Spectral analysis as applied to a set of gyrocompass transit times

Spectral analysis by least squares as developed by Vaníček is applied to a series of transit times measurements obtained with a suspended gyrocompass (Wild) electronically equipped with three photocells and a printing chronograph. Instead of being the Fourier transform of the autocovariance function as in the usual spectral analysis of time series (Wiener theory), the spectral function used here is a function of an estimator of the variance factor obtained after a least squares fitting of a sinusoid to the data. That function is normalized to values between zero and one. For step-by-step spectral analysis by least squares each time a significant frequency appears in the spectrum it is removed by least squares fitting of the corresponding sinusoid including a damping coefficient, the residuals being again examined by spectral analysis by least squares. We find four significant frequencies: the well known principal period of about 7^min in the spinning case; a very strong component with a period nearly exactly half the principal period and an amplitude of about 70″, explained by taking into account the second-order term in the theory developed by Jeudy, and two remaining periods with much smaller amplitudes (2″.9 and 0″.9). It is shown that the shortest period (0^s.021), predicted by theory, exists in the measurements and cannot be neglected. The smallest component is considered to correspond to the wobble which can easily be observed in the perturbed motion.     

Evaluation of deflections of the vertical using topographic-isostatic and astrogeodetic data for the area of Greece

The evaluation of deflections of the vertical for the area of Greece is attempted using a combination of topographic and astrogeodetic data. Tests carried out in the area bounded by 35°≤ϕ≤42°, 19°≤λ≤27° indicate that an accuracy of ±3″.3 can be obtained in this area for the meridian and prime vertical deflection components when high resolution topographic data in the immediate vicinity of computation points are used, combined with high degree spherical harmonic expansions of the geopotential and isostatic reduction potential. This accuracy is about 25% better than the corresponding topographic-Moho deflection components which are evaluated using topographic and Moho data up to 120 km around each station, without any combination with the spherical harmonic expansion of the geopotential or isostatic reduction potential. The accuracy in both cases is increased to about 2″.6 when the astrogeodetic data available in the area mentioned above are used for the prediction of remaining values. Furthermore the estimation of datum-shift parameters is attempted using least squares collocation.     

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.     

Space orientation and geodetic azimuths of long lines from observations of the ANNA Satellite

The space orientation and geodetic azimuths of lines ranging from 300 km to 1400 km have been determined from simultaneous optical observations of the ANNA Flashing Satellite. The results of this test prove that the azimuth and the space direction between two stations can be achieved to an accuracy of 0.5″ and 0.8″ second respectively with only a limited amount of data. The reason for the high accuracy is attributed to two factors: [1] the metric quality of the PC-1000's stellar cameras, and [2] the “perfect” simultaneity in the observations provided by the ANNA flashing light. Much of this work was accomplished by the writer while employed by the Geodesy and Gravity Branch of Cambridge Research Laboratories.     

The effect of topography on the determination of the geoid using analytical downward continuation

The method of analytical downward continuation has been used for solving Molodensky’s problem. This method can also be used to reduce the surface free air anomaly to the ellipsoid for the determination of the coefficients of the spherical harmonic expansion of the geopotential. In the reduction of airborne or satellite gradiometry data, if the sea level is chosen as reference surface, we will encounter the problem of the analytical downward continuation of the disturbing potential into the earth, too. The goal of this paper is to find out the topographic effect of solving Stoke’sboundary value problem (determination of the geoid) by using the method of analytical downward continuation. It is shown that the disturbing potential obtained by using the analytical downward continuation is different from the true disturbing potential on the sea level mostly by a −2πGρh 2/p. This correction is important and it is very easy to compute and add to the final results. A terrain effect (effect of the topography from the Bouguer plate) is found to be much smaller than the correction of the Bouguer plate and can be neglected in most cases. It is also shown that the geoid determined by using the Helmert’s second condensation (including the indirect effect) and using the analytical downward continuation procedure (including the topographic effect) are identical. They are different procedures and may be used in different environments, e.g., the analytical downward continuation procedure is also more convenient for processing the aerial gravity gradient data. A numerical test was completed in a rough mountain area, 35°<ϕ<38°, 240°<λ<243°. A digital height model in 30″×30″ point value was used. The test indicated that the terrain effect in the test area has theRMS value ±0.2−0.3 cm for geoid. The topographic effect on the deflections of the vertical is around1 arc second.     

联合多种测高数据确定中国边缘海及全球海域的垂线偏差

On the basis of gravity field model （EIGEN_CG01C）, together with multi-altimeter data, the improved deflection of the vertical gridded in 2＇×2＇ in China marginal sea and gridded in 5＇×5＇ in the global sea was determined by using the weighted method of along-track least squares, and the accuracy is better than 1.2^# in China marginal sea. As for the quality of the deflection of the vertical, it meets the challenge for the gravity field of high resolution and accuracy, it shows that, compared with the shipboard gravimetry in the sea, the accuracy of the gravity anomalies computed with the marine deflection of the vertical by inverse Vening-Meinesz formula is 7.75 m.s ^-2.     

Evaluation of deflections of the vertical on the sphere and the plane: a comparison of FFT techniques

This paper presents a set of efficient formulas to evaluate the deflections of the vertical on the sphere using gridded data. The Vening-Meinesz formula, the topographic indirect effect on the deflections of the vertical as well as the terrain corrections are expressed as both 2D and 1D convolutions on the sphere, and consequently can be evaluated by the 2D and the 1D fast Fourier transform (FFT). When compared with the results obtained from pointwise integration, the use of the 1D FFT gives identical results, and therefore these results were used as control values in this paper. The use of the spherical 2D FFT improves significantly the computational efficiency with little sacrifice of accuracy (0.6^″ rms difference from the 1D FFT results). The planar 2D FFT, which is as efficient as the spherical 2D FFT, gives worse results (1.2^″ rms difference from the 1D FFT results) because of the extra approximations. Received: 27 February 1996; Accepted: 24 January 1997     

Die absolute Lotabweichung in Postdam und die Geod?tischen ausgangswerte des Gesamteurop?ischen netzes auf dem Hayfordschen Ellipsoid

Summary The absolute deviations of the vertical detectable at Potsdam from astronomical-gravity comparisons differ significantly from earlier values obtained from the astronomic-geodetic observations which form the starting elements of the European triangulation network. It is possible to show, by the use of values of the deviation of the vertical extending over the whole of Europe as far as longitude 30°, and referred to the undulations of the geoid ofTanni, that the starting elements adopted at Potsdam are too large by 3″.0 in latitude and 1″.4 in longitude.

---

Resumen Las desviaciones absolutas de la vertical obtenidas en Potsdam por procedimiento astronómico-gravimétrico, se alejan sensiblemente de los valores encontrados enteriormente por procedimiento astronómicogeodésico, y que constituyen la base de partida de la red de conjunto europea. Es posible demostrar, basándose en el material de desviaciones de la vertical extendido a toda Europa hasta la longitud de 30°, partiendo de las ondulaciones del geoide deTanni, que los datos de partida adoptados en Potsdam son demasiado grandes en 3″,0 de latitud y 1″,4 de longitud.

---

Résumé Les déviations absolues de la verticale relevées à Potsdam par voie astronomico-gravimétrique s’écartent sensiblement des valeurs trouvées autrefois par voie astronomico-géodésique, et qui forment la base de départ du réseau d’ensemble de l’Europe. Il est possible de démontrer, en s’appuyant sur le matériel de déviations de la verticale étendu sur toute l’Europe jusqu’à la longitude de 30°, en partant des ondulations du géo?de deTanni, que les données de départ adoptées à Potsdam sont trop grandes de 3″.0 en latitude et de 1″.4 en longitude.

---

Sommario La deviazione assoluta della verticale determinata a Postdam per via astronomico-gravimetrica, si scosta sensibilmente dal valore trovato in precedenza per via astronomico-geodetica, che forma la base della compensazione delle reti europee. Si può dimostrare, appoggiandosi sulle deviazioni della verticale conosciute in Europa fino alla longitudine di 30°, e partendo dalle ondulazioni del geoide diTanni, che i dati di partenza adottati a Postdam sono troppo grandi di 3″,0 in latitudine e di 1″,4 in longitudine.

---

---

Communication présentée à l’Assemblée Générale de Bruxelles     

Least-squares prediction of horizontal coordinate distortions in Canada

A least-squares prediction method is described to estimate horizontal coordinate distortions at lower order points of a network using known coordinate differences (NAD27 coordinate distortions Δϕ′s and Δλ′s) at higher order points between NAD27 coordinates and coordinates derived from a recent (MAY 76), relatively distortion free, adjustment of these points. Empirical autocovariance functions of Δϕ and Δλ and crosscovariance function between Δϕ and Δλ are derived from some 5,250 data points and modelled using series of exponential functions. Empirical mean square values of Δϕ and Δλ, which are a measure of the distortions in NAD27 ϕ and λ, are 0.051 and 0.645 arcsecs² respectively. The corresponding mean value of the product ΔϕΔλ, which is a measure of the correlation between Δϕ and Δλ, is 0.056 arcsecs². The accuracy obtainable for predicted Δϕ and Δλ at an arbitrary point (e.g., lower order station) is a function of the accuracy and configuration of known Δϕ′s and Δλ′s in the surrounding area. Accuracies obtainable for various types of data configuration are given. Under favorable conditions taking place in about 60% of cases, accuracies in terms of ms agreement with known values of 0″.02 (0.6 m) and 0″.01 (0.2 m along parallel at latitude 50°) for the predicted latitude and longitude distortions are obtainable. Finally, a comparison with a method based on the use of complex polynomials is made. Presented at International Symposium on Geodetic Networks and Computations, Munich, August–September 1981.     

GPS-GRAVIMETRIC GEOID DETERMINATION IN EGYPT

1　ＩｎｔｒｏｄｕｃｔｉｏｎＤｉｆｆｅｒｅｎｔｇｅｏｉｄｓｏｌｕｔｉｏｎｓｗｅｒｅｃａｒｒｉｅｄｏｕｔｆｏｒＥ ｇｙｐｔｕｓｉｎｇｈｅｔｅｒｏｇｅｎｅｏｕｓｄａｔａａｎｄｄｉｆｆｅｒｅｎｔｍｅｔｈｏｄｏｌｏｇｉｅｓ (Ｅｌ＿Ｔｏｋｈｅｙ ,1 993) .ＴｈｅｍａｉｎｇｏａｌｏｆｔｈｉｓｐａｐｅｒｉｓｔｏｄｅｔｅｒｍｉｎｅａｍｏｓｔａｃｃｕｒａｔｅｎｅｗｇｅｏｉｄｆｏｒＥｇｙｐｔｔａｋｉｎｇａｄｖａｎｔａｇｅｏｆａｎｅｗｕｐｄａｔｅｄｇｒａｖｉｔｙｄａｔａｂａｓｅ,ｔｈｅｉｎｆｏｒｍａｔｉｏｎｇｉｖｅｎｂｙ…     

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.     

Local characteristics of the gravity anomaly covariance function

Summary Using a data set of 260 000 gravity anomalies it is shown that common characteristics for a local covariance function exist in an area as large as Canada excluding the Rocky Mountains. After eliminating global features by referencing the data to the GEM-10 satellite solution, the shape of the covariance function is remarkably consistent from one sample area to the next. The determination of the essential parameters and the fitting of the covariance function are discussed in detail. To test the reliability of the derived function, deflections of the vertical are estimated at about 230 stations where astrogeodetic data are available. Results show that the standard error obtained from the discrepancies is about1″ for each component and that the error covariance matrix of least-squares collocation reflects this accuracy remarkably well.     

Azimuth by the method of multiple timed observations of a gyrotheodolite without electronic registration

A method of determining azimuth by gyrotheodolite without electronic registration is described. The method requires observations of time at each instant the moving mark passes a scale division. Thus many observations of time may be achieved in a single oscillation of the moving mark. The observations when used in the appropriate observation equation may determine azimuth with a standard error of ±3″ with 2 hours of observations. This assumes knowledge of the additive constant to about 1″.3 and neglects the effects of dislevelment in the prime vertical. For practical application of the method a time recording device and microcomputer, such as the Hewlett Packard HP41CV and HP85, are recommended.     

Ultra-high degree spherical harmonic analysis and synthesis using extended-range arithmetic

We present software for spherical harmonic analysis (SHA) and spherical harmonic synthesis (SHS), which can be used for essentially arbitrary degrees and all co-latitudes in the interval (0°, 180°). The routines use extended-range floating-point arithmetic, in particular for the computation of the associated Legendre functions. The price to be paid is an increased computation time; for degree 3,000, the extended-range arithmetic SHS program takes 49 times longer than its standard arithmetic counterpart. The extended-range SHS and SHA routines allow us to test existing routines for SHA and SHS. A comparison with the publicly available SHS routine GEOGFG18 by Wenzel and HARMONIC SYNTH by Holmes and Pavlis confirms what is known about the stability of these programs. GEOGFG18 gives errors <1 mm for latitudes [-89°57.5′, 89°57.5′] and maximum degree 1,800. Higher degrees significantly limit the range of acceptable latitudes for a given accuracy. HARMONIC SYNTH gives good results up to degree 2,700 for almost the whole latitude range. The errors increase towards the North pole and exceed 1 mm at latitude 82° for degree 2,700. For a maximum degree 3,000, HARMONIC SYNTH produces errors exceeding 1 mm at latitudes of about 60°, whereas GEOGFG18 is limited to latitudes below 45°. Further extending the latitudinal band towards the poles may produce errors of several metres for both programs. A SHA of a uniform random signal on the sphere shows significant errors beyond degree 1,700 for the SHA program SHA by Heck and Seitz.     

The determination of absolute directions in space with artificial satellites

The Smithsonian Astrophysical Observatory has modified the Baker-Nunn cameras to perform simultaneous observations. When only two stations are involved in simultaneous observing, the directions in an absolute system of reference of the line connecting the two stations can be determined. Fifty-six pairs of simultaneous observations between stations Villa Dolores, Argentina; Arequipa, Peru; Cura?ao, Netherlands Antilles; Jupiter, Florida; and Organ Pass, New Mexico, indicate that an accuracy of better than 1″ of arc can be expected. This work was supported in part by grant NsG 87-60 of the National Aeronautics and Space Administration.     

Geoid determination in central Spain from gravity and height data

Summary The least-squares collocation method has been used for the computation of a geoid solution in central Spain, combining a geopotential model complete to degree and order 360, gravity anomalies and topographic information. The area has been divided in two 1°× 1° blocks and predictions have been done in each block with gravity data spacing about 5 × 5 within each block, extended 1/2°. Topographic effects have been calculated from 6 × 9 heights using an RTM reduction with a reference terrain model of 30 × 30 mean heights.     

Downward continuation of the free-air gravity anomalies to the ellipsoid using the gradient solution and terrain correction—An attempt of global numerical computations

The formulas for the determination of the coefficients of the spherical harmonic expansion of the disturbing potential of the earth are defined for data given on a sphere. In order to determine the spherical harmonic coefficients, the gravity anomalies have to be analytically downward continued from the earth's surface to a sphere—at least to the ellipsoid. The goal of this paper is to continue the gravity anomalies from the earth's surface downward to the ellipsoid using recent elevation models. The basic method for the downward continuation is the gradient solution (theg ₁ term). The terrain correction has also been computed because of the role it can play as a correction term when calculating harmonic coefficients from surface gravity data. Theg ₁ term and the terrain correction were expanded into the spherical harmonics up to180 ^th order. The corrections (theg ₁ term and the terrain correction) have the order of about 2% of theRMS value of degree variance of the disturbing potential per degree. The influences of theg ₁ term and the terrain correction on the geoid take the order of 1 meter (RMS value of corrections of the geoid undulation) and on the deflections of the vertical is of the order 0.1″ (RMS value of correction of the deflections of the vertical).     

Convergence and optimal truncation of binomial expansions used in isostatic compensations and terrain corrections

 A binomial expansion is a powerful tool in geodetic research. It is often used in terrain correction and isostatic compensation. The behaviour, convergence and truncation of the binomial expansion are investigated. The relation of the topographic height H (or the compensation depth), spherical harmonic degree n and the binomial series term m is discussed using theoretical and numerical results. According to the relation, a truncation number M is determined for obtaining an accuracy of 1%, i.e. it can be found how many terms (or power numbers of the topography) should be used in practical calculations. Received: 24 February 1999 / Accepted: 28 June 2000     

Methodology for the combination of sub-daily Earth rotation from GPS and VLBI observations

A combination procedure of Earth orientation parameters from Global Positioning System (GPS) and Very Long Baseline Interferometry (VLBI) observations was developed on the basis of homogeneous normal equation systems. The emphasis and purpose of the combination was the determination of sub-daily polar motion (PM) and universal time (UT1) for a long time-span of 13 years. Time series with an hourly resolution and a model for tidal variations of PM and UT1-TAI (dUT1) were estimated. In both cases, 14-day nutation corrections were estimated simultaneously with the ERPs. Due to the combination procedure, it was warranted that the strengths of both techniques were preserved. At the same time, only a minimum of de-correlating or stabilizing constraints were necessary. Hereby, a PM time series was determined, whose precision is mainly dominated by GPS observations. However, this setup benefits from the fact that VLBI delivered nutation and dUT1 estimates at the same time. An even bigger enhancement can be seen for the dUT1 estimation, where the high-frequency variations are provided by GPS, while the long term trend is defined by VLBI. The estimated combined tidal PM and dUT1 model was predominantly determined from the GPS observations. Overall, the combined tidal model for the first time completely comprises the geometrical benefits of VLBI and GPS observations. In terms of root mean squared (RMS) differences, the tidal amplitudes agree with other empirical single-technique tidal models below 4 μas in PM and 0.25 μs in dUT1. The noise floor of the tidal ERP model was investigated in three ways resulting in about 1 μas for diurnal PM and 0.07 μs for diurnal dUT1 while the semi-diurnal components have a slightly better accuracy.