errors have been considered. Some arguments are given to justify these shortcuts, which may not affect seriously the validity of the results. An extention of the theory to non-polar orbits is given. [Full text of abstract available in document.]Show more Item A simulation study to test the prediction of 1° x 1° mean gravity anomalies using least squares collocation from the Gravsat mission(Ohio State University. Division of Geodetic Science, 1981-09) Hajela, D. P.Show more A simulation analysis is described for the determination of an improved gravity field from the GRAVSAT mission by examining the resolution achievable in terms of mean anomaly block size, the accuracy of the predicted anomaly, and testing, economical data reduction procedures in terms of computer time. [Full text of abstract available in document.]Show more Item The estimation of free-air anomalies(Ohio State University. Division of Geodetic Science, 1981-09) Sünkel, HansShow more A prediction model of point and/or mean free-air anomalies, based on point values, is proposed. It relies on the concept of least-squares collocation with parameters. The model parameters are the two regression parameters of a linear model, which assumes a linear correlation between free-air anomaly and elevation. Particular emphasis is put on the role of the topography in mountainous areas.Show more Item Simulation studies on the computation of the gravity vector in space from surface data considering the topography of the earth(Ohio State University. Division of Geodetic Science, 1981-06) Katsambalos, Kostas E.Show more Three approaches are investigated for the computation of the components of the gravity vector in space considering the topography of the earth: a numerical integration approach based on the application of Green's third identity, the Discrete Dirac approach, and the Least-Squares Collocation approach. Under a spherical approximation, the surface of the earth is assumed to be known through the elevations of its points above a reference sphere. The first technique requires as data gravity disturbances and disturbing potentials on the earth's surface, while the other two techniq require as data surface gravity anomalies. Two point masses located on the axes of symmetry of two simple terrain models (a cone and a sphere), generate on their surfaces the synthetic data needed for the simulations. The agreement between the rigorously computed vectors (from the models), and those from the three techniques, is analyzed in terms of factors such as the inclination of the model's surface, data density, altitude of the space point etc. The application of the Dirac approach is questionable due to its limited accuracy for large data spacing. The Green's approach is recommended for computations above 10 km altitude, while the Collocation approach is suitable for computations at points between the earth's surface and the 10 km level. Comparison of these techniques with the classical approach (the Direct Integration Method), indicate that the consideration of the topography improves significantly the accuracy of the computations.Show more Item The use of finite elements in physical geodesy(Ohio State University. Division of Geodetic Science, 1981-04) Meissl, PeterShow more Currently used methods of computational physical geodesy are compared with respect to their efficiency during production runs on a computer. These methods include: (1) Least Squares adjustment with respect to spherical harmonics, (2) Surface layers, buried masses and related methods, (3) Least squares collocation, (4) Representation of the potential by spline functions, (5) Explicit integral formulas. As an alternative, the feasibility of applying the finite element method to the fundamental problems of physical geodesy is investigated. The methods listed under (1)-(4) can be dramatically speeded up if the distribution of data and weights satisfies certain symmetry-requirements which are rather stringent. Method (5) relies altogether on a special type and distribution of data. In the absence of data homogeneity and regularity, the finite element method is asymptotically superior with respect to computational efficiency. Let N denote the number of parameters necessary to describe the variation of the potential on the reference surface. The computational effort associated with methods (1)-(4) grows proportional to N3 . That one resulting from finite elements grows proportional to N3/2. The constants of proportionality are, however, unfavorable for the finite element method. Hence its superiority comes through only for large values of N , which, in case of a global solution, corresponds to data averaged over 2° x 2° blocks. [Some mathematical expressions are not fully represented in the metadata. Full text of abstract available in document.]Show more Item Integral formulas and collocation(Ohio State University. Division of Geodetic Science, 1975-12) Moritz, Helmut, 1933-Show more The report deals with various theoretical and numerical aspects of the interplay between least-squares collocation and classical integral formulas. It is proved that geodetic integral formulas may be considered as limiting cases of collocation for homogeneous and regularly and densely distributed data. Collocation methods can he employed for adjusting continuous data and combining them with other measurements, before using them with integral formulas. With respect to numerical computation, integral formulas and collocation techniques mutually complement each other, so that in many practical cases a judicious combination of the two procedures may be practically most convenient.Show more Item Cardinal interpolation(Ohio State University. Division of Geodetic Science, 1981-03) Sünkel, HansShow more Base functions of various kind can be used for prediction problems. All of them are interrelated to each other. In particular, the family of splines, the Gaussian function, the sinπx/πx function, the Hirvonen covariance functions, and the Bjerhammar interpolation functions are considered here. Particular emphasis is put on the important role of the correlation length of the base (covariance) function; its strong relations to the data sampling rate and to the maximum prediction error should advantageously be used as a guideline for the design of gravity field data-collection and data-processing programs.Show more Item Feasibility studies for the prediction of the gravity disturbance vector in high altitudes(Ohio State University. Division of Geodetic Science, 1981-03) Sünkel, HansShow more The accuracy of the gravity disturbance vector in high altitude (30 000 - 200 000), predicted from a surface-covering set of mean gravity anomalies, is estimated. Two methods are used and found to provide estimates which differ by less than 10%, the least-squares collocation and the integral solution; for the integral solution, the estimation of the representation error has been performed in the frequency domain. For the collocation solution an optimal algorithm has been developed which takes advantage of the regular data distribution and is up to 64 times faster than a non-optimized solution. The results indicate that the radial component of the gravity disturbance vector can be estimated with an accuracy of ±1 mgal at an altitude of about 50 000 ft. on the basis of the available data sets; in order to achieve the same accuracy at 30 000 ft. altitude, the data error, particularly that of 5' x 5' anomalies, has to be reduced by some 60%; the available data distributions are adequate. The prediction error drops quickly with increasing altitude. The situation is considerably different for the horizontal component: with the best available data distribution an accuracy of ±2.3 mgal at 30 000 ft. altitude can be achieved; (this corresponds to ±0."5 in the direction of the gravity vector). An accuracy of ±1 mgal requires a block size reduction by a factor or 2 not only in the innermost zone, but also up to a spherical distance of about 30°; in addition, the overall data error needs to be reduced by some 30%. The prediction error decreases only slowly with increasing altitude. [Some mathematical expressions are not fully represented in the metadata. Full text of abstract available in document.]Show more Item Numerical methods for harmonic analysis on the sphere(Ohio State University. Division of Geodetic Science, 1981-03) Colombo, Oscar L.Show more Item Theories of nutation and polar motion I(Ohio State University. Division of Geodetic Science, 1980-12) Moritz, Helmut, 1933-Show more The present report attempts a systematics presentation and review of modern theories of the earth's rotation (precession, nutation, and polar motion) for a rigid earth, a purely elastic earth, and the Poincare model consisting of a rigid mantle and a liquid core. Emphasis is on the treatment of an elastic earth on the basis of Liouville's equation and on the consideration of the earth's rotation as an eigenvalue problem.Show more Item Ellipsoidal corrections for geoid undulation computations(Ohio State University. Division of Geodetic Science, 1981-03) Rapp, Richard H.Show more 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 errors that are investigated in this paper. The equations dealing with ellipsoidal corrections developed by Lelgemann and by Moritz are 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 have been 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° the correction can reach -33 cm. Ellipsoidal corrections were also computed for the Marsh/Chang geoids. These corrections reach -45 cm for a cap size of 20°. Global maps are given showing the distribution of the corrections so that more accurate geoid undulations can be found.Show more Item Accuracy of the determination of mean anomalies and mean geoid undulations from a satellite gravity field mapping mission(Ohio State University. Division of Geodetic Science, 1980-08) Jekeli, Christopher, 1953-; Rapp, Richard H.Show more Improved knowledge of the earth's gravity field can be obtained from new and improved satellite measurements such as satellite to satellite tracking and gradiometry. This improvement has been 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 has been related to the spectrum of the disturbing potential of the earth's gravity field. Equations were derived for a low-low (radial or horizontal seperation) mission and a gradiometer mission. For a low-low mission of six month's duration, at an altitude of 160km, with a data noise of ± 1 μm/sec for a four second integration time, we would expect to determine 1° x 1° mean anomalies to an accuracy of ±2.3 mgals and 1° x 1° mean geoid undulations to ± 4.3 cm. A very fast Fortran program is available to study various mission configurations and block sizes.Show more Item Geoid heights, geoid height differences, and mean gravity anomalies from "low-low" satellite-to-satellite tracking: an error analysis(Ohio State University. Division of Geodetic Science, 1980-06) Rummel, R. (Reiner), 1945-Show more The mathematical model for a simultaneous estimation of improved orbital parameters and an approximation of the earth's gravity field from range rate observations in an SST "low-low" experiment is described. In a somewhat simplified model an error analysis for the estimation of geoid heights, geoid height differences 1° x 1° mean gravity anomalies is performed employing the least squares collocation method. Investigated is the dependence of the estimated parameter upon the measurement precision, the spatial configuration of the two satellites, the intersatellite distance, and the experiment altitude. In an optimal situation - assuming a range rate precision of ±10-s ms-1, an intersatellite distance of 250km, and an experiment altitude of 200km - the estimated a posteriori std. dev. are ±0.9 m for point geoid heights, ±0.7m for geoid height differences (point seperation 150 km), and ±6 to 7 mgal for 1° x 1° mean gravity anomalies. These numbers compare very well with the results obtained from GEOS-3 altimetry for the seasurface topography. Unmodelled short-wavelength uncertainties in the orbit have thereby to be controlled down to 1 cm in radial direction, whereas the requirements for the control of long-wavelength error effects are moderate. [Some mathematical expressions are not fully represented in the metadata. Full text of abstract available in document.]Show more Item Post-mission adjustment techniques for inertial surveys(Ohio State University. Division of Geodetic Science, 1980-10) Hannah, John; Pavlis, Despina E.Show more Inertial surveying systems present an entirely new method for the determination of geodetic positions, heights, deflection components and gravity anomalies at a large number of points in a relatively short time. This report first details in brief the existing inertial survey systems and then proceeds to outline and compare various suggested approaches to the post-mission adjustment procedures used for smoothing the raw positional output from the real time Kalman filter. A simulation of an area survey is made in which the error model is held fixed and the number, location of control points, and method of traversing is varied. From these simulations, recommendations for traverse procedures involving these variables are suggested. Finally, recommendations are made for future investigations into post-mission adjustment procedures and traverse configurations.Show more