The Stokes and Vening-Meinesz functionals in a moving tangent space

The regularized solution of the external sphericalStokes boundary value problem as being used for computations of geoid undulations and deflections of the vertical is based upon theGreen functions S ₁(₀, ₀, , ) ofBox 0.1 (R = R ₀) andV ₁(₀, ₀, , ) ofBox 0.2 (R = R ₀) which depend on theevaluation point {₀, ₀} S _R0 ² and thesampling point {, } S _R0 ² ofgravity anomalies (, ) with respect to a normal gravitational field of typegm/R (free air anomaly). If the evaluation point is taken as the meta-north pole of theStokes reference sphere S _R0 ² , theStokes function, and theVening-Meinesz function, respectively, takes the formS() ofBox 0.1, andV ₂() ofBox 0.2, respectively, as soon as we introduce {meta-longitude (azimuth), meta-colatitude (spherical distance)}, namely {A, } ofBox 0.5. In order to deriveStokes functions andVening-Meinesz functions as well as their integrals, theStokes andVening-Meinesz functionals, in aconvolutive form we map the sampling point {, } onto the tangent plane T₀S _R0 ² at {₀, ₀} by means ofoblique map projections of type(i) equidistant (Riemann polar/normal coordinates),(ii) conformal and(iii) equiareal.Box 2.1.–2.4. andBox 3.1.– 3.4. are collections of the rigorously transformedconvolutive Stokes functions andStokes integrals andconvolutive Vening-Meinesz functions andVening-Meinesz integrals. The graphs of the correspondingStokes functions S ₂(),S ₃(r),,S ₆(r) as well as the correspondingStokes-Helmert functions H ₂(),H ₃(r),,H ₆(r) are given byFigure 4.1–4.5. In contrast, the graphs ofFigure 4.6–4.10 illustrate the correspondingVening-Meinesz functions V ₂(),V ₃(r),,V ₆(r) as well as the correspondingVening-Meinesz-Helmert functions Q ₂(),Q ₃(r),,Q ₆(r). The difference between theStokes functions / Vening-Meinesz functions andtheir first term (only used in the Flat Fourier Transforms of type FAST and FASZ), namelyS ₂() – (sin /2)^–1,S ₃(r) – (sinr/2R ₀)^–1,,S ₆(r) – 2R ₀/r andV ₂() + (cos /2)/2(sin² /2),V ₃(r) + (cosr/2R ₀)/2(sin² r/2R ₀),, illustrate the systematic errors in theflat Stokes function 2/ or flatVening-Meinesz function –2/². The newly derivedStokes functions S ₃(r),,S ₆(r) ofBox 2.1–2.3, ofStokes integrals ofBox 2.4, as well asVening-Meinesz functionsV ₃(r),,V ₆(r) ofBox 3.1–3.3, ofVening-Meinesz integrals ofBox 3.4 — all of convolutive type — pave the way for the rigorousFast Fourier Transform and the rigorousWavelet Transform of theStokes integral / theVening-Meinesz integral of type equidistant, conformal and equiareal.     

The oblique azimuthal projection of geodesic type for the biaxial ellipsoid: Riemann polar and normal coordinates

Summary Riemann polar/normal coordinates are the constituents to generate the oblique azimuthal projection of geodesic type, here applied to the reference ellipsoid of revolution (biaxial ellipsoid).Firstly we constitute a minimal atlas of the biaxial ellipsoid built on {ellipsoidal longitude, ellipsoidal latitude} and {metalongitude, metalatitude}. TheDarboux equations of a 1-dimensional submanifold (curve) in a 2-dimensional manifold (biaxial ellipsoid) are reviewed, in particular to represent geodetic curvature, geodetic torsion and normal curvature in terms of elements of the first and second fundamental form as well as theChristoffel symbols. The notion of ageodesic anda geodesic circle is given and illustrated by two examples. The system of twosecond order ordinary differential equations of ageodesic (Lagrange portrait) is presented in contrast to the system of twothird order ordinary differential equations of ageodesic circle (Proofs are collected inAppendix A andB). A precise definition of theRiemann mapping/mapping of geodesics into the local tangent space/tangent plane has been found.Secondly we computeRiemann polar/normal coordinates for the biaxial ellipsoid, both in theLagrange portrait (Legendre series) and in theHamilton portrait (Lie series).Thirdly we have succeeded in a detailed deformation analysis/Tissot distortion analysis of theRiemann mapping. The eigenvalues — the eigenvectors of the Cauchy-Green deformation tensor by means of ageneral eigenvalue-eigenvector problem have been computed inTable 3.1 andTable 3.2 (₁, ₂ = 1) illustrated inFigures 3.1, 3.2 and3.3. Table 3.3 contains the representation ofmaximum angular distortion of theRiemann mapping. Fourthly an elaborate global distortion analysis with respect toconformal Gau-Krüger, parallel Soldner andgeodesic Riemann coordinates based upon theAiry total deformation (energy) measure is presented in a corollary and numerically tested inTable 4.1. In a local strip [-l _E,l _E] = [-2°, +2°], [b _S,b _N] = [-2°, +2°]Riemann normal coordinates generate the smallest distortion, next are theparallel Soldner coordinates; the largest distortion by far is met by theconformal Gau-Krüger coordinates. Thus it can be concluded that for mapping of local areas of the biaxial ellipsoid surface the oblique azimuthal projection of geodesic type/Riemann polar/normal coordinates has to be favored with respect to others.     

Validation of ETOPO5U in the Hellenic area

Mean 5 × 5 heights and depths from ETOPO5U (Earth Topography at 5 spacing Updated) Digital Terrain Model (DTM) were compared with corresponding quantities of a local DTM in the test area [38° 40°, 21° 24°]. From this comparison a shift of ETOPO5U with respect to the local DTM in the longitudinal direction equal to 5 min was found after applying an efficient fast Fourier transform (FFT) technique. Furthermore, sparse mean height differences larger than 1,000 m were observed between ETOPO5U and the local DTM due rather to errors of ETOPO5U. The effect of these errors on gravity and height anomalies was computed in a subregion of the area under consideration.     

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.     

Interannual variability of low-degree gravitational change, 1980–2002

Time variations in the Earths gravity field at periods longer than 1 year, for degree-two spherical harmonics, C₂₁, S₂₁, and C₂₀, are estimated from accurately measured Earth rotational variations. These are compared with predictions of atmospheric, oceanic, and hydrologic models, and with independent satellite laser ranging (SLR) results. There is remarkably good agreement between Earth rotation and model predictions of C₂₁ and S₂₁ over a 22-year period. After decadal signals are removed, Earth-rotation-derived interannual C₂₀ variations are dominated by a strong oscillation of period about 5.6 years, probably due to uncertainties in wind and ocean current estimates. The model-predicted C₂₀ agrees reasonably well with SLR observations during the 22-year period, with the exception of the recent anomaly since 1997/1998.     

Elliptical harmonic series and the original Stokes problem with the boundary of the reference ellipsoid

Since the earth is closer to a revolving ellipsoid than a sphere, it is very important to study directly the original model of the Stokes' BVP on the reference ellipsoid, where denotes the reference ellipsoid, is the Somigliana normal gravity, andh is the outer normal direction of. This paper deals with: 1) simplification of the above BVP under preserving accuracy to , 2) derivation of computational formula of the elliptical harmonic series, 3) solving the BVP by the elliptical harmonic series, and 4) providing a principle for finding the elliptical harmonic model of the earth's gravity field from the spherical harmonic coefficients ofg. All results given in the paper have the same accuracy as the original BVP, that is, the accuracy of the BVP is theoretically preserved in each derivation step.     

Geoid determination in Turkey (TG-91)

It is considered that precise geoid determination is one of the main current geodetic problems in Turkey since GPS defined coordinates require geoidal heights in practice. In order to determine the geoid by least squares collocation (LSC) the area covering Turkey was divided into 114 blocks of size 1° × 1°. LSC approximation to the geoid based upon the tailored geopotential model GPM2-T1 is constructed within each block. The model GPM2-T1 complete to degree and order 200 has been developed by tailoring of the model GPM2 to mean free-air anomalies and mean heights of one degree blocks in Turkey. Terrain effect reduced point gravity data spaced 5 × 5 within each block which the sides extended 0°.5 were used in LSC. Residual terrain model (RTM) depends on point heights at 15×20 griding and 5×5 and 15×15 mean heights has been carried out in terrain effect reduction. Indirect effect of RTM on geoid is also taken into account. The geoid, called Turkish Geoid 1991 (TG-91), referenced to GRS-80 ellipsoid has been computed at 3 × 3 griding nodes within each block. The quality of the TG-91 is also evaluated by comparing computed and GPS derived geoidal height differences, and 2.1 – 2.6 ppm accuracy for average baseline lenght of 45 km is obtained.     

On the separation of gravitation and inertia and the determination of the relativistic gravity field in the case of free motion

Summary The authors explored the possibility of separating gravitation from inertia in the frame of general relativity. The Riemann tensor is intimately related with gravitational fields and has nothing to do with inertial effects. One can judge the existence or nonexistence of a gravitational field according as the Riemann tensor does not vanish or vanishes. In the free fall case, by using a gradiometer on a satellite, gravitational effects can be separated from inertia completely. Furthermore, the authors put forward a general method of determining the relativistic gravity field by using gradiometers mounted on satellites. At the same time the following two statements are proved: in the case of using gradiometers on a satellite, with some kind of approximation the Riemann tensorR can be found; in the case of free motion, if the measured Riemannian componentsR _(i0j0) are equal to zero, the Riemann tensorR equals zero.     

A comparison of a Hi/Lo GPS constellation with a populated conventional GPS constellation in support of RTK: A covariance analysis

This study makes an initial comparison of three GPS-like constellations. Starting with a simplified constellation of 25 GPS satellites as a reference, GPS(25), we determine what kinematic positioning improvements would result from a constellation comprising a Hi component of 16 GPS satellites (at roughly 16.8 earth radii) coupled with a Lo component of 49 GPS satellites (at roughly 2.1 earth radii). We also include a GPS constellation of 49 GPS satellites, GPS(49), which comprises orbits like the GPS(25) constellation. The GPS(49) and the Hi(16)/Lo(49) constellations have semi-major axes selected so that they have exactly the same average number of satellites above 7.5 degrees elevation (averaged over 24 hours). What motivated this study was a need to measure the benefits, to precision differential kinematic positioning methods (i.e., RTK), which result from the higher Doppler shifts (hence speedier integrated Doppler) generated by the Lo component. Quicker initial convergence was anticipated, of course.     

Intuitive derivation of loop inverses and array algebra

Array algebra forms the general base of fast transforms and multilinear algebra making rigorous solutions of a large number (millions) of parameters computationally feasible. Loop inverses are operators solving the problem of general matrix inverses. Their derivation starts from the inconsistent linear equations by a parameter exchangeX→L ₀, where X is a set of unknown observables,A ₀ forming a basis of the so called “problem space”. The resulting full rank design matrix of parameters L₀ and its ℓ-inverse reveal properties speeding the computational least squares solution expressed in observed values . The loop inverses are found by the back substitution expressing ∧X in terms ofL through . Ifp=rank (A) ≤n, this chain operator creates the pseudoinverseA ⁺. The idea of loop inverses and array algebra started in the late60's from the further specialized case,p=n=rank (A), where the loop inverse A ₀ ⁻¹ (AA ₀ ⁻¹ )^ℓ reduces into the ℓ-inverse A^ℓ=(A^TA)⁻¹A^T. The physical interpretation of the design matrixA A ₀ ⁻¹ as an interpolator, associated with the parametersL ₀, and the consideration of its multidimensional version has resulted in extended rules of matrix and tensor calculus and mathematical statistics called array algebra.     

Groebner-basis solution of the three-dimensional resection problem (P4P)

The three-dimensional (3-D) resection problem is usually solved by first obtaining the distances connecting the unknown point P{X,Y,Z} to the known points P_i{X_i,Y_i,Z_i}i=1,2,3 through the solution of the three nonlinear Grunert equations and then using the obtained distances to determine the position {X,Y,Z} and the 3-D orientation parameters {,, }. Starting from the work of the German J. A. Grunert (1841), the Grunert equations have been solved in several substitutional steps and the desire as evidenced by several publications has been to reduce these number of steps. Similarly, the 3-D ranging step for position determination which follows the distance determination step involves the solution of three nonlinear ranging (`Bogenschnitt') equations solved in several substitution steps. It is illustrated how the algebraic technique of Groebner basis solves explicitly the nonlinear Grunert distance equations and the nonlinear 3-D ranging (`Bogenschnitt') equations in a single step once the equations have been converted into algebraic (polynomial) form. In particular, the algebraic tool of the Groebner basis provides symbolic solutions to the problem of 3-D resection. The various forward and backward substitution steps inherent in the classical closed-form solutions of the problem are avoided. Similar to the Gauss elimination technique in linear systems of equations, the Groebner basis eliminates several variables in a multivariate system of nonlinear equations in such a manner that the end product normally consists of a univariate polynomial whose roots can be determined by existing programs e.g. by using the roots command in Matlab.Acknowledgments.The first author wishes to acknowledge the support of JSPS (Japan Society of Promotion of Science) for the financial support that enabled the completion of the write-up of the paper at Kyoto University, Japan. The author is further grateful for the warm welcome and the good working atmosphere provided by his hosts Professors S. Takemoto and Y. Fukuda of the Department of Geophysics, Graduate School of Science, Kyoto University, Japan.     

Horizontal strain and precision of EDM from repeated surveys of the Precise Geodetic Net of Japan

A simple statistical approach has been applied to the repeated electro-optical distance measurements (EDM) of 1,358 lines in the Tohoku district of Japan to obtain knowledge about the precision of EDM and the possible accumulation of strain. The average time interval between measurements is about seven or eight years. It is shown that the whole data of the difference between distance measurements repeated over a given lineD are interpreted in terms of EDM errors comprising distance proportional systematic errors and standard errors expressed by the usual form . The rate of horizontal deformation must therefore be much smaller than the strain rates of about 0.7 0.8 ppm over 7 to 8 years which have been hitherto expected.     

High-resolution regional geoid computation without applying Stokes’s formula: a case study of the Iranian geoid

A new theory for high-resolution regional geoid computation without applying Stokess formula is presented. Operationally, it uses various types of gravity functionals, namely data of type gravity potential (gravimetric leveling), vertical derivatives of the gravity potential (modulus of gravity intensity from gravimetric surveys), horizontal derivatives of the gravity potential (vertical deflections from astrogeodetic observations) or higher-order derivatives such as gravity gradients. Its algorithmic version can be described as follows: (1) Remove the effect of a very high degree/order potential reference field at the point of measurement (POM), in particular GPS positioned, either on the Earths surface or in its external space. (2) Remove the centrifugal potential and its higher-order derivatives at the POM. (3) Remove the gravitational field of topographic masses (terrain effect) in a zone of influence of radius r. A proper choice of such a radius of influence is 2r=4×10⁴ km/n, where n is the highest degree of the harmonic expansion. (cf. Nyquist frequency). This third remove step aims at generating a harmonic gravitational field outside a reference ellipsoid, which is an equipotential surface of a reference potential field. (4) The residual gravitational functionals are downward continued to the reference ellipsoid by means of the inverse solution of the ellipsoidal Dirichlet boundary-value problem based upon the ellipsoidal Abel–Poisson kernel. As a discretized integral equation of the first kind, downward continuation is Phillips–Tikhonov regularized by an optimal choice of the regularization factor. (5) Restore the effect of a very high degree/order potential reference field at the corresponding point to the POM on the reference ellipsoid. (6) Restore the centrifugal potential and its higher-order derivatives at the ellipsoidal corresponding point to the POM. (7) Restore the gravitational field of topographic masses ( terrain effect) at the ellipsoidal corresponding point to the POM. (8) Convert the gravitational potential on the reference ellipsoid to geoidal undulations by means of the ellipsoidal Bruns formula. A large-scale application of the new concept of geoid computation is made for the Iran geoid. According to the numerical investigations based on the applied methodology, a new geoid solution for Iran with an accuracy of a few centimeters is achieved.Acknowledgments. The project of high-resolution geoid computation of Iran has been support by National Cartographic Center (NCC) of Iran. The University of Tehran, via grant number 621/3/602, supported the computation of a global geoid solution for Iran. Their support is gratefully acknowledged. A. Ardalan would like to thank Mr. Y. Hatam, and Mr. K. Ghazavi from NCC and Mr. M. Sharifi, Mr. A. Safari, and Mr. M. Motagh from the University of Tehran for their support in data gathering and computations. The authors would like to thank the comments and corrections made by the four reviewers and the editor of the paper, Professor Will Featherstone. Their comments helped us to correct the mistakes and improve the paper.     

Theory of integer equivariant estimation with application to GNSS

Carrier phase ambiguity resolution is the key to high-precision global navigation satellite system (GNSS) positioning and navigation. It applies to a great variety of current and future models of GPS, modernized GPS and Galileo. The so-called fixed baseline estimator is known to be superior to its float counterpart in the sense that its probability of being close to the unknown but true baseline is larger than that of the float baseline, provided that the ambiguity success rate is sufficiently close to its maximum value of one. Although this is a strong result, the necessary condition on the success rate does not make it hold for all measurement scenarios. It is discussed whether or not it is possible to take advantage of the integer nature of the ambiguities so as to come up with a baseline estimator that is always superior to both its float and its fixed counterparts. It is shown that this is indeed possible, be it that the result comes at the price of having to use a weaker performance criterion. The main result of this work is a Gauss–Markov-like theorem which introduces a new minimum variance unbiased estimator that is always superior to the well-known best linear unbiased (BLU) estimator of the Gauss–Markov theorem. This result is made possible by introducing a new class of estimators. This class of integer equivariant estimators obeys the integer remove–restore principle and is shown to be larger than the class of integer estimators as well as larger than the class of linear unbiased estimators. The minimum variance unbiased estimator within this larger class is referred to as the best integer equivariant (BIE) estimator. The theory presented applies to any model of observation equations having both integer and real-valued parameters, as well as for any probability density function the data might have. AcknowledgementsThis contribution was finalized during the authors stay, as a Tan Chin Tuan Professor, at the Nanyang Technological Universitys GPS Centre (GPSC) in Singapore. The hospitality of the GPSCs director Prof Law Choi Look and his colleagues is greatly appreciated.     

A general model for modifying Stokes’ formula and its least-squares solution

Today the combination of Stokes formula and an Earth gravity model (EGM) for geoid determination is a standard procedure. However, the method of modifying Stokes formula varies from author to author, and numerous methods of modification exist. Most methods modify Stokes kernel, but the most widely applied method, the remove compute restore technique, removes the EGM from the gravity anomaly to attain a residual gravity anomaly under Stokes integral, and at least one known method modifies both Stokes kernel and the gravity anomaly. A general model for modifying Stokes formula is presented; it includes most of the well-known techniques of modification as special cases. By assuming that the error spectra of the gravity anomalies and the EGM are known, the optimum model of modification is derived based on the least-squares principle. This solution minimizes the expected mean square error (MSE) of all possible solutions of the general geoid model. A practical formula for estimating the MSE is also presented. The power of the optimum method is demonstrated in two special cases. AcknowledgementsThis paper was partly written whilst the author was a visiting scientist at The University of New South Wales, Sydney, Australia. He is indebted to Professor W. Kearsley and his colleagues, and their hospitality is acknowledged.     

Monitoring Earth orientation using space-geodetic techniques: state-of-the-art and prospective

Earth orientation parameters (EOPs) provide the transformation between the International Terrestrial Reference Frame (ITRF) and the International Celestial Reference Frame (ICRF). The different EOP series computed at the Earth Orientation Centre at the Paris Observatory are obtained from the combination of individual EOP series derived from the various space-geodetic techniques. These individual EOP series contain systematic errors, generally limited to biases and drifts, which introduce inconsistencies between EOPs and the terrestrial and celestial frames. The objectives of this paper are first to present the various combined EOP solutions made available at the EOP Centre for the different users, and second to present analyses concerning the long-term consistency of the EOP system with respect to both terrestrial and celestial reference frames. It appears that the present accuracy in the EOP combined IERS C04 series, which is at the level of 200 as for pole components and 20 s for UT1, does not match its internal precision, respectively 100 as and 5 s, because of propagation errors in the realization of the two reference frames. Rigorous combination methods based on a simultaneous estimation of station coordinates and EOPs, which are now being implemented within the International Earth Rotation Service (IERS), are likely to solve this problem in the future.     

A calibration system for superconducting gravimeters

A new method for the calibration of a superconducting gravity meter is described, in which a 273 Kg annular mass is placed around the meter and is moved up and down. The geometry of the apparatus is easy to model and the accuracy in the computation of the gravity variation induced by the mass, 6.7µgal, is limited only by the accuracy in the knowledge of value of the gravitational constant. Measurements done in 91 and 92 for the calibration of the instrument GWR-T015 are described. The calibration factor has been determined with a precision of about 0.3%.     

Atmospheric tides and rotation of the Earth

The six-hourly values of the atmospheric angular momentum (AAM) functions computed by the U.S. National Meteorological Center (NMC) were used to estimate the effects of the atmospheric tides on the Earth's rotation. Variations of the equatorial components ₁ and ₂ of the AAM have periods close to gravitational tidesP ₁ andK ₁.The amplitudes of the detected variations in ₁ and ₂ functions have been found to be much larger than the theoretical ones, the reason of this amplification remains unexplained. According to theoretical formulations, these waves can be expressed only as retrograde motions. Because of frame effects, there is a correspondance between diurnal retrograde polar motion and precession-nutations and the atmospheric effect on polar motion cannot be detected from observations.The second part of this paper deals the effects of atmospheric tides in Earth rotation. High-frequency UT1 variations have been derived from VLBI and GPS techniques during the SEARCH'92 campaign (Study ofEarth-AtmosphereRapidCHanges) (Dickey et al. 1994). They have been compared to values derived by Ray et al. (1994) from global ocean tide model. The results obtained in the present paper show the existence of variations of thermal origin with an amplitude of about 1µs in Universal Time UT1. The agreement between observed and theoretical values is better when the determined thermal atmospheric tides are taken into account.Oceanic tidal signal explains a large part (60% of the signal variance) of the diurnal and sub-diurnal variations. Our results show that only a small part of the residuals (5%) accounts for the atmospheric tidal effects. The residual signal remains unexplained; it might be due to mismodelization of oceanic or atmospheric tides or effect of other geophysical phenomena.     

Marine gravity surveying line system adjustment

The general theories and methods of marine surveying line system adjustment were introduced in (1979) and Tang (1991) . According to the characteristics of marine gravity measurement, this paper presents a new method of combined adjustment which takes into account both direct and indirect influence of position errors. The method is particularly suitable to be used in the post- processing of marine gravity observation data. With some practical applications, it is proved to be effective in improving the quality of marine gravity data.     

The common short periodic oscillations in the Earth's rate of rotation,atmospheric angular momentum and solar activity

In order to find short periodic oscillations in the Earth's rate of rotation, atmospheric angular momentum, solar activity the Maximum Entropy Spectral Analysis — MESA (Burg, 1967) has been applied. The MESA with moving autoregressive order has been introduced in order to detect more accurately periods of very weak short periodic variations. Oscillations with periods of about 75, 50, 27 and 18 days have been found in length of day — LOD, from which tidal oscillations were removed up to 35 days — LODR computed by the Center for Space Research — CSR from Lageos Laser Ranging data, in the axial component of atmospheric angular momentum — ₃ determined by the U.S. National Meteorological Center — NMC and in the geomagnetic activity represented by the geomagnetic index —A _p (Lincoln, 1967). These oscillations computed by Ormsby band pass filter (Ormsby, 1961) are in a very good phase agreement in the case of oscillations with periods of 50 and 18 days in these 3 series. The MESA of the cross covariance estimations between LODR- ₃, ₃-A _p,A _p-LODR, LODR-FLUX, ₃-FLUX, andA _p-FLUX has confirmed the existence of common oscillations with periods of 70, 50, 27 and 18 days. This indicates a possible relationship between solar activity and the short periodic exchange of angular momentum between the atmosphere and the solid Earth.