Spectral accuracy of Jgm3 from satellite crossover altimetry

A detailed accuracy assessment of the geopotential model Jgm3 is made based on independent single- and dual-satellite sea-height differences at crossovers from altimetry with Jgm3-based orbits. These differences, averaged over long time spans and in latitude bands, are converted to spectra (latitude-lumped coefficients) by least-squares estimation. The observed error spectra so obtained are then compared directly to error projections for them from the Jgm3 variance–covariance matrix. It is found from these comparisons that Jgm3 is generally well calibrated with respect to the crossover altimetry of and between Geosat, TOPEX/Poseidon (T/P), and Ers 1. Some significant discrepancies at a few lower orders (namely m=1 and 3) indicate a need for further improvement of Jgm3. A companion calibration (by order) of the geopotential model Jgm2 shows its variance–covariance matrix also to be generally well calibrated for the same single- and dual-satellite altimeter data sets (but based on Jgm2 orbits), except that the error projections for Geosat are too pessimistic. The analysis of the dual-satellite crossovers reveals possible relative coordinate system offsets (particularly for Geosat with respect to T/P) which have been discussed previously. The long-term detailed seasonally averaged Geosat sea level with respect to T/P (covering 1985–1996) should be useful in gauging the relative change in sea level between different parts of the ocean over the single 4-year gap between these missions (1988–1992). Received: 16 February 1998 / Accepted: 25 November 1998     

A study on the reference frame consistency in recent Earth gravitational models

All gravity field functionals obtained from an Earth gravitational model (EGM) depend on the underlying terrestrial reference frame (TRF), with respect to which the EGM’s spherical harmonic coefficients refer to. In order to maintain a coherent framework for the comparison of current and future EGMs, it is thus important to investigate the consistency of their inherent TRFs, especially when their use is intended for high precision studies. Following the methodology described in an earlier paper by Kleusberg (1980), the similarity transformation parameters between the associated reference frames for several EGMs (including the most recent CHAMP/GRACE models at the time of writing this paper) are estimated in the present study. Specifically, the differences between the spherical harmonic coefficients for various pairs of EGMs are parameterized through a 3D-similarity spatial transformation model that relates their underlying TRFs. From the least-squares adjustment of such a parametric model, the origin, orientation and scale stability between the EGMs’ reference frames can be identified by estimating their corresponding translation, rotation and scale factor parameters. Various aspects of the estimation procedure and its results are highlighted in the paper, including data weighting schemes, the sensitivity of the results with respect to the selected harmonic spectral band, the correlation structure and precision level of the estimated transformation parameters, the effect of the estimated differences of the EGMs’ reference frames on their height anomaly signal, and the overall feasibility of Kleusberg’s formulae for the assessment of TRF inconsistencies among global geopotential models.     

10 Hz or 10 s?

Data sampling frequencies in many kinematic GNSS applications are often in the range of 2–10 Hz or even higher. In contrast, the sampling frequency of standard reference stations is usually not higher than 1 Hz, and many stations even deliver data at sampling intervals as large as 30 s. An easily implemented algorithm for data interpolation will be presented and it will be demonstrated that interpolation of pseudorange and carrier phase reference station data is possible at a high level of accuracy. This technique—which has not received proper attention so far—is helpful to reduce data storage capacity for postprocessing applications, but and can also be beneficial for real-time applications suffering from slow data links. Results of test trials indicate that a standard deviation better than 2 mm can be reached for interpolated carrier phases collected at reference sites sampling data with 5-s intervals in double difference mode. These interpolated double difference data obviously still follow a Gaussian distribution. A trend function for the expected standard deviation of interpolated double difference carrier phase measurements will be presented. From this function, a recommendation for an optimal sampling rate of reference station data can be derived which is close to 10 s.     

Estimation of dynamic ocean topography in the Gulf Stream area using the Hotine formula and altimetry data

Two modifications of the Hotine formula using the truncation theory and marine gravity disturbances with altimetry data are developed and used to compute a marine gravimetric geoid in the Gulf Stream area. The purpose of the geoid computation from marine gravity information is to derive the absolute dynamic ocean topography based on the best estimate of the mean surface height from recent altimetry missions such as Geosat, ERS-1, and Topex. This paper also tries to overcome difficulties of using Fast Fourier Transformation (FFT) techniques to the geoid computation when the Hotine kernel is modified according to the truncation theory. The derived absolute dynamic ocean topography is compared with that from global circulation models such as POCM4B and POP96. The RMS difference between altimetry-derived and global circulation model dynamic ocean topography is at the level of 25cm. The corresponding mean difference for POCM4B and POP96 is only a few centimeters. This study also shows that the POP96 model is in slightly better agreement with the results derived from the Hotine formula and altimetry data than POCM4B in the Gulf Stream area. In addition, Hotine formula with modification (II) gives the better agreement with the results from the two global circulation models than the other techniques discussed in this paper. Received: 10 October 1996 / Accepted: 16 January 1998     

Accuracy analysis of vertical deflection data observed with the Hannover Digital Zenith Camera System TZK2-D

This paper analyses the accuracy of vertical deflection measurements carried out with the Digital Zenith Camera System TZK2-D, an astrogeodetic state-of-the-art instrumentation developed at the University of Hannover. During 107 nights over a period of 3.5 years, the system was used for repeated vertical deflection observations at a selected station in Hannover. The acquired data set consists of about 27,300 single measurements and covers 276 h of observation time, respectively. For the data collected at an earlier stage of development (2003 to 2004), the accuracy of the nightly mean values has been found to be about 0′′.10−0′′.12. Due to applying a refined observation strategy since 2005, the accuracy of the vertical deflection measurements was enhanced into the unprecedented range of 0′′.05 − 0′′.08. Accessing the accuracy level of 0′′.05 requires usually 1 h of observational data, while the 0′′.08 accuracy level is attained after 20 min measurement time. In comparison to the analogue era of geodetic astronomy, the accuracy of vertical deflection observations is significantly improved by about one order of magnitude.     

Interpretation of the tidal residuals during the 11 July 1991 total solar eclipse

Observations of gravity and atmospheric pressure variations during the total solar eclipse of 11 July 1991 in Mexico City are presented. An LCR-G402 gravimeter equipped with a feedback system and a digital data acquisition system scanned gravity and pressure every second around the totality. On the pressure record an oscillation, starting at the totality, with a peak to peak amplitude of 0.5 hPa and a periodicity of 40 to 50 min, can clearly be seen. This oscillation results from the thermal shock wave produced by the Moon shadow travelling at supersonic speed. At the 0.1 μGal (1 nm · s⁻²) level all gravity perturbations are explained by the atmospheric pressure effect. Received: 10 February 1995 / Accepted: 7 June 1996     

On the multivariate total least-squares approach to empirical coordinate transformations. Three algorithms

The multivariate total least-squares (MTLS) approach aims at estimating a matrix of parameters, Ξ, from a linear model (Y−E _Y = (X−E _X ) · Ξ) that includes an observation matrix, Y, another observation matrix, X, and matrices of randomly distributed errors, E _Y and E _X . Two special cases of the MTLS approach include the standard multivariate least-squares approach where only the observation matrix, Y, is perturbed by random errors and, on the other hand, the data least-squares approach where only the coefficient matrix X is affected by random errors. In a previous contribution, the authors derived an iterative algorithm to solve the MTLS problem by using the nonlinear Euler–Lagrange conditions. In this contribution, new lemmas are developed to analyze the iterative algorithm, modify it, and compare it with a new ‘closed form’ solution that is based on the singular-value decomposition. For an application, the total least-squares approach is used to estimate the affine transformation parameters that convert cadastral data from the old to the new Israeli datum. Technical aspects of this approach, such as scaling the data and fixing the columns in the coefficient matrix are investigated. This case study illuminates the issue of “symmetry” in the treatment of two sets of coordinates for identical point fields, a topic that had already been emphasized by Teunissen (1989, Festschrift to Torben Krarup, Geodetic Institute Bull no. 58, Copenhagen, Denmark, pp 335–342). The differences between the standard least-squares and the TLS approach are analyzed in terms of the estimated variance component and a first-order approximation of the dispersion matrix of the estimated parameters.     

GPS network monitors the Western Alps' deformation over a five-year period: 1993–1998

  The Western Alps are among the best studied collisional belts with both detailed structural mapping and also crustal geophysical investigations such as the ECORS and EGT seismic profile. By contrast, the present-day kinematics of the belt is still largely unknown due to small relative motions and the insufficient accuracy of the triangulation data. As a consequence, several tectonic problems still remain to be solved, such as the amount of N–S convergence in the Occidental Alps, the repartition of the deformation between the Alpine tectonic units, and the relation between deformation and rotation across the Alpine arc. In order to address these problems, the GPS ALPES group, made up of French, Swiss and Italian research organizations, has achieved the first large-scale GPS surveys of the Western Alps. More than 60 sites were surveyed in 1993 and 1998 with a minimum observation of 3 days at each site. GPS data processing has been done by three independent teams using different software. The different solutions have horizontal repeatabilities (N–E) of 4–7 mm in 1993 and 2–3 mm in 1998 and compare at the 3–5-mm level in position and 2-mm/yr level in velocity. A comparison of 1993 and 1998 coordinates shows that residual velocities of the GPS marks are generally smaller than 2 mm/yr, precluding a detailed tectonic interpretation of the differential motions. However, these data seem to suggest that the N–S compression of the Western Alps is quite mild (less than 2 mm/yr) compared to the global convergence between the African and Eurasian plate (6 mm/yr). This implies that the shortening must be accomodated elsewhere by the deformation of the Maghrebids and/or by rotations of Mediterranean microplates. Also, E–W velocity components analysis supports the idea that E–W extension exists, as already suggested by recent structural and seismotectonic data interpretation. Received: 27 November 2000 / Accepted: 17 September 2001     

Assessing Greenland ice mass loss by means of point-mass modeling: a viable methodology

Greenland ice mass loss is one of the most serious phenomena of present-day global climate change. In this context, both the quantification of overall deglaciation rates and its spatial localization are highly significant. We have thoroughly investigated the technique of point-mass modeling in order to derive mass-balance patterns from GRACE (Gravity Recovery And Climate Experiment) gravimetry. The method infers mass variations on the Earth’s surface from gravitational signals at satellite altitude. In order to solve for point-mass changes, we applied least-squares adjustment. Due to downward continuation, numerical stabilization of the inversion process gains particular significance. We stabilized the ill-posed problem by Tikhonov regularization. Our simulation and real data experiments show that point-mass modeling provides both rational deglaciation rates and high-resolution spatial mass variation patterns.     

GPS sensitivity analysis applied to non-permanent deformation control networks

This paper illustrates the surveys and the results obtained in an experiment whose goal is to evaluate the Global Positioning System (GPS) sensitivity and accuracy for deformation control on non-permanent network of different extensions. To this aim a high-precision device was properly built to set up known displacements along three orthogonal axes of a GPS antenna. One of the antennas in the considered GPS networks was moved according to centimeter and sub-centimeter displacements; after careful GPS data processing, it was evaluated whether these simulated deformations were correctly a posteriori detected and at which probability level. This experiment was carried out both on a local (baselines ranging between 3 and 30 km) and on a regional (baselines ranging between 300 and 600 km) GPS network. The results show that in the local network it is possible to identify the displacements at a level of 10 mm in height and at a level of 5 mm in horizontal position. The analysis of the regional network showed that it is fundamental to investigate new strategies to model the troposphere; in fact, it is necessary to improve the precision of the height in order to correctly identify displacements lower than 60–80 mm; on the contrary, horizontal displacements can be evidenced at the level of 20 mm. Received: 27 April 1998 / Accepted: 14 December 1998     

Making the connection between Geosat and TOPEX/Poseidon

We can presently construct two independent time series of sea level, each at a precision of a few centimeters, from Geosat (1985–1988) and TOPEX/Poseidon (1992–1995) collinear altimetry. Both are based on precise satellite orbits computed using a common geopotential model, JGM-2 (Nerem et al. 1994). We have attempted to connect these series using Geosat-T/P crossover differences in order to assess long-term ocean changes between these missions. Unfortunately, the observed result are large-scale sea level differences which appear to be due to a combination of geodetic and geopotential error sources. The most significant geodetic component seems to be a coordinate system bias for Geosat sea level (relative to T/P) of −7 to −9 cm in the y-axis (towards the Eastern Pacific). The Geosat-T/P sea height differences at crossovers (with JGM-2 orbits) probably also contain stationary geopotential-orbit error of about the same magnitude which also distort any oceanographic interpretation of the observed changes. We also found JGM-3 Geosat orbits have not resolved the datum errors evident from the JGM-2 Geosat -T/P results. We conclude that the direct altimetric approach to accurate determination of sea level change using Geosat and T/P data still depends on further improvement in the Geosat orbits, including definition of the geocenter. Received: 11 March 1996; Accepted: 19 September 1996     

Atmospheric Angular Momentum Time-Series: Characterization of their Internal Noise and Creation of a Combined Series

The atmosphere induces variations in Earth rotation. These effects are classically computed using the “angular momentum approach”. In this method, the variations in Earth rotation are estimated from the variations in the atmospheric angular momentum (AAM). Several AAM time-series are available from different meteorological centers. However, the estimation of atmospheric effects on Earth rotation differs when using one atmospheric model or the other. The purpose of this work is to build an objective criterion that justifies the use of one series in particular. Because the atmosphere is not the only cause of Earth rotation variations, this criterion cannot rely only on a comparison of AAM series with geodetic data. Instead, we determine the quality of each series by making an estimation of their noise level, using a generalized formulation of the “three-cornered hat method”. We show the existence of a link between the noise of the AAM series and their correlation with geodetic data: a noisy series is usually less correlated with Earth orientation data. As the quality of the series varies in time, we construct a combined AAM series, using time-dependent weights chosen so that the noise level of the combined series is minimal. To determine the influence of a minimal noise level on the correlation with geodetic data, we compute the correlation between the combined series and Earth orientation data. We note that the combined series is always amongst the best correlated series, which confirms the link established before. The quality criterion, while totally independent of Earth orientation observations, appears to be physically convincing when atmospheric and geodetic data are compared     

Impact of FORMOSAT-3/COSMIC GPS radio occultation and dropwindsonde data on regional model predictions during the 2007 Mei-yu season

A study of the impact of FORMOSAT-3/COSMIC GPS radio occultation (RO) and dropwindsonde data on regional model simulations for a 11-day period during the 2007 Mei-yu season is presented. The Weather Research and Forecasting (WRF) model and its three-dimensional variation component, WRF-Var, are used for regional model predictions of heavy rainfall events in Taiwan. Without the use of GPS RO and dropwindsonde data, pressure and relative humidity are, in general, underestimated by the model; temperature predictions have a warm bias at the low level and a cold bias at the high level; and the east–west and north–south component winds have positive and negative biases, respectively. Incorporating GPS RO data tends to improve the prediction for longer integration. The assimilation of dropwindsonde data improves the forecast at the earlier time and at higher levels, and the improvement decreases over time. The reason the dropwindsonde data produce a positive impact earlier and the GPS RO data later is that there are few GPS RO observations in the fine domain. The large-scale simulation is first improved using the GPS RO observations, and the resulting changes can have a positive impact on the mesoscale at the later time. The dropwindsonde observations were taken inside the fine domain such that their impact can be detected early in the simulation. With both types of observation included, the prediction shows even greater improvement. At the earlier forecast time, there is nearly no impact from GPS and dropwindsonde data on rainfall forecasts. However, at the later integration time, the GPS data start to significantly improve the rainfall forecast. The dropwindsonde data also provide a positive impact on rainfall forecasts, but it is not as significant as that of the GPS data.     

Precise ionosphere-free single-frequency GNSS positioning

Ionospheric delays can be efficiently eliminated from single-frequency data using a combination of carrier phases and code ranges. Unfortunately, GPS and GLONASS ranges are relatively noisy which can limit the use of the positioning method. Nevertheless, position standard deviations are in the range of 6–8 cm (horizontal) and 7–9 cm (3d) obtained from diurnal data batches from selected IGS reference stations can be further reduced to 2–3 cm (3d) for weekly smoothed averages. GPS data sets collected in Ghana (Africa) reveal a typical level of 10 cm of deviation that must be anticipated under average conditions. Looking at the future of GNSS, the European Galileo system will, in contrast to GPS, provide the broadband signal E5 that is by far less affected by multipath thus providing rather precise range measurements. Simulated processing runs featuring both high ionospheric and tropospheric delay variations show a 3d position precision of 4 cm even for a data batch as short as just 1 h, whereas GPS L1/Galileo E1 performance is close to 13 cm for the same data set.     

Terrain complexity and reduction of topographic data

Digital terrain data are useful for a variety of applications in mapping and spatial analysis. Most available terrain data are organized in a raster format, among them being the most extensively-used Digital Elevation Models (DEM) of the U.S. Geological Survey. A common problem with DEM for spatial analysis at the landscape scale is that the raster encoding of topography is subject to data redundancy and, as such, data volumes may become prohibitively large. To improve efficiency in both data storage and information processing, the redundancy of the terrain data must be minimized by eliminating unnecessary elements. To what extent a set of terrain data can be reduced for improving storage and processing efficiency depends on the complexity of the terrain. In general, data elements for simpler, smoother surfaces can be substantially reduced without losing critical topographic information. For complex terrains, more data elements should be retained if the topography is to be adequately represented. In this paper, we present a measure of terrain complexity based on the behavior of selected data elements in representing the characteristics of a surface. The index of terrain complexity is derived from an estimated parameter which denotes the relationship between terrain representation (percentage surface representation) and relative data volume (percentage DEM elements). The index can be used to assess the required volume of topographic data and determine the appropriate level of data reduction. Two quadrangles of distinct topographic characteristics were examined to illustrate the efficacy of the developed methodology.     

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.     

Accurate absolute GPS positioning through satellite clock error estimation

 An algorithm for very accurate absolute positioning through Global Positioning System (GPS) satellite clock estimation has been developed. Using International GPS Service (IGS) precise orbits and measurements, GPS clock errors were estimated at 30-s intervals. Compared to values determined by the Jet Propulsion Laboratory, the agreement was at the level of about 0.1 ns (3 cm). The clock error estimates were then applied to an absolute positioning algorithm in both static and kinematic modes. For the static case, an IGS station was selected and the coordinates were estimated every 30 s. The estimated absolute position coordinates and the known values had a mean difference of up to 18 cm with standard deviation less than 2 cm. For the kinematic case, data obtained every second from a GPS buoy were tested and the result from the absolute positioning was compared to a differential GPS (DGPS) solution. The mean differences between the coordinates estimated by the two methods are less than 40 cm and the standard deviations are less than 25 cm. It was verified that this poorer standard deviation on 1-s position results is due to the clock error interpolation from 30-s estimates with Selective Availability (SA). After SA was turned off, higher-rate clock error estimates (such as 1 s) could be obtained by a simple interpolation with negligible corruption. Therefore, the proposed absolute positioning technique can be used to within a few centimeters' precision at any rate by estimating 30-s satellite clock errors and interpolating them. Received: 16 May 2000 / Accepted: 23 October 2000     

Results of three year observation with a superconducting gravimeter at the GeoForschungsZentrum Potsdam

The superconducting gravimeter (SG) TT70 has been continuously recording gravity data at the GeoForschungsZentrum (GFZ) Potsdam absolute gravity site since July 1992. The recorded data are edited and preprocessed by spike and step detection and elimination and gap filling. An atmospheric pressure correction is carried out on gravity data in the time domain with a complex admittance before tidal fitting. The atmospheric pressure admittance is calculated from tide free output of SG data and local atmospheric pressure using the cross spectral method. The ground water level admittance is determined by a single coefficient. Improvements with these corrections are shown in analysis results. New tidal parameters for Potsdam site are determined and compared with recordings of an Askania spring gravimeter at a nearby site. Deviations against the Wahr-Dehant-model are shown. Polar motion data of the IERS (International Earth Rotation Service, Paris) are used to calculate variations of centrifugal acceleration caused by polar motion (pole tide). These are compared with the corrected tide free output of SG series. For drift determination the polar motion correction is applied on SG data. The nutation period equivalent to the Earth's Nearly Diurnal Free Wobble is calculated from the SG data with a value of T_FCN = (437.4 ± 1.5) sidereal days. This result is compared with those obtained from other SG stations. Received 19 December 1995; Accepted 13 September 1996     

Recent Trends of Arctic and Antarctic Summer Sea-Ice Cover Observed from Space-Borne Scatterometer

Long term variations in Sea ice distribution strongly influence the atmosphere and ocean in the polar regions. In the recent period significant variations in sea ice cover have been observed in both the hemispheres. In the past, studies have been carried out that report the trends either at the Arctic/Antarctic level or at sector level. However, only a few studies have concentrated on the investigation of trends at grid level using scatterometer data. The present study focuses on the investigations of the sea ice trend at 1 × 1 degree grid level over the period 2000–2007 using QuickSCAT 0.2-degree resolution Scatterometer data. It was observed that in the Arctic overall monthly trend is negative in all the sectors, with the Arctic level decline of 3.26% per year. In the Antarctic, region-wise different trends have been observed. Negative trend is observed in the Amundsen- Bellingshausen Seas and also in the Indian Ocean sector near the continental Ice shelves. It was highlighted that significant trends exists within the pockets of marginal seas.