The Hammer projection of the ellipsoid of revolution (azimuthal, transverse, rescaled equiareal)

The classical Hammer projection of the sphere, which is azimuthal transverse rescaled equiareal, is generalized to the ellipsoid of revolution. Its first constituent, the azimuthal transverse equiareal projection of the biaxial ellipsoid, is derived giving equations for an equiareal transverse azimuthal projection. The second constituent, the equiareal mapping of the biaxial ellipsoid with respect to a transverse frame of reference and a change of scale, is then considered; results give collections of the general mapping equations generating the ellipsoidal Hammer projection, which lead to a world map. Received: 14 May 1996 / Accepted: 21 February 1997     

Mixed cylindric map projections of the ellipsoid of revolution

The mixed spherical map projections of equiareal, cylindric type are based upon the Lambert projection and the sinusoidal Sanson–Flamsteed projection. These cylindric and pseudo-cylindric map projections of the sphere are generalized to the ellipsoid of revolution (biaxial ellipsoid). They are used in consequence by two lemmas to generate a horizontal and a vertical weighted mean of equiareal cylindric map projections of the ellipsoid of revolution. Its left–right deformation analysis via further results leads to the left–right principal stretches/eigenvalues and left–right eigenvectors/eigenspace, as well as the maximal left–right angular distortion for these new mixed cylindric map projections of ellipsoidal type. Detailed illustrations document the cartographic synergy of mixed cylindric map projections. Received: 23 April 1996 / Accepted: 19 April 1997     

Estimation and modelling of the local empirical covariance function using gravity and satellite altimeter data

The estimation of a local empirical covariance function from a set of observations was done in the Faeroe Islands region. Gravity and adjusted Seasat altimeter data relative to theGPM2 spherical harmonic approximation were selected holding one value in celles of1/8°×1/4° covering the area. In order to center the observations they were transformed into a locally best fitting reference system having a semimajor axis1.8 m smaller than the one ofGRS80. The variance of the data then was273 mgal ² and0.12 m ² respectively. In the calculations both the space domain method and the frequency domain method were used. Using the space domain method the auto-covariances for gravity anomalies and geoid heights and the cross-covariances between the quantities were estimated. Furthermore an empirical error estimate was derived. Using the frequency domain method the auto-covariances of gridded gravity anomalies was estimated. The gridding procedure was found to have a considerable smoothing effect, but a deconvolution made the results of the two methods to agree. The local covariance function model was represented by a Tscherning/Rapp degree-variance model,A/((i−1)(i−2)(i+24))(R _B /R _E )²ⁱ⁺², and the error degree-variances related to the potential coefficient setGPM2. This covariance function was adjusted to fit the empirical values using an iterative least squares inversion procedure adjusting the factor A, the depth to the Bjerhammar sphere(R _E −R _B ), and a scale factor associated with the error degree-variances. Three different combinations of the empirical covariance values were used. The scale factor was not well determined from the gravity anomaly covariance values, and the depth to the Bjerhammar sphere was not well determined from geoid height covariance values only. A combination of the two types of auto-covariance values resulted in a well determined model.     

The optimal Mercator projection and the optimal polycylindric projection of conformal type – case-study Indonesia

As a conformal mapping of the sphere S ² _R or of the ellipsoid of revolution E ² _A , _B the Mercator projection maps the equator equidistantly while the transverse Mercator projection maps the transverse metaequator, the meridian of reference, with equidistance. Accordingly, the Mercator projection is very well suited to geographic regions which extend east-west along the equator; in contrast, the transverse Mercator projection is appropriate for those regions which have a south-north extension. Like the optimal transverse Mercator projection known as the Universal Transverse Mercator Projection (UTM), which maps the meridian of reference Λ₀ with an optimal dilatation factor &ρcirc;=0.999 578 with respect to the World Geodetic Reference System WGS 84 and a strip [Λ₀−Λ _W ,Λ₀ + Λ _E ]×[Φ _S ,Φ _N ]= [−3.5^∘,+3.5^∘]×[−80^∘,+84^∘], we construct an optimal dilatation factor ρ for the optimal Mercator projection, summarized as the Universal Mercator Projection (UM), and an optimal dilatation factor ρ₀ for the optimal polycylindric projection for various strip widths which maps parallel circles Φ₀ equidistantly except for a dilatation factor ρ₀, summarized as the Universal Polycylindric Projection (UPC). It turns out that the optimal dilatation factors are independent of the longitudinal extension of the strip and depend only on the latitude Φ₀ of the parallel circle of reference and the southern and northern extension, namely the latitudes Φ _S and Φ _N , of the strip. For instance, for a strip [Φ _S ,Φ _N ]= [−1.5^∘,+1.5^∘] along the equator Φ₀=0, the optimal Mercator projection with respect to WGS 84 is characterized by an optimal dilatation factor &ρcirc;=0.999 887 (strip width 3^∘). For other strip widths and different choices of the parallel circle of reference Φ₀, precise optimal dilatation factors are given. Finally the UPC for the geographic region of Indonesia is presented as an example. Received: 17 December 1997 / Accepted: 15 August 1997     

A new method for computing the ellipsoidal correction for Stokes's formula

 This paper generalizes the Stokes formula from the spherical boundary surface to the ellipsoidal boundary surface. The resulting solution (ellipsoidal geoidal height), consisting of two parts, i.e. the spherical geoidal height N ₀ evaluated from Stokes's formula and the ellipsoidal correction N ₁, makes the relative geoidal height error decrease from O(e ²) to O(e ⁴), which can be neglected for most practical purposes. The ellipsoidal correction N ₁ is expressed as a sum of an integral about the spherical geoidal height N ₀ and a simple analytical function of N ₀ and the first three geopotential coefficients. The kernel function in the integral has the same degree of singularity at the origin as the original Stokes function. A brief comparison among this and other solutions shows that this solution is more effective than the solutions of Molodensky et al. and Moritz and, when the evaluation of the ellipsoidal correction N ₁ is done in an area where the spherical geoidal height N ₀ has already been evaluated, it is also more effective than the solution of Martinec and Grafarend. Received: 27 January 1999 / Accepted: 4 October 1999     

Ellipsoidal terrain correction based on multi-cylindrical equal-area map projection of the reference ellipsoid

An operational algorithm for computation of terrain correction (or local gravity field modeling) based on application of closed-form solution of the Newton integral in terms of Cartesian coordinates in multi-cylindrical equal-area map projection of the reference ellipsoid is presented. Multi-cylindrical equal-area map projection of the reference ellipsoid has been derived and is described in detail for the first time. Ellipsoidal mass elements with various sizes on the surface of the reference ellipsoid are selected and the gravitational potential and vector of gravitational intensity (i.e. gravitational acceleration) of the mass elements are computed via numerical solution of the Newton integral in terms of geodetic coordinates {,,h}. Four base- edge points of the ellipsoidal mass elements are transformed into a multi-cylindrical equal-area map projection surface to build Cartesian mass elements by associating the height of the corresponding ellipsoidal mass elements to the transformed area elements. Using the closed-form solution of the Newton integral in terms of Cartesian coordinates, the gravitational potential and vector of gravitational intensity of the transformed Cartesian mass elements are computed and compared with those of the numerical solution of the Newton integral for the ellipsoidal mass elements in terms of geodetic coordinates. Numerical tests indicate that the difference between the two computations, i.e. numerical solution of the Newton integral for ellipsoidal mass elements in terms of geodetic coordinates and closed-form solution of the Newton integral in terms of Cartesian coordinates, in a multi-cylindrical equal-area map projection, is less than 1.6×10^–8 m²/s² for a mass element with a cross section area of 10×10 m and a height of 10,000 m. For a mass element with a cross section area of 1×1 km and a height of 10,000 m the difference is less than 1.5×10^–4m²/s². Since 1.5× 10^–4 m²/s² is equivalent to 1.5×10^–5m in the vertical direction, it can be concluded that a method for terrain correction (or local gravity field modeling) based on closed-form solution of the Newton integral in terms of Cartesian coordinates of a multi-cylindrical equal-area map projection of the reference ellipsoid has been developed which has the accuracy of terrain correction (or local gravity field modeling) based on the Newton integral in terms of ellipsoidal coordinates.Acknowledgments. This research has been financially supported by the University of Tehran based on grant number 621/4/859. This support is gratefully acknowledged. The authors are also grateful for the comments and corrections made to the initial version of the paper by Dr. S. Petrovic from GFZ Potsdam and the other two anonymous reviewers. Their comments helped to improve the structure of the paper significantly.     

Somigliana–Pizzetti gravity: the international gravity formula accurate to the sub-nanoGal level

 The Somigliana–Pizzetti gravity field (the International gravity formula), namely the gravity field of the level ellipsoid (the International Reference Ellipsoid), is derived to the sub-nanoGal accuracy level in order to fulfil the demands of modern gravimetry (absolute gravimeters, super conducting gravimeters, atomic gravimeters). Equations (53), (54) and (59) summarise Somigliana–Pizzetti gravity Γ(φ,u) as a function of Jacobi spheroidal latitude φ and height u to the order ?(10⁻¹⁰ Gal), and Γ(B,H) as a function of Gauss (surface normal) ellipsoidal latitude B and height H to the order ?(10⁻¹⁰ Gal) as determined by GPS (`global problem solver'). Within the test area of the state of Baden-Württemberg, Somigliana–Pizzetti gravity disturbances of an average of 25.452 mGal were produced. Computer programs for an operational application of the new international gravity formula with (L,B,H) or (λ,φ,u) coordinate inputs to a sub-nanoGal level of accuracy are available on the Internet. Received: 23 June 2000 / Accepted: 2 January 2001     

Generalized inverses of nonlinear mappings and the nonlinear geodetic datum problem

Motivated by the existing theory of the geometric characteristics of linear generalized inverses of linear mappings, an attempt is made to establish a corresponding mathematical theory for nonlinear generalized inverses of nonlinear mappings in finite- dimensional spaces. The theory relies on the concept of fiberings consisting of disjoint manifolds (fibers) in which the domain and range spaces of the mappings are partitioned. Fiberings replace the quotient spaces generated by some characteristic subspaces in the linear case. In addition to the simple generalized inverse, the minimum-distance and the x ₀-nearest generalized inverse are introduced and characterized, in analogy with the least-squares and the minimum-norm generalized inverses of the linear case. The theory is specialized to the geodetic mapping from network coordinates to observables and the nonlinear transformations (Baarda's S-transformations) between different solutions are defined with the help of transformation parameters obtained from the solution of nonlinear equations. In particular, the transformations from any solution to an x ₀-nearest solution (corresponding to Meissl's inner solution) are given for two- and three-dimensional networks for both the similarity and the rigid transformation case. Finally the nonlinear theory is specialized to the linear case with the help of the singular-value decomposition and algebraic expressions with specific geometric meaning are given for all possible types of generalized inverses. Received: 11 April 1996 / Accepted: 19 April 1997     

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.     

Digitizing the thermal and hydrological parameters of land surface in subtropical China using AMSR-E brightness temperatures

ABSTRACT

Digitizing the land surface temperature (T_s) and surface soil moisture (m_v) is essential for developing the intelligent Digital Earth. Here, we developed a two parameter physical-based passive microwave remote sensing model for jointly retrieving T_s and m_v using the dual-polarized T_b of Aqua satellite advanced microwave scanning radiometer (AMSR-E) C-band (6.9 GHz) based on the simplified radiative transfer equation. Validation using in situ T_s and m_v in southern China showed the average root mean square errors (RMSE) of T_s and m_v retrievals reach 2.42 K (R² = 0.61, n = 351) and 0.025 g cm^?3 (R² = 0.68, n = 663), respectively. The results were also validated using global in situ T_s (n = 2362) and m_v (n = 1657) of International Soil Moisture Network. The corresponding RMSE are 3.44 k (R² = 0.86) and 0.039 g cm^?3 (R² = 0.83), respectively. The monthly variations of model-derived Ts and mv are highly consistent with those of the Moderate Resolution Imaging Spectroradiometer T_s (R² = 0.57; RMSE = 2.91 k) and ECV_SM m_v (R² = 0.51; RMSE = 0.045 g cm^?3), respectively. Overall, this paper indicates an effective way to jointly modeling T_s and m_v using passive microwave remote sensing.     

National height datum,the Gauss–Listing geoid level value w0 and its time variation 0 (Baltic Sea Level Project: epochs 1990.8, 1993.8, 1997.4)

 A methodology for precise determination of the fundamental geodetic parameter w ₀, the potential value of the Gauss–Listing geoid, as well as its time derivative ₀, is presented. The method is based on: (1) ellipsoidal harmonic expansion of the external gravitational field of the Earth to degree/order 360/360 (130 321 coefficients; http://www.uni-stuttgard.de/gi/research/ index.html projects) with respect to the International Reference Ellipsoid WGD2000, at the GPS positioned stations; and (2) ellipsoidal free-air gravity reduction of degree/order 360/360, based on orthometric heights of the GPS-positioned stations. The method has been numerically tested for the data of three GPS campaigns of the Baltic Sea Level project (epochs 1990.8,1993.4 and 1997.4). New w ₀ and ₀ values (w ₀=62 636 855.75 ± 0.21 m²/s², ₀=−0.0099±0.00079 m²/s² per year, w ₀/&γmacr;=6 379 781.502 m,₀/&γmacr;=1.0 mm/year, and &γmacr;= −9.81802523 m²/s²) for the test region (Baltic Sea) were obtained. As by-products of the main study, the following were also determined: (1) the high-resolution sea surface topography map for the Baltic Sea; (2) the most accurate regional geoid amongst four different regional Gauss–Listing geoids currently proposed for the Baltic Sea; and (3) the difference between the national height datums of countries around the Baltic Sea. Received: 14 August 2000 / Accepted: 19 June 2001     

Estimation of higher chlorophylla concentrations using field spectral measurement and HJ-1A hyperspectral satellite data in Dianshan Lake,China

Based on in situ water sampling and field spectral measurements in Dianshan Lake, a semi-analytical three-band algorithm was used to estimate Chlorophylla (Chla) content in case II waters. The three bands selected to estimate Chla for high concentrations included 653, 691 and 748 nm. An equation, based on the difference in reciprocal reflectance between 653 and 691 nm, multiplied by reflectance at 748 nm as [R_rs⁻¹(653) − R_rs⁻¹ (691)] R_rs(748), explained 85.57% of variance in Chla concentration with a root mean square error (RMSE) of <6.56 mg/m³. In order to test the utility of this model with satellite data, HJ-1A Hyperspectral Imager (HSI) data were analyzed using comparable wavelengths selected from the in situ data [B₆₇⁻¹(656) − B₈₀⁻¹(716)] B₈₇(753). This model accounted for 84.3% of Chla variation, estimating Chla concentrations with an RMSE of <4.23 mg/m³. The results illustrate that, based on the determined wavelengths, the spectrum-based model can achieve a high estimation accuracy and can be applied to hyperspectral satellite imagery especially for higher Chla concentration waters.     

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 geopotential value <Emphasis Type="Italic">W</Emphasis><Subscript>0</Subscript> for specifying the relativistic atomic time scale and a global vertical reference system

The TOPEX/Poseidon (T/P) satellite alti- meter mission marked a new era in determining the geopotential constant W ₀. On the basis of T/P data during 1993–2003 (cycles 11–414), long-term variations in W ₀ have been investigated. The rounded value W ₀ = 62636856.0 ± 0.5) m ² s ⁻² has already been adopted by the International Astronomical Union for the definition of the constant L _G = W ₀/c ² = 6.969290134 × 10⁻¹⁰ (where c is the speed of light), which is required for the realization of the relativistic atomic time scale. The constant L _G , based on the above value of W ₀, is also included in the 2003 International Earth Rotation and Reference Frames Service conventions. It has also been suggested that W ₀ is used to specify a global vertical reference system (GVRS). W ₀ ensures the consistency with the International Terrestrial Reference System, i.e. after adopting W ₀, along with the geocentric gravitational constant (GM), the Earth’s rotational velocity (ω) and the second zonal geopotential coefficient (J ₂) as primary constants (parameters), then the ellipsoidal parameters (a,α) can be computed and adopted as derived parameters. The scale of the International Terrestrial Reference Frame 2000 (ITRF2000) has also been specified with the use of W ₀ to be consistent with the geocentric coordinate time. As an example of using W ₀ for a GVRS realization, the geopotential difference between the adopted W ₀ and the geopotential at the Rimouski tide-gauge point, specifying the North American Vertical Datum 1988 (NAVD88), has been estimated.     

Effect of cover management factor in quantification of soil loss: case study of Sungai Akah subwatershed,Baram River basin Sarawak,Malaysia

The present study evaluates the effectiveness and suitability of cover management factors (C factor) generated through different techniques like land use/land cover-based arbitrary value (C_LULC), Normalised Different Vegetation Index-based methods C_NDVI1 and C_NDVI2 and Modified Soil Adjusted Vegetation Index 2-based method (C_MSAVI2). The C factors generated using these four methods were tested in the calculation and assessment of annual average soil loss from an upland forested subwatershed in the Baram river basin using the Revised Universal Soil Loss Equation (RUSLE). The four cover management factor maps generated by this analysis show some variation among the results. The LULC method uses a single arbitrary value for each LULC type mapped in the subwatershed. The other three methods show a range of C values within each mapped LULC type. The effects of these variations were tested in the RUSLE by keeping the factors such as rainfall erosivity (R), soil erodibility (K), slope-length and steepness (LS) constant. The maximum annual average soil loss is 1191 t. ha^?1. y^?1 based on the C_LULC. Soil losses estimated with other three methods are very different compared to those estimated with the C_LULC method. The highest calculated soil loss values were 1832, 1674 and 1608 t. ha^?1. y^?1 in the study area based, respectively, on C_NDVI1, C_NDVI2 and C_MSAVI2 C factors. These maximum values represent the worst pixel scenario values of soil loss in the region. The statistical analysis performed indicates different relationship between the parameters and suggests the acceptance of the methodology based on C_NDVI2 for the study area, instead of a single value method such as C_LULC. Among the other two methods, the C_MSAVI2 was found to be more consistent than the C_NDVI1 method, but both methods lead to over-prediction of annual soil loss rate and therefore need to be reconsidered before applied in the RUSLE.     

Harmonic maps

Harmonic maps are generated as a certain class of optimal map projections. For instance, if the distortion energy over a meridian strip of the International Reference Ellipsoid is minimized, we are led to the Laplace–Beltrami vector-valued partial differential equation. Harmonic functions x(L,B), y(L,B) given as functions of ellipsoidal surface parameters of Gauss ellipsoidal longitude L and Gauss ellipsoidal latitude B, as well as x(,q), y(,q) given as functions of relative isometric longitude =L–L₀ and relative isometric latitude q=Q–Q₀ gauged to a vector-valued boundary condition of special symmetry are constructed. The easting and northing {x(b,),y(b,)} of the new harmonic map is then given. Distortion energy analysis of the new harmonic map is presented, as well as case studies for (1) B[–40°,+40°], L[–31°,+49°], B₀= ±30°, L₀=9° and (2) B[46°,56°], L{[4.5°, 7.5°]; [7.5°, 10.5°]; [10.5°,13.5°]; [13.5°,16.5°]}, B₀= 51°, L₀ {6°,9°,12°,15°}.     

Comparative study of the Ferranti,Honeywell and Litton inertial surveying systems testnet Ebersberger Forst

In order to assess the accuracy, reliability and efficiency of current inertial surveying systems a joint project has been started at the University of the Bundeswehr Munich. As a part of this project the testnet Ebersberger Forst has been established. It consists of 60 monumented points located at cross-roads in a flat area and it makes up a grid pattern of6 N-S and9 E-W traverses with a spacing of1–2 km. The points are determined by classical surveying techniques with a standard deviation of less than1 cm forE, N andH. Observation campaigns were carried through with a Ferranti, a Honeywell and a Litton system. Each campaign consisted of three independent missions performed under identical observation schemes. The preliminary evaluation of the data sets leads to standard deviations of between8 and16 cm for each coordinate if determined in a single mission with a Honeywell or Litton system. The correlations along a traverse follow approximately the series ρ, ρ², ρ³, ... with ρ⋟0.9. Cross correlation is only present betweenE orN, respectively, andH. The positions observed with the Ferranti system are less accurate, which might be explainable by the applied software and by two gross input errors during the missions. A rigorous post-mission adjustment of the data considerably improved the results.     

Construction of Green's function for the Stokes boundary-value problem with ellipsoidal corrections in the boundary condition

Green's function for the boundary-value problem of Stokes's type with ellipsoidal corrections in the boundary condition for anomalous gravity is constructed in a closed form. The `spherical-ellipsoidal' Stokes function describing the effect of two ellipsoidal correcting terms occurring in the boundary condition for anomalous gravity is expressed in O(e <sup>2</sup> <sub>0</sub>)-approximation as a finite sum of elementary functions analytically representing the behaviour of the integration kernel at the singular point ψ=0. We show that the `spherical-ellipsoidal' Stokes function has only a logarithmic singularity in the vicinity of its singular point. The constructed Green function enables us to avoid applying an iterative approach to solve Stokes's boundary-value problem with ellipsoidal correction terms involved in the boundary condition for anomalous gravity. A new Green-function approach is more convenient from the numerical point of view since the solution of the boundary-value problem is determined in one step by computing a Stokes-type integral. The question of the convergence of an iterative scheme recommended so far to solve this boundary-value problem is thus irrelevant. Received: 5 June 1997 / Accepted: 20 February 1998     

Earth＇s Temporal Principal Moments of Inertia and Variable Rotation

Based on the gravity field models EGM96 and EIGEN-GL04C, the Earth＇s time-dependent principal moments of inertia A, B, C are obtained, and the variable rotation of the Earth is determined. Numerical results show that A, B, and C have increasing tendencies; the tilt of the rotation axis increases 2.1×10^ 8 mas/yr; the third component of the rotational angular velocity, ω3 , has a decrease of 1.0×10^ 22 rad/s^2, which is around 23% of the present observed value. Studies show in detail that both 0 and ω3 experience complex fluctuations at various time scales due to the variations of A, B and C.     

Helmert's projection of a ground point onto the rotational reference ellipsoid in topocentric cartesian coordinates

Summary In order to derive the ellipsoidal height of a point P_t, on the physical surface of the earth, and the direction of ellipsoidal normal through P_t, we presente here an iterative procedure rapidly convergent to compute, in a topocentric Cartesian system, the coordinates of Helmert's projection of the ground point P_t onto the reference ellipsoid of revolution .We derive as well the cofactor matrix of total vector of the topocentric coordinates of the above ground point and of its Helmert's projection so that to compute the variance of ellipsoidal height.