Yunzhong Shen

Outlier separability analysis with a multiple alternative hypotheses test

Outlier separability analysis is a fundamental component of modern geodetic measurement analysis, positioning, navigation, and many other applications. The current theory of outlier separability is based on using two alternative hypotheses—an assumption that may not necessarily be valid. In this paper, the current theory of outlier separability is statistically analysed and then extended to the general case, where there are multiple alternative hypotheses. Taking into consideration the complexity of the critical region and the probability density function of the outlier test, the bounds of the associated statistical decision probabilities are then developed. With this theory, the probabilities of committing type I, II, and III errors can be controlled so that the probability of successful identification of an outlier can be guaranteed when performing data snooping. The theoretical findings are then demonstrated using a simulated GPS point positioning example. Detailed analysis shows that the larger the correlation coefficient, between the outlier statistics, the smaller the probability of committing a type II error and the greater the probability of committing a type III error. When the correlation coefficient is greater than 0.8, there is a far greater chance of committing a type III error than committing a type II error. In addition, to guarantee successful identification of an outlier with a set probability, the minimal detectable size of the outlier (often called the Minimal Detectable Bias or MDB) should dramatically increase with the correlation coefficient.

Geometry‐specified troposphere decorrelation for subcentimeter real‐time kinematic solutions over long baselines

Real‐time kinematic (RTK) GPS techniques have been extensively developed for applications including surveying, structural monitoring, and machine automation. Limitations of the existing RTK techniques that hinder their applications for geodynamics purposes are twofold: (1) the achievable RTK accuracy is on the level of a few centimeters and the uncertainty of vertical component is 1.5–2 times worse than those of horizontal components and (2) the RTK position uncertainty grows in proportional to the base‐torover distances. The key limiting factor behind the problems is the significant effect of residual tropospheric errors on the positioning solutions, especially on the highly correlated height component. This paper develops the geometry‐specified troposphere decorrelation strategy to achieve the subcentimeter kinematic positioning accuracy in all three components. The key is to set up a relative zenith tropospheric delay (RZTD) parameter to absorb the residual tropospheric effects and to solve the established model as an ill‐posed problem using the regularization method. In order to compute a reasonable regularization parameter to obtain an optimal regularized solution, the covariance matrix of positional parameters estimated without the RZTD parameter, which is characterized by observation geometry, is used to replace the quadratic matrix of their “true” values. As a result, the regularization parameter is adaptively computed with variation of observation geometry. The experiment results show that new method can efficiently alleviate the model’s ill condition and stabilize the solution from a single data epoch. Compared to the results from the conventional least squares method, the new method can improve the longrange RTK solution precision from several centimeters to the subcentimeter in all components. More significantly, the precision of the height component is even higher. Several geosciences applications that require subcentimeter real‐time solutions can largely benefit from the proposed approach, such as monitoring of earthquakes and large dams in real‐time, high‐precision GPS leveling and refinement of the vertical datum. In addition, the high‐resolution RZTD solutions can contribute to effective recovery of tropospheric slant path delays in order to establish a 4‐D troposphere tomography.

Efficient Estimation of Variance and Covariance Components: A Case Study for GPS Stochastic Model Evaluation

The variance and covariance component estimation (VCE) has been extensively investigated. However, in real application, the bottleneck problem is the huge computation burden, particularly when many variance and covariance components are involved for many heterogeneous observations. The objective of this paper is to develop a new method allowing the efficient estimation of variance and covariance components. The core of the new method is to construct an orthogonal complement matrix of the coefficient matrix in a Gauss-Markov model using only the coefficient matrix itself. Therefore, the constructed matrix and the computed discrepancies of measurements with each other, which are the essential inputs for the VCE, are invariant in the iterative procedure of computing the variance and covariance components. As a result, the computation efficiency is significantly improved. As a case study, we apply the new method to evaluate the GPS stochastic model with 15 variance and covariance components demonstrating its superior performance. Comparing with the traditional VCE method, the equivalent results are achievable, and the computation efficiency is improved by 34.2%. In the future, much more sensors will be available, and plentiful data can be acquired. Therefore, the new method will be very promising to efficiently estimate the variance and covariance components of the measurements from the different sensors and reasonably balance their contributions to the fused solution, benefiting the higher time-resolution solutions.

Seamless multivariate affine error-in-variables transformation and its application to map rectification

Affine transformation that allows the axis-specific rotations and scalars to capture the more transformation details has been extensively applied in a variety of geospatial fields. In tradition, the computation of affine parameters and the transformation of non-common points are individually implemented, in which the coordinate errors only of the target system are taken into account although the coordinates in both target and source systems are inevitably contaminated by random errors. In this article, we propose the seamless affine error-in-variables (EIV) transformation model that computes the affine parameters and transforms the non-common points simultaneously, importantly taking into account the errors of all coordinates in both datum systems. Since the errors in coefficient matrix are involved, the seamless affine EIV model is nonlinear. We then derive its least squares iterative solution based on the Euler–Lagrange minimization method. As a case study, we apply the proposed seamless affine EIV model to the map rectification. The transformation accuracy is improved by up to 40%, compared with the traditional affine method. Naturally, the presented seamless affine EIV model can be applied to any application where the transformation estimation of points fields in the different systems is involved, for instance, the geodetic datum transformation, the remote sensing image matching, and the LiDAR point registration.

Monthly gravity field models derived from GRACE Level 1B data using a modified short‐arc approach

In this study, a new time series of Gravity Recovery and Climate Experiment (GRACE) monthly solutions, complete to degree and order 60 spanning from January 2003 to August 2011, has been derived based on a modified short-arc approach. Our models entitled Tongji-GRACE01 are available on the website of International Centre for Global Earth Models (http://icgem.gfz-potsdam.de/ICGEM/). The traditional short-arc approach, with no more than 1 h arcs, requires the gradient corrections of satellite orbits in order to reduce the impact of orbit errors on the final solution. Here the modified short-arc approach has been proposed, which has three major differences compared to the traditional one: (1) All the corrections of orbits and range rate measurements are solved together with the geopotential coefficients and the accelerometer biases using a weighted least squares adjustment; (2) the boundary position parameters are not required; and (3) the arc length can be extended to 2 h. The comparisons of geoid degree powers and the mass change signals in the Amazon basin, the Antarctic, and Antarctic Peninsula demonstrate that our model is comparable with the other existing models, i.e., the Centre for Space Research RL05, Jet Propulsion Laboratory RL05, and GeoForschungsZentrum RL05a models. The correlation coefficients of the mass change time series between our model and the other models are better than 0.9 in the Antarctic and Antarctic Peninsula. The mass change rates in the Antarctic and Antarctic Peninsula derived from our model are −92.7 ± 38.0 Gt/yr and −23.9 ± 12.4 Gt/yr, respectively, which are very close to those from other three models and with similar spatial patterns of signals.

Effects of errors-in-variables on weighted least squares estimation

Although total least squares (TLS) is more rigorous than the weighted least squares (LS) method to estimate the parameters in an errors-in-variables (EIV) model, it is computationally much more complicated than the weighted LS method. For some EIV problems, the TLS and weighted LS methods have been shown to produce practically negligible differences in the estimated parameters. To understand under what conditions we can safely use the usual weighted LS method, we systematically investigate the effects of the random errors of the design matrix on weighted LS adjustment. We derive the effects of EIV on the estimated quantities of geodetic interest, in particular, the model parameters, the variance–covariance matrix of the estimated parameters and the variance of unit weight. By simplifying our bias formulae, we can readily show that the corresponding statistical results obtained by Hodges and Moore (Appl Stat 21:185–195, 1972) and Davies and Hutton (Biometrika 62:383–391, 1975) are actually the special cases of our study. The theoretical analysis of bias has shown that the effect of random matrix on adjustment depends on the design matrix itself, the variance–covariance matrix of its elements and the model parameters. Using the derived formulae of bias, we can remove the effect of the random matrix from the weighted LS estimate and accordingly obtain the bias-corrected weighted LS estimate for the EIV model. We derive the bias of the weighted LS estimate of the variance of unit weight. The random errors of the design matrix can significantly affect the weighted LS estimate of the variance of unit weight. The theoretical analysis successfully explains all the anomalously large estimates of the variance of unit weight reported in the geodetic literature. We propose bias-corrected estimates for the variance of unit weight. Finally, we analyze two examples of coordinate transformation and climate change, which have shown that the bias-corrected weighted LS method can perform numerically as well as the weighted TLS method.

Bias-corrected regularized solution to inverse ill-posed models

A regularized solution is well-known to be biased. Although the biases of the estimated parameters can only be computed with the true values of parameters, we attempt to improve the regularized solution by using the regularized solution itself to replace the true (unknown) parameters for estimating the biases and then removing the computed biases from the regularized solution. We first analyze the theoretical relationship between the regularized solutions with and without the bias correction, derive the analytical conditions under which a bias-corrected regularized solution performs better than the ordinary regularized solution in terms of mean squared errors (MSE) and design the corresponding method to partially correct the biases. We then present two numerical examples to demonstrate the performance of our partially bias-corrected regularization method. The first example is mathematical with a Fredholm integral equation of the first kind. The simulated results show that the partially bias-corrected regularized solution can improve the MSE of the ordinary regularized function by 11%. In the second example, we recover gravity anomalies from simulated gravity gradient observations. In this case, our method produces the mean MSE of 3.71 mGal for the resolved mean gravity anomalies, which is better than that from the regularized solution without bias correction by 5%. The method is also shown to successfully reduce the absolute maximum bias from 13.6 to 6.8 mGal.

GNSS Elevation-Dependent Stochastic Modeling and Its Impacts on the Statistic Testing

AbstractOnly the correct stochastic model can be applied to derive the optimal parameter estimation and then realize the precision global navigation satellite system (GNSS) positioning. The key for refining the GNSS stochastic model is to establish the easy-to-use stochastic model that should capture the error characteristics adequately based on the estimated precisions from the real observations. In this paper, the authors study the GNSS elevation-dependent precision modeling and analyze its impact on the statistic testing involved in the adjustment reliability. With the zero-baseline dual-frequency Global Positioning System (GPS) data, the authors first estimate the elevation-dependent precisions and establish the stochastic models by fitting them with three predefined functions, including the unique precision function and the sine and exponential types of elevation-dependent functions. Three established models are then evaluated by their performance in the overall and w-statistic testing. The results i...

Global Earth’s gravity field solution with GRACE orbit and range measurements using modified short arc approach

Traditionally, the Earth’s gravity field model is computed from GRACE orbit and range rate measurements, e.g., in a short arc approach where both the position and the velocity vectors are integrated from a force model. In this contribution, we use the GRACE orbit and range measurements to recover the Earth’s gravity field model, thus we only need to integrate the position vectors. We use the range differences between two adjacent epochs to eliminate the range ambiguities. Using GRACE Level-1B RL02 data released by Jet Propulsion laboratory, the gravity field model TJGRACE02O complete to degree and order 90 is developed from 7 years of reduced dynamic orbits covering the period 2004–2010, and the gravity field model TJGRACE02K complete to degree and order 120 is computed from 1 month of kinematic orbits and K-band range data of January. Comparing the degree geoid errors of our new models with recent gravity field models such as the CHAMP-only models EIGEN-CHAMP05S, AIUB-CHAMP03S, ULUX-CHAMP2013S and the GRACE-only models GGM05S, Tongji-GRACE01 as well as a monthly model from the ITG-GRACE2010 time series, and validating these models with GPS-leveling data sets in the USA, we can conclude that the TJGRACE02O model is more accurate than all the CHAMP-only models and TJGRACE02K is comparable in quality to the corresponding GRACE monthly model from ITG-GRACE2010.

Improving BeiDou precise orbit determination using observations of onboard MEO satellite receivers

In recent years, the precise orbit determination (POD) of the regional Chinese BeiDou Navigation Satellite System (BDS) has been a hot spot because of its special constellation consisting of five geostationary earth orbit (GEO) satellites and five inclined geosynchronous satellite orbit (IGSO) satellites besides four medium earth orbit (MEO) satellites since the end of 2012. GEO and IGSO satellites play an important role in regional BDS applications. However, this brings a great challenge to the POD, especially for the GEO satellites due to their geostationary orbiting. Though a number of studies have been carried out to improve the POD performance of GEO satellites, the result is still much worse than that of IGSO and MEO, particularly in the along-track direction. The major reason is that the geostationary characteristic of a GEO satellite results in a bad geometry with respect to the ground tracking network. In order to improve the tracking geometry of the GEO satellites, a possible strategy is to mount global navigation satellite system (GNSS) receivers on MEO satellites to collect the signals from GEO/IGSO GNSS satellites so as that these observations can be used to improve GEO/IGSO POD. We extended our POD software package to simulate all the related observations and to assimilate the MEO-onboard GNSS observations in orbit determination. Based on GPS and BDS constellations, simulated studies are undertaken for various tracking scenarios. The impact of the onboard GNSS observations is investigated carefully and presented in detail. The results show that MEO-onboard observations can significantly improve the orbit precision of GEO satellites from metres to decimetres, especially in the along-track direction. The POD results of IGSO satellites also benefit from the MEO-onboard data and the precision can be improved by more than 50% in 3D direction.