Least-squares Variance Component Estimation Research Articles

Analysis of the fault slip, creep, and coupling characteristics of the Maomaoshan-Laohushan-Haiyuan Fault using InSAR and GNSS measurements

Line-of-sight Interferometric Synthetic Aperture Radar (InSAR) observations are affected by long-wavelength errors and the inaccurate estimation of vertical deformation, thereby restricting horizontal slip rate estimations for strike-slip faults, such as the Maomaoshan-Laohushan-Haiyuan Fault (MMS-LHS-HY). To overcome this limitation, herein, InSAR (2016–2022) and Global Navigation Satellite System (GNSS) observations were combined and the Least Squares Variance Component Estimation (LS-VCE) method was used to obtain the Three-Dimensional (3-D) deformation field of the Haiyuan fault zone; subsequently, the slip-rate and coupling distributions were inverted. The results showed that: (1) Compared with the GNSS and leveling deformation velocities, the residuals of the 3-D deformation field obtained using the LS-VCE method combined with the GNSS and InSAR observations reached an RMS of 1 mm/a, significantly better than the deformation field obtained without GNSS data. (2) Vertical deformation from InSAR observations led to over- or under-estimation of the fault strike-slip rate, with the maximum influence on the shallow creep rate of 2.1 mm/a. (3) Based on the 3-D deformation field, fault slip rate, locking depth, and fault creep characteristics obtained from the elastic dislocation and elastic block models were consistent with each other. (4) The strike-slip rate of the MMS-LHS-HY fault gradually decreased from west to east, whereas the thrust rate gradually increased in the same direction. (5) Based on the 3-D deformation field, the creep range of the LHS fault was ∼40 km, with the mean shallow creep rate reaching 2.5 ± 0.1 mm/a, notably larger than the creep rate obtained by ascending InSAR observation alone.

ARAIM Stochastic Model Refinements for GNSS Positioning Applications in Support of Critical Vehicle Applications.

Integrity monitoring (IM) is essential if GNSS positioning technologies are to be fully trusted by future intelligent transport systems. A tighter and conservative stochastic model can shrink protection levels in the position domain and therefore enhance the user-level integrity. In this study, the stochastic models for vehicle-based GNSS positioning are refined in three respects: (1) Gaussian bounds of precise orbit and clock error products from the International GNSS Service are used; (2) a variable standard deviation to characterize the residual tropospheric delay after model correction is adopted; and (3) an elevation-dependent model describing the receiver-related errors is adaptively refined using least-squares variance component estimation. The refined stochastic models are used for positioning and IM under the Advanced Receiver Autonomous Integrity Monitoring (ARAIM) framework, which is considered the basis for multi-constellation GNSS navigation to support air navigation in the future. These refinements are assessed via global simulations and real data experiments. Different schemes are designed and tested to evaluate the corresponding enhancements on ARAIM availability for both aviation and ground vehicle-based positioning applications.

Combination of GNSS orbits using least-squares variance component estimation

Over the past years, the International GNSS Service (IGS) has been putting efforts into extending its service by setting up and running the Multi-GNSS experiment and pilot project (MGEX). Several MGEX analysis centers (ACs) contribute by providing solutions containing not only GPS and GLONASS but also Galileo, BeiDou, and QZSS. As the current IGS combination software can only handle the orbits of one constellation at a time, it requires substantial modifications to obtain a consistent MGEX orbit product. In this contribution, we present a least-squares framework for a Multi-GNSS orbit combination, where the weights used to combine the ACs’ orbits are determined by least-squares variance component estimation. We introduce and compare two weighting strategies, where either AC-specific weights or AC and constellation-specific weights are used. An automated Z-score test is implemented yielding a common set of core satellites that are used to determine the weights. Both strategies are tested using MGEX orbit solutions for a period of two and a half years. They yield similar results with an agreement with the ACs’ orbits at the one centimeter level for GPS and up to a few centimeters for the other constellations. The 3D-RMS is generally slightly better with the AC and constellation weighting. A comparison of our combination approach with the official IGS combination using three years of GPS and GLONASS orbits shows an agreement of better than 5 mm and 12 mm for GPS and GLONASS, respectively, while the agreement of the official IGS combination with the ACs’ GPS solutions is only around 15,textrm{mm}. An external validation using satellite laser ranging shows that the mean residuals of our combined products are around -3,textrm{mm} for Galileo, 6,textrm{mm} for GLONASS, -8,textrm{mm} for BeiDou, and -31,textrm{mm} for QZSS.

Locally weighted total least-squares variance component estimation for modeling urban air pollution.

Land use regression (LUR) models are one of the standard methods for estimating air pollution concentration in urban areas. These models are usually low accurate due to inappropriate stochastic models (weight matrix). Furthermore, the measurement or modeling of dependent and independent variables used in LUR models is affected by various errors, which indicates the need to use an efficient stochastic and functional model to achieve the best estimation. This study proposes a locally weighted total least-squares variance component estimation (LW-TLS-VCE) for modeling urban air pollution. In the proposed method, in the first step, a locally weighted total least-squares (LW-TLS) regression is developed to simultaneously considers the non-stationary effects and errors of dependent and independent variables. In the second step, the variance components of the stochastic model are estimated to achieve the best linear unbiased estimation of unknowns. The efficiency of the proposed method is evaluated by modeling PM2.5 concentrations via meteorological, land use, and traffic variables in Isfahan, Iran. The benefits provided by the proposed method, including considering non-stationary effects and random errors of all variables, besides estimating the actual variance of observations, are evaluated by comparing four consecutive methods. The obtained results demonstrate that using a suitable stochastic and functional model will significantly increase the proposed method's efficiency in PM2.5 modeling.

Improved deformation modelling of structures by least-squares variance component estimation based on multi-sensor data integration

In this contribution, to improve the deformation modelling based on data integration, the LS-VCE algorithm is proposed by obtaining a stochastic model of input multi-sensor data. So, one can achieve the accurate variance-covariance matrix of multi-sensor observations to participate in iterative least-squares. A practical application was made for the settlement observations from geotechnical settlement-meters and geodetic levelling (respectively known as internal and external sensors) to model the surface settlement variation of the Karkhe earth-dam. The determined variance component shows less contribution of the geotechnical settlements in the deformation modelling. An achievement of this paper is that the LS-VCE method improves the integration of the geotechnical with geodetic data by estimating an optimal stochastic model resulting in deformation model optimization. Validation results of estimated surface settlements on the check-points show an RMSE of about 3 cm and a relative-error of about 14%, which indicates the success of the modelling.

Assessment of noise in InSAR timeseries using least squares variance component estimation

In recent decades, Interferometric Synthetic Aperture Radar (InSAR) has progressed as an effective and reliable tool for monitoring the surface deformations of the earth. Despite the potential of this method for deformation monitoring, the quality description of InSAR timeseries in terms of precision and noise structure and, consequently, the precision description of the InSAR-derived parameters (e.g., displacement and its velocity) are still somewhat ambiguous. In this paper, we propose a data-derived methodology that directly estimates the precision and noise structure of the final InSAR products, using Least Squares Variance Component Estimation (LS-VCE). Note that due to the spatial correlation among adjacent coherent pixels and adjacent acquisitions, a multivariate LS-VCE model should be applied. We used the proposed method on deformation timeseries derived from the Sentine-l data over city of Tehran, Iran. The results show that applying the multivariate LS-VCE method in our case study improves the results by about 50% compared with the case where the noise parameters are not considered. In addition, the results confirm that InSAR timeseries are highly correlated in time and space. Particularly, the spatial correlation between a series of neighbouring targets for the noise components is significant and gradually decreases with increasing arc length. It should be noted that the observed spatial correlation should be differentiated from the well-known spatial correlation imposed by atmospheric components. In fact, due to the atmosphere filtering step, the noise structure of the final results will be different from the statistical characteristics of a raw atmospheric signal. The proposed methodology is not case study dependent and can be used as an appropriate approach to provide the precision (as a quality descriptor) of the timeseries InSAR products.

LS-VCE Applied to Stochastic Modeling of GNSS Observation Noise and Process Noise

Stochastic models play a crucial role in global navigation satellite systems (GNSS) data processing. Many studies contribute to the stochastic modeling of GNSS observation noise, whereas few studies focus on the stochastic modeling of process noise. This paper proposes a method that is able to jointly estimate the variances of observation noise and process noise. The method is flexible since it is based on the least-squares variance component estimation (LS-VCE), enabling users to estimate the variance components that they are specifically interested in. We apply the proposed method to estimate the variances for the dual-frequency GNSS observation noise and for the process noise of the receiver code bias (RCB). We also investigate the impact of the stochastic model upon parameter estimation, ambiguity resolution, and positioning. The results show that the precision of GNSS observations differs in systems and frequencies. Among the dual-frequency GPS, Galileo, and BDS code observations, the precision of the BDS B3 observations is highest (better than 0.2 m). The precision of the BDS phase observations is better than two millimeters, which is also higher than that of the GPS and Galileo observations. For all three systems, the RCB process noise ranges from 0.5 millimeters to 1 millimeter, with a data sampling rate of 30 s. An improper stochastic model of the observation noise results in an unreliable ambiguity dilution of precision (ADOP) and position dilution of precision (PDOP), thus adversely affecting the assessment of the ambiguity resolution and positioning performance. An inappropriate stochastic model of RCB process noise disturbs the estimation of the receiver clock and the ionosphere delays and is thus harmful for timing and ionosphere retrieval applications.

Improvement of Multi-GNSS Precision and Success Rate Using Realistic Stochastic Model of Observations

With the advancement of multi-constellation and multi-frequency global navigation satellite systems (GNSSs), more observations are available for high precision positioning applications. Although there is a lot of progress in the GNSS world, achieving realistic precision of the solution (neither too optimistic nor too pessimistic) is still an open problem. Weighting among different GNSS systems requires a realistic stochastic model for all observations to achieve the best linear unbiased estimation (BLUE) of unknown parameters in multi-GNSS data processing mode. In addition, the correct integer ambiguity resolution (IAR) becomes crucial in shortening the Time-To-Fix (TTF) in RTK, especially in challenging environmental conditions. In general, it is required to estimate various variances for observation types, consider the correlation between different observables, and compensate for the satellite elevation dependence of the observable precision. Quality control of GNSS signals, such as GPS, GLONASS, Galileo, and BeiDou can be performed by processing a zero or short baseline double difference pseudorange and carrier phase observations using the least-squares variance component estimation (LS-VCE). The efficacy of this method is investigated using real multi-GNSS data sets collected by the Trimble NETR9, SEPT POLARX5, and LEICA GR30 receivers. The results show that the standard deviation of observations depends on the system and the observable type in which a particular receiver could have the best performance. We also note that the estimated variances and correlations among different observations are also dependent on the receiver type. It is because the approaches utilized for the recovery techniques differ from one type of receiver to another kind. The reliability of IAR will improve if a realistic stochastic model is applied in single or multi-GNSS data processing. According to the results, for the data sets considered, a realistic stochastic model can increase the computed empirical success rate to 100% in multi-GNSS as well as a single system. As mentioned previously, the realistic precision of the solution can be achieved with a realistic stochastic model. However, using the estimated stochastic model, in fact, leads to better precision and accuracy for the estimated baseline components, up to 39% in multi-GNSS.

Combination of regional and global geoid models at continental scale: application to Iranian geoid

High precision geoid determination is a challenging task at the national scale. Many efforts have been conducted to determine precise geoid, locally or globally. Geoid models have different precision depending on the type of information and the strategy employed when calculating the models. This contribution addresses the challenging problem of combining different regional and global geoid models, possibly combined with the geometric geoid derived from GNSS/leveling observations. The ultimate goal of this combination is to improve the precision of the combined model. We employ fitting an appropriate geometric surface to the geoid heights and estimating its (co)variance components. The proposed functional model uses the least squares 2D bi-cubic spline approximation (LS-BICSA) theory, which approximates the geoid model using a 2D spline surface fitted to an arbitrary set of data points in the region. The spline surface consists of third- order polynomial pieces that are smoothly connected together, imposing some continuity conditions at their boundaries. In addition, the least-squares variance component estimation (LS- VCE) is used to estimate precise weights and correlation among different models. We apply this strategy to the combined adjustment of the high-degree global gravitational model EIGEN-6C4, the regional geoid model IRG2016, and the Iranian geometric geoid derived from GNSS/leveling data. The accuracy of the constructed surface is investigated with five randomly selected subsamples of check points. The optimal combination of the two geoid models along with the GNSS/leveling data shows a reduction of 21 mm (~20%) in the RMSE values of discrepancies at the check points.

GPS-BDS-Galileo double-differenced stochastic model refinement based on least-squares variance component estimation

AbstractStochastic models are essential for precise navigation and positioning of the global navigation satellite system (GNSS). A stochastic model can influence the resolution of ambiguity, which is a key step in GNSS positioning. Most of the existing multi-GNSS stochastic models are based on the GPS empirical model, while differences in the precision of observations among different systems are not considered. In this paper, three refined stochastic models, namely the variance components between systems (RSM1), the variances of different types of observations (RSM2) and the variances of observations for each satellite (RSM3) are proposed based on the least-squares variance component estimation (LS-VCE). Zero-baseline and short-baseline GNSS experimental data were used to verify the proposed three refined stochastic models. The results show that, compared with the traditional elevation-dependent model (EDM), though the proposed models do not significantly improve the ambiguity resolution success rate, the positioning precision of the three proposed models has been improved. RSM3, which is more realistic for the data itself, performs the best, and the precision at elevation mask angles 20°, 30°, 40°, 50° can be improved by 4⋅6%, 7⋅6%, 13⋅2%, 73⋅0% for L1-B1-E1 and 1⋅1%, 4⋅8%, 16⋅3%, 64⋅5% for L2-B2-E5a, respectively.

Nonlinear total least-squares variance component estimation for GM(1,1) model

The solution of the grey model (GM(1,1) model) generally involves equal-precision observations, and the (co)variance matrix is established from the prior information. However, the data are generally available with unequal-precision measurements in reality. To deal with the errors of all observations for GM(1,1) model with errors-in-variables (EIV) structure, we exploit the total least-squares (TLS) algorithm to estimate the parameters of GM(1,1) model in this paper. Ignoring that the effect of the improper prior stochastic model and the homologous observations may degrade the accuracy of parameter estimation, we further present a nonlinear total least-squares variance component estimation approach for GM(1,1) model, which resorts to the minimum norm quadratic unbiased estimation (MINQUE). The practical and simulative experiments indicate that the presented approach has significant merits in improving the predictive accuracy in comparison with control methods.

Improved stochastic modeling of multi-GNSS single point positioning with additional BDS-3 observations

The accuracy of Global Navigation Satellite System (GNSS) observations is affected by many factors, such as different systems, frequencies, carriers and pseudoranges, all of which also vary with different situations. Therefore, it is challenging to establish an accurate stochastic model for multi-GNSS positioning in theory, particularly for the additional BeiDou-3 Global Navigation Satellite System (BDS-3). In practical applications, the real stochastic model needs to be estimated based on the characteristics of the observations themselves. We evaluated the influence of BDS-3 on the positioning results using 46 sites distributed around the world and proposed an improved stochastic model for multi-GNSS single point positioning (SPP) based on the least-squares variance component estimation (LS-VCE). The results show that when the BDS-3 observations are added, the positioning precision and accuracy are significantly improved. By using the improved LS-VCE method in GPS/BDS dual system positioning, the accuracy of E, N and U directions are 0.373, 0.498 and 1.044 m, respectively, when compared to the traditional method with 0.502, 0.533 and 1.333 m. The proposed stochastic model improves the multi-GNSS SPP accuracy without significantly increasing the calculation time. Furthermore, reliable results are obtained for all epochs with the improved LS-VCE model.

A constrained extended Kalman filter based on LS-VCE formulated by condition equations with prediction of cross-covariances

A constrained extended Kalman filter (CEKF) based on least-squares variance component estimation (LS-VCE) is generally developed by condition equations since the proper prediction of dispersion matrices is one of the main bottlenecks in the KF algorithms. Here we investigate four problems which have not been simultaneously considered yet. These problems are examination of non-linearty of dynamic model, VCE, general non-linear state constraints and fairly general stochastic model. Although a few contributions proposed some adaptive KF in particular based on Helmert’s VCE method, they developed their filters for special problems with some restrictive conditions such as independence of all variables and/or linearity of the dynamic model. Also some of these filters did not apply VCE methods to all parts of the dynamic model. In this contribution, we try to overcome all of these restrictions. Moreover, LS-VCE method gives some added advantages over other VCE methods. First the new formulation of CEKF is developed by condition equations with prediction of all possible cross-covariances as algorithm 1. Then the LS-VCE method is applied to it after some modifications which results in an adaptive constrained extended Kalman filter (ACEKF) as the second algorithm.

Automatic Calibration of Process Noise Matrix and Measurement Noise Covariance for Multi-GNSS Precise Point Positioning

The Expectation-Maximization algorithm is adapted to the extended Kalman filter to multiple GNSS Precise Point Positioning (PPP), named EM-PPP. EM-PPP considers better the compatibility of multiple GNSS data processing and characteristics of receiver motion, targeting to calibrate the process noise matrix Qt and observation matrix Rt, having influence on PPP convergence time and precision, with other parameters. It is possibly a feasible way to estimate a large number of parameters to a certain extent for its simplicity and easy implementation. We also compare EM-algorithm with other methods like least-squares (co)variance component estimation (LS-VCE), maximum likelihood estimation (MLE), showing that EM-algorithm from restricted maximum likelihood (REML) will be identical to LS-VCE if certain weight matrix is chosen for LS-VCE. To assess the performance of the approach, daily observations from a network of 14 globally distributed International GNSS Service (IGS) multi-GNSS stations were processed using ionosphere-free combinations. The stations were assumed to be in kinematic motion with initial random walk noise of 1 mm every 30 s. The initial standard deviations for ionosphere-free code and carrier phase measurements are set to 3 m and 0.03 m, respectively, independent of the satellite elevation angle. It is shown that the calibrated Rt agrees well with observation residuals, reflecting effects of the accuracy of different satellite precise product and receiver-satellite geometry variations, and effectively resisting outliers. The calibrated Qt converges to its true value after about 50 iterations in our case. A kinematic test was also performed to derive 1 Hz GPS displacements, showing the RMSs and STDs w.r.t. real-time kinematic (RTK) are improved and the proper Qt is found out at the same time. According to our analysis despite the criticism that EM-PPP is very time-consuming because a large number of parameters are calculated and the first-order convergence of EM-algorithm, it is a numerically stable and simple approach to consider the temporal nature of state-space model of PPP, in particular when Qt and Rt are not known well, its performance without fixing ambiguities can even parallel to traditional PPP-RTK.

A single-receiver geometry-free approach to stochastic modeling of multi-frequency GNSS observables

The proper choice of stochastic model is of great importance to global navigation satellite system (GNSS) data processing. Whereas extensive investigations into stochastic modeling are mainly based on the relative (or differential) method employing zero and/or short baselines, this work proposes an absolute method that relies upon a stand-alone receiver and works by applying the least-squares variance component estimation to the geometry-free functional model, thus facilitating the characterization of stochastic properties of multi-frequency GNSS observables at the undifferenced level. In developing the absolute method, special care has been taken of the code multipath effects by introducing ambiguity-like parameters to the code observation equations. By means of both the relative and absolute methods, we characterize the precision, cross and time correlation of the code and phase observables of two newly emerging constellations, namely the Chinese BDS and the European Galileo, collected by a variety of receivers of different types at multiple frequencies. Our first finding is that so far as the precision is concerned, the absolute method yields nearly the same numerical values as those derived by the zero-baseline-based relative method. However, the two methods give contradictory results with regard to the cross correlation, which is found (not) to occur between BDS phase observables when use has been made of the relative (absolute) method. Our explanation to this discrepancy is that the cross correlation found in the relative method originates from the parts (antenna, cable, low noise amplifier) shared by two receivers creating a zero baseline. The time correlation is only of significance when the multipath effects are present, as is the case with the short-baseline-based relative method; this correlation turns out to be largely weaker (or ideally absent) in the absolute (or zero-baseline-based relative) method. Moreover, with the absolute method, the stochastic properties determined for two receivers of the same type but subject to different multipath effects are virtually the same. We take this as a convincing evidence that the absolute method is robust against multipath effects. Hence, the absolute method proposed in the present work represents a promising complement to the relative method and appears to be particularly beneficial to GNSS positioning, navigation and timing technologies based on the undifferenced observables, typically the precise point positioning.

Assessment of Tide Gauge Biases and Precision by the Combination of Multiple Collocated Time Series

AbstractThis study proposes a method for the cross calibration of tide gauges. Based on the combination of at least three collocated sea level time series, it takes advantage of the least squares variance component estimation (LS-VCE) method to assess both sea level biases and uncertainties in real conditions. The method was applied to a multi-instrument experiment carried out on Aix Island, France, in 2016. Six tide gauges were deployed to carry out simultaneous sea level recordings for 11 h. The best results were obtained with an electrical contact probe, which reaches a 3-mm uncertainty. The method allows us to assess both the biases and the precision—that is, the full accuracy—for each instrument. The results obtained with the proposed combination method have been compared to that of a buddy-checking method. It showed that the combination of all the time series also provides more precise bias estimates.

ADJUSTMENT OF CADASTRAL NETWORK USING LEAST-SQUARES VARIANCE COMPONENT ESTIMATION

Abstract. The role of the stochastic model very important in data processing of geodetic network since it describes the accuracy of the measurements and their correlation with each other. Knowledge of weights of the observables is required to provide a better understanding of the sources of errors and to model the error, hence the weights need to be determined correctly. This study concentrates on the estimation of variance components from different types of instruments used in the cadastral survey. The ideas are to combine the conventional and advanced instruments in a traverse network to enhance the estimated variance component in the stochastic model. Thus, Least Squares Variance Component Estimation (LS-VCE) method was used in this study because the method is simple, flexible and attractive due to the precision of variance estimators that can be directly obtained. Observation data come with several types of instruments such as chain measurement, Electronic Distance Measurement and total station were utilized. The findings showed that LS-VCE method was very reliable in cadastral network application. Besides, the results revealed that the estimated variance components for distance scale error, σp seem to become unrealistic for each data tested. It was found that the traverse network which included chain survey, showed the insignificant result to the precision of station coordinates when the measurements were combined.

Variance-covariance component estimation for structured errors-in-variables models with cross-covariances

In this contribution, an iterative algorithm for variance-covariance component estimation based on the structured errors-in-variables (EIV) model is proposed. We introduce the variable projection principle and derive alternative formulae for the structured EIV model by applying Lagrange multipliers, which take the form of a least-squares solution and are easy to implement. Then, least-squares variance component estimation (LS-VCE) is applied to estimate different (co)variance components in a structured EIV model. The proposed algorithm includes the estimation of covariance components, which is not considered in other recently proposed approaches. Finally, the estimability of the (co)variance components of the EIV stochastic model is discussed in detail. The efficacy of the proposed algorithm is demonstrated through two applications: multiple linear regression and auto-regression, on simulated datasets or on a real dataset with some assumptions.

Three-Dimensional Displacement Fields from InSAR through Tikhonov Regularization and Least-Squares Variance Component Estimation

Synthetic aperture radar interferometry (InSAR) measures the projection of three-dimensional (3D) ground displacement in the range direction and in the azimuth direction through image processing. The incapability of InSAR in measuring the 3D displacements restricts its capability for assessing real Earth surface deformation. The near-polar orbiting characteristics of InSAR missions reduce the sensitivity of line-of-sight (LOS) displacements significantly to the north-south components of the real 3D displacement fields and weaken the geometric strength of a given configuration. Applying range measurements from various missions to address 3D displacement leads to an ill-posed inverse problem that needs to be regularized. Moreover, it needs appropriate weighting of the observations to give proper estimates of the parameters. In this study, we propose Tikhonov regularization (TR) and least-squares variance component estimation (LS-VCE) methods for retrieving 3D displacement vectors from range and azimuth displacements. Depending on the functional degree of freedom (DoF) of the inverse problem, the TR and LS-VCE methods are applied in determined and overdetermined equation systems, respectively, to stabilize the ill-conditioned models and estimate the variance components of observations. These methods were evaluated by two synthetic data sets and a real data set from the Sentinel-1 terrain observation by progressive scan (TOPS) and ALOS-2 phased array type L-band synthetic aperture radar (PALSAR-2) missions in 2015 of the MW = 8.3 Illapel earthquake in Chile. Results indicate more than 40% improvement in both the precision and accuracy of retrieving 3D deformation fields when the regularized LS-VCE (RLS-VCE) is adopted instead of the conventional method (CM) that considers primary weighting for observations. Applying the range and azimuth InSAR displacements together with adopting the LS-VCE method reveal a north-south convergent borderline near 31.2° S in the 2015 Illapel earthquake.

Stochastic modeling of BDS2/3 observations with application to RTD/RTK positioning

The Chinese BeiDou navigation satellite system (BDS) is gradually transitioning from a regional system (BDS2) to a global system (BDS3) to provide its positioning, navigation and timing (PNT) services to users worldwide. Real-time differencing (RTD) and real-time kinematic (RTK) positioning play important roles in the applications of BDS, and the optimal performance of these two techniques relies upon the proper choice of stochastic model. In this paper, the least-squares variance component estimation (LS-VCE) method is adopted to establish a realistic stochastic model considering the elevation-dependent precision, cross-correlation and time correlation for BDS2/3 observations collected in zero and short baselines. The factors affecting the stochastic model, including the receiver multipath mitigation technique and antenna type, are investigated. The results show that BDS3 B1C and B2a observations possess relatively high precision among all observations collected in the zero baseline, which may be attributed to the advanced signal structure and modulation technique, whereas this superior performance is absent in the short baseline due to the presence of multipath effects. A positive correlation of approximately 0.8 is found between BDS3 B1I and B3I phase observations, and similar correlations exist between BDS2 phase observations. The time correlations are weakened when applying the multipath mitigation technique, and the stochastic characteristics of observations are specific to the type of antenna. The BDS3-only RTD/RTK positioning precision with our established stochastic model is comparable to the BDS2-only positioning precision, and the BDS2 + BDS3 RTD/RTK positioning precision is improved by 40% to 65% with respect to the BDS2-only case.