The adjustment of Global Navigation Satellite System (GNSS) observations, e.g., in GNSS relative positioning or precise point positioning, is based on functional and stochastic models [1
]. The functional model describes the relations between observations and unknown parameters, which is sufficiently known and well-documented in the literature. The stochastic model, represented by the (co)variance matrix, reflects the precisions, cross correlations between observation types and time correlations. Only with a realistic stochastic model can one obtain minimum variance estimators of the parameters in a linear model and arrive at a proper description of the estimator’s quality [2
]. Therefore, various studies have been conducted to improve our knowledge of the GNSS stochastic model by estimating unknown components of the (co)variance matrix, which is generally known as variance component estimation (VCE).
Earlier studies of Eueler and Goad [3
], Gerdan [4
] and Jin and de Jong [5
] investigated the elevation dependence of observation variances. Jonkman [6
] and Wang et al. [7
] stepped further and took cross correlations and time correlations of observations into consideration. Bona [8
] examined the precisions, cross correlations and time correlations of Global Positioning System (GPS) phase and code observations using seven types of GPS receivers. Tiberius and Kenselaar [9
] presented reasonably simple VCE formulas and analyzed stochastic models of GPS observables on a zero baseline. The elevation dependence of precisions, cross correlations between observation types and time correlations are found to be significant, while the correlation between different channels/satellites turns out to be quite small, indicating that a tracking loop is dedicated to one satellite and operates autonomously. Li et al. [10
] assessed the stochastic models of GPS measurements collected by different types of receivers on ultra-short baselines, and identified the existence of elevation dependence of precisions, time correlations and cross correlations between observation types. They pointed out that the stochastic model should be specified for the receiver and observation types. Amiri-Simkooei et al. [11
] assessed the stochastics of GPS observables on zero baselines and found significant cross correlations between C1 and P2 code observations and between L1 and L2 phase observations. Li [12
] presented a comprehensive study on stochastic modeling of triple-frequency signals of the Chinese BeiDou navigation satellite system with four types of BeiDou receivers. For more studies on GNSS stochastic models, the readers can refer to Teunissen and Amiri-Simkooei [13
Although GNSS stochastic models have been extensively studied, only GNSS data with low sampling rates, i.e., 1 Hz or lower, have been analyzed up to present. With the recent development of receiver technology, very high-rate (10–50 Hz, or even up to 100 Hz) GNSS data have become readily available, leading to numerous very high-rate GNSS applications, such as very high-rate GNSS seismology [14
], earthquake/tsunami early warning [18
], and structural monitoring [20
]. Very high-rate data may have a different sampling mechanism from low sampling-rate data, e.g., with different parameters of signal processing in receivers (such as phase-locked loop bandwidth) adapted to fast sampling [23
], leading to different stochastic models. Therefore, precisions and cross correlations estimated from low sampling-rate data may not hold true for high-rate data, and time correlations within very short intervals (less than 1 s) need to be specially investigated. The lack of realistic stochastic models for very high-rate data may have us misunderstand high-rate solutions and decrease the data value. To fully exploit the benefits brought by very high-rate GNSS data, detailed studies of stochastic models for very high-rate data have become an urgent issue.
Over the past decades, the GPS has made remarkable contributions to scientific applications and engineering services as a standalone global navigation satellite system. In recent years, new constellations have emerged and been put into operation, resulting in significant improvement in terms of satellite visibility, spatial geometry and dilution of precision. The BeiDou Navigation Satellite System, independently established and operated by China, began to offer positioning and navigation services in the Asia-Pacific region at the end of 2012. It is expected to provide global services by about 2020 [24
]. The rapid development of multi-constellation GNSS can bring about wider and more precise applications, e.g., for positioning, timing and remote sensing [25
This paper aims to study the stochastic models of very high-rate data from two GNSS constellations, namely GPS and BeiDou. It is organized as follows: Section 2
introduces the method of least-squares variance component estimation, describes the functional model used in this study, which is a geometry-based model using GNSS single-differenced observations, and describes the stochastic model, namely the precisions, cross correlations and time correlations of GNSS observations. Section 3
describes the very high-rate (50 Hz) data used in this study along with the VCE procedure, and presents the results of GPS/BeiDou stochastic models. Section 4
gives the conclusions.
In this study, we assess the stochastic models of very high-rate (50 Hz) GPS/BeiDou phase and code observations. Since previous studies on stochastic models of GNSS code and phase observations are limited to sampling rates not higher than 1 Hz, this study provides significant results of very high-rate (50 Hz) data, especially the time correlations at very small time lags 0.02–1 s. This helps to enrich our knowledge about very high-rate GNSS data and support very high-rate GNSS applications.
The geometry-based functional model is adopted, which has a larger redundancy and is more suitable for VCE than the geometry-free model. We use the between-receiver SD observations rather than DD observations in the functional model because in this way we can estimate the stochastic properties of individual satellites. Three stochastic properties are considered: the precisions of individual satellites, cross correlations between observation types and time correlations of each observation type. The (co)variance matrices considering these properties are elaborately constructed following the three assumptions (1) there is no correlation between different channels/satellites or between different constellations; (2) the correlation between each observation-type pair is the same for all satellites; and (3) the time correlation of each observation type is the same for all satellites. The method of least-squares variance component estimation is applied to estimate the precisions, cross and time correlations in a step-wise way. Estimating the time correlations is time-consuming because of too many unknowns involved. To improve the efficiency, we present the double sampling-rate procedure to estimate the time correlations, namely using 50 Hz data to calculate time correlations at time lags of 0.02–5 s, and using decimated 1 Hz data to calculate time correlations at time lags of 1–180 s. The 50 Hz GPS/BeiDou phase and code observations spanning 24 h collected on a zero baseline are analyzed in this study.
When estimating precisions and cross correlations iteratively, the convergence is so fast that estimates after 2–4 iterations are almost invariant. Even the estimates of the first iteration are already very close to the final convergent values. In our study, precisions and cross correlations are estimated by performing four iterations and one iteration, respectively. However, the time correlations hardly achieve convergence through iterations, which may be caused by the ill-posedness of the model. Therefore, the time correlations are obtained by performing only one iteration.
We have presented the precisions of cross and time correlation estimates. Within one group, the precisions of cross correlation estimates are generally higher than 0.007. The precisions of time correlation estimates depend on the time lags, getting lower as the time lag increases. At the time lag of 1 s, the precisions of phase time correlations are generally higher than 0.008, and the precisions of code time correlations are generally higher than 0.02 within one group. When the estimates are averaged over all groups, the precisions can be improved by with the number of groups. This precision level, together with the consistence of estimates from different groups, has validated the reliability of our cross and time correlation estimates.
The precisions of undifferenced GNSS observations are shown to be elevation-dependent at satellite elevation angles below 40°. Above 40°, they are relatively stable and can be assumed as constants. Therefore, the precisions can be modeled using a two-piece function. The first piece describes the low-elevation precisions with an elevation-dependent function, while the second piece represents high-elevation precisions with a constant value. Note, however, the precision patterns differ for different observation types and satellites, especially for BeiDou because different types of satellites are involved. Therefore, when modeling the precisions the parameters should be individually considered for each observation type and each satellite. GPS and BeiDou have comparable precisions at high satellite elevation angles, reaching 0.91–1.26 mm and 0.13–0.17 m for phase and code, respectively, while at low satellite elevation angles, GPS precisions are generally lower than BeiDou ones. At satellite elevation angles above 30°, the GPS L2 phase precisions are only slightly lower than L1 phase precisions, while at satellite elevation angles below 30°, the GPS L2 phase precisions are significantly lower than L1 phase precisions. For BeiDou, the B2 phase precisions are slightly lower than B1 ones over all satellite elevation angles. The B1 code precisions are slightly lower than B2 ones at satellite elevation angles lower than 40°.
Cross correlations are very significant between GPS L1 and L2 phase and between BeiDou B1 and B2 phase, reaching 0.773 and 0.927, respectively. Cross correlations between GPS L1 and L2 code and between BeiDou B1 and B2 code are less significant, reaching 0.141 and 0.111, respectively. Cross correlations of other observation type pairs, mixed with phase and code, are less than 0.03, indicating that phase and code observations are essentially not correlated.
Time correlations are significant at very small time lags of 0.02–0.12 s, especially for code observations. The maximum time correlations reach 0.175–0.293 and 0.858–0.882 for GPS phase and code, respectively, and reach 0.041–0.051 and 0.904–0.945 for BeiDou phase and code, respectively. The turning point of time correlations at the time lag of 0.12 s has been observed for both GPS and BeiDou phase observations. At the time lag of 1 s, the time correlations decrease a lot and reach 0.000–0.141 and 0.014–0.177 for phase and code, respectively. All of them reach zero values and show no time correlations at all starting from some larger time lags, which are dependent on the constellations and observation types.
In high-precision GNSS positioning applications, the satellite elevation angle-dependent and satellite-specific precisions should be applied to weight the observations. Cross correlations between phase observations on different frequencies and between code observations on different frequencies are significant and must be taken into account. In traditional applications with data sampling rate equal to or less than 1 Hz, time correlations of phase can be safely neglected. However, in very high-rate applications, time correlations of phase at very small time lags must be taken into account in order to achieve the most precise positioning.
More comprehensive researches are needed by including more receiver types, more observation types, e.g., triple-frequency observations, and more constellations, e.g., Russia’s GLONASS and Europe’s Galileo. Data collected on a short baseline will be analyzed in the future to investigate the stochastic models of external errors such as multipath. We also need to study how to further increase the computation efficiency of variance component estimation when large amounts of data and large numbers of unknown (co)variance components are involved. Besides, the impact analysis of an improved stochastic model of very high-rate GNSS observations on estimating coordinates and zenith tropospheric delays also needs to be conducted in the future.