Determination of Differential Code Bias of GNSS Receiver Onboard Low Earth Orbit Satellite

The uncertainty of differential code bias (DCB) is one of the main error sources in the low Earth orbit (LEO) based total electron content (TEC) retrieval, whereas the derivation of the LEO DCB is not systematically studied. In this paper, we propose an improved DCB estimation method (ZERO method) based on the assumption that the LEO-based TEC can reach zero and also optimize the parameter configuration in the commonly used least square method (LSQ method). In the improved ZERO method, the combination of the lower quartile minimum relative TEC during each orbital revolution with the daily minimum relative TEC gives a stable and reliable DCB estimation. For the LSQ method, the 3-TECU cutoff vertical TEC with 10° cutoff elevation is considered to offer a reasonable DCB estimation. Subsequently, Global Positioning System (GPS) observations from multiple LEO satellites at different altitudes are used to study the variability of the LEO DCBs. Our results revealed that the LEO DCBs underwent obvious long-term variation and periodic oscillations of months. Moreover, the CHAMP data illustrated that the long-term variation of LEO DCBs is partly associated with the GPS satellite replacement, and the periodic variation can be attributed to the variation of the hardware thermal status, represented by the receiver CPU temperature in this study.

SECTION I. Introduction

The total electron content (TEC) derived from the Global Navigation Satellite System (GNSS) dual-frequency observations of ground-based receivers has shown valuable applications in studying the ionosphere [1]–​ [3]. Recently, low Earth orbit (LEO) satellites equipped with GNSS receivers for either precise orbit determination (POD) or radio occultation purpose have been new tools for TEC measurements [4]–​ [7]. Since the LEO-based TEC covers hardly accessible regions, it is playing an important role in space weather research, especially in revealing the underlying physical mechanisms of the variation in the topside ionosphere and plasmasphere [8]–​ [11].

In the LEO-based TEC retrieval, the differential code bias (DCB) caused by GNSS satellite transmitter and receiver hardware is still one of the main error sources. To determine the absolute TEC, the DCBs of the GNSS satellite and receiver must be removed. With the well-developed methods, the Global Positioning System (GPS) satellite DCBs released by GNSS communities are reliable [3]. For ground-based receivers, the derivation and variation of the DCBs are widely examined [12]. The common method for ground-based receiver network is to model the ionospheric vertical TEC (vTEC) by polynomial or spherical harmonic expansion in latitude and local time, and then, the DCBs and the coefficients of the polynomial or expansion are estimated using Kalman filtering approach or least square fitting [13], [14].

As the LEO satellite moves very fast and its sampling locations change quickly, the traditional approach on ground-based receivers cannot be directly applied to the limited observations of the LEO satellite. There are three common approaches for LEO DCB estimation. The first method is based on the assumption that the observed TEC above the LEO satellite can approach zero at high latitudes or during nighttime. Using this method, Lee et al. [15] tried to determine the DCB of Jason-1 satellite by assuming that the minimum observed TEC value is equal to zero. The advantage of this method is simple and fast. However, the cycle slip and multipath effect of LEO satellites are usually serious [16], so that the minimum observed TEC might be the outlier. Stephens et al. [17] estimated the LEO DCB by subtracting the background TEC of 0.5 TECU from the averaged minimal TEC obtained at high latitude and high elevation angle measurements for COSMIC satellites (800 km). However, how to determine the background TEC for LEO satellites at different altitudes is not well addressed. The second method is based on the spherical symmetry ionosphere assumption that, as a least square method (LSQ method) described in Yue et al. [7], the simultaneous paired observations with different elevation angles are assumed to have the same vTEC. For this method, the parameter configuration, which has a significant influence on DCB, is not well evaluated. For the third method, Heise et al. [18] assumed that the vTEC obtained from the parameterized ionospheric model is close to the actual vTEC, and then, the DCB can be extracted with the model outputs. However, this method strongly depends on the accuracy of the ionosphere/plasmasphere model. In addition to the LEO DCB estimation, the variations of LEO DCBs are still not systematically investigated.

In this paper, we propose an improved DCB estimation method (ZERO method) for LEO satellites based on the zero TEC assumption. The parameter configuration of cutoff elevation and vTEC in the LSQ method is further studied. GNSS observations from multiple LEO satellites at different altitudes are used to study the variations of the LEO DCBs. Finally, the long-term and periodic variations of the LEO DCBs are particularly discussed.

SECTION II. Data and sTEC Extraction

The GPS POD observations from CHAMP, GRACE, TerraSAR-X, SAC-C, MetOp-A, and Jason-1 are used to examine the variations of the LEO DCBs in our study. The orbit information and time coverage for these LEO satellites are given in Table I and Fig. 1.

TABLE I Information for the LEO Satellites That Provided POD Data in This Study

Fig. 1.

Variations of orbital altitudes for multiple missions including CHAMP, GRACE, TerraSAR-X, SAC-C, MetOp-A, and Jason-1 from 2002 to 2013. The horizontal lines represent the availability of data. The solar activity proxy F107 and sketch of a typical electron density profile are also shown.

The pseudorange and carrier phase observations are recorded at the two GPS frequencies ( $f_{1} = 1575.42\ \text{MHz}$ and $f_{2} = 1227.60\ \text{MHz}$). Before processing the data, the outliers are eliminated, and cycle slips are detected and corrected to create continuous arcs using the algorithm described by Blewitt [19]. Slant TEC (sTEC) information can be obtained from the frequency difference of observations as follows: $$\begin{align*} \text{sTEC}_{P} =&\ \alpha(P_{2} - P_{1}) \\ =&\ \text{sTEC}^{\mathrm{abs}} - \text{DCB}_{s}^{i} - \text{DCB}_{r} + \varepsilon_{P}\tag{1} \\ \text{sTEC}_{L} =&\ \alpha(L_{1} - L_{2}) \\ =&\ \text{sTEC}^{\mathrm{abs}} + \frac{1}{\alpha}(N_{1} \lambda_{1} - N_{2} \lambda_{2}) + \varepsilon_{L}\tag{2} \end{align*}$$ where $P_{1}$ and $P_{2}$ are pseudoranges, $L_{1}$ and $L_{2}$ are carrier phase observations at the two frequencies, $\alpha$ is the factor of $((f_{1}^{2} \times f_{2}^{2})/40.3(f_{1}^{2} - f_{2}^{2}))\ \text{m}^{3}\text{s}^{-2}$, $\text{sTEC}^{\mathrm{abs}}$ is the actual absolute sTEC, $\text{DCB}_{s}$ and $\text{DCB}_{r}$ are the GPS satellite DCB and receiver DCB, the superscript $i$ represents the pseudorandom number of the GPS satellite, $N_{1}\lambda_{1}$ and $N_{2}\lambda_{2}$ are the integer cycle ambiguity terms, and $\varepsilon_{P}$ and $\varepsilon_{L}$ are the error terms. As the pseudorange TEC $(\text{sTEC}_{P})$ is biased by the DCBs of the GPS satellite and receiver and as the phase TEC $(\text{sTEC}_{L})$ is affected by integer cycle ambiguities, both of these TECs cannot be directly used. Since the noise of the carrier phase data is much lower than the pseudorange data, the pseudorange TEC is usually utilized to adjust the level of the phase TEC [13], [20]. The level-based constant $B_{{rs}}$ can be obtained in each phase-connected arc as follows: $$\begin{equation*} B_{{rs}} = \frac{\sum\limits_{i} (\text{sTEC}_{Pi} - \text{sTEC}_{Li}) \times w_{i}^{2}}{\sum\limits_{i} w_{i}^{2}}\tag{3} \end{equation*}$$ where $w_{i}$ is the weighting factor of the signal-to-noise ratio or the sine of observed elevation in our study. The TEC observable is then obtained from the phase TEC and level-based constant $$\begin{equation*} \text{sTEC}^{\mathrm{level}} = \text{sTEC}_{L} + B_{{rs}}.\tag{4} \end{equation*}$$

Note that the leveled phase TEC $(\text{sTEC}^{\mathrm{level}})$ is still biased by the DCBs, and the relationship between the leveled phase TEC and absolute TEC $(\text{sTEC}^{\mathrm{abs}})$ can be written as follows: $$\begin{equation*} \text{sTEC}^{\mathrm{abs}} = \text{sTEC}^{\mathrm{level}} + \text{DCB}_{s}^{i} + \text{DCB}_{r} + \varepsilon_{\mathrm{level}}\tag{5} \end{equation*}$$ where $\varepsilon_{\mathrm{level}}$ is the leveling error term. The DCBs of the GPS satellites and receiver are assumed to be constant on the daily basis in our study. The GPS satellite DCBs from the International GNSS Service (IGS) are used to calibrate the leveled phase TEC. Note that the zero-mean condition is imposed on the estimated GPS DCBs in the IGS as follows [14]: $$\begin{equation*} \frac{1}{n}\sum_{i = 1}^{n}\text{DCB}_{s}^{i}=0\tag{6} \end{equation*}$$ where $n$ is the number of observed GPS satellites. After the GPS DCBs are calibrated, (5) can be rewritten as follows: $$\begin{equation*} \text{sTEC}^{\mathrm{abs}} = \text{sTEC}^{r} + \text{DCB}_{r} + \varepsilon_{\mathrm{level}}\tag{7} \end{equation*}$$ where $\text{sTEC}^{r}$ is the relative TEC biased with the receiver DCB, i.e., $\text{sTEC}^{r} = \text{sTEC}^{\mathrm{level}} + \text{DCB}_{s}^{i}$.

SECTION III. LEO DCB Determination

The applications of the LEO-based TEC have been widely studied, but the LEO DCB derivation is not well addressed. In this section, we describe an improved ZERO method to enhance the reliability of the derived DCB. Meanwhile, we discuss the sensitivity of the LEO DCB to the parameter setting in the LSQ method. Moreover, the DCBs of both methods for six LEO satellites are presented for reference.

A. Improved ZERO TEC Method

Since the LEO satellite can measure the TEC above the orbital altitude, the TEC at high latitudes or during nighttime can be very small, even close to zero. Some studies directly used the minimum $\text{sTEC}^{r}$ ( $\text{sTEC}^{r}$ is the relative TEC with the GPS satellite DCB calibrated but still biased with the receiver DCB) to derive the receiver DCB if $\text{sTEC}^{\mathrm{abs}}$ in (7) is assumed to be zero [15], [21]. Generally, the $\text{DCB}_{0}^{d}$ from the ZERO TEC method can be expressed as follows: $$\begin{equation*} \text{DCB}_{0}^{d} = 0 - \text{sTEC}_{\mathrm{dailymin}}^{r}\tag{8} \end{equation*}$$ where $\text{sTEC}_{\mathrm{dailymin}}^{r}$ is the daily minimum of the relative TEC. This method is simple and fast. However, the $\text{sTEC}_{\mathrm{dailymin}}^{r}$ might be an outlier sometimes, and thus, the derived $\text{DCB}_{0}^{d}$ is not reliable sometimes. Outliers are mainly caused by the leveling errors, which are associated with cycle slip, multipath effect, and observational noise [22].

In order to overcome the shortcoming of the general ZERO method, we propose an improved ZERO method as follows. The improved ZERO method is to find a suitable minimum $\text{sTEC}^{r}$. The processed data are restricted to elevation angles larger than 40° to improve the data quality. Then, the GPS DCBs from the IGS are used to calibrate the leveled phase TEC to get the $\text{sTEC}^{r}$ (i.e., the relative sTEC calibrated with GPS DCB but still biased with receiver DCB). Since the time of one revolution around the Earth of the LEO satellite is short (e.g., ∼95 min), the LEO satellite can complete about 15 revolutions in one day. The passed orbit in one revolution is divided into ascending and descending orbits. The minimum $\text{sTEC}^{r}$ of the ascending (or descending) orbit in each revolution in one day is obtained, and then, the lower quartile (25%) minimum $(\text{sTEC}_{\mathrm{lqmin}}^{r})$ among these minimum relative TECs of the ascending (or descending) orbit is calculated. The smaller $\text{sTEC}_{\mathrm{lqmin}}^{r}$ between those of ascending and descending orbits is selected because the smaller $\text{sTEC}_{\mathrm{lqmin}}^{r}$ is usually from the nighttime orbit. The suitable minimum $\text{sTEC}^{r}$ value should be the following: 1) stable and less affected by outliers and 2) satisfied with the zero TEC assumption to a certain extent. When the selected percentile is lower, it will be more obviously affected by outliers; when the selected percentile is too high, it will deviate from the zero TEC assumption. It is found that the lower quartile (25%) minimum generally gives stable and reasonable minimum $\text{sTEC}^{r}$ based on our LEO satellite data analysis. Note that the determination of the selected percentile also partly depends on the LEO satellite data. Then, the $\text{DCB}_{0}^{q}$ obtained from the selected $\text{sTEC}_{\mathrm{lqmin}}^{r}$ (i.e., $\text{DCB}_{0}^{q}=-\text{sTEC}_{\mathrm{lqmin}}^{r}$) can be treated as a stable receiver DCB estimation.

Since the plasmasphere can be a significant contributor to the sTEC observations of the LEO satellite [23], the zero TEC assumption is not fully valid, and the actual observed minimum TEC value would include TEC with few TECU. In addition, $\text{sTEC}_{\mathrm{lqmin}}^{r}$ could be a little higher than the actual observed minimum TEC for the selecting method of lower quartile minimum value. Hence, if $-\text{sTEC}_{\mathrm{lqmin}}^{r}$ is referred to as $\text{DCB}_{0}^{q}$, it would underestimate the receiver DCB. To further improve the accuracy of DCB from the improved ZERO method, an offset value is needed to compensate the contributions from the topside ionosphere and plasmasphere and the effect of the lower quartile selecting method on the estimation of $\text{DCB}_{0}^{q}$.

Here, we describe a method to calculate the offset value based on the hypothesis as follows: 1) the daily minimum relative TEC $(\text{sTEC}_{\mathrm{dailymin}}^{r})$ is satisfied the zero TEC assumption but with deviation; 2) the deviations of $\text{sTEC}_{\mathrm{dailymin}}^{r}$ from the actual value follow the Gaussian distribution; and 3) a correction value is applicable for the whole period data. In each day, we first obtain the difference $(\mu)$ between $\text{sTEC}_{\mathrm{lqmin}}^{r}$ and $\text{sTEC}_{\mathrm{dailymin}}^{r}$ (i.e., $\mu=\text{sTEC}_{\mathrm{lqmin}}^{r}-\text{sTEC}_{\mathrm{dailymin}}^{r}$, or $\mu=\text{DCB}_{0}^{d}-\text{DCB}_{0}^{q}$, equivalently). Then, we get all daily $\mu$ when the satellite data are available. The Gaussian fitting function is used to fit the numerical distribution of the daily $\mu$ (as the histogram bar shown in the lower right corner in Fig. 2). Note that the right tail of the numerical distribution is possibly due to the outliers, and these values should not greatly affect the Gaussian fitting result. As the impact of outliers is mitigated, the fitting center $(\mu_{0})$ of the Gaussian fitting result is used to be the offset (i.e., the approximate actual observed TEC for $\text{sTEC}_{\mathrm{lqmin}}^{r}$), so (8) can be rewritten as $$\begin{equation*} \text{DCB} = \mu_{0} - \text{sTEC}_{\mathrm{lqmin}}^{r}.\tag{9} \end{equation*}$$ Equivalently, the derived DCB of our improved ZERO method is given by the following formula: $$\begin{equation*} \text{DCB} = \mu_{0} + \text{DCB}_{0}^{q}.\tag{10} \end{equation*}$$ It is worth mentioning that the same offset value $\mu_{0}$ is applied to all daily $\text{DCB}_{0}^{q}$ during the whole studying period for each LEO satellite in our study.

Fig. 2.

Variations of $\text{DCB}_{0}^{d}$ and $\text{DCB}_{0}^{q}$ representing the estimated DCBs from the daily minimum relative TEC and the lower quartile minimum relative TEC, respectively. The histogram bar of numerical distribution of their difference is displayed at the lower right corner. The green line represents the Gaussian fitting of the histogram bar, and the vertical red line represents the Gaussian fitting center $\mu_{0}$.

To explain the improved ZERO TEC method explicitly, Fig. 2 illustrates the variations of $\text{DCB}_{0}^{d}$ and $\text{DCB}_{0}^{q}$ for CHAMP during 2002–2010 and for GRACE during 2002–2013. It is clear that $\text{DCB}_{0}^{d}$ had larger noise and some of them might be the outliers, while $\text{DCB}_{0}^{q}$ was continuous and stable. For CHAMP, as shown in Fig. 2(a), the average value of $\text{DCB}_{0}^{q}$ was about −17.5 TECU. There are obvious long-term and periodic (about four-month) variations, which will be addressed in detail in Section IV. $\text{DCB}_{0}^{q}$ was about 0.5 TECU smaller than $\text{DCB}_{0}^{d}$, except during 2009–2010 when there were larger differences between them. These large differences are attributed to the low quality of data so that $\text{DCB}_{0}^{d}$ became outliers. Nevertheless, $\text{DCB}_{0}^{q}$ did not show such sharp change in 2009.

In Fig. 2(b), $\text{DCB}_{0}^{d}$ and $\text{DCB}_{0}^{q}$ of GRACE also showed long-term variations. They increased monotonously from 2002 to 2010 but started to drop after 2010. The average value for $\text{DCB}_{0}^{q}$ was about −55 TECU. During high solar activity, $\text{DCB}_{0}^{d}$ showed an obvious discontinuity. In addition, the differences between $\text{DCB}_{0}^{d}$ and $\text{DCB}_{0}^{q}$ were small during 2005–2010 when the solar activity was low, whereas they became much greater as the increase of solar activity.

From Fig. 2, it is clear that $\text{DCB}_{0}^{q}$ had a more stable variation than $\text{DCB}_{0}^{d}$ for both CHAMP and GRACE. However, as discussed previously, if $\text{DCB}_{0}^{q}$ is used to represent the LEO DCB, the derived DCB would be underestimated. We applied the Gaussian function fitting to the numerical distribution of the daily differences between $\text{DCB}_{0}^{d}$ and $\text{DCB}_{0}^{q}$ to estimate the offset, as shown in the lower right corner in Fig. 2. Therefore, the DCB from our improved ZERO method is more stable and reliable, compared with the DCB directly derived from the minimum $\text{sTEC}^{r}$. The improved ZERO method is also suitable to the other LEO satellites in our study, and the corresponding DCB results are given in Section III-C.

B. LSQ Method

The LSQ method was previously used to estimate the LEO DCB [7], [11]. The basic assumption of this method is that the vTECs from simultaneous observations with different elevation angles are equal (i.e., regional spherical symmetry ionosphere assumption). To convert sTEC to vTEC, the geometric mapping function proposed by Foelsche and Kirchengast [24] is adopted $$\begin{equation*} m(e) = \frac{\sin e+\sqrt{\left(\frac{R_{\mathrm{shell}}}{R_{\mathrm{orbit}}}\right)^{2}-(\cos e)^{2}}}{1+R_{\mathrm{shell}}/R_{\mathrm{orbit}}}\tag{11} \end{equation*}$$ where $e$ is the elevation angle of the GPS ray and $R_{\mathrm{shell}}$ and $R_{\mathrm{orbit}}$ are the ionospheric effective height and satellite orbital altitude from the Earth center, respectively. The ionospheric effective height is selected as a function of the orbital altitude of the LEO satellite according to Zhong et al. [25], which have demonstrated that the mapping function proposed by Foelsche and Kirchengast [24] along with the ionospheric effective height from the centroid method is more suitable for the LEO-based TEC conversion. Using the mapping function, the sTEC can be converted to vTEC as follows: $$\begin{equation*} \text{vTEC}^{\mathrm{abs}} = \text{sTEC}^{\mathrm{abs}} \times m(e)\tag{12} \end{equation*}$$ where $\text{vTEC}^{\mathrm{abs}}$ is the absolute vTEC. Using (7), the $\text{vTEC}^{\mathrm{abs}}$ can be rewritten as follows: $$\begin{equation*} \text{vTEC}^{\mathrm{abs}} = (\text{sTEC}^{r} + \text{DCB}_{r}) \times m(e).\tag{13} \end{equation*}$$ Then, under the regional spherical symmetry assumption, the simultaneous absolute (true) sTEC observations with different elevation angles are assumed to have the same vTEC $$\begin{equation*} \!\,\left(\text{sTEC}_{1}^{r} \!+ \!\text{DCB}_{r}\right) \!\times\! m(e_{1}) \!=\! \left(\text{sTEC}_{2}^{r}\! +\! \text{DCB}_{r}\right) \!\times\! m(e_{2}).\!\!\!\!\tag{14} \end{equation*}$$ Thousands of observational pairs during one day are used to create linear equations. Then, the receiver DCB can be obtained by solving the linear equations with the LSQ method. To evaluate the performance of the estimated DCB, the root-mean-square error (RMSE) of the result in one day is calculated as follows: $$\begin{equation*} \text{RMSE}= \frac{\sqrt{\sum\limits_{i}^{n}(\text{DCB}_{i} - \overline{\text{DCB}})^{2}}}{n}\tag{15} \end{equation*}$$ where $\text{DCB}_{i}$ is the DCB estimated by the $i$th observational pair in (12) and $\overline {\text{DCB}}$ is the least square solution using all data pairs.

In order to satisfy the regional spherical symmetry assumption, restrictions should be imposed on processed observations. The restricted conditions can be set with the elevation, vTEC, latitude, local time, and separation angle between two GNSS satellite signal rays. Data of high elevations help to mitigate the error caused by the slant to vertical conversion. The selection of latitude, local time, and vTEC values can avoid the region of the equatorial ionization anomaly or other regions with great horizontal gradient. In addition, simultaneous paired observations can become closer when the separation angle between them is smaller.

The strict restrictions can make it satisfy the spherical symmetry assumption but also significantly reduce the number of processed data. How to maintain a balance between them through the parameter setting is not well addressed. Since small vTEC corresponds to that at high latitudes or during nighttime to some extent, the restriction on vTEC should be more effective. Next, we will discuss the effect of the setting of the cutoff elevation and vTEC in detail. The DCBs from our improved ZERO method are used as references to determine whether the derived vTEC is negative or not. Subsequently, the Gaussian fitting center $(\delta)$ of the numerical distribution of the differences between the DCBs from LSQ and improved ZERO methods during the data period is calculated. Additionally, the median of daily RMSE during the data period is used as an estimation of the accuracy. Note that the averaged DCBs within ±10 days centered on each day are displayed in our plots.

Data from CHAMP and GRACE are utilized to illustrate the impact of the cutoff elevation and vTEC on the LEO DCB derivation. Fig. 3 shows the variations of the estimated DCBs for 10°, 30°, and 50° cutoff elevations. For CHAMP, the DCBs of all cases had similar periodic variations, but they were different in magnitude. The DCB of 30° cutoff elevation was usually smaller than those of 10° and 50°, and it was averagely 0.29 TECU lower than that from the improved ZERO method. Moreover, the median RMSE was lower when the cutoff elevation was smaller, and the median RMSE of 10° cutoff elevation was the smallest. For GRACE [ Fig. 3(b)], it is clear that the DCB of 50° cutoff elevation had a different periodic variation and was about 1 TECU greater than the other two cases. While the DCBs of 10° and 30° cutoff elevations were similar. Again, the RMSE was smaller as the cutoff elevation decreased.

Fig. 3.

Variations of the LEO DCBs obtained from the LSQ method for different cutoff elevations. DCB differences ( $\delta$ in unit of TECU) between the LSQ and improved ZERO methods and the median RMSE are also given in each panel. The top and bottom panels represent CHAMP and GRACE, respectively.

Fig. 4 shows the variation of estimated DCB for different cutoff vTECs. The initial DCBs from the improved ZERO method are used in (11) to constrain the processed data in the cutoff vTECs. For CHAMP, the DCBs of four cases were consistent in terms of the long-term or periodic variation, while the DCB was greater when the cutoff vTEC was lower. The RMSE also increased when the cutoff vTEC decreased. The DCB of 3-TECU cutoff vTEC had a greater RMSE than the other cases. For GRACE, it is clear that the DCB of 20-TECU cutoff vTEC was lower than the other cases. The DCBs of 3- and 5-TECU cutoff vTECs were similar, and they were averagely 0.16 and 0.07 TECU greater than that from the improved ZERO method, respectively. Also, the RMSE increased when the cutoff vTEC decreased.

Fig. 4.

Variations of the LEO DCBs obtained from the LSQ method for different cutoff vTECs. DCB differences ( $\delta$ in unit of TECU) between the LSQ and improved ZERO methods and the median RMSE are also given in each panel. The top and bottom panels represent CHAMP and GRACE, respectively.

C. DCB Variations of Multiple LEO Satellites

Using the data from CHAMP and GRACE, the effects of the cutoff elevation and vTEC are demonstrated as in the previous discussion. The combined effects of these two parameters on DCB estimation are further studied as follows with data of six LEO satellites. It is worth noting again that the $\delta$ represents the averaged DCB difference between the LSQ and improved ZERO methods and is used to determine whether the derived vTEC from the LSQ method is negative or not. The RMSE measures the differences between the estimated DCB from the LSQ method and the DCB values actually observed in each observational pair, used to assess the quality of the estimated DCB.

Fig. 5.

DCB differences ( $\delta$ in unit of TECU) between the LSQ and improved ZERO methods and the median RMSE versus cutoff elevation for different cutoff vTECs. The results for six LEO satellites are presented in this figure.

The variations of Gaussian fitting center $\delta$ and median of daily RMSE under different cutoff elevation and vTEC combinations are shown in Fig. 5. For CHAMP, the cutoff elevation of 20° or 30° gave the smallest $\delta$ value. When the cutoff vTEC was greater, the $\delta$ usually became lower. The RMSE of CHAMP was greater with the increase of the cutoff elevation. The patterns of $\delta$ and RMSE for GRACE and TerraSAR-X were similar to those of CHAMP. For the cases of SAC-C and MetOp-A, the cutoff elevation of 30° gave the smaller $\delta$. For Jason-1, both the $\delta$ and RMSE became greater as the cutoff elevation increased. The $\delta$ values of the cases of 3- and 5-TECU cutoff vTECs were obviously greater than those of 8 and 20 TECU, which suggests that the daily maximum TEC for Jason-1 is usually between 5 and 8 TECU [15]. Based on the aforementioned results, the common characteristics of cutoff elevation and vTEC setting can be summarized as follows. First, the cutoff elevation less than 20° or greater than 40° gives the greater $\delta$ value. Second, the smaller cutoff vTEC gives the greater $\delta$ value. Third, the RMSE is smaller when the cutoff elevation is lower.

The parameter setting in the LSQ method can be determined according to the aforementioned results. If the $\delta$ value is lower than 0 TECU, then it means that some of the derived TEC might be negative. The smaller cutoff vTEC usually had a greater $\delta$ value. Moreover, a small vTEC corresponds to data at high latitudes or during nighttime, which can be more in tune with the regional spherical symmetry assumption. Thus, the setting of small cutoff vTEC is suggested, although the small cutoff vTEC will significantly reduce the number of processed data.

Additionally, when the RMSE is lower, the DCB is considered better. The RMSE of the DCB decreased apparently when the cutoff elevation was smaller. Also, the setting of lower or higher cutoff elevation can give the greater $\delta$ value. Although a lower cutoff elevation will make the spherical symmetry assumption less satisfied, the setting of low cutoff elevation can be used to keep the number of data in DCB estimation. In addition, according to the work of Banville et al. [26], the leveling errors are mitigated when divided by the mapping function especially for data of low elevation. To maintain a balance between the spherical symmetry ionosphere assumption and the number of processed data, the cutoff vTEC of 3 TECU with a cutoff elevation of 10° is considered better.

Fig. 6.

Variations of the LEO DCBs from the improved ZERO method and LSQ methods for six LEO satellites. DCB differences ( $\delta$ in unit of TECU) between the LSQ and improved ZERO methods and the median RMSE are also given in each panel.

Fig. 6 shows the variations of the DCBs obtained from the LSQ method and the improved ZERO method for six LEO satellites. Two cases of the parameter settings in the LSQ method are 3-TECU cutoff vTECs with 10° and 40° cutoff elevations, respectively. The DCBs from the improved ZERO method and LSQ method were consistent. It is obvious that all of the DCBs showed a long-term variation. Periodic variation with a few months can also be clearly seen in the DCBs of CHAMP and Jason-1. The DCBs from the LSQ method with the setting of 10° cutoff elevation had a better continuity than those of 40° cutoff elevation. This might be because the cutoff elevation of 10° gives more data in DCB estimation, and thus, the obtained DCB is more stable.

Fig. 7.

Variations of CHAMP DCBs and CHAMP receiver CPU temperature. Blue dots denote the reconstructed CHAMP DCB under the zero-mean condition imposed on the GPS DCBs of 13 continuously operating satellites. The V-shaped temperature excursion of the receiver is mainly associated with the orbital phases in the periods with no eclipse (i.e., full sun orbit). See text for details.

For SAC-C and MetOp-A, the DCBs from the improved ZERO method were obviously greater than the LSQ method. Because both SAC-C and MetOp-A are sun-synchronous circular polar orbit satellites, the local times at the equator are 10:20/22:20 (09:20/21:20, after 2006) for SAC-C and 09:30/21:30 for MetOp-A. Their sampling data are mainly in sunrise/sunset time, so they are much noisier due to the sharp variation of the ionosphere. For this case, the improved ZERO method might be not reliable. Thus, both the improved ZERO method and the LSQ methods are suggested to be used together to crosscheck the obtained DCB.

SECTION IV. Long-Term and Periodic Variations of LEO DCBs

As shown in Fig. 6, the LEO DCBs underwent obvious long-term and periodic variations. In this section, we examine the possible causes for the long-term and periodic variations of the LEO DCB using the CHAMP observations.

Fig. 7 shows the variations of the CHAMP DCBs obtained from the improved ZERO method during 2002–2008. The trend of the CHAMP DCB (gray dots) was monotonously increasing from 2002 to 2008 with a cumulative increment of about 7 TECU. It is interesting to investigate whether this long-term variation of the LEO DCBs is also associated with the GPS satellite replacement. Zhong et al. [27] have assessed the long-term variation of the GPS satellite DCBs. Since only the combined satellite–receiver DCB can be actually determined, if the satellite and receiver DCBs need to be further separated from the combined satellite–receiver DCB, an additional constraint condition is required. The zero-mean condition imposed on all satellite DCBs (i.e., the daily mean of all GPS DCBs is zero) is generally introduced to separate satellite and receiver DCBs. When a new satellite replaces a decommissioned one with difference satellite type (note that the GPS DCBs of different satellite types are obviously different), the daily mean value of GPS DCBs should change. However, under the zero-mean condition imposed on all satellite DCBs, the daily mean value is still set as zero. Then, the GPS DCB values change relatively to satisfy the zero-mean condition. Thus, the long-term trend seen in the GPS DCBs is attributed to the GPS satellite replacement and the zero-mean condition imposed on all GPS satellites (please refer to [27] for more details). In order to remove this long-term variation associated with the GPS replacement, the receiver DCB can be reconstructed under the zero-mean condition imposed on continuously operating GPS satellites during data period [27]. For individual day, the mean value for the IGS DCBs of those continuously operating GPS satellites is calculated, and then, this offset is added to the original receiver DCB. We called this obtained DCB as the reconstructed DCB.

As shown in Fig. 7, the reconstructed DCB of CHAMP only showed a slight variation (less than 2 TECU) on the long-term scale. Clearly, the variations of the reconstructed DCB are in-phase with the receiver CPU temperature, which represents the hardware thermal status of the LEO satellite. Coster et al. [28] demonstrated a clear environmental temperature dependence on the ground-based receiver DCB. Our estimation indicated that a 1 °C change in the receiver CPU temperature corresponded to about 0.3-TECU change in the receiver DCB for CHAMP. Note that the DCB is usually assumed to be constant in one day; however, the receiver CPU temperature can change within one day as the LEO satellite moves fast. If the LEO satellites are equipped with excellent temperature controllers, the resultant DCB change will be reduced.

It should be noted that Yue et al. [7] and Lin et al. [29] attributed the periodic and long-term variations of CHAMP DCB to the variation of thermospheric temperature. However, the kinetic temperature of thermospheric gas should not have an obvious impact on the LEO DCB, given that the thermosphere is a very high vacuum (at the altitudes of LEO satellites). As revealed in Fig. 7, the periodic oscillations of CHAMP DCB could be associated with the changes of receiver CPU temperature. The period of the DCB is about 130 days, which is consistent with the orbital cycle of CHAMP (261 days), and the change of orbital local time of the satellite affects the hardware thermal status.

After the effect of the GPS replacement is removed, the long-term variation can be still seen in the reconstructed DCBs of other LEO satellites, especially for GRACE and Jason-1 (not shown). For GRACE, it dropped during 2002–2004 with 1 TECU and increased during 2004–2013 with 2.5 TECU. For Jason-1, it decreased during 2002–2006 with 4 TECU and then increased from 2006 to 2009 with 1.5 TECU. Nevertheless, the reconstructed DCB of MetOp-A remained the same in the long-term scale. Also, periodic variations can be seen in other LEO satellites, but they are less evident compared with that of CHAMP. This might be associated with the fact that the local times remain unchanged in the sun-synchronous circular polar orbit for MetOp-A. In particular, the orbital cycle of GRACE is similar to that of CHAMP, but their periodic variations are different, which is suggestive of a better thermal controller for the GRACE receiver. It is interesting to further investigate whether the changes of satellite hardware thermal status contribute to the long-term and periodic variations of the DCBs for other LEO satellites if the receiver temperature data for other satellites become available.

SECTION V. Concluding Remarks

In this paper, we have proposed an improved ZERO method to estimate the LEO DCB by combining the lower quartile minimum relative TEC during each orbital revolution with the minimum relative TEC in one day. This improved ZERO method makes LEO DCBs more stable and reliable as compared with the results calculated from the minimum relative TEC only. We have also discussed the sensitivity of the LSQ method to cutoff elevation and vTEC. The LEO DCBs obtained from the LSQ method depend greatly on the cutoff elevation and vTEC. In principle, higher cutoff elevation and smaller cutoff vTEC could give a better estimation for the LEO DCB. Our results demonstrated that, due to the limited observations, the combination of 3-TECU cutoff vTEC and 10° cutoff elevation is suitable for the LEO DCB derivation. Furthermore, the LSQ method is recommended to estimate the LEO DCB if the quality of the LEO satellite data is not high, albeit the improved ZERO method generally gives reliable results.

The calculated LEO DCBs of multiple satellites obtained from both the improved ZERO and LSQ methods showed obvious long-term and periodic variations. In this paper, we have explored the causes for the variations of CHAMP DCB. Our preliminary result revealed that the long-term variation of CHAMP DCB is generally associated with the GPS satellite replacement, while the periodic variation is mainly attributed to the variation of the CHAMP hardware thermal status. Definitely, further investigation is required to address the causes responsible for the long-term and periodic variations of the DCBs for other LEO satellites.

The DCB should be estimated accurately in order to process high-precision vTEC, given that the LEO-based TEC is usually much smaller than the ground-based TEC. Our results demonstrate the potential to further enhance the reliability of the LEO DCB and to provide the high-accuracy LEO-based TEC in space weather application. In the future, these LEO-based TEC observations can be utilized to explore the climatology of the topside ionosphere and the plasmasphere, and the topside ionospheric response to space weather events.

ACKNOWLEDGMENT

The satellite data were provided by UCAR CDAAC ( http://cdaac-www.cosmic.ucar.edu/cdaac/) and NASA ( ftp://podaac-ftp.jpl.nasa.gov/), and the CHAMP GPS receiver CPU temperature data were provided by J. Wickert and W. Koehler from GFZ.