Minimum mean squared error (MSE) adjustment and the optimal Tykhonov–Phillips regularization parameter via reproducing best invariant quadratic uniformly unbiased estimates (repro-BIQUUE)

In a linear Gauss–Markov model, the parameter estimates from BLUUE (Best Linear Uniformly Unbiased Estimate) are not robust against possible outliers in the observations. Moreover, by giving up the unbiasedness constraint, the mean squared error (MSE) risk may be further reduced, in particular when the problem is ill-posed. In this paper, the α-weighted S-homBLE (Best homogeneously Linear Estimate) is derived via formulas originally used for variance component estimation on the basis of the repro-BIQUUE (reproducing Best Invariant Quadratic Uniformly Unbiased Estimate) principle in a model with stochastic prior information. In the present model, however, such prior information is not included, which allows the comparison of the stochastic approach (α-weighted S-homBLE) with the well-established algebraic approach of Tykhonov–Phillips regularization, also known as R-HAPS (Hybrid APproximation Solution), whenever the inverse of the “substitute matrix” S exists and is chosen as the R matrix that defines the relative impact of the regularizing term on the final result. The delay in publishing this paper is due to a number of unfortunate complications. It was first submitted as a multi-author paper in two parts. Due to some miscommunication among the original authors, it was reassigned to one of the J Geod special issues, but later reassigned at this author’s request to a standard issue of J Geod. This compounded with a difficulty to find willing reviewers to slow the process. We apologize to the author.     

The convergence problem of collocation solutions in the framework of the stochastic interpretation

The problem of the convergence of the collocation solution to the true gravity field was defined long ago (Tscherning in Boll Geod Sci Affini 39:221–252, 1978) and some results were derived, in particular by Krarup (Boll Geod Sci Affini 40:225–240, 1981). The problem is taken up again in the context of the stochastic interpretation of collocation theory and some new results are derived, showing that, when the potential T can be really continued down to a Bjerhammar sphere, we have a quite general convergence property in the noiseless case. When noise is present in data, still reasonable convergence results hold true.

“Democrito che ’l mondo a caso pone” “Democritus who made the world stochastic” Dante Alighieri, La Divina Commedia, Inferno, IV – 136     

Unbiased vs biased estimation of GPS phase ambiguities from dual-frequency code and phase observables

With access to dual-frequency pseudorange and phase Global Positioning System (GPS) data, the wide-lane ambiguity can easily be fixed. Advantage is taken of this information in the linear combination of the above four observables for base ambiguity estimation (i.e. of N ₁ and N ₂). Starting points for our analysis are the Best Linear Unbiased Estimators BLUE₁ and BLUE₂. BLUE₁ is the best one (with minimum mean square error, MSE) if the ionosphere effect is negligible. If this is not the case, BLUE₂ has the smallest variance, but not necessarily the least mean square error. Hence, both estimators may suffer from a non-optimal treatment of the ionosphere bias. BLUE₁ ignores possible ionosphere bias, while BLUE₂ compensates for this bias in a less favourable way by eliminating it at the price of increased noise. As an alternative, linear estimators are derived, which make a compromise between the ionosphere bias and the random observation errors. This leads to the derivation of the Best Linear Estimator (BLE) and the Restricted Best Linear Estimator (RBLE) with minimum MSE. The former is generally not very useful, while the RBLE is recommended for practical use. It is shown that the MSE of the RBLE is limited by the variances of BLUE₁ and BLUE₂, i.e.

Interpolating atmospheric water vapor delay by incorporating terrain elevation information

In radio signal-based observing systems, such as Global Positioning System (GPS) and Interferometric Synthetic Aperture Radar (InSAR), the water vapor in the atmosphere will cause delays during the signal transmission. Such delays vary significantly with terrain elevation. In the case when atmospheric delays are to be eliminated from the measured raw signals, spatial interpolators may be needed. By taking advantage of available terrain elevation information during spatial interpolation process, the accuracy of the atmospheric delay mapping can be considerably improved. This paper first reviews three elevation-dependent water vapor interpolation models, i.e., the Best Linear Unbiased Estimator in combination with the water vapor Height Scaling Model (BLUE + HSM), the Best Linear Unbiased Estimator coupled with the Elevation-dependent Covariance Model (BLUE + ECM), and the Simple Kriging with varying local means based on the Baby semi-empirical model (SKlm + Baby for short). A revision to the SKlm + Baby model is then presented, where the Onn water vapor delay model is adopted to substitute the inaccurate Baby semi-empirical model (SKlm + Onn for short). Experiments with the zenith wet delays obtained through the GPS observations from the Southern California Integrated GPS Network (SCIGN) demonstrate that the SKlm + Onn model outperforms the other three. The RMS of SKlm + Onn is only 0.55 cm, while those of BLUE + HSM, BLUE + ECM and SKlm + Baby amount to 1.11, 1.49 and 0.77 cm, respectively. The proposed SKlm + Onn model therefore represents an improvement of 29–63% over the other known models.     

Calibration of satellite gradiometer data aided by ground gravity data

Parametric least squares collocation was used in order to study the detection of systematic errors of satellite gradiometer data. For this purpose, simulated data sets with a priori known systematic errors were produced using ground gravity data in the very smooth gravity field of the Canadian plains. Experiments carried out at different satellite altitudes showed that the recovery of bias parameters from the gradiometer “measurements” is possible with high accuracy, especially in the case of crossing tracks. The mean value of the differences (original minus estimated bias parameters) was relatively large compared to the standard deviation of the corresponding second-order derivative component at the corresponding height. This mean value almost vanished when gravity data at ground level were combined with the second-order derivative data set at satellite altitude. In the case of simultaneous estimation of bias and tilt parameters from ∂² T/∂z ²“measurements”, the recovery of both parameters agreed very well with the collocation error estimation. Received: 10 October 1996 / Accepted 25 May 1998     

Global height datum unification: a new approach in gravity potential space

The problem of “global height datum unification” is solved in the gravity potential space based on: (1) high-resolution local gravity field modeling, (2) geocentric coordinates of the reference benchmark, and (3) a known value of the geoid’s potential. The high-resolution local gravity field model is derived based on a solution of the fixed-free two-boundary-value problem of the Earth’s gravity field using (a) potential difference values (from precise leveling), (b) modulus of the gravity vector (from gravimetry), (c) astronomical longitude and latitude (from geodetic astronomy and/or combination of (GNSS) Global Navigation Satellite System observations with total station measurements), (d) and satellite altimetry. Knowing the height of the reference benchmark in the national height system and its geocentric GNSS coordinates, and using the derived high-resolution local gravity field model, the gravity potential value of the zero point of the height system is computed. The difference between the derived gravity potential value of the zero point of the height system and the geoid’s potential value is computed. This potential difference gives the offset of the zero point of the height system from geoid in the “potential space”, which is transferred into “geometry space” using the transformation formula derived in this paper. The method was applied to the computation of the offset of the zero point of the Iranian height datum from the geoid’s potential value W ₀=62636855.8 m²/s². According to the geometry space computations, the height datum of Iran is 0.09 m below the geoid.     

Nonlinear least-squares method via an isomorphic geometrical setup

The resolution of a nonlinear parametric adjustment model is addressed through an isomorphic geometrical setup with tensor structure and notation, represented by a u-dimensional “model surface” embedded in a flat n-dimensional “observational space”. Then observations correspond to the observational-space coordinates of the pointQ, theu initial parameters correspond to the model-surface coordinates of the “initial” pointP, and theu adjusted parameters correspond to the model-surface coordinates of the “least-squares” point . The least-squares criterion results in a minimum-distance property implying that the vector Q must be orthogonal to the model surface. The geometrical setup leads to the solution of modified normal equations, characterized by a positive-definite matrix. The latter contains second-order and, optionally, thirdorder partial derivatives of the observables with respect to the parameters. This approach significantly shortens the convergence process as compared to the standard (linearized) method.     

Defining the basic entities in a geodetic data base

The term “entity” covers, when used in the field of electronic data processing, the meaning of words like “thing”, “being”, “event”, or “concept”. Each entity is characterized by a set of properties. An information element is a triple consisting of an entity, a property and the value of a property. Geodetic information is sets of information elements with entities being related to geodesy. This information may be stored in the form ofdata and is called ageodetic data base provided (1) it contains or may contain all data necessary for the operations of a particular geodetic organization, (2) the data is stored in a form suited for many different applications and (3) that unnecessary duplications of data have been avoided. The first step to be taken when establishing a geodetic data base is described, namely the definition of the basic entities of the data base (such as trigonometric stations, astronomical stations, gravity stations, geodetic reference-system parameters, etc...). Presented at the “International Symposium on Optimization of Design and Computation of Control Networks”, Sopron, Hungary, July 1977.     

Mixed Integer-Real Valued Adjustment (IRA) Problems: GPS Initial Cycle Ambiguity Resolution by Means of the LLL Algorithm

In order to achieve to GPS solutions of first-order accuracy and integrity, carrier phase observations as well as pseudorange observations have to be adjusted with respect to a linear/linearized model. Here the problem of mixed integer-real valued parameter adjustment (IRA) is met. Indeed, integer cycle ambiguity unknowns have to be estimated and tested. At first we review the three concepts to deal with IRA: (i) DDD or triple difference observations are produced by a properly chosen difference operator and choice of basis, namely being free of integer-valued unknowns (ii) The real-valued unknown parameters are eliminated by a Gauss elimination step while the remaining integer-valued unknown parameters (initial cycle ambiguities) are determined by Quadratic Programming and (iii) a RA substitute model is firstly implemented (real-valued estimates of initial cycle ambiguities) and secondly a minimum distance map is designed which operates on the real-valued approximation of integers with respect to the integer data in a lattice. This is the place where the integer Gram-Schmidt orthogonalization by means of the LLL algorithm (modified LLL algorithm) is applied being illustrated by four examples. In particular, we prove that in general it is impossible to transform an oblique base of a lattice to an orthogonal base by Gram-Schmidt orthogonalization where its matrix enties are integer. The volume preserving Gram-Schmidt orthogonalization operator constraint to integer entries produces “almost orthogonal” bases which, in turn, can be used to produce the integer-valued unknown parameters (initial cycle ambiguities) from the LLL algorithm (modified LLL algorithm). Systematic errors generated by “almost orthogonal” lattice bases are quantified by A. K. Lenstra et al. (1982) as well as M. Pohst (1987). The solution point of Integer Least Squares generated by the LLL algorithm is = (L')⁻¹[L'◯] ∈ ℤ ^m where L is the lower triangular Gram-Schmidt matrix rounded to nearest integers, [L], and = [L'◯] are the nearest integers of L'◯, ◯ being the real valued approximation of z ∈ ℤ ^m , the m-dimensional lattice space Λ. Indeed due to “almost orthogonality” of the integer Gram-Schmidt procedure, the solution point is only suboptimal, only close to “least squares.” ? 2000 John Wiley & Sons, Inc.     

The Abel-Poisson kernel and the Abel-Poisson integral in a moving tangent space

The upward-downward continuation of a harmonic function like the gravitational potential is conventionally based on the direct-inverse Abel-Poisson integral with respect to a sphere of reference. Here we aim at an error estimation of the “planar approximation” of the Abel-Poisson kernel, which is often used due to its convolution form. Such a convolution form is a prerequisite to applying fast Fourier transformation techniques. By means of an oblique azimuthal map projection / projection onto the local tangent plane at an evaluation point of the reference sphere of type “equiareal” we arrive at a rigorous transformation of the Abel-Poisson kernel/Abel-Poisson integral in a convolution form. As soon as we expand the “equiareal” Abel-Poisson kernel/Abel-Poisson integral we gain the “planar approximation”. The differences between the exact Abel-Poisson kernel of type “equiareal” and the “planar approximation” are plotted and tabulated. Six configurations are studied in detail in order to document the error budget, which varies from 0.1% for points at a spherical height H=10km above the terrestrial reference sphere up to 98% for points at a spherical height H = 6.3×10⁶km. Received: 18 March 1997 / Accepted: 19 January 1998     

The spheroidal fixed–free two-boundary-value problem for geoid determination (the spheroidal Bruns' transform)

The target of the spheroidal Gauss–Listing geoid determination is presented as a solution of the spheroidal fixed–free two-boundary value problem based on a spheroidal Bruns' transformation (“spheroidal Bruns' formula”). The nonlinear spheroidal Bruns' transform (nonlinear spheroidal Bruns' formula), the spheroidal fixed part and the spheroidal free part of the two-boundary value problem are derived. Four different spheroidal gravity models are treated, in particular to determine whether they pass the test to fit to the postulate of a level ellipsoidal gravity field, namely of Somigliana–Pizzetti type. Received: 4 May 1999 / Accepted: 21 May 1999     

Finite covariance functions

Because of the full covariance matrices and the computer storage limitations the number of measurements which can be handled by the collocation method simultaneously, is limited. This paper presents a method to compute covariance functions with a finite support yielding sparse covariance matrices. The theoretical background is pointed out and, for the one- and two-dimensional case, special functions are developed which can be combined with the usually used covariance functions to get a “finite covariance function”. Simulated examples to demonstrate the behaviour of different solution methods to solve these special, sparse covariance matrices supplement our investigations.     

Spatial and Temporal Variability of Net Primary Productivity (NPP) over Terrestrial Biosphere of India Using NOAA-AVHRR Based GloPEM Model

The monitoring of terrestrial carbon dynamics is important in studies related with global climate change. This paper presents results of the inter-annual variability of Net Primary Productivity (NPP) from 1981 to 2000 derived using observations from NOAA-AVHRR data using Global Production Efficiency Model (GloPEM). The GloPEM model is based on physiological principles and uses the production efficiency concept, in which the canopy absorption of photosynthetically active radiation (APAR) is used with a conversion “efficiency” to estimate Gross Primary Production (GPP). NPP derived from GloPEM model over India showed maximum NPP about 3,000 gCm⁻²year⁻¹ in west Bengal and lowest up to 500 gCm⁻²year⁻¹ in Rajasthan. The India averaged NPP varied from 1,084.7 gCm⁻²year⁻¹ to 1,390.8 gCm⁻²year⁻¹ in the corresponding years of 1983 and 1998 respectively. The regression analysis of the 20 year NPP variability showed significant increase in NPP over India (r = 0.7, F = 17.53, p < 0.001). The mean rate of increase was observed as 10.43 gCm⁻²year⁻¹. Carbon fixation ability of terrestrial ecosystem of India is increasing with rate of 34.3 TgC annually (t = 4.18, p < 0.001). The estimated net carbon fixation over Indian landmass ranged from 3.56 PgC (in 1983) to 4.57 PgC (in 1998). Grid level temporal correlation analysis showed that agricultural regions are the source of increase in terrestrial NPP of India. Parts of forest regions (Himalayan in Nepal, north east India) are relatively less influenced over the study period and showed lower or negative correlation (trend). Finding of the study would provide valuable input in understanding the global change associated with vegetation activities as a sink for atmospheric carbon dioxide.     

Changes of spheroid and of datum

Summary A datum change between two geodetic systems with points in common may be derived in three stages; slight adjustments of coordinates to make the networks of common points geometrically similar in the two systems; a scale factor to make them geometrically congruent; finally, an orthogonal transformation to swing them into coincidence. The geometrical concept is developed of a “datum screw”, not arbitrarily chosen as is the “origin” or “datum point” of a geodetic survey, but intrinsic to the geometry. The conditions under which it degenerates to a simple “datum shift” are discussed. Differential and other formulae for changes of spheroid and of datum are given, together with a set of tables of coefficients.     

Advances in the numerical solution of the linear molodensky problem

The solution of the linear Molodensky problem by analytical continuation to point level is numerically the most convenient of all the theoretically equivalent solutions. It is obtained by successively applying the same integral operator and it does not depend explicitly on the terrain inclination. However, its dependence on the computation point restricts somehow the computational efficiency. The use of the Fourier transform for the evaluation of the integral operator in planar approximation lessens significantly the burden of computations. Using this spectral approach, the problem has been reformulated and solved in the frequency domain. Moreover, it is shown that the solution can be easily split into two steps: (a) “downward” continuation to sea level, which is independent of the computation point, and (b) “upward” continuation from sea to point level, using the values computed at sea level. Such a treatment not only simplifies the formulas and increases the numerical efficiency but also clarifies the physical interpretation and the theoretical equivalence of the continuation solution with respect to the other solution types. Numerical tests have been performed to investigate which terms in the Molodensky series are of significance for geoid and deflection computations. The practical difficulty of differences in the grid spacings of gravity and height data has been overcome by frequency domain interpolation. Presented at theXIX IUGG General Assembly, Vancouver, B.C., August 9–22, 1987.     

From kinematical geodesy to inertial positioning

WhenH. Moritz (1967, 1971) studied “kinematical geodesy” for the purpose of separation of gravitation and inertia, especially within combined accelerometer-gradiometer systems, it was hard to believe that within five years time inertial survey systems would be available, exactly operating according to his theoretical design. Here, we attempt to give a geodetic introduction into the fundamental equation of inertial positioning materialized by inertial survey systems with emphasis on a careful error model, including 36 parameters of type time interval, initial positions, initial gravity, varying acceleration, varying gravity gradients, accelerometer bias, accelerometer random uncertainty, accelerometer non-orthogonality, initial misalignment angles, accelerometer scale factor uncertainty. The notion of “multipoint” boundary value problem and initial value problem of inertial positioning is reviwed. So-called “post-mission” adjustment techniques for inertial surveys are discussed.

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Sommaire QuandH. Moritz (1967, 1971) a étudié la géodésie cinématique dans le but de séparer la gravitation et l’inertie, spécialement en combinant accéléromètres et gradiomètres, il était difficile de croire qu’en cinq ans les systèmes d’arpentage inertiels seraient disponibles et fonctionneraient exactement selon ses prévisions théoriques. Ici, nous allons tenter de donner une introduction géodésique à l’équation fondamentale du positionnement inertiel, matérialisée par un système d’arpentage inertiel en soulignant l’importance d’un modèle d’erreur incluant 36 paramètres du genre intervalle de temps, positions initiales, gravité initiale, accélération variable, gradient de la gravité variable, déviation des accéléromètres, incertitude aléatoire des accéléromètres, non-orthogonalité des accéléromètres, angles initiaux des défauts d’alignement, incertitude du facteur d’échelle des accéléromètres. La notion de problème “multipoint” aux limites et du problème de la valeur initiale du positionnement inertiel y est revue. Les techniques de compensation “après la mission” y sont discutées.

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Presented at the 2nd International Symposium on Inertial Technology for Surveying and Geodesy, Banff, Canada, June 1–5, 1981.     

The spacetime gravitational field of a deformable body

The high-resolution analysis of orbit perturbations of terrestrial artificial satellites has documented that the eigengravitation of a massive body like the Earth changes in time, namely with periodic and aperiodic constituents. For the space-time variation of the gravitational field the action of internal and external volume as well as surface forces on a deformable massive body are responsible. Free of any assumption on the symmetry of the constitution of the deformable body we review the incremental spatial (“Eulerian”) and material (“Lagrangean”) gravitational field equations, in particular the source terms (two constituents: the divergence of the displacement field as well as the projection of the displacement field onto the gradient of the reference mass density function) and the `jump conditions' at the boundary surface of the body as well as at internal interfaces both in linear approximation. A spherical harmonic expansion in terms of multipoles of the incremental Eulerian gravitational potential is presented. Three types of spherical multipoles are identified, namely the dilatation multipoles, the transport displacement multipoles and those multipoles which are generated by mass condensation onto the boundary reference surface or internal interfaces. The degree-one term has been identified as non-zero, thus as a “dipole moment” being responsible for the varying position of the deformable body's mass centre. Finally, for those deformable bodies which enjoy a spherically symmetric constitution, emphasis is on the functional relation between Green functions, namely between Fourier-/ Laplace-transformed volume versus surface Love-Shida functions (h(r),l(r) versus h ^′(r),l ^′(r)) and Love functions k(r) versus k ^′(r). The functional relation is numerically tested for an active tidal force/potential and an active loading force/potential, proving an excellent agreement with experimental results. Received: December 1995 / Accepted: 1 February 1997     

＂Digital Region＂ in the Context of the ＂Grid Computing＂

This paper is to construct a “digital local, regional, region“ information framework based on the technology of “SIG“ and its significance and application to the regional sustainable development evaluation system. First, the concept of the “grid computing“ and “SIG“ is interpreted and discussed, then the relationship between the “grid computing“ and “digital region“ is analyzed, and the framework of the “digital region“ is put forward. Finally, the significance and application of “grid computing“ to the “region sustainable development evaluation system“ are discussed.     

On some problems of the downward continuation of the 5′×5′ mean Helmert gravity disturbance

This research deals with some theoretical and numerical problems of the downward continuation of mean Helmert gravity disturbances. We prove that the downward continuation of the disturbing potential is much smoother, as well as two orders of magnitude smaller than that of the gravity anomaly, and we give the expression in spectral form for calculating the disturbing potential term. Numerical results show that for calculating truncation errors the first 180^∘ of a global potential model suffice. We also discuss the theoretical convergence problem of the iterative scheme. We prove that the 5^′×5^′ mean iterative scheme is convergent and the convergence speed depends on the topographic height; for Canada, to achieve an accuracy of 0.01 mGal, at most 80 iterations are needed. The comparison of the “mean” and “point” schemes shows that the mean scheme should give a more reasonable and reliable solution, while the point scheme brings a large error to the solution. Received: 19 August 1996 / Accepted: 4 February 1998     

Arcsecond-level pointing calibration for ICESat laser altimetry of ice sheets

 Arcsecond-level accuracy of NASA's ICESat (Ice, Cloud, and land Elevation Satellite) satellite laser altimeter beam pointing angle is required to satisfy the scientific goal of detecting centimeter-level elevation changes, over time, in the Greenland and Antarctic ice sheets. Two different approaches, termed “topographic inferred” and “direct detection”, were examined for calibrating the laser pointing angle (that is, detecting and removing pointing determination bias) at the 1.5-arcsec level, using information independent of the onboard pointing instrumentation. Both approaches entail estimating the beam pointing by differencing the three-dimensional position of the altimeter instrument and the laser-beam spot (or “footprint”) location on the ground. Analytical assessments of the two approaches are discussed, along with recommendations for the ICESat pointing determination calibration strategy. Received: 28 April 2000 / Accepted: 6 November 2000