Showing papers in "Journal of Geodesy in 2003"

The effect of coloured noise on the uncertainties of rates estimated from geodetic time series

Abstract: Until recently, it was typically assumed that only white noise was present in geodetic time series. However, several data sets have now provided evidence for the presence of power-law noise. The uncertainty of any rates estimated from such data sets is dependent on the error model assumed for the data. Here a general form for the covariance matrix for any power-law noise model is introduced and simple equations that relate the rate uncertainty to the noise amplitude, sampling frequency and length of the time series are empirically derived. In addition, equations to analyse data sets and obtain a rate uncertainty when computational speed is at a premium are provided. These equations are tested against previously published geodetic data sets.

GPS network monitors the Arabia-Eurasia collision deformation in Iran

Abstract: The rate of crustal deformation in Iran due to the Arabia–Eurasia collision is estimated. The results are based on new global positioning system (GPS) data. In order to address the problem of the distribution of the deformation in Iran, Iranian and French research organizations have carried out the first large-scale GPS survey of Iran. A GPS network of 28 sites (25 in Iran, two in Oman and one in Uzbekistan) has been installed and surveyed twice, in September 1999 and October 2001. Each site has been surveyed for a minimum observation of 4 days. GPS data processing has been done using the GAMIT-GLOBK software package. The solution displays horizontal repeatabilities of about 1.2 mm in 1999 and 2001. The resulting velocities allow us to constrain the kinematics of the Iranian tectonic blocks. These velocities are given in ITRF2000 and also relative to Eurasia. This last kinematic model demonstrates that (1) the north–south shortening from Arabia to Eurasia is 2–2.5 cm/year, less than previously estimated, and (2) the transition from subduction (Makran) to collision (Zagros) is very sharp and governs the different styles of deformation observed in Iran. In the eastern part of Iran, most of the shortening is accommodated in the Gulf of Oman, while in the western part the shortening is more distributed from south to north. The large faults surrounding the Lut block accommodate most of the subduction–collision transition.

Development and assessment of GPS virtual reference stations for RTK positioning

Abstract: The past few years have seen substantial growth in multiple-reference-station networks which are used to overcome the limitations of standard real-time kinematic (RTK) systems. The use of a multiple-reference-station network, as opposed to a single reference station, results in a larger service area coverage, increased robustness, and a higher positioning accuracy. However, real-time application is still a difficult task to implement in practice. The virtual reference station (VRS) concept is an efficient method of transmitting corrections through a data link to the network users for RTK positioning. A novel method of creating VRS for high-precision RTK positioning has been developed and tested at the Nanyang Technological University (NTU). The emphasis has been on real-time implementation. A number of tests were conducted using the Singapore Integrated Multiple Reference Station Network (SIMRSN). The tests were done at different locations in Singapore to assess the achievable accuracy and initialization times for VRS RTK positioning using the NTU method. The results confirmed that VRS RTK positioning can be achieved to within 3-cm accuracy in horizontal position. Height accuracy is in the range of 1 to 5 cm. The average initialization time is within 2 min.

A Monte-Carlo technique for weight estimation in satellite geodesy

Abstract: Precise orbit determination, and satellite-geodetic applications such as gravity field modelling or satellite altimetry, rely on different observation types and groups that have to be processed in a common parameter estimation scheme. Naturally, the choice of the relative weights for these data sets as well as for added prior information is of importance for obtaining reliable estimates of the unknown parameters and their associated covariance matrices. If the observations are predominantly affected by random errors and systematic errors play a minor role, variance component models can be applied. However, most of the methods proposed so far for variance component estimation involve repeated inversion of large matrices, resulting in intensive computations and large storage requirements if more than a few hundred unknowns are to be determined. In addition, these matrices are not necessarily provided as standard output from common geodetic least-squares estimation software. Therefore, amethod is proposed which is based on Monte-Carlo estimation of the redundancy contributions of disjunctive observation groups. The method can handle unknown variance components without the need for repeated inversion of matrices. It is computationally simple, numerically stable and easy to implement. Its application is demonstrated in an experiment concerning low-medium-degree gravity field recovery from simulated orbit perturbations of the GOCE mission, and compared in performance with Lerch’s method of subset solutions.

Estimation of elevation-dependent satellite antenna phase center variations of GPS satellites

Abstract: A method for the estimation of the phase center variations of GPS satellite antennas using global GPS data is presented. First estimations have shown an encouraging repeatability from day to day and between satellites of the same block. Thus, two different satellite antenna patterns for Block II/IIA and for Block IIR with a range of about 4 cm and an accuracy of less than 1 mm could be found. The present approach allows the creation of a consistent set of receiver and satellite antenna patterns and phase center offsets. Thereby, it is possible to switch from relative to absolute phase center variations without a scale problem in global networks. This changeover has an influence on troposphere parameters, reduces systematic effects due to uncorrect antenna modeling and should diminish the elevation dependence of GPS results.

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

Abstract: 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.

A computational scheme to model the geoid by the modified Stokes formula without gravity reductions

Abstract: In a modern application of Stokes’ formula for geoid determination, regional terrestrial gravity is combined with long-wavelength gravity information supplied by an Earth gravity model. Usually, several corrections must be added to gravity to be consistent with Stokes’ formula. In contrast, here all such corrections are applied directly to the approximate geoid height determined from the surface gravity anomalies. In this way, a more efficient workload is obtained. As an example, in applications of the direct and first and second indirect topographic effects significant long-wavelength contributions must be considered, all of which are time consuming to compute. By adding all three effects to produce a combined geoid effect, these long-wavelength features largely cancel. The computational scheme, including two least squares modifications of Stokes’ formula, is outlined, and the specific advantages of this technique, compared to traditional gravity reduction prior to Stokes’ integration, are summarised in the conclusions and final remarks.

Energy Integral Method For Gravity Field Determination From Satellite Orbit Coordinates

Abstract: A fast iterative method for gravity field determination from low Earth satellite orbit coordinates has been developed and implemented successfully. The method is based on energy conservation and avoids problems related to orbit dynamics and initial state. In addition, the particular geometry of a repeat orbit is exploited by using a very efficient iterative estimation scheme, in which a set of normal equations is approximated by a sparse block-diagonal equivalent. Recovery experiments for spherical harmonic gravity field models up to degree and order 80 and 120 were conducted based on a 29-day simulated data set of orbit coordinates. The method was found to be very flexible and could be easily adapted to include observations of non-conservative accelerations, such as (to be) provided by satellites like CHAMP, GRACE, and GOCE. A serious drawback of the method is its large sensitivity to satellite velocity errors. Existing orbit determination strategies need to be altered or augmented to include algorithms that focus on optimizing the accuracy of estimated velocities.

Satellite antenna phase center offsets and scale errors in GPS solutions

Abstract: ITRF2000 solutions have shown that there are parts-per-billion (ppb)-level scale differences between GPS and other techniques, as well as among various GPS analysis centers (ACs). The trends of the scale differences reach 0.2 ppb per year. It is demonstrated that the uncertainty in the satellite antenna phase center offset (z-direction) is one of the major reasons for the scale differences. The scale change caused by an error in the z_offset can be approximately expressed as: scale change (in ppb)=7.8 × error of z_offset (in meters). Changing the z_offset value of the BLOCK II and IIA satellites from 102 to 95 cm could produce a 0.5-ppb scale variation. For BLOCK IIR satellites the uncertainty of the z_offset is much larger. The number of these satellites increases each year. The accuracy of IGS products could be significantly affected if this problem is not properly solved. Besides scale, satellite antenna phase center uncertainties have perceptible effects on also the clock, zenith path delay, and other solved-for parameters.

How to handle colored observation noise in large least-squares problems

Abstract: An approach to handling colored observation noise in large least-squares (LS) problems is presented. The handling of colored noise is reduced to the problem of solving a Toeplitz system of linear equations. The colored noise is represented as an auto regressive moving-average (ARMA) process. Stability and invertability of the ARMA model allows the solution of the Toeplitz system to be reduced to two successive filtering operations using the inverse transfer function of the ARMA model. The numerical complexity of the algorithm scales proportionally to the order of the ARMA model times the number of observations. This makes the algorithm particularly suited for LS problems with millions of observations. It can be used in combination with direct and iterative algorithms for the solution of the normal equations. The performance of the algorithm is demonstrated for the computation of a model of the Earth’s gravity field from simulated satellite-derived gravity gradients up to spherical harmonic degree 300.

Theory of integer equivariant estimation with application to GNSS

Abstract: 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.

Nonlinear analysis of the three-dimensional datum transformation[conformal group C7(3)]

Abstract: The weighted Procrustes algorithm is presented as a very effective tool for solving the three-dimensional datum transformation problem In particular, the weighted Procrustes algorithm does not require any initial datum parameters for linearization or any iteration procedure As a closed-form algorithm it only requires the values of Cartesian coordinates in both systems of reference Where there is some prior information about the variance–covariance matrix of the two sets of Cartesian coordinates, also called pseudo-observations, the weighted Procrustes algorithm is able to incorporate such a quality property of the input data by means of a proper choice of weight matrix Such a choice is based on a properly designed criterion matrix which is discussed in detail Thanks to the weighted Procrustes algorithm, the problem of incorporating the stochasticity measures of both systems of coordinates involved in the seven parameter datum transformation problem [conformal group ℂ7(3)] which is free of linearization and any iterative procedure can be considered to be solved Illustrative examples are given

Harmonic analysis of the Earth's gravitational field by means of semi-continuous ephemerides of a low Earth orbiting GPS-tracked satellite. Case study: CHAMP

Abstract: An algorithm for the determination of the spherical harmonic coefficients of the terrestrial gravitational field representation from the analysis of a kinematic orbit solution of a low earth orbiting GPS-tracked satellite is presented and examined. A gain in accuracy is expected since the kinematic orbit of a LEO satellite can nowadays be determined with very high precision, in the range of a few centimeters. In particular, advantage is taken of Newton's Law of Motion, which balances the acceleration vector with respect to an inertial frame of reference (IRF) and the gradient of the gravitational potential. By means of triple differences, and in particular higher-order differences (seven-point scheme, nine-point scheme), based upon Newton's interpolation formula, the local acceleration vector is estimated from relative GPS position time series. The gradient of the gravitational potential is conventionally given in a body-fixed frame of reference (BRF) where it is nearly time independent or stationary. Accordingly, the gradient of the gravitational potential has to be transformed from spherical BRF to Cartesian IRF. Such a transformation is possible by differentiating the gravitational potential, given as a spherical harmonics series expansion, with respect to Cartesian coordinates by means of the chain rule, and expressing zero- and first-order Ferrer's associated Legendre functions in terms of Cartesian coordinates. Subsequently, the BRF Cartesian coordinates are transformed into IRF Cartesian coordinates by means of the polar motion matrix, the precession–nutation matrices and the Greenwich sidereal time angle (GAST). In such a way a spherical harmonic representation of the terrestrial gravitational field intensity with respect to an IRF is achieved. Numerical tests of a resulting Gauss–Markov model document not only the quality and the high resolution of such a space gravity spectroscopy, but also the problems resulting from noise amplification in the acceleration determination process.

On Helmert's methods of condensation

Abstract: Helmert’s first and second method of condensation are reviewed and generalized in two respects: First, the point at which the effects of topographical and condensation masses are calculated may be situated on or outside the topographical surface; second, the depth of the condensation layer below the geoid is arbitrary. While the first extension permits the application of the generalized model to the evaluation of airborne and satellite data, the second one gives an additional degree of freedom which can be used to provide a smooth gravity field after reducing the observation data. The respective formulae are derived for the generalized condensation model in both planar and spherical approximation. A comparison of the planar and the spherical model shows some structural differences, which are primarily visible in the out-of-integral terms. Considering the respective formulae for the combined topographic–condensation reduction on the background of the density structure of the Earth’s lithosphere, the consequences for the residual gravity field are investigated; it is shown that the residual field after applying Helmert’s second model of reduction is very rough, making this procedure unfavourable for downward continuation. Further considerations refer to the question of which sets of formulae should be used in geoid and quasigeoid determination. It is concluded that for high-precision applications the generalized spherical model, involving a depth of the condensation layer of between 20 and 30 km, should be superior to Helmert’s second model of condensation, although it requires the direct calculation of the indirect effect, which is larger than in the case of Helmert’s second method of condensation.

Predicting atmospheric biases for real-time ambiguity resolution in GPS/GLONASS reference station networks

Abstract: When using multiple reference station networks to support real-time kinematic positioning, the global positioning system (GPS) and/or GLONASS carrier-phase ambiguities associated with between-reference-receiver processing have to be resolved. At the initialisation stage it is not difficult to reliably resolve these ambiguities, in the post-processing mode, using data sets of several hours or days in length. However, due to significant residual atmospheric delays in the double-differenced measurements, it is a big challenge to resolve the ambiguities within the reference station network in real time, particularly in the case of the ambiguities of the newly risen satellites. Both temporal and spatial correlation characteristics for the atmospheric delays are discussed, and appropriate atmospheric delay prediction models are proposed. The experimental results show that the proposed bias prediction models can be used to reliably and efficiently resolve the integer ambiguities within reference station ne tworks in real time, on a single-epoch basis.

Green's function solution to spherical gradiometric boundary-value problems

Abstract: Three independent gradiometric boundary-value problems (BVPs) with three types of gradiometric data, {Γrr}, {Γrθ,Γrλ} and {Γθθ−Γλλ,Γθλ}, prescribed on a sphere are solved to determine the gravitational potential on and outside the sphere. The existence and uniqueness conditions on the solutions are formulated showing that the zero- and the first-degree spherical harmonics are to be removed from {Γrθ,Γrλ} and {Γθθ−Γλλ,Γθλ}, respectively. The solutions to the gradiometric BVPs are presented in terms of Green's functions, which are expressed in both spectral and closed spatial forms. The logarithmic singularity of the Green's function at the point ψ=0 is investigated for the component Γrr. The other two Green's functions are finite at this point. Comparisons to the paper by van Gelderen and Rummel [Journal of Geodesy (2001) 75: 1–11] show that the presented solution refines the former solution.

A solution to the downward continuation effect on the geoid determined by Stokes' formula

Abstract: The analytical continuation of the surface gravity anomaly to sea level is a necessary correction in the application of Stokes' formula for geoid estimation. This process is frequently performed by the inversion of Poisson's integral formula for a sphere. Unfortunately, this integral equation corresponds to an improperly posed problem, and the solution is both numerically unstable, unless it is well smoothed, and tedious to compute. A solution that avoids the intermediate step of downward continuation of the gravity anomaly is presented. Instead the effect on the geoid as provided by Stokes' formula is studied directly. The practical solution is partly presented in terms of a truncated Taylor series and partly as a truncated series of spherical harmonics. Some simple numerical estimates show that the solution mostly meets the requests of a 1-cm geoid model, but the truncation error of the far zone must be studied more precisely for high altitudes of the computation point. In addition, it should be emphasized that the derived solution is more computer efficient than the detour by Poisson's integral.

Computation of spherical harmonic coefficients from gravity gradiometry data to be acquired by the GOCE satellite: regularization issues

Abstract: The issue of optimal regularization is investigated in the context of the processing of satellite gravity gradiometry (SGG) data that will be acquired by the GOCE (Gravity Field and Steady-State Ocean Circulation Explorer) satellite. These data are considered as the input for determination of the Earth’s gravity field in the form of a series of spherical harmonics. Exploitation of a recently developed fast processing algorithm allowed a very realistic setup of the numerical experiments to be specified, in particular: a non-repeat orbit; 1-s sampling rate; half-year duration of data series; and maximum degree and order set to 300. The first goal of the study is to compare different regularization techniques (regularization matrices). The conclusion is that the first-order Tikhonov regularization matrix (the elements are practically proportional to the degree squared) and the Kaula regularization matrix (the elements are proportional to the fourth power of the degree) are somewhat superior to other regularization techniques. The second goal is to assess the generalized cross-validation method for the selection of the regularization parameter. The inference is that the regularization parameter found this way is very reasonable. The time expenditure required by the generalized cross-validation method remains modest even when a half-year set of SGG data is considered. The numerical study also allows conclusions to be drawn regarding the quality of the Earth’s gravity field model that can be obtained from the GOCE SGG data. In particular, it is shown that the cumulative geoid height error between degrees 31 and 200 will not exceed 1 cm.

A Study On The Combination of Satellite, Airborne and Terrestrial Gravity Data

Abstract: Satellite gravity missions, such as CHAMP, GRACE and GOCE, and airborne gravity campaigns in areas without ground gravity will enhance the present knowledge of the Earth’s gravity field. Combining the new gravity information with the existing marine and ground gravity anomalies is a major task for which the mathematical tools have to be developed. In one way or another they will be based on the spectral information available for gravity data and noise. The integration of the additional gravity information from satellite and airborne campaigns with existing data has not been studied in sufficient detail and a number of open questions remain. A strategy for the combination of satellite, airborne and ground measurements is presented. It is based on ideas independently introduced by Sjoberg and Wenzel in the early 1980s and has been modified by using a quasi-deterministic approach for the determination of the weighting functions. In addition, the original approach of Sjoberg and Wenzel is extended to more than two measurement types, combining the Meissl scheme with the least-squares spectral combination. Satellite (or geopotential) harmonics, ground gravity anomalies and airborne gravity disturbances are used as measurement types, but other combinations are possible. Different error characteristics and measurement-type combinations and their impact on the final solution are studied. Using simulated data, the results show a geoid accuracy in the centimeter range for a local test area.

Spurious periodic horizontal signals in sub-daily GPS position estimates

Abstract: Periodic horizontal signals have been identified in sub-daily estimates of position obtained from GPS data using batch least-squares solutions. The estimation of sub-daily position is commonly used for ocean loading and ice movement studies, for example. These spurious motions are artefacts of the processing methodology due to the presence of unmodelled, tidally-induced vertical signals in the GPS data. The periodic horizontal signals have magnitudes of about 40–50% of the amplitude of the vertical periodic signal. The shape of the periodic horizontal signals is related to the first derivative and the square of the vertical signal. The artefacts are mostly evident in the east–west component and can be removed by fixing the carrier phase ambiguities to integer values. We introduce a sensitivity analysis to study in detail the effects of ambiguity resolution and latitudinal variation of the artefacts. The overall conclusions are that for high precision batch processing of time series of geodetic positions, using sub-daily data spans, care needs to be taken to avoid aliasing of horizontal coordinates due to the vertical motion of the site.

Geoid determination with density hypotheses from isostatic models and geological information

Abstract: Geoid determination by Stokes's formula requires a complete knowledge of the topographical mass density distribution in order to perform gravity reductions to the geoid boundary. However, deeper masses are also of interest, in order to produce a smooth field of gravity anomalies which will improve results from interpolation procedures. Until now, in most cases a constant mass density has been considered, which is a very rough approximation of reality. The influence on the geoid height coming from different mass density hypotheses given by the isostatic models of Pratt/Hayford, Airy/Heiskanen and Vening Meinesz is studied. Apart from a constant mass density value, additional density information deduced from geological maps and thick sedimentary layers is considered. An overview of how mass density distributions act within Stokes's theory is given. The isostatic models are considered in spherical and planar approximation, as well as with constant and lateral variable mass density of the topographical and deeper masses. Numerical results in a test area in south-west Germany show that the differences in the geoid height due to different density hypotheses can reach a magnitude of more than 1 decimetre, which is not negligible in a precise geoid determination with centimetre accuracy.

Effectiveness of robust methods in heterogeneous linear models

Abstract: In outlier detection using robust estimation methods, if random errors in a linear model are independent and identically distributed (IID), the model is called homoscedastic. In cases where random errors' variances may vary depending on a parameter, the model is called heteroscedastic. However, random errors in a linear model are a combination of homoscedastic and heteroscedastic random errors. This type of model is called heterogeneous. How does the effectiveness of robust methods change in a contaminated heterogeneous linear model when outliers are small? This question is investigated by simulation using two robust methods in a contaminated linear heterogeneous regression model where homoscedastic and heteroscedastic linear regressions are combined. One simple and two different multiple regressions are employed. When the stochastic model is true, the effectiveness of two robust methods in a contaminated heterogeneous linear regression changes depending on the strength of the heteroscedasticity, the number of outliers, the number of unknowns, and the number of heteroscedastic observations. In addition, the robust methods behave differently against outliers in some cases. If the stochastic model is not true, the effectiveness of the robust methods decreases significantly.

Fast spherical collocation: theory and examples

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

Time transfer using GPS carrier phase: error propagation and results

Abstract: A joint time-transfer project between the Astronomical Institute of the University of Berne (AIUB) and the Swiss Federal Office of Metrology and Accreditation (METAS) was initiated to investigate the power of the time transfer using GPS carrier phase observations. Studies carried out in the context of this project are presented. The error propagation for the time-transfer solution using GPS carrier phase observations was investigated. To this purpose a simulation study was performed. Special interest was focussed on errors in the vertical component of the station position, antenna phase-center variations and orbit errors. A constant error in the vertical component introduces a drift in the time-transfer results for long baselines in east–west directions. The simulation study was completed by investigating the profit for time transfer when introducing the integer carrier phase ambiguities from a double-difference solution. This may reduce the drift in the time-transfer results caused by constant vertical error sources. The results from the present time-transfer solution are shown in comparison to results obtained with independent time-transfer techniques. The interpretation of the comparison benefits from the investigations of the error propagation study. Two types of solutions are produced on a regular basis at AIUB: one based on the rapid orbits from CODE, the other on the CODE final orbits. The rapid solution is available the day after the observations and has nearly the same quality as the final solution, which has a latency of about one week. The differences between these two solutions are below the nanosecond level. The differences from independent time-transfer techniques such as TWSTFT (two-way satellite time and frequency transfer) are a few nanoseconds for both products.

Rank-defect integer estimation and phase-only modernized GPS ambiguity resolution

Abstract: Carrier-phase ambiguity resolution is usually based on the assumption that the underlying model of observation equations is of full rank. In this contribution the model of observation equations is assumed to be of less than full rank. The well-known three-step procedure of integer least squares is generalized and it is shown how the solution can be affected by the rank deficiency. Although the theory is of interest in its own right, a prime application is found in the problem of phase-only ambiguity resolution in the presence of ionospheric delays. The impact of the third global positioning system (GPS) frequency is therefore studied and it is shown by means of suitable ambiguity transformation which ambiguities are integer estimable and which are not in the case of phase-only modernized GPS. A pitfall when using ionosphere-free linear phase combinations is identified. It is shown that only a particular class of such linear phase combinations permits a parameterization in terms of integer-estimable ambiguities. This pitfall does not manifest itself so clearly with the current dual-frequency GPS system.

Fast and accurate computation of spherical harmonic coefficients from satellite gravity gradiometry data

Abstract: A very efficient approach for computing spherical harmonic coefficients from satellite gravity gradiometry (SGG) data has been developed. The core of the proposed approach is a combination of the method of conjugate gradients with preconditioning on the one hand and an extremely fast algorithm for synthesis (application of the design matrix to a vector) and co-synthesis (application of the transposed design matrix to a vector) on the other. The high performance of the synthesis and co-synthesis is achieved by introducing an intermediate step, where computations are made on a regular three-dimensional (3-D) spherical grid. As a result, the Legendre functions can be computed for all the points at a given latitude only once and 1-D fast-Fourier techniques can be fully exploited. Transition to the true observation points is carried out by means of a 3-D spline interpolation. It is expected that the technique will be able to invert the full set of SGG data from the GOCE satellite mission (12-month data, four tensor components, 1-s sampling) in only a few hours on an SGI Origin 3800 computer with 16 processing elements. This corresponds approximately to 1 or 2 days of computation on a Pentium-IV PC. The choice of a relatively coarse 3-D spherical grid improves the efficiency even further, at the cost of minor errors in the solution. In this mode, the proposed approach can be used for quick-look data analysis.

Applying differential InSAR to orbital dynamics: a new approach for estimating ERS trajectories

Abstract: A new approach for tuning the trajectories of the European remote sensing (ERS) satellites is developed and assessed. Differential dual-pass interferometry is applied to calculate interferograms from the phase difference of synthetic aperture radar (SAR) images acquired by the ERS satellites over the site of the 1992 earthquake in Landers, California. These interferograms contain information about orbital trajectories and geophysical deformation. Beginning with good prior estimates of the orbital trajectories, a radial and an across-track orbital adjustment is estimated at each epoch. The data are the fringe counts along distance and azimuth. Errors in the across-track and radial components of the orbit estimates produce fringes in the interferograms. The spacing between roughly parallel fringes gives the gradients in distance and azimuth coordinates. The approach eliminates these fringes from interferometric pairs spanning relatively short time intervals containing few topographic residuals or atmospheric artefacts. An optimum interferometric path with six SAR acquisitions is selected to study post- and inter-seismic deformation fields. In order to regularize the problem, it is assumed that the radial and across-track adjustments both sum to zero. Applying the adjustment approach to the prior estimates of trajectory from the Delft Institute for Earth-Orientated Space Research (DEOS), root mean squares of 7.3 cm for the across-track correction components and 2.4 cm for the radial ones are found. Assuming 0.1 fringes for the a priori standard deviation of the measurement, the approach yields mean standard deviations of 2.4 cm for the across-track and 4.5 cm for the radial components. The approach allows an ‘interval by interval’ improvement of a set of orbital estimates from which post-fit interferograms of different time intervals spanning a total 3.8-year inter-seismic time interval can be created. The interferograms calculated with the post-fit orbital estimates compare favorably with those corrected with a conventional orbital tuning approach. Using the adjustment approach, it is possible to distinguish between orbital and deformation contributions to interferometric SAR (InSAR) phase gradients. Surface deformation changes over an inter-seismic time interval longer than one year can be measured. This approach is, however, limited to well-correlated interferograms where it is possible to measure the fringe gradient.

On geoid determination from airborne gravity

Abstract: Recent advances in the performance of scalar airborne gravimetry, routinely yielding gravity data with an accuracy as high as 1 mGal with a minimum spatial resolution (full wavelength) of 10 km, allow for the use of airborne gravity in precise geoid computations. Theoretical issues related to geoid determination from discrete samples of band-limited airborne gravity and practical geoid computations based on high-frequency synthetic data and actual observations are discussed. The mathematical model for geoid determination from band-limited airborne gravity which is introduced is based on Helmert's reduction of gravity/co-geoid and the application of the discretized integral formula developed specifically for airborne gravity data. Computational formulae are derived for the following specifications of airborne gravity data: almost-white observation noise, a band-limited frequency content of processed observations, and a smooth regular surface on which gravity observations are collected. Two major applications of airborne gravity data for geoid determination are discussed. Airborne gravimetry can be used over areas with a sparse or no ground gravity coverage to provide the medium- to high-frequency components of the Earth's gravity field. These data can be used in combination with global gravity models for the computation of the gravimetric geoid. Airborne gravimetry can also be used to fill in gaps in existing ground gravity coverage (mainly in inaccessible areas). Both approaches are tested with actual airborne data observed using an inertially referenced airborne gravimeter at the Alexandria test range near Ottawa, Canada. Observed airborne gravity disturbances, combined with either ground or global gravity data, are used for the determination of the gravimetric geoid in the two applications outlined above. The combined solutions are compared to the latest official Canadian gravimetric geoid and to available GPS/levelling data in the area. Numerical results show that airborne data can be used for geoid determination with centimetre-level accuracy (medium- and high-frequency information) over areas with negligible topographical effects on gravity and the geoid.

Orthometric corrections from leveling, gravity, density and elevation data: a case study in Taiwan

Abstract: A new orthometric correction (OC) formula is presented and tested with various mean gravity reduction methods using leveling, gravity, elevation, and density data. For mean gravity computations, the Helmert method, a modified Helmert method with variable density and gravity anomaly gradient, and a modified Mader method were used. An improved method of terrain correction computation based on Gaussian quadrature is used in the modified Mader method. These methods produce different results and yield OCs that are greater than 10 cm between adjacent benchmarks (separated by ∼2 km) at elevations over 3000 m. Applying OC reduces misclosures at closed leveling circuits and improves the results of leveling network adjustments. Variable density yields variation of OC at millimeter level everywhere, while gravity anomaly gradient introduces variation of OC of greater than 10 cm at higher elevations, suggesting that these quantities must be considered in OC. The modified Mader method is recommended for computing OC.

A bilinear approximation of the surface relief in terrain correction computations

Abstract: A new method is presented for the computation of the gravitational attraction of topographic masses when their height information is given on a regular grid. It is shown that the representation of the terrain relief by means of a bilinear surface not only offers a serious alternative to the polyhedra modeling, but also approaches even more smoothly the continuous reality. Inserting a bilinear approximation into the known scheme of deriving closed analytical expressions for the potential and its first-order derivatives for an arbitrarily shaped polyhedron leads to a one-dimensional integration with - apparently - no analytical solution. However, due to the high degree of smoothness of the integrand function, the numerical computation of this integral is very efficient. Numerical tests using synthetic data and a densely sampled digital terrain model in the Bavarian Alps prove that the new method is comparable to or even faster than a terrain modeling using polyhedra.