On robust estimation of the location parameter

Matrix Differential Calculus with Applications in Statistics and Econometrics

Generalized Linear Models (2nd ed.)

For more information on the USGS—the Federal source for science about the Earth, its natural and living resources, natural hazards, and the environment, visit http://www.usgs.gov or call 1–888–ASK–USGS. For an overview of USGS information products, including maps, imagery, and publications, Any use of trade, product, or firm names is for descriptive purposes only and does not imply endorsement by the U.S. Government. Although this report is in the public domain, permission must be secured from the individual copyright owners to reproduce any copyrighted materials contained within this report. 10. Diagram showing the GMTED2010 layer extents (minimum and maximum latitude and longitude) are a result of the coordinate system inherited from the 1-arc-second SRTM

Global Multi-resolution Terrain Elevation Data 2010 (GMTED2010)

The Multi-GNSS Experiment (MGEX) of the International GNSS Service (IGS) - Achievements, prospects and challenges

China completed a basic COMPASS navigation network with three Geostationary and three Inclined Geosynchronous satellites in orbit in April 2011. The network has been able to provide preliminary positioning and navigation functions. We first present a quality analysis using 1-week COMPASS measurements collected in Wuhan. Satellite visibility and validity of measurements, carrier-to-noise density ratio and code noise are analyzed. The analysis of multipath combinations shows that the noise level of COMPASS code measurements is higher than that of GPS collected using the same receiver. Second, the results of positioning are presented and analyzed. For the standalone COMPASS solutions, an accuracy of 20 m can be achieved. An accuracy of 3.0 m for the vertical, 1.5 m for the North and about 0.6---0.8 m for the East component is obtained using dual-frequency code only measurements for a short baseline. More importantly, code and phase measurements of the short baseline are processed together to obtain precise relative positioning. Kinematic solutions are then compared with the ground truth. The precision of COMPASS only solutions is better than 2 cm for the North component and 4 cm for the vertical. The standard deviation of the East component is smaller than 1 cm, which is even better than that of the East component of GPS solutions. The accuracy of GPS/COMPASS combination solutions is at least 20 % better than that of GPS alone. Furthermore, the geometry-based residuals of double differenced phase and code measurements are analyzed. The analysis shows that the noise level of un-differenced phase measurements is about 2---4 mm on both B1 and B2 frequencies. For the code measurements, the noise level is less than 0.45 m for B1 CA and about 0.35 m for B2 P code. Many of the COMPASS results presented are very promising and have been obtained for the first time.

Precise relative positioning using real tracking data from COMPASS GEO and IGSO satellites

This contribution summarizes the strategy used by Wuhan University (WHU) to determine precise orbit and clock products for Multi-GNSS Experiment (MGEX) of the International GNSS Service (IGS). In particular, the satellite attitude, phase center corrections, solar radiation pressure model developed and used for BDS satellites are addressed. In addition, this contribution analyzes the orbit and clock quality of the quad-constellation products from MGEX Analysis Centers (ACs) for a common time period of 1 year (2014). With IGS final GPS and GLONASS products as the reference, Multi-GNSS products of WHU (indicated by WUM) show the best agreement among these products from all MGEX ACs in both accuracy and stability. 3D Day Boundary Discontinuities (DBDs) range from 8 to 27 cm for Galileo-IOV satellites among all ACs’ products, whereas WUM ones are the largest (about 26.2 cm). Among three types of BDS satellites, MEOs show the smallest DBDs from 10 to 27 cm, whereas the DBDs for all ACs products are at decimeter to meter level for GEOs and one to three decimeter for IGSOs, respectively. As to the satellite laser ranging (SLR) validation for Galileo-IOV satellites, the accuracy evaluated by SLR residuals is at the one decimeter level with the well-known systematic bias of about $$-5$$
  cm for all ACs. For BDS satellites, the accuracy could reach decimeter level, one decimeter level, and centimeter level for GEOs, IGSOs, and MEOs, respectively. However, there is a noticeable bias in GEO SLR residuals. In addition, systematic errors dependent on orbit angle related to mismodeled solar radiation pressure (SRP) are present for BDS GEOs and IGSOs. The results of Multi-GNSS combined kinematic PPP demonstrate that the best accuracy of position and fastest convergence speed have been achieved using WUM products, particularly in the Up direction. Furthermore, the accuracy of static BDS only PPP degrades when the BDS IGSO and MEO satellites switches to orbit-normal orientation, particularly for COM products, whereas the WUM show the slightest degradation.

Precise orbit determination for quad-constellation satellites at Wuhan University: strategy, result validation, and comparison

The development of the COMPASS satellite system is introduced, and the regional tracking network and data availability are described. The precise orbit determination strategy of COMPASS satellites is presented. Data of June 2012 are processed. The obtained orbits are evaluated by analysis of post-fit residuals, orbit overlap comparison and SLR (satellite laser ranging) validation. The RMS (root mean square) values of post-fit residuals for one month’s data are smaller than 2.0 cm for ionosphere-free phase measurements and 2.6 m for ionosphere-free code observations. The 48-h orbit overlap comparison shows that the RMS values of differences in the radial component are much smaller than 10 cm and those of the cross-track component are smaller than 20 cm. The SLR validation shows that the overall RMS of observed minus computed residuals is 68.5 cm for G01 and 10.8 cm for I03. The static and kinematic PPP solutions are produced to further evaluate the accuracy of COMPASS orbit and clock products. The static daily COMPASS PPP solutions achieve an accuracy of better than 1 cm in horizontal and 3 cm in vertical. The accuracy of the COMPASS kinematic PPP solutions is within 1–2 cm in the horizontal and 4–7 cm in the vertical. In addition, we find that the COMPASS kinematic solutions are generally better than the GPS ones for the selected location. Furthermore, the COMPASS/GPS combinations significantly improve the accuracy of GPS only PPP solutions. The RMS values are basically smaller than 1 cm in the horizontal components and 3–4 cm in the vertical component.

Initial results of precise orbit and clock determination for COMPASS navigation satellite system

Chinese Beidou satellite navigation system constellation currently consists of eight Beidou satellites and can provide preliminary service of navigation and positioning in the Asia-Pacific Region. Based on the self-developed software Position And Navigation Data Analysis(PANDA) and Beidou Experimental Tracking Stations (BETS), which are built by Wuhan University, the study of Beidou precise orbit determination, static precise point positioning (PPP), and high precision relative positioning, and differential positioning are carried out comprehensively. Results show that the radial precision of the Beidou satellite orbit determination is better than 10 centimeters. The RMS of static PPP can reach several centimeters to even millimeters for baseline relative positioning. The precision of kinematic pseudo-range differential positioning and RTK mode positioning are 2–4 m and 5–10 cm respectively, which are close to the level of GPS precise positioning. Research in this paper verifies that, with support of ground reference station network, Beidou satellite navigation system can provide precise positioning from several decimeters to meters in the wide area and several centimeters in the regional area. These promising results would be helpful for the implementation and applications of Beidou satellite navigation system.

Precise orbit determination of Beidou Satellites with precise positioning

The weighted total least squares (TLS) method has been developed to deal with observation equations, which are functions of both unknown parameters of interest and other measured data contaminated with random errors. Such an observation model is well known as an errors-in-variables (EIV) model and almost always solved as a nonlinear equality-constrained adjustment problem. We reformulate it as a nonlinear adjustment model without constraints and further extend it to a partial EIV model, in which not all the elements of the design matrix are random. As a result, the total number of unknowns in the normal equations has been significantly reduced. We derive a set of formulae for algorithmic implementation to numerically estimate the unknown model parameters. Since little statistical results about the TLS estimator in the case of finite samples are available, we investigate the statistical consequences of nonlinearity on the nonlinear TLS estimate, including the first order approximation of accuracy, nonlinear confidence region and bias of the nonlinear TLS estimate, and use the bias-corrected residuals to estimate the variance of unit weight.

Jingnan Liu

Papers

Precise relative positioning using real tracking data from COMPASS GEO and IGSO satellites

Precise orbit determination for quad-constellation satellites at Wuhan University: strategy, result validation, and comparison

Initial results of precise orbit and clock determination for COMPASS navigation satellite system

Precise orbit determination of Beidou Satellites with precise positioning

Total least squares adjustment in partial errors-in-variables models: algorithm and statistical analysis