On the reliability of the VCV Matrix: A case study based on GAMIT and Bernese GPS Software

Commonly, the variance-covariance (VCV) matrix derived from GPS processing software underestimates the magnitude of the error, mainly due to the fact that physical correlations are normally neglected. The GAMIT and Bernese software packages serve the scientific community as important tools for GPS measurement processing and analyzing, especially in precise applications. Therefore, the reliability of the VCV matrices derived by the GAMIT and Bernese packages is of great importance. Formal accuracies derived from both software need to be scaled by applying a scaling factor (SF) that multiplies the software-derived formal errors. However, to the best of our knowledge, no standard approach approved by the GPS community exists. In this report, an analysis is carried out in order to test the reliability and the validity of the VCV matrices in both software, and to provide SFs needed to calculate the realistic accuracies reflecting the actual error levels. The method applied in this study allows deriving SFs for formal accuracies obtained from GAMIT and Bernese. The results attained from the time series of eight days for eight baselines (lengths of 20–415 km) indicate that the overall SF for GAMIT is more than 10 times smaller than for Bernese (1.9 and 23.0, respectively). Although no distance-dependent SF was detected in either case, the session-duration dependence was detected for the Bernese software, while no clear session-duration dependence was observed for the GAMIT. Furthermore, no receiver/antenna dependence could be deduced from the results of this analysis.     

The ellipsoidal corrections to the topographic geoid effects

The topographic effects by Stokes formula are typically considered for a spherical approximation of sea level. For more precise determination of the geoid, sea level is better approximated by an ellipsoid, which justifies the consideration of the ellipsoidal corrections of topographic effects for improved geoid solutions. The aim of this study is to estimate the ellipsoidal effects of the combined topographic correction (direct plus indirect topographic effects) and the downward continuation effect. It is concluded that the ellipsoidal correction to the combined topographic effect on the geoid height is far less than 1 mm. On the contrary, the ellipsoidal correction to the effect of downward continuation of gravity anomaly to sea level may be significant at the 1-cm level in mountainous regions. Nevertheless, if Stokes formula is modified and the integration of gravity anomalies is limited to a cap of a few degrees radius around the computation point, nor this effect is likely to be significant.AcknowledgementsThe author is grateful for constructive remarks by J Ågren and the three reviewers.     

Position and velocity reliability testing in degraded GPS signal environments

Reliability testing, namely receiver autonomous integrity monitoring (RAIM), consists of statistical testing of least-squares residuals of observations, e.g., on an epoch-by-epoch basis aiming towards reliable navigation fault detection and exclusion (FDE). In this paper, classic RAIM and FDE methods are extended with testing of range-rate residuals to find inconsistent velocity solutions in order to contribute to the reliability of the system with special focus on degraded signal environments. Reliability enhancement efforts discussed include a Backward-FDE scheme based on statistical outlier detection and an iteratively reweighted robust estimation technique, a modified Danish method. In addition, measurement weighting assigned to code and Doppler observations is assessed in the paper in order to allow fitting a priori variance models to the estimation processes. The schemes discussed are also suitable in terms of computational convenience for a combined GPS/Galileo system. The objective of this paper is to assess position and velocity reliability testing and enhancement in urban and indoor conditions and to analyze the navigation accuracy conditions with high sensitivity GPS (HSGPS) tests. The results show the necessity of weighted estimation and FDE for reliability enhancement in degraded signal-environment navigation.     

Verification and mitigation of the power-induced measurement errors for airport pseudolites in LAAS

Certain GPS receivers exhibit a power-induced measurement error when the input signal power exceeds a certain threshold that is a function of the receiver architecture. The characterization of this error is important in the case of airport pseudolites (APLs) due to receiver saturation at ranges close to the transmission antenna. This paper details an effort using laboratory test to characterize the power-induced measurement error in NovAtel Millennium and Beeline receivers that are widely used for APL development. The results from the laboratory tests show that this effect can lead to errors as large as 3 m on pseudorange measurement for the dynamic range typical for APLs. The carrier-phase error however is seen to be limited to 5 cm. On the verification and characterization of the error, a cost-effective mitigation technique for LAAS is proposed. A solution for the ground subsystem is sought by employing power-control measures to limit the amount of APL power that LGF GPS/APL receivers are exposed to. In order to keep the integration cost low, the solution for the airborne subsystem does not involve hardware modifications or additions, but rather relies on the transition to a differential carrier-phase positioning algorithm prior to the onset of the power-induced error along the approach path. The proposed architecture was flight-tested to verify successful mitigation of the power-induced error on both the ground and air, and the results from these tests are presented in the paper.     

Reliability checking for GNSS baseline and network processing

Reliability analysis is inseparably connected with the formulation of failure scenarios, and common test statistics are based on specific assumptions. This is easily overlooked when processing observation differences. Poor failure identification performance and misleading pre-analysis results, mainly meaningless minimum detectable biases and external reliability measures, are the consequence. A reasonable failure scenario for use with differenced GNSS observations is formulated which takes into account that individual outliers in the original data affect more than one processed observation. The proper test statistics and reliability indicators are given for use with correlated observations and both batch processing and Kalman filtering. It is also shown that standardized residuals and redundancy numbers fail completely when used with double differenced observations.

Analysis of temporal sar and optical data for rice mapping

This study investigates the potential of multi-temporal signature analysis of satellite imagery to map rice area in South 24 Paraganas district of West Bengal. Two optical data (IRS ID LISS III) and three RADARSAT SAR data of different dates were acquired during 2001. Multi-temporal SAR backscatter signatures of different landcovers were incorporated into knowledge based decision rules and kharif landcover map was generated. Based on the spectral variation in signature, the optical data acquired during rabi (January) and summer (March) season were classified using supervised maximum likelihood classifier. A co-incidence matrix was generated using logical approach for a combined “rabi-summer” and “kharif-rabi-summer” landcover mapping. The major landcovers obtained in South 24 Paraganas using remote sensing data are rice, water, aquaculture ponds, homestead, mangrove, and urban area. The classification accuracy of rice area was 98.2% using SAR data. However, while generating combined “kharif-rabi-summer” landcovers, the classification accuracy of rice area was improved from 81.6% (optical data) to 96.6% (combined SAR-Optical). The primary aim of the study is to achieve better accuracy in classifying rice area using the synergy between the two kinds of remotely sensed data.     

A hyperspectral experiment over tropical forests based on the EO-1 orbit change and PROSAIL simulation

ABSTRACT

We designed a unique hyperspectral experiment from the Earth Observing One (EO-1) orbit change to evaluate solar illumination effects over tropical forests in Brazil. Ten nadir-viewing Hyperion images collected over a fixed site and period of the year (July to August) were selected for analysis. We evaluated variations in reflectance and in 16 narrowband vegetation indices (VIs) with increasing solar zenith angle (SZA) from the pre-drift (2004–2008) to the EO-1 drift period (2011–2016). To detect changes in reflectance and shadows, we applied spectral mixture analysis (SMA) and principal component analysis (PCA) and calculated the similarity spectral angle (θ) between the vegetation spectra measured with variable SZA. The magnitude of the illumination effects was also evaluated from change-point analysis and nonparametric Mann-Whitney U tests applied over the time series. Finally, we complemented our experiment using the PROSAIL model to simulate the VIs variation with increasing SZA resultant from satellite drift. The results showed significant changes in Hyperion reflectance and VIs, especially when the EO-1 crossed the study area at earlier times and larger SZA in 2015 (9:05 a.m.; SZA = 59°) and 2016 (8:30 a.m.; SZA = 67°). Compared to the pre-drift period (10:30 a.m.; SZA = 45°), the SZA differences of 14° (2015) and 22° (2016) increased the shade fractions and decreased the vegetation brightness. PCA separated the pre-drift and drift reflectance datasets, showing shifts in scores due to changes in brightness. θ increased with SZA, indicating changes in the shape of the vegetation spectra with drift. For most VIs, the change-point analysis indicated 2015 (SZA = 59°) as the predominant year of detected changes. Compared to the EO-1 original orbit, the Plant Senescence Reflectance Index (PSRI), Anthocyanin Reflectance Index (ARI) and Structure Insensitive Pigment Index (SIPI) presented the largest positive changes during drift, while the Photochemical Reflectance Index (PRI), Visible Atmospherically Resistant Index (VARI) and Enhanced Vegetation Index (EVI) had the largest negative changes. The effect size of the illumination geometry on these VIs was large, as indicated by increasing values of the Cohen’s r metric toward 2016. The anisotropy of the Hyperion VIs was generally consistent with that from PROSAIL in the simulated pre-drift and drift periods. Focusing on structural indices, it affected the relationships between VIs and simulated leaf area index (LAI) at large SZA.     

Geoid, topography, and the Bouguer plate or shell

 Topography plays an important role in solving many geodetic and geophysical problems. In the evaluation of a topographical effect, a planar model, a spherical model or an even more sophisticated model can be used. In most applications, the planar model is considered appropriate: recall the evaluation of gravity reductions of the free-air, Poincaré–Prey or Bouguer kind. For some applications, such as the evaluation of topographical effects in gravimetric geoid computations, it is preferable or even necessary to use at least the spherical model of topography. In modelling the topographical effect, the bulk of the effect comes from the Bouguer plate, in the case of the planar model, or from the Bouguer shell, in the case of the spherical model. The difference between the effects of the Bouguer plate and the Bouguer shell is studied, while the effect of the rest of topography, the terrain, is discussed elsewhere. It is argued that the classical Bouguer plate gravity reduction should be considered as a mathematical construction with unclear physical meaning. It is shown that if the reduction is understood to be reducing observed gravity onto the geoid through the Bouguer plate/shell then both models give practically identical answers, as associated with Poincaré's and Prey's work. It is shown why only the spherical model should be used in the evaluation of topographical effects in the Stokes–Helmert solution of Stokes' boundary-value problem. The reason for this is that the Bouguer plate model does not allow for a physically acceptable condensation scheme for the topography. Received: 24 December 1999 / Accepted: 11 December 2000