An Optimal Carrier-Phase Smoothing Code Algorithm for Low-Cost Single-Frequency Receivers

The single-frequency pseudorange and carrier phase observations can be expressed as follows:where $P(i)$ and $\mathrm{\Phi }(i)$ = measured pseudorange and carrier phase, respectively, at $i$th epoch at $L1$ frequency; $\phi (i)$ and $\lambda$ = measured carrier phase at $i$th epoch and wavelength, respectively (cycles); $c$ = speed of light in vacuum; $dtr(i)$ and $dts(i)$ = receiver clock offset and satellite clock offset, respectively, at $i$th epoch; $T(i)$ = tropospheric delay at $i$th epoch; $I(i)$ = ionosphere delay at $i$th epoch; ${\mathrm{TGD}}_P$ and ${\mathrm{TGD}}_{\phi }$ = time group delay (TGD) of measured pseudorange and carrier phase, respectively; and ${\epsilon }_P(i)$ and ${\epsilon }_{\phi }(i)$ are noise items for pseudorange and carrier phase observations, respectively, at $i$th epoch.

This section introduces the principle of the CPSC. As mentioned previously, this method is used to reduce noise for pseudorange observations. The smoothing procedure can be described as follows (Hatch 1982):where $P(i)$ = original pseudorange observation; and $\overline{P}(k)$ = smoothed code, where $k$ = number of smoothing epochs (i.e., width of smoothing window)

Eq. (3) can be expressed by averaging Eq. (4)

Because the carrier phase noise is much smaller than the code noise, the variance of the carrier phase can be ignored. The variance of the smoothed code noise can be expressed by the error propagation lawwhere ${\sigma }_{{\epsilon }_P}^2$ and ${\sigma }_{{\epsilon }_{\overline{P}}}^2$ = variance of code noise and of smoothed code noise, respectively.

Because the influence of ionospheric error has the opposite sign to the code and carrier phase observations, the CPSC gradually diverges as the smoothing windows moves forward. Therefore, the ionospheric error difference between successive epochs can be estimated using

Because the variation of ionospheric error between two consecutive epochs can be estimated by GIM, it can be expressed as

Assuming that the variance of ionospheric error [$I(k)$] is ${\sigma }_I^2$, the variance of ionospheric error between two consecutive epochs can be expressed using the error propagation law as

Accumulations of ionospheric error $\overline{I}(k)$ and variance ${\sigma }_{\overline{I}}^2(k)$ can be expressed as Eqs. (9) and (10), respectively

When $k\ge 1(k\in Z)$, ${\sigma }_{\overline{I}}^2(k)\ge 0$ and increases monotonically [Eq. (10)]. Therefore, a wider smoothing window increases the variation of ionospheric errors, as described previously.

Therefore, the variance of the code ${\sigma }_{\overline{P}}^2(k)$ is obtained as

According to Eq. (11), with a large smoothing window $k$, the noise error largely can be reduced; however, the ionospheric error largely remains. In generally, we can only strike a balance between them. To achieve better performance, this trade-off can be optimized. To minimize ${\sigma }_{\overline{P}}^2(k)$, we take partial derivative of ${\sigma }_{\overline{P}}^2(k)$ as follows:

Although the OCPSC algorithm can adjust the width smoothing window adaptively, in the quadratic Eq. (13), computation time is increased exponentially

We used only positive real roots to solve Eq. (13). Therefore, the solution ($k_o$) can be expressed as

Solution $k_o$ is the adaptive optimal solution to Eq. (11), and must be an integer value, so we rounded $k_o$ to the nearest integer.

To verify the influence of the TDCS algorithm on suppressing the cycle slips, three instances of simulated cycle slips were added to the real measured GPS data. The ellipses in Fig. 4 indicate the 3 simulated cycle slip events, which included 3 cycles at UTC 00:33:01, 10 cycles at UTC 1:23:01, and 20 cycles at UTC 1:56:21. The OCPSC algorithm with TDCS and without TDCS effectively smoothed the pseudorange, and observation noise was reduced significantly (Fig. 4). However, when there are cycle slips in observations, a systematic bias, which is proportion to the magnitude of the cycle slips, will be introduced to the smoothed pseudorange until the smoothing window is reset (Chang et al. 2019). When a cycle slip of three cycles was added at UTC 00:33:01, the smoothed pseudorange residual was increased by about 0.5 m. When a cycle slip of 20 cycles was added at UTC 1:56:21, the smoothed pseudorange residual was increased by about 3.5 m. Unfortunately, the smoothed pseudorange residuals of the subsequent epochs will be affected until the smoothing window is reset. Therefore, it is necessary to integrate TDCS into the OCPSC algorithm to suppress the influence of cycle slips. Fortunately, our OCPSC algorithm with the TDCS can sensitively detect cycle slips at all epochs and automatically reset the smoothing window.

Although CPSC performs well in noise reduction and is easy to implement, the variation of ionospheric error cannot be ignored. Dual-frequency (Kee et al. 1997) observations and single-frequency observations can be used to estimate the variation of ionosphere error between consecutive epochs [Eq. (10)]. The difference in ionosphere error between consecutive epochs can be calculated using Eqs. (22) and (23) for DF and SF observations, respectively

The performance of the traditional CPSC depends on the standard deviation of code noise and the variation of ionosphere error between consecutive epochs [Eq. (11)]. A small smoothing window is sensitive to the variation of ionosphere errors, but does not reduce noise sufficiently [Eq. (11)]. In contrast, a larger smoothing window is more effective in reducing noise, but cannot suppress the variation of ionosphere error, and may even lead to a divergence of the traditional CPSC. Therefore, the selection of smoothing window width is a kind of trade-off problem. Furthermore, we used dual-frequency observations from the JFNG site to analyze the variation of ionosphere error between consecutive epochs.

Fig. 5 shows the estimation of the variation of ionospheric error using the dual-frequency carrier phase and GIM for Satellite PRN02. The variation of ionospheric error calculated by GIM agreed well with average values calculated by the dual-frequency phase at all epochs, which verifies that the GIM performs well in estimating ionospheric error for single-frequency receivers (Fig. 5). Because the variation of ionospheric errors estimated using dual-frequency observations includes carrier-phase noise [Eq. (19)], it is much larger than the variation of ionospheric errors estimated using single-frequency observations

The measurement noise depends on the elevation angle and observation environment of the receiver antenna (Kee et al. 1997). We modeled the receivers’ code noise effects with the elevation angle ($El$) as follows:

To estimate the parameters $(x_0,x_1,x_2)$, the standard deviation of the noise from dual-frequency observations and single-frequency observations were curve-fitted. When using ${\sigma }_{{\epsilon }_P}$ and $El$ for curve fitting, ${\sigma }_{{\epsilon }_P}$ is the mean value including the noise standard deviation of all observed satellites at the elevation angle. The parameters $(x_0,x_1,x_2)$ are listed in Table 1. There was almost no significant difference between the noise parameters of the dual-frequency observations and the single-frequency observations (Table 1). When the elevation angle decreased from 65° to 15°, the standard deviations of the observed noise for the single-frequency observations and the dual-frequency observations varied from 0 to 1.6 m, and the fitted standard deviations of noise varied from 0.2 to 0.4 m (Fig. 6). The results indicate that the fitting performance was the same for both the dual-frequency and the single-frequency observations.

In the proposed OCPSC algorithm, the adaptive smoothing window width can be determined using the standard deviation of the code noise and the variation of ionospheric error between consecutive epochs [Eq. (11)]. The standard deviation of the code noise depends on the elevation angle in Eq. (25), and the variation of ionospheric error between consecutive epochs is estimated using the GIM in Eqs. (16) and (17).

To prove that the optimal smoothing window width $k_o$ is flexible for each satellite and each epoch, we analyzed the relationship between the optimal smoothing window width $k_o$ and elevation angles of Satellites PRN02 and PRN05 (Fig. 7). The optimal smoothing window width varied from 110 to 15 s as the elevation angle varied from 65° to 15° for Satellite PRN02, which was similar to that for Satellite PRN05, but the optimal smoothing window width of Satellites PRN02 and PRN05 was different at the same epoch. In the Wullschleger et al. (2000), in the case of traditional CPSC, a smoothing window width greater than 100 s will reduce the noise level but simultaneously cause a large divergence error (Park et al. 2008). On the other hand, for some satellites at some epochs, when the smoothing window is as large as 1,000 s, the OCPSC algorithm can reduce the code noise without a serious divergence error.

The dual-frequency observations were used to verify the OCPSC algorithm, and the true coordinates in the sinex file of the IGS product were used as reference values to assess the positioning performance of the raw code and the CPSC and OCPSC algorithms. In previous studies, the researchers proposed a fixed smoothing window of 100 s to obtain the best performance of traditional CPSC (Park et al. 2017). Therefore, the results derived using a fixed smoothing window of 100 s were used as a reference. A small cycle slip occurred in Satellite PRN13 at about UTC 00:44 (Fig. 4), which was not what we had simulated. This cycle slip had a serious impact on the performance of the traditional CPSC until the smoothing window was reset [Fig. 8(b)]. Our OCPSC algorithm can detect cycle slips in a timely and effective manner, and suppress the impact of cycle slips on positioning performance. Therefore, the OCPSC is more robust than traditional CPSC. Compared with the positioning accuracy of raw SPP, the horizontal positioning accuracy improved by about 0.04 m, and the vertical accuracy improved by about 0.1 m when applying the traditional CPSC with a smoothing window of 100 s (Fig. 8 and Table 2). The OCPSC algorithm with adaptive window width had better performance. It improved the horizontal accuracy and vertical accuracy by 0.05 and 0.3 m, respectively. Compared with the traditional CPSC, the positioning accuracy was not improved significantly by applying the OCPSC algorithm (Table 2). The horizontal positioning accuracy improved from 0.49 to only 0.48 m. This mainly was because the increase of the variation of ionospheric errors was not significant for traditional CPSC during the continuous observation period of about 2 h (Fig. 5).

To illustrate more clearly that the OCPSC algorithm effectively can solve the problem between the reduction of noise and the increase of ionospheric error variation, a longer period of observation data were processed by the proposed algorithm. The DAEJeon (DAEJ) site data from GPS times 00:00:00 to 7:30:00 on March 1, 2022 were collected and resampled at an interval of 1 s. The positioning results of the DAEJ site are presented in Table 2 and Fig. 9. The CPSC with 100-s smoothing windows improved only 2.3% horizontally and 4.3% vertically compared with the raw SPP. Because the DAEJ site has longer observation times than Station JFNG, the increase of ionospheric error variation between consecutive epochs was larger for the DAEJ station than for the JFNG site. Therefore, when applying traditional CPSC with a smoothing window of 100 s, the improvement in horizontal positioning accuracy (2.3%) at the DAEJ site was smaller than that at the JFNG site (7.5%). Fortunately, the proposed OCPSC algorithm can reduce the ionospheric error variation between consecutive epochs that increases with the observation time. When applying the OCPSC, the improvement in horizontal positioning accuracy (12.6%) at the DAEJ site was larger than that at the JFNG site (9.4%).

We investigated the positioning performance of the OCPSC during ionospheric storms. The 3-h-interval Kp-index can be used to measure global ionospheric disturbances (Bartels et al. 1939). The level of a geomagnetic storm is defined as Kp-index 5 for minor geomagnetic storms (G1), Kp-index 7 for moderate geomagnetic storms (G2), Kp-index 8 for stronger geomagnetic storms (G4), and Kp-index 9 for extreme geomagnetic storms (G5) (Hanslmeier 2002). The Kp-index is provided by NASA’s OMNIWeb Data Explorer (Papitashvili 2020). Fig. 10 shows the value of Kp-index from November 3 to 5, 2021. On November 4, 2021, there was an ionospheric storm between moderate and strong, with a Kp-index greater than 70.

To assess the positioning performance of the OCPSC algorithm during ionospheric storms, the algorithm was used to process observation data with high sampling intervals. The papeete (THTG) site data from GPS times 00:00:00 to 23:59:59 from November 3 to 5, 2021 were collected and resampled at an interval of 1 s. In this experiment, only $L1$ frequency observation data were used. GIM estimated the mean variation of the ionospheric error for all visible satellites on November 3, 4, and 5, 2021 (Fig. 11). It is obvious that the variation of ionospheric error during the ionospheric storm (Fig. 11) was much higher than that in Fig. 5. Before 2:00 on November 4, 2011, more points deviated from zero in the variation of ionospheric error than on November 3 and 5, 2021, because the corresponding Kp-index was greater than that on the other two days. Between 8:00 and 17:00, with the decrease of the Kp-index, the variation of the ionospheric error also decreased and was closer to zero on November 3, 4, and 5, 2021. This illustrates that ionospheric storms significantly affect the variation of ionospheric errors.

The distribution of positioning errors in the horizontal and vertical directions during ionospheric storms is shown in Fig. 12 based on GPS data of the THTG station using the OCPSC algorithm. When many points in the variation of ionospheric error deviated from zero at UTC 5:27:52, the positioning error of the traditional CPSC increased by about 38 m, whereas the positioning error of the OCPSC was only about 10 m (Fig. 12). At UTC 19:35:50, when the variation of ionospheric error increased by 1.5 m, the residual of the traditional CPSC increased by about 15 m. the residual of the OCPSC increases by only about 10 m. Compared with the traditional CPSC, the OCPSC algorithm can suppress the influence of the variation of ionospheric error on positioning performance. Compared with raw SPP, the traditional CPSC did not improve the horizontal and vertical position accuracy, because the traditional CPSC does not consider the variation of the impact of ionospheric error on positioning performance (Fig. 12 and Table 3). However, compared with the raw SPP, the positioning errors of the OCPSC algorithm in the horizontal and vertical directions both were reduced by 0.01 m. Compared with the original SPP, the positioning accuracy of the OCPSC algorithm does not seem to be improved significantly. This mainly is because the OCPSC algorithm depends on the variation of ionospheric error to give an optimal window, which can constrain the impact of the variation of ionospheric error on positioning performance, rather than completely eliminating it.

As mentioned previously, we conducted static experiments on dual-frequency and survey-grade receivers to verify that the OCPSC algorithm is effective. Then the OCPSC algorithm was applied to a u-blox EVK-M8T receiver (Thalwil, Switzerland), which is a low-cost single-frequency receiver. The u-blox receiver is an automotive-grade module that can track the signals from multiple GNSS constellations and provide raw pseudorange, carrier-phase, and Doppler observation (Odolinski et al. 2015). Because the u-blox receiver is connected to a patch antenna, it is applied widely to navigation, transport, traffic, and so forth. The positioning accuracy of low-cost single-frequency receivers for SPP usually is at the meter level (Pan et al. 2016), and at the decimeter level for real-time kinematics (Odolinski and Teunissen 2016). Therefore, in this paper, the RTK results were used for comparison to evaluate the positioning performance of the OCPSC algorithm with u-blox receivers.

To analyze the SPP positioning performance of low-cost single-frequency receivers using the OCPSC algorithm, static and kinematic experiments were carried out on Dengzhuang South Road, Haidian District, Beijing (Fig. 13). The observations from the u-blox receiver were recorded at a sampling interval of 1 s, and the satellite cutoff elevation angle was set to 10° to reduce the impact of obstacles. The static and kinematic data sets were collected on June 3, 2019 from local time 8:55:45 to 11:00:45, and on September 4, 2018 from local time 20:02:42 to 20:49:43. Fig. 14 shows the PDOP and satellite numbers of low-cost single-frequency receivers in static and kinematic experiments. In static experiments, the number of satellites was more than 7, and the PDOP was less than 3. In the kinematics experiment, which was affected by the obstacles in the observation environment, the number of satellites was between 4 and 6, and the PDOP was greater than 7.

In the static test, the code observation residuals of each satellite fluctuated around zero [Fig. 15(a)]. However, in the kinematics test, because the observation was affected by the environment obstacles, the code observation residuals of many satellites had many multipath errors [Fig. 15(b)]. Both the traditional CPSC and the OCPSC algorithm can reduce the horizontal and vertical position errors (Figs. 16 and 17). In static testing, compared with the raw SPP, the horizontal and vertical errors of the traditional CPSC with a smoothing window of 100 s were reduced by 0.7 and 1 m, and the positioning accuracy improvement rates were 24% and 22.4%, respectively. This improvement in positioning accuracy is due mainly to the fact that low-cost single-frequency receivers include a great deal of noise when collecting data, whereas CPSC performs well in noise reduction. Compared with the traditional CPSC, the positioning error of the OCPSC algorithm was reduced by 0.15 and 0.21 m in the horizontal and vertical directions, respectively. The positioning accuracy improvement rate of the OCPSC algorithm was 28.9% and 27.4% compared with raw SPP. In the kinematics test, compared with the raw SPP, the positioning accuracy of the traditional CPSC improved by 0.02 m (0.5%) and 0.03 m (0.6%) (Table 4). Compared with the traditional CPSC, the positioning performance of the OCPSC algorithm was better, but the difference was not large. This mainly was because some observations, such as PRN24 and PRN32, may have a certain multipath error [Fig. 15(b)], resulting in little improvement in the positioning performance of the OCPSC algorithm.