The solution of mixed boundary value problems with the reference ellipsoid as boundary

In this paper we study the following mixed boundary value problem

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.     

Rectilinear minimax hub location problems

The minimax hub location problem sites a facility to minimize the maximum weighted interaction cost between pairs of fixed nodes. In this paper, distances are represented by a rectilinear norm and may be suited to factory layout or street network problems. The problem is already well known (in 2-D) as the round trip location problem and is extended to 3-D in this paper. One rationale for the solution method is based on an extension of the geometric arguments used to solve the minimax single facility location problem. Suppose a budget is provided for interactions, and that each interaction must be accomplished for no more than this cost. The algorithm uses a bi-section search for the feasible budget until it finds the expenditure needed to provide for these flows. The extension in the present paper is that the nodes are permitted to be on different layers (levels). This 3-D version of the problem appears to be a new variant of the hub model. The models and solution techniques developed in the paper are illustrated using a small 55 node problem. Because of a relatively efficient implementation of the bi-section search, the algorithm in 2-D and 3-D is also applied successfully to a 550 node problem.

Ambiguity resolution strategies using the results of the International GPS Geodynamics Service (IGS)

Resolving the initial phase ambiguities of GPS carrier phase observations was always considered an important aspect of GPS processing techniques. Resolution of the so-called wide-lane ambiguities using a special linear combination of theL ₁ andL ₂ carrier and code observations has become standard. New aspects have to be considered today: (1) Soon AS, the so-called Anti-Spoofing, will be turned on for all Block II spacecrafts. This means that precise code observations will be no longer available, which in turn means that the mentioned approach to resolve the wide-lane ambiguities will fail. (2) Most encouraging is the establishment of the new International GPS Geodynamics Service (IGS), from where high quality orbits, earth rotation parameters, and eventually also ionospheric models will be available. We are reviewing the ambiguity resolution problem under these new aspects: We look for methods to resolve the initial phase ambiguities without using code observations but using high quality orbits and ionospheric models from IGS, and we study the resolution of the narrow-lane ambiguities (after wide-lane ambiguity resolution) using IGS orbits.     

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.     

An algorithm for the inverse of a multivariate homogeneous polynomial of degreen

Summary Various geodetic problems (the free nonlinear geodetic boundary value problem, the computation of Gauß-Krüger coordinates or UTM coordinates, the problem of nonlinear regression) demand theinversion of an univariate, bivariate, trivariate, in generalmultivariate homogeneous polynomial of degree n. The new algorithm which is oriented towardsSymbolic Computer Manipulation is based upon the algebraic power base computation with respect toKronecker-Zehfu product structure leading to the solution of a system oftriangular matrix equations: Only the first row of the inverse triangular matrix has to be computed. TheSymbolic Computer Manipulation program of the GKS algorithm is available from the authors.     

Surface gravitational field and topography changes induced by the Earth’s fluid core motions

Using a Love number formalism, the elastic deformations of the mantle and the mass redistribution gravitational potential within the Earth induced by the fluid pressure acting at the core–mantle boundary (CMB) are computed. This pressure field changes at a decadal time scale and may be estimated from observations of the surface magnetic field and its secular variation. First, using a spherical harmonic expansion, the poloidal and toroidal part of the fluid velocity field at the CMB for the last 40 years is computed, under the hypothesis of tangential geostrophy. Then the associated geostrophic pressure, whose order of magnitude is about 1000 Pa, is computed. The surface topography induced by this pressure field is computed using Love numbers, and is a few millimetres. The mass redistribution gravitational potential induced by these deformations and, in particular, the zonal components of the related surface gravitational potential perturbation (J₂, J₃ and J₄ coefficients), are calculated. Overall perturbations for the J₂ coefficient of about 10^–10, for J₃ of about 10^–11 and for J₄ are found of about 0.3×10^–11. Finally, these theoretical results are compared with recent observations of the decadal variation of J₂ from satellite laser ranging. Results concerning J₂ can be described as follows: first, they are one order of magnitude too small to explain the observed decadal variation of J₂ and, second, they show a significant linear trend over the last 40 years, whose rate of decrease amounts to 7% of the observed value.     

Computer algebra solution of the GPS N-points problem

A computer algebra solution is applied here to develop and evaluate algorithms for solving the basic GPS navigation problem: finding a point position using four or more pseudoranges at one epoch (the GPS N-points problem). Using Mathematica 5.2 software, the GPS N-points problem is solved numerically, symbolically, semi-symbolically, and with Gauss–Jacobi, on a work station. For the case of N > 4, two minimization approaches based on residuals and distance norms are evaluated for the direct numerical solution and their computational duration is compared. For N = 4, it is demonstrated that the symbolic computation is twice as fast as the iterative direct numerical method. For N = 6, the direct numerical solution is twice as fast as the semi-symbolic, with the residual minimization requiring less computation time compared to the minimization of the distance norm. Gauss–Jacobi requires eight times more computation time than the direct numerical solution. It does, however, have the advantage of diagnosing poor satellite geometry and outliers. Besides offering a complete evaluation of these algorithms, we have developed Mathematica 5.2 code (a notebook file) for these algorithms (i.e., Sturmfel’s resultant, Dixon’s resultants, Groebner basis, reduced Groebner basis and Gauss–Jacobi). These are accessible to any geodesist, geophysicist, or geoinformation scientist via the GPS Toolbox () website or the Wolfram Information Center ().

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.     

Variational formulation of the geodetic boundary value problem

A variational principle for the Stokesian boundary value problem is derived using the Euler-Lagrange theory. The resulting variational principle is then transformed into an equation determining the semi-major axis of the best fitting ellipsoid which fulfills the conditionU ₀ =W ₀ . The computations using three different geopotential models yields the semi-major axis of the earth ellipsoid asa=6378145.4 metres for the flatteningf=1/298.2564. The corresponding equatorial gravity and the geopotential number are computed as γ_a=978029.59 mgals andU ₀=W ₀=6.26367371 10⁶ kgalmeters respectively.     

Solution to the spectral filter problem of residual terrain modelling (RTM)

In physical geodesy, the residual terrain modelling (RTM) technique is frequently used for high-frequency gravity forward modelling. In the RTM technique, a detailed elevation model is high-pass-filtered in the topography domain, which is not equivalent to filtering in the gravity domain. This in-equivalence, denoted as spectral filter problem of the RTM technique, gives rise to two imperfections (errors). The first imperfection is unwanted low-frequency (LF) gravity signals, and the second imperfection is missing high-frequency (HF) signals in the forward-modelled RTM gravity signal. This paper presents new solutions to the RTM spectral filter problem. Our solutions are based on explicit modelling of the two imperfections via corrections. The HF correction is computed using spectral domain gravity forward modelling that delivers the HF gravity signal generated by the long-wavelength RTM reference topography. The LF correction is obtained from pre-computed global RTM gravity grids that are low-pass-filtered using surface or solid spherical harmonics. A numerical case study reveals maximum absolute signal strengths of \(\sim 44\) mGal (0.5 mGal RMS) for the HF correction and \(\sim 33\) mGal (0.6 mGal RMS) for the LF correction w.r.t. a degree-2160 reference topography within the data coverage of the SRTM topography model (\(56^{\circ }\hbox {S} \le \phi \le 60^{\circ }\hbox {N}\)). Application of the LF and HF corrections to pre-computed global gravity models (here the GGMplus gravity maps) demonstrates the efficiency of the new corrections over topographically rugged terrain. Over Switzerland, consideration of the HF and LF corrections reduced the RMS of the residuals between GGMplus and ground-truth gravity from 4.41 to 3.27 mGal, which translates into \(\sim 26\)% improvement. Over a second test area (Canada), our corrections reduced the RMS of the residuals between GGMplus and ground-truth gravity from 5.65 to 5.30 mGal (\(\sim 6\)% improvement). Particularly over Switzerland, geophysical signals (associated, e.g. with valley fillings) were found to stand out more clearly in the RTM-reduced gravity measurements when the HF and LF correction are taken into account. In summary, the new RTM filter corrections can be easily computed and applied to improve the spectral filter characteristics of the popular RTM approach. Benefits are expected, e.g. in the context of the development of future ultra-high-resolution global gravity models, smoothing of observed gravity data in mountainous terrain and geophysical interpretations of RTM-reduced gravity measurements.     

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.     

Finite element analysis of quasi-static earthquake displacement fields observed by GPS

In order to investigate the ability of numerical techniques in computing seismic displacement fields, a forward problem of dislocation theory is first solved by both analytical and numerical techniques. Convergency of the numerical solution and a comparison with the results of a well-known analytical solution are presented for a particular problem. The results are then used when the numerical technique is applied to the same problem without any information on the dislocation source. The latter results may be applied in an inverse dislocation problem. Numerical techniques are in particular expected to be flexible enough to (1) include internal discontinuities, (2) apply geodetic observations as boundary conditions and (3) model lateral, in addition to radial, material heterogeneities. A model with lateral variation of elastic parameters is considered in the last problem. Acknowledgments.The authors are most grateful to Y. Okada, National Research Institute for Earth Science and Disaster Prevention, Japan, for providing the Fortran code of his analytical solution.     

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.     

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°}.     

Questioning emissions-based approaches for the definition of REDD+ deforestation baselines in high forest cover/low deforestation countries

Background

REDD+?is being questioned by the particular status of High Forest/Low Deforestation countries. Indeed, the formulation of reference levels is made difficult by the confrontation of low historical deforestation records with the forest transition theory on the one hand. On the other hand, those countries might formulate incredibly high deforestation scenarios to ensure large payments even in case of inaction.

Results

Using a wide range of scenarios within the Guiana Shield, from methods involving basic assumptions made from past deforestation, to explicit modelling of deforestation using relevant socio-economic variables at the regional scale, we show that the most common methodologies predict huge increases in deforestation, unlikely to happen given the existing socio-economic situation. More importantly, it is unlikely that funds provided under most of these scenarios could compensate for the total cost of avoided deforestation in the region, including social and economic costs.

Conclusion

This study suggests that a useful and efficient international mechanism should really focus on removing the underlying political and socio-economic forces of deforestation rather than on hypothetical result-based payments estimated from very questionable reference levels.

     

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 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.     

Gibbs sampler for computing and propagating large covariance matrices

The use of sampling-based Monte Carlo methods for the computation and propagation of large covariance matrices in geodetic applications is investigated. In particular, the so-called Gibbs sampler, and its use in deriving covariance matrices by Monte Carlo integration, and in linear and nonlinear error propagation studies, is discussed. Modifications of this technique are given which improve in efficiency in situations where estimated parameters are highly correlated and normal matrices appear as ill-conditioned. This is a situation frequently encountered in satellite gravity field modelling. A synthetic experiment, where covariance matrices for spherical harmonic coefficients are estimated and propagated to geoid height covariance matrices, is described. In this case, the generated samples correspond to random realizations of errors of a gravity field model. AcknowledgementsThe authors are indebted to Pieter Visser and Pavel Ditmar for providing simulation output that was used in the GOCE error generation experiments. Furthermore, the NASA/NIMA/OSU team is acknowledged for providing public ftp access to the EGM96 error covariance matrix. The two anonymous reviewers are thanked for their valuable comments.