On the computation and approximation of ultra-high-degree spherical harmonic series

Spherical harmonic series, commonly used to represent the Earth’s gravitational field, are now routinely expanded to ultra-high degree (> 2,000), where the computations of the associated Legendre functions exhibit extremely large ranges (thousands of orders) of magnitudes with varying latitude. We show that in the degree-and-order domain, (ℓ,m), of these functions (with full ortho-normalization), their rather stable oscillatory behavior is distinctly separated from a region of very strong attenuation by a simple linear relationship: , where θ is the polar angle. Derivatives and integrals of associated Legendre functions have these same characteristics. This leads to an operational approach to the computation of spherical harmonic series, including derivatives and integrals of such series, that neglects the numerically insignificant functions on the basis of the above empirical relationship and obviates any concern about their broad range of magnitudes in the recursion formulas that are used to compute them. Tests with a simulated gravitational field show that the errors in so doing can be made less than the data noise at all latitudes and up to expansion degree of at least 10,800. Neglecting numerically insignificant terms in the spherical harmonic series also offers a computational savings of at least one third.     

TRIGONOMETRICAL LEVELLING ACROSS THE INLAND ICE IN NORTH GREENLAND

Abstract

As part of the scientific work of the British North Greenland Expedition (1952–1954), a programme of trigonometrical levelling was carried out from the east to the west coast of Greenland, along a line across the inland ice between latitudes 76° 40′ N., and 78° 10′ N. The primary purpose of the work was to determine accurately the heights above sea level of a series of gravity stations, the gravity measurements being made in connection with determinations of ice thickness. For meteorological purposes it was necessary to know also the altitude of the Expedition's central station, situated in latitude 78° 04′ N., longitude 38° 29′ W. The accuracy necessary for the purpose of the gravity survey was a few metres for the altitudes, while the latitude of each gravity station had to be determined with an accuracy of ± 0.1 minute.     

Fast spherical collocation: theory and examples

 It has long been known that a spherical harmonic analysis of gridded (and noisy) data on a sphere (with uniform error for a fixed latitude) gives rise to simple systems of equations. This idea has been generalized for the method of least-squares collocation, when using an isotropic covariance function or reproducing kernel. The data only need to be at the same altitude and of the same kind for each latitude. This permits, for example, the combination of gravity data at the surface of the Earth and data at satellite altitude, when the orbit is circular. Suppose that data are associated with the points of a grid with N values in latitude and M values in longitude. The latitudes do not need to be spaced uniformly. Also suppose that it is required to determine the spherical harmonic coefficients to a maximal degree and order K. Then the method will require that we solve K systems of equations each having a symmetric positive definite matrix of only N × N. Results of simulation studies using the method are described. Received: 18 October 2001 / Accepted: 4 October 2002 Correspondence to: F. Sansò     

Tectonic plate motion and co-seismic steps surveyed by DORIS field beacons: the New Hebrides experiment

 The New Hebrides experiment consisted of setting up a pair of DORIS beacons in remote tropical islands in the southwestern Pacific, between 1993 and 1997. Because of orbitography requirements on TOPEX/Poséidon, the beacons were only transmitting to SPOT satellites. Root-mean-square (RMS) scatters at the centimeter level on the latitude and vertical components were achieved, but 2-cm RMS scatters affected the longitude component. Nevertheless, results of relative velocity (123 mm/year N250°) are very consistent with those obtained using the global positioning system (GPS) (126 mm/yr N246°). The co-seismic step (12 mm N60°) related to the Walpole event (M _W = 7.7) is consistent with that derived from GPS (10 mm N30°) or from the centroid moment tensor (CMT) of the quake (12 mm N000°). Received: 19 November 1999 / Accepted: 17 May 2000     

Statistics of GPS ionospheric scintillation and irregularities over polar regions at solar minimum

A statistical study of the occurrence characteristic of GPS ionospheric scintillation and irregularity in the polar latitude is presented. These measurements were made at Ny-Alesund, Svalbard [78.9°N, 11.9°E; 75.8°N corrected geomagnetic latitude (CGMLat)] and Larsemann Hills, East Antarctica (69.4°S, 76.4°E; 74.6°S CGMLat) during 2007–2008. It is found that the GPS phase scintillation and irregularity activity mainly takes place in the months 10, 11 and 12 at Ny-Alesund, and in the months 5, 6 at Larsemann Hills. The seasonal pattern of phase scintillation with respect to the station indicates that the GPS phase scintillation occurrence is a local winter phenomenon, which shows consistent results with past studies of 250 MHz satellite beacon measurements. The occurrence rates of GPS amplitude scintillation at the two stations are below 1%. A comparison with the interplanetary magnetic field (IMF) B _y and B _z components shows that the phase scintillation occurrence level is higher during the period from later afternoon to sunset (16–19 h) at Ny-Alesund, and from sunset to pre-midnight (18–23 h) at Larsemann Hills for negative IMF components. The findings seem to indicate that the dependence of scintillation and irregularity occurrence on geomagnetic activity appears to be associated with the magnetic local time (MLT).     

Computation of spherical harmonic coefficients and their error estimates using least-squares collocation

 Equations expressing the covariances between spherical harmonic coefficients and linear functionals applied on the anomalous gravity potential, T, are derived. The functionals are the evaluation functionals, and those associated with first- and second-order derivatives of T. These equations form the basis for the prediction of spherical harmonic coefficients using least-squares collocation (LSC). The equations were implemented in the GRAVSOFT program GEOCOL. Initially, tests using EGM96 were performed using global and regional sets of geoid heights, gravity anomalies and second-order vertical gravity gradients at ground level and at altitude. The global tests confirm that coefficients may be estimated consistently using LSC while the error estimates are much too large for the lower-order coefficients. The validity of an error estimate calculated using LSC with an isotropic covariance function is based on a hypothesis that the coefficients of a specific degree all belong to the same normal distribution. However, the coefficients of lower degree do not fulfil this, and this seems to be the reason for the too-pessimistic error estimates. In order to test this the coefficients of EGM96 were perturbed, so that the pertubations for a specific degree all belonged to a normal distribution with the variance equal to the mean error variance of the coefficients. The pertubations were used to generate residual geoid heights, gravity anomalies and second-order vertical gravity gradients. These data were then used to calculate estimates of the perturbed coefficients as well as error estimates of the quantities, which now have a very good agreement with the errors computed from the simulated observed minus calculated coefficients. Tests with regionally distributed data showed that long-wavelength information is lost, but also that it seems to be recovered for specific coefficients depending on where the data are located. Received: 3 February 2000 / Accepted: 23 October 2000     

Comparison of mean gravity anomalies at elevation with corresponding ground anomalies

In support of requirements for the U.S. Air Force Cambridge Research Laboratories, gravity anomalies have been upward continued to several elevations in different areas of the United States. One area was 34⁰ to 40⁰ N in latitude and 96⁰ to 103⁰ W in longitude, generally called the Oklahoma area. The computations proceeded from 26, 032 point anomalies to the prediction of mean anomalies in 14, 704, 2.5′×2.5′ blocks and 9,284, 5′×5′ blocks. These anomalies were upward continued along 28 profiles at 5′ intervals for every 30′ in latitude and longitude. These anomalies at elevations were meaned in various patterns to form mean 30′×30″, 1⁰×1⁰, 5⁰×5⁰ blocks. Comparisons were then made to the corresponding ground values. The results of these comparisons lead to practical recommendations on the arrangement of flight profiles in airborne gravimetry.     

Hydromorphogeology and Remote Sensing applications for groundwater exploration in agnigundala mineralised belt, Andhra Pradesh, India

Hydromorphogeological studies have been carried out around Agnigundala mineralised belt (longitude 70°.39′ - 16°.51′ and latitude 16°.2′ - 16°.15′) using remote sensing IRS-IB and SPOT data for ground water exploration. Based on erosional and depositional characters of various geomorphic units like Hills (Structural and denudational) Pediment, Buried pediment, plains and valley fills have been identified in various lithologies like granite, granite gneiss, biotite schist, phyllite,. quartzite and dolomite. The acclamations of individual geomorphic units through visual interpretation are verified from field data. The groundwater potentials of the individual geomorphologic units have been evaluated to obtain a complete hvdrogcological picture of the area. The field data have further helped in quanlifying various geomorphological units with reference to their potential for ground water occurrence.     

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.     

Efficient and accurate high-degree spherical harmonic synthesis of gravity field functionals at the Earth’s surface using the gradient approach

Spherical harmonic synthesis (SHS) of gravity field functionals at the Earth’s surface requires the use of heights. The present study investigates the gradient approach as an efficient yet accurate strategy to incorporate height information in SHS at densely spaced multiple points. Taylor series expansions of commonly used functionals quasigeoid heights, gravity disturbances and vertical deflections are formulated, and expressions of their radial derivatives are presented to arbitrary order. Numerical tests show that first-order gradients, as introduced by Rapp (J Geod 71(5):282–289, 1997) for degree 360 models, produce cm- to dm-level RMS approximation errors over rugged terrain when applied with EGM2008 to degree 2190. Instead, higher-order Taylor expansions are recommended that are capable of reducing approximation errors to insignificance for practical applications. Because the height information is separated from the actual synthesis, the gradient approach can be applied along with existing highly efficient SHS routines to compute surface functionals at arbitrarily dense grid points. This confers considerable computational savings (above or well above one order of magnitude) over conventional point-by-point SHS. As an application example, an ultra-high resolution model of surface gravity functionals (EurAlpGM2011) is constructed over the entire European Alps that incorporates height information in the SHS at 12,000,000 surface points. Based on EGM2008 and residual topography data, quasigeoid heights, gravity disturbances and vertical deflections are estimated at ~200m resolution. As a conclusion, the gradient approach is efficient and accurate for high-degree SHS at multiple points at the Earth’s surface.     

Global optimization of the Gauss conformal mappings of an ellipsoid to a sphere

The Gauss conformal mappings (GCMs) of an oblate ellipsoid of revolution to a sphere are those that transform the meridians into meridians, and the parallels into parallels of the sphere. The infinitesimal-scale function associated with these mappings depends on the geodetic latitude and contains three parameters, including the radius of the sphere. Gauss derived these constants by imposing local optimum conditions on certain parallel. We deal with the problem of finding the constants to minimize the Chebyshev or maximum norm of the logarithm of the infinitesimal-scale function on a given ellipsoidal segment (the region contained between two parallels). We show how to solve this minimax problem using the intrinsic function fminsearch of Matlab. For a particular ellipsoidal segment, we get the solution and show the alternation property characteristic of best Chebyshev approximations. For a pair of points relatively close in the ellipsoid at different latitudes, the best minimax GCM on the segment defined by these points is used to approximate the geodesic distance between them by the spherical distance between their projections on the corresponding sphere. This approach, combined with the best locally GCM if the points are on the same parallel, is illustrated by applying it to some case studies but specially to a 10° × 10° region contained between portions of two parallels and two meridians. In this case, the maximum absolute error of this spherical approximation is equal to 2.9 mm occurring at a distance about 1,360 km. This error decreases up to 0.94 mm on an 8° × 8° region of this type. So, the spherical approximation to the solution of the inverse geodesic problem by best GCM can be acceptable in many practical geodetic activities.     

Analysis of inversion errors of ionospheric radio occultation

The retrieved electron density profile of ionospheric radio occultation (RO) simulation data can be compared with the background model value during the simulation and the inversion error can be obtained exactly. This paper studies the inversion error of ionospheric RO through simulation. The sources of the inversion errors are analyzed. The impacts of measurement errors, such as the errors in phase measurements and satellite orbits, are very small and can be neglected. The approximation of straight-line propagation introduces errors at the height of the F1 layer under solar maximum condition. The spherical symmetry approximation of the electron density distribution is found to be the main source of the inversion error. The statistical results reveal some characteristics of the inversion errors. (1) The relative error increases with enhanced solar activity. (2) It is larger in winter than in equinox season, and it is smallest in summer. (3) For all seasons, it is smaller at middle latitude than at other latitudes. (4) For all seasons and geomagnetic latitudes, it is smaller at daytime than at other times. The NmF2 of the ROs from COSMIC are compared with the measurements of ionosondes, and the relative differences show the same dependencies on season, geomagnetic latitude and local time, as the relative errors of the simulated ionospheric ROs.

Simple and highly accurate formulas for the computation of Transverse Mercator coordinates from longitude and isometric latitude

A conformal approximation to the Transverse Mercator (TM) map projection, global in longitude λ and isometric latitude q, is constructed. New formulas for the point scale factor and grid convergence are also shown. Assuming that the true values of the TM coordinates are given by conveniently truncated Gauss–Krüger series expansions, we use the maximum norm of the absolute error to measure globally the accuracy of the approximation. For a Universal Transverse Mercator (UTM) zone the accuracy equals 0.21  mm, whereas for the region of the ellipsoid bounded by the meridians  ±20° the accuracy is equal to 0.3  mm. Our approach is based on a four-term perturbation series approximation to the radius r(q) of the parallel q, with a maximum absolute deviation of 0.43  mm. The small parameter of the power series expansion is the square of the eccentricity of the ellipsoid. This closed approximation to r(q) is obtained by solving a regularly perturbed Cauchy problem with the Poincaré method of the small parameter. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Algorithm for Harmonic Tidal Analysis along T/P Tracks： Taking Differences of Observed Sea Surface Heights at Adjacent Points as Observations

An algorithm （differential mode） is presented for the improvement of harmonic tidal analysis along T/P tracks, in which the differences between the observed sea surface heights at adjacent points are taken as observations. Also, the observation equations are constrained with the results of the crossover analysis; the parameter estimations are performed at 0.1° latitude intervals by the least squares. Cycle 10 to 330 T/P altimeter data covering the China Sea and the Northwest Pacific Ocean （2°-50° N,105°-150° E） are adopted for a refined along-track harmonic tidal analysis, and harmonic constants of 12 constituents in 8 474 points are obtained, which indicates that the algorithm can efficiently remove non-tidal effects in the altimeter observations, and improve the precision of tide parameters. Moreover, parameters along altimetry tracks represent a smoother distribution than those obtained by traditional algorithms. The root mean squares of the fitting errors between the tidal height model and the observations reduce from 11 cm to 1.3 cm.     

LONG LINES ON THE EARTH: VARIOUS FORMULAE

Abstract

Extensions were given for all these formulae, so that precise results may now be obtained even for lines of 500 miles in latitudes above 45°. The present instalment gives the extension of the Clarke approximate (sic) formulae to lines of 500miles, with a practical example and general conclusions: the great advantage of the method is that 8-figuretables sufficeto give rigorous results.     

A NOTE ON THE TRIGONOMETRICAL SURVEY OF SOUTHERN RHODESIA

Abstract

After the completion of Simms's Geodetic Chain in 1901 and the publication of the results in 1905—Volume iii of the Geodetic Survey of South Africa—nothing further of a geodetic nature was done until 1928 when a short chain was run westwards from Simms's chain, at about latitude 17° 10′, to fix the Copper Queen mining area. The Eastern Circuit was commenced shortly after this; it runs from Salisbury eastwards to the Portuguese Boundary, southwards through Umtali to about latitude 20° and then westwards, joining Simms's chain again to the east of Bulawayo. Another chain running north from Simms's work has been commenced near Bulawayo. The several series are exhibited on the outline map attached.     

Analysis on the global morphology of stratospheric gravity wave activity deduced from the COSMIC GPS occultation profiles

Monthly mean global morphologies of potential energy density E _p from stratospheric gravity waves are revealed by observations of COSMIC GPS radio occultation. The E _p is obtained from vertical wavelengths ranging from 2 to 10 km over cells of 1° × 2° in latitude and longitude. The computed values confirm previous results and obtain new ones. The large gravity wave E _p values found in the tropics between 25°N and 25°S could be mainly due to the strong tropical cumulus convection; July values are larger than those for January (2007). In mid and high latitudes, the most prominent features of the northern winter hemisphere are the enhanced densities above the Eurasian continent and the North Atlantic and the depressed E _p values above the North Pacific and North America for which topography, wind sources and wind filtering may be responsible. In southern winter hemisphere, large E _p values are found around 180° and 300° longitudes that are likely due to the topography of the Antarctic plateau, the Antarctic Peninsula and South America. Enhanced E _p values are found over Scandinavia. However, there is no clear evidence to show that gravity waves are localized over the Rocky Mountains, the Himalayas and the Andes. Topography and planetary wave modulations are proposed to interpret the large-scale longitudinal variations and inter-hemisphere asymmetry of the GW activity.     

A first look at the effects of ionospheric signal bending on a globally processed GPS network

This study provides a first attempt at quantifying potential signal bending effects on the GPS reference frame, coordinates and zenith tropospheric delays (ZTDs). To do this, we homogeneously reanalysed data from a global network of GPS sites spanning 14 years (1995.0–2009.0). Satellite, Earth orientation, tropospheric and ground station coordinate parameters were all estimated. We tested the effect of geometric bending and dTEC bending corrections, which were modelled at the observation level based, in part, on parameters from the International Reference Ionosphere 2007 model. Combined, the two bending corrections appear to have a minimal effect on site coordinates and ZTDs except for low latitude sites. Considering five days (DOY 301–305, 28 October–1 November 2001) near ionospheric maximum in detail, they affect mean ZTDs by up to ~1.7 mm at low latitudes, reducing to negligible levels at high latitudes. Examining the effect on coordinates in terms of power-spectra revealed the difference to be almost entirely white noise, with noise amplitude ranging from 0.3 mm (high latitudes) to 2.4 mm (low latitudes). The limited effect on station coordinates is probably due to the similarity in the elevation dependence of the bending term with that of tropospheric mapping functions. The smoothed z-translation from the GPS reference frame to ITRF2005 changes by less than 2 mm, though the effect combines positively with that from the second order ionospheric refractive index term. We conclude that, at the present time, and for most practical purposes, the geometric and dTEC bending corrections are probably negligible at current GPS/reference frame precisions.     

GEOGRAPHICAL POSITIONS IN MALAYA AND SIAM

Abstract

Malaya.—The geographical positions of points in the “Primary Triangulation of Malaya”, published in 1917, depend upon latitude and azimuth determinations at Bukit Asa and on the longitude of Fort Cornwallis Flagstaff, Penang, the latter being supposed to be 100° 20′ 44″.4 E. This value was obtained by Commander (later Admiral) Mostyn Field in H.M.S. Egeria 1893, by the exchange of telegraphic signals with Mr Angus Sutherland at Singapore, Old Transit Circle. The longitude, 103° 51′ 15″.75 E., accepted for Singa- pore in order to arrive at this determination of Fort Cornwallis Flagstaff, was based upon that of an Observation Spot, 103° 51′ 15″.00 E., fixed in 1881 by Lieutenant Commander Green, United States Navy, by meridian distance from the transit circle ofMadras Observatory, the corresponding longitude of the latter being taken as 80° 14′ 51″.51 E.