Satellite dependency of GNSS phase biases between receivers and between signals

Transcription

1 J. Geod. Sci. 2017; 7: Research Article Open Access M. Håkansson* Satellite dependency of GNSS phase biases between receivers and between signals Received May 11, 2017; accepted October 3, 2017 Abstract: The existence of hardware induced phase biases might influence the accuracy in precise positioning if not handled properly. This is extra problematic if the biases are dependent on the satellite tracked, as these biases no longer will be common between the satellites, and thus will not be absorbed by the receiver clock term of the positioning solution. In this paper, we carried out two studies to investigate whether there exists a satellite dependency of the relative phase biases. Even though small in size, satellite dependent variations were found in both cases. In the first case, relative receiver phase biases were studied, while relative phase biases between signals (e.g. between carrier phases from C/A-code and P-code tracking) were investigated in the second case. The biases in the first case had a size of 0.8 mm between the satellites with the largest and smallest values, and additionally showed temporal variations that were consistent over time. The corresponding sizes of the biases second case were 2 mm and 3.5 mm for GPS L1 and L2 respectively, and no temporal variations were found. Keywords: GNSS, hardware delays, phase biases, satellite dependency 1 Introduction Global Navigation Satellite System (GNSS) hardware biases appear as a result of differences in hardware designs and signal structures. These biases affect the code and phase measurements, and might influence the accuracy of GNSS positioning and timing if not properly accounted for. For a review of code and phase biases in multi- GNSS positioning the reader is referred to Håkansson, et al. (2017). *Corresponding Author: M. Håkansson: KTH Royal Institute of Technology Stockholm, Swede, martin.hakansson@lm.se Hardware induced biases are present in both code and phase measurements. In the case of code biases, they may for instance occur between the transmission and the reception of codes at the satellite and the receiver, respectively. This kind of bias is commonly referred to as differential code biases (DCBs) (Montenbruck, et al. 2014) or group delay differentials (IS-GPS-200H 2013). In positioning with GPS this bias needs to be considered in all cases where the ionosphere-free linear combination of P-code on L1 and L2 are not employed (IS-GPS-200H 2013). This bias is also important in timing and GNSS based modeling of the ionosphere (Jensen, et al. 2007, Lanyi, et al. 1988, Sardon, et al. 1997). Other instances where code biases need to be considered are positioning involving Globalnaya Navigatsionnaya Sputnikovaya Sistema (GLONASS) (Chuang, et al. 2013) and positioning involving multiple GNSS systems together (Odijk, et al. 2012, Paziewski, et al. 2014). The position accuracy may also be affected by phase biases, which is the primary topic of this paper. Phase biases are especially problematic for precise positioning, which is heavily dependent on the phase observables to achieve its high level of accuracy. Hardware induced phase biases exist both at the receiver and satellite side. At the receiver side they affect precise positioning with GLONASS via inter-frequency biases (IFBs) (Leick, et al. 1998, Raby, et al. 1993, Wanninger, et al. 2007), and precise positioning with multiple GNSS systems via inter-system biases (ISBs) (Odijk, et al. 2012, Paziewski, et al. 2014), while phase biases at the satellite side are most problematic when trying to resolve the phase ambiguity as an integer in Precise Point Positioning (PPP) (Teunissen, et al. 2014). It should be noted that biases with a satellite dependency will have an effect on the estimated position, while biases that are common for all satellites only will affect estimation of the receiver clock error. Of the receiver biases with a satellite dependency, the GLONASS IFB is most notable and affects all positioning methods where GLONASS is employed. This bias occurs because GLONASS has adopted Frequency Division Multiple Access (FDMA), where different satellites are distinguished with unique carrier frequencies. At the receiver, the delays in the receiver channels are dependent on the Open Access M. Håkansson, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution- NonCommercial-NoDerivs 4.0 License.

2 Satellite dependency of GNSS phase biases between receivers and between signals 131 carrier frequency, which in fact leads to biases with a satellite dependency. Receiver phase biases for GNSS systems employing Code Division Multiple Access (CDMA) are often assumed to be satellite independent, meaning that they are assumed to be the same for all tracked satellites for a given signal. This assumption is based on the fact that the double differenced phase ambiguity most often can be resolved as an integer, which increases the accuracy of the positioning solution, also between receivers of different type. This observation, however, only asserts that receiver phase biases are similar between satellites to a degree that allows integer resolution of the phase ambiguity together with an uncertainty estimate of the positioning solution. This paper investigates and characterizes the potential satellite dependency of receiver phase biases that may exist as an effect of signal distortions or other effects that are associated with individual satellites. Signal distortions that arise at transmission from the satellites have shown to be problematic for the code observables, as these lead to receiver code biases that exhibit a satellite dependency (Hauschild, et al. 2016, Lestarquit, et al. 2012). The reason for this is that the signal distortions tend to be specific for each satellite. As a result, the signal distortions induce code biases in the signal processing of the receiver that are specific for each satellite. Receiver code biases due to signal distortions affect positioning for both FDMA and CDMA based satellite positioning systems. As these biases are satellite-dependent they will have an effect on the positioning accuracy, and can therefore be of significant influence in PPP and relative positioning (Hauschild, et al. 2016). It is not clear whether signal distortions also cause receiver phase biases that are satellite-dependent, but as phase tracking depends on the results from the code tracking loop (Enge, et al. 2006), the existence of such a connection would not be inconceivable. To the author s knowledge such investigation has not previously been performed. The investigation is crucial as a satellite dependency would affect the accuracy in ambiguity resolved relative positioning if the satellite dependency also varies between receivers of different type. It can further lead to contamination of the satellite phase bias corrections distributed for PPP with ambiguity resolution, and thereby indirectly influence the accuracy of this positioning technique. Even though we do not know any studies where the satellite dependency of receiver biases is investigated specifically, studies exist where receiver phase biases are studied in general. For instance, in (Banville, et al. 2008) a GPS signal generator is used for the estimation of receiver code and phase biases. In this study, a small discrepancy between receiver phase biases of different satellites are observed that are assumed to be channel-dependent. Any further investigation of these discrepancies is however not performed. The paper will also investigate phase biases between signals of the same carrier frequency (e.g. between carrier phases from C/A-code and P-code tracking). These may originate both from the receiver and from the satellite hardware, and could potentially be problematic in relative positioning between receivers employing different modes of tracking. 2 Method To determine receiver phase biases, we connected a number of receivers in a zero-baseline configuration. In addition to being connected to the same antenna, the receivers were also connected to a common external rubidium oscillator. With this setup, most error sources, such as atmospheric errors and satellite clocks, will be common between the receivers and thereby cancel out when forming single differences. The error sources that remain are the relative receiver clock error, the phase ambiguities, and possibly residuals of multipath between the receivers. Thus, the receiver biases to be investigated can in this way be isolated from most other error sources. In theory, one could also assume that the drift of the receiver clocks should be identical as they are corrected from the same external frequency source. This would lead to a relative receiver clock error that would remain constant over time. In reality however, as will be clear later, in this experiment the receivers tended to slightly deviate from their external frequency input. In order to distinguish actual biases from local effects, such as multipath, the experiment was performed on two different sites: a main site, at the headquarters of Lantmäteriet in Gävle; and a secondary site in Mårtsbo, which is located in the outskirt of Gävle, approximately 10 km from the main site. The experimental setup of the main and the secondary site are depicted in Figure 1 and 2, respectively. Figures 3 and 4 shows photos of the antenna sites in Gävle and Mårtsbo. We characterized receiver phase biases for three different receiver types: Javad TRE_G3T Sigma, Trimble NetR9, and Septentrio PolaRx5. As only single differenced biases can be estimated in the described setup, two receivers from each of the examined types were used to get bias references between receivers of the same type in the experiment. At the main site in Gävle, all six receivers

3 132 M. Håkansson Figure 1: Zero-baseline setup at the main site in Gävle Figure 3: Photo of the antenna site in Gävle 2.1 Estimation of satellite dependent receiver phase biases In the first part of the experiment, single differenced (SD) receiver phase biases were estimated. The purpose of the first part was to investigate whether a satellite dependency existed in the receiver phase biases, and thus one phase bias parameter per satellite was estimated. A slightly modified version of the phase observation equation that can be found in Hofmann-Wellenhof, et al. (2008) is Figure 2: Zero-baseline setup at the secondary site in Mårtsbo were connected to the same choke ring antenna of type, JAVRINGANT_DM. Also at the secondary site a choke ring antenna of the same type was used. Receiver biases have shown a dependency on the receiver firmware (Wanninger 2011). It is however not the purpose of this study to examine how biases vary in relation to various receiver firmwares. Nonetheless, as the firmware certainly affects some of the results in this study, they are listed for each one of the receivers in Table 1. In the case of the Javad TRE_G3T Sigma receivers, the user has the option to use either the feature stable or the latest version of the firmware. In this experiment, all receivers had the feature stable firmware versions installed. Additionally, Table 1 defines a designated label for each one of the receivers. Henceforth, receivers mentioned will be referred to using this label. Φ s r = 1 ( ρ s λ r + Tr s Ir s ) ( +f δr δ s + φ r φ s + b s r b s) +Nr s +ε (1) It has been modified to also include the receiver and satellite bias terms b s r and b s where the receiver and satellite dependencies are denoted r and s. As can be noted, the receiver bias term is not assumed to be independent of the satellite tracked and it is therefore also labeled by s. The other terms in the equation are the geometric range ρ s r between the receiver and the satellite, the tropospheric error Tr s, the ionospheric error Ir s, the receiver and satellite clock error δ r and δ s, the initial receiver and satellite phases φ r and φ s, and the integer valued phase ambiguity Nr s. ε contains noise and the remaining error sources, including multipath. The carrier wavelength and frequency are denoted λ and f, respectively. By single differences, error sources that are common between the receivers can be eliminated. The single differenced phase observation equation, derived from Eq. (1), which are formed from observations of the receivers A and B with a zero-baseline receiver setup, thus becomes Φ s AB = f ( δ AB + φ AB + b s AB) + N s AB + ε AB (2)

4 Satellite dependency of GNSS phase biases between receivers and between signals 133 Table 1: Equipment used in the zero-baseline experiment Site Receiver type Receiver firmware Label Javad TRE_G3T Sigma (Stable) J1 Javad TRE_G3T Sigma (Stable) J2 Gävle Trimble NetR T1 Trimble NetR T2 Septentrio PolaRx S1 Septentrio PolaRx S2 Mårtsbo Javad TRE_G3T Sigma (Stable) J3 Trimble NetR T3 clocks against drift in relation to some proper time frame. It is thus in this case generally true that δ AB = 0, while δ AB 0. Equation (2) is constructed for all satellites. As this equation system is rank deficient, it has to be reparametrized in order to reduce the number of unknowns. In the following, two approaches for estimation of relative phase biases will be introduced. They differ in whether they regard relative clocks and phase biases as constant over time or not. 2.2 Estimation of relative phase biases without temporal variations Figure 4: Photo of the antenna site in Mårtsbo where A, B {J1, J2, J3, T1, T2, T3, S1, S2}, A = B, according to Table 1. The terms common between the receivers that have been canceled out are the geometric range, the satellite clock error, the satellite phase bias, the initial phase at the satellite, the tropospheric error, and the ionospheric error. The remaining terms are the between receivers relative clock error δ AB, the relative initial phase of the receivers φ AB, the relative receiver phase bias b s AB with its satellite dependency denoted by s, the integer phase ambiguity NAB s, and ε AB containing relative noise and un-modeled error sources such as multipath residuals. It is not an uncommon misconception that connecting two receivers to the same external frequency input will cancel out the relative clock error δ AB between them. The reason why this is not true is that the external frequency does not provide absolute time, but only stabilizes the The observations were first analyzed with the assumption that the relative clocks and phase biases are constant over time. This is a probable assumption as all the receivers are connected to the same external oscillator, and the biases furthermore might be assumed to be constant over time. With this assumption, relative clocks and biases will be estimated as one single value for all epochs. Equation (2) can in this case be re-parametrized with the following substitutions. Φ s AB = δ AB + Ñ s AB + ε AB (3) where and δ AB = f ( ) δ AB + φ AB + b 1 AB + NAB 1 (4) Ñ s AB = f ( δ AB + φ AB + b s AB) + N s AB f ( δ AB + φ AB + b 1 AB) N 1 AB = f ( b s AB b1 AB) + N s AB N 1 AB (5) In cases when a satellite dependency of the receiver phase biases exists, the term f ( b s AB b1 AB) will not generally be equal to zero. This means that neither will Ñ s AB

5 134 M. Håkansson be exactly an integer. It should be noted that the reparametrization of Eq. (2) leads to ÑAB 1 = 0, and thus eliminates the rank deficiency of this equation. Equation (3) can thereby be solved in a least squares sense over one or several epochs of observations. According to Eq. (5), ÑAB s is the relative phase bias in cycles shifted by an ambiguous integer NAB s N1 AB. According to the earlier reasoning that satellite dependent variations are small if they even exist, the relative phase biasf ( b s AB AB) b1 is here expected to assume either a positive or negative number much closer to zero than any other positive or negative integers. We can thereby also expect ( ) f b s AB b 1 AB = ÑAB s ÑAB s (6) where ÑAB s = Ns AB N1 AB, to be true. In this equation, is the rounding to nearest integer operator. The final relative phase bias estimates ˆb s AB were calculated from Eq. (6) and centered around zero (instead of letting ˆb 1 AB = 0) by subtracting the common mean value according to ˆb s AB = Ñ s AB Ñ s AB 1 n n [Ñi AB ÑAB ] i i=1 where n is the total number of satellites. 2.3 Estimation of relative phase biases with temporal variations As will later be apparent, the assumptions about constant relative clock errors and biases are not true at the levels of precision where phase bias satellite dependencies can be observed. In order to investigate the temporal characteristics of the relative phase biases, a Kalman filter was designed. In this filter, the dynamics of both relative clock error and biases where modeled as random walk parameters, i.e. the filter s transition matrix is equal to the identity matrix. All observations were pre-processed to remove ambiguity differences and potential cycle slips between different epochs of observations. The ambiguity differences and potential cycle slips were removed by Φ s AB(t) = Φ s AB (t) + Φ1 AB (1) Φs AB (t) = Φ s AB (t) + N1 AB (1) Ns AB (t) = f ( δ AB (t) + φ AB + b s AB (t)) + N 1 AB (1) + ε AB(t) where Φ s AB (t) is the SD phase at time t given by Eq. (2), and f ( b 1 AB AB) bs is assumed to be much closer to zero than any other positive or negative integers. (7) (8) Eq. (8) is rank deficient and thus needs to be reparametrized to be solvable. Also, if the equation were to be reparametrized into one relative clock error and one relative phase bias per satellite it would be rank deficient. Thus, an extra constraint is needed. If one considers for some value v(t) f n b s AB(t) nv(t) = 0 (9) i=1 where n is the total number of satellites, Eq. (8) can be re-parametrized as where and Φ s AB(t) = δ AB (t) + b s AB(t) (10) δ AB (t) = f ( δ AB (t) + φ AB ) + N 1 AB (1) + v(t) (11) b s AB(t) = fb s AB(t) v(t) (12) The observation model for each epoch t then becomes Φ 1 AB(t) = δ AB (t) + b 1 AB (t) Φ 2 AB(t) = δ AB (t) + b 2 AB (t). Φ n AB(t) = δ AB (t) + b n AB (t) 0 = n b i AB (t) i=1 (13) where the last equation is an additional pseudoobservation constraining the mean value of the biases to zero. Because of this constraint, the estimated parameters δ AB (t) and b s AB (t) will be biased by the unknown quantity v(t) according to Eq. (11) and (12). The measurement covariance matrix used in the Kalman-filter was derived from measured uncertainties of the SD observations, while the covariance matrix of the filter itself derived from a trial and error approach. 2.4 Estimation of satellite dependent phase biases between different phase observables In the second part of the experiment the satellite dependency of phase bias differences between different phase observables S 1 and S 2 were investigated. The differences were formed between observables from the same receiver and associated with the same carrier frequency.

6 Satellite dependency of GNSS phase biases between receivers and between signals 135 S 1 and S 2 can for instance refer to the L1 carrier measurements from GPS associated with C/A-code and P-code, respectively. The differenced observation equation derived from Equation (1) becomes Φ s S 1 S 2,r = f ( φ S1 S 2,r φ s S 1 S 2 + b s S 1 S 2,r b s S 1 S 2 ) +N s S1 S 2,r+ε S1 S 2 (14) where all error sources, except the initial phases and the biases at the satellite and the receiver, cancel out. The relative initial phases from the receiver and the satellite are remaining in the equation as it was not clear whether this term would cancel out or not. In theory, one could expect these terms to be equal to zero as both carrier signals are tracked by the same receiver relying on the same internal or external clock, while at the satellite side the same frequency source is used when generating the signals. Equation (14) was re-parametrized into where Φ s S 1 S 2,r = b s S 1 S 2,r + N s S 1 S 2,r + ε S1 S 2 (15) b s S 1 S 2,r = f ( φ S1 S 2,r φ s S 1 S 2 + b s S 1 S 2,r b s S 1 S 2 ) (16) In contrast to the between-receivers biases previously mentioned, the term b s S 1 S 2,r in Eq. (15) does not necessarily originate from the receiver, but can also be the cause of biases at the satellite side. Equation (15) also includes the phase ambiguity to be handled. With the assumption that b s S 1 S 2,r is much smaller than the carrier wavelength this is easily done by We can thus consider Φ s S 1 S 2,r = Φ s S 1 S 2,r Φ s S 1 S 2,r (17) Φ s S 1 S 2,r = b s S 1 S 2,r + ε S1 S 2 (18) to be true. This equation system is non-singular and thereby solvable in a least squares sense for one or several epochs of observations. Also, these bias estimates were adjusted with their average value to increase their comparability between different receivers. The estimated bias quantity was thus determined as ˆb s S 1 S 2,r = b s S 1 S 2,r 1 n b i S n 1 S 2,r (19) i=1 where n is the total number of satellites. 2.5 Separation of receiver phase biases from other error sources It is our hypothesis that if a satellite dependency of receiver phase biases exists, it might be a consequence of signal distortions that are produced in the satellite payload at transmission. The reasoning behind this is that a systematic satellite-dependent effect would originate from some property of the received signal that is typically produced by the transmitting satellite. Effects that influence the signals after transmission from the satellite can thereby be ruled out as these would vary randomly with regards to which satellite is transmitting. This includes ionospheric scintillation, which also could affect single differenced zero-baseline observations. The remaining possible effects at transmission can be divided into designdependent properties that are unique for each satellite, and unintentional properties that are characteristic for a specific satellite. Of the design-dependent properties only the pseudo-random noise (PRN) codes are consistent over time. Of the unintentional properties, only signal distortions that arise at the satellite would be characteristic. Of these two, the latter has already shown to give rise to satellite-dependent receiver code biases (Hauschild, et al. 2016, Lestarquit, et al. 2012). Under the assumption that satellite-dependent receiver phase biases exist as a result of signal distortions, it is important to separate between signal distortions that arise in the satellite and distortions that arise locally in the vicinity of the receiver. Mulitpath is a local error source that distorts the received signals and causes errors in the code and phase observables. In our investigation of satellite-dependent receiver phase biases, we needed a way to rule out the effect of multipath in the estimation process. To achieve this, the experiment was also performed at a secondary site with a different multipath signature. The idea was that similar results from both sites would ensure that multipath was not a dominant effect on the phase bias estimates. It should be noted that both the sites in Gävle and Mårtsbo are built with the purpose to optimize the conditions for collection of GNSS measurements. This means that both antennas are mounted on high locations where the expected impact of multipath are low. The use of choke ring antennas on both sites further reduces the influence of multipath and other error sources on the collected measurements. 2.6 Data processing GNSS observations were logged in the memory on each of the receivers included in the experiment. The data was stored in each receiver s proprietary binary format and later converted to the Receiver Independent Exchange Format (RINEX), version 3, with a logging interval of one second. The full range of signals supported by RINEX 3 was

7 136 M. Håkansson Figure 5: Hourly phase biases of various receiver combinations at DOY 2 thereby available for analysis. The first part of this study will however be limited to L1C (carrier phase of L1 C/A code tracking). The later part analyzes civil and encrypted code on both L1 and L2. Both parts are delimited to GPS only. After extraction and conversion of the logged data into RINEX, all processing was performed using scripts developed in the Matlab software. 3 Results 3.1 Satellite dependency of SD receiver phase biases SD receiver phase biases were calculated for phase observations collected at the main site in Gävle for day of year (DOY) by solving Eq. (3) in a least squares sense. The observations were collected with a logging interval of one second, and one receiver phase bias was calculated with the observations from each of the hours during the day. Figure 5 shows the estimated SD phase biases with the integer part removed and adjusted by the common mean in accordance with Eq. (7). For the sake of brevity only results from four hourly estimates of DOY 2 are shown. These diagrams show results that are representative for every hour of the day. Results are shown for each of the five receiver combinations: Trimble NetR9 -Trimble NetR9, Javad TRE_G3T Sigma -Javad TRE_G3T Sigma, Septentrio PolaRx5 - Septentrio PolaRx5, Trimble NetR9 - Javad TRE_G3T Sigma, Trimble NetR9 - Septentrio PolaRx5. In Figure 5 the listed receivers are denoted by the labels as provided in Table 1. In the diagrams of Figure 5, three major observations can be made: 1. The variations of the SD receiver phase biases between satellites tend to be very small, all variations are below 0.8 mm between the largest and smallest values. These variations will thereby have a negligible effect on the positioning accuracy in relative positioning and PPP with ambiguity resolution, also when receivers of different types are employed. In a majority of applications these bias variations can be ignored without causing any serious effect on the end-results. 2. The bias variations of combinations consisting of different receiver types tend to be larger than those of combinations consisting of the same types. This observation supports the notion that there actually exist receiver phase biases with a satellite-dependency typical for a specific receiver architecture, as systematic satellite-dependent variations would cancel out between receivers with the same architecture and thus show lower variations. Table 2 shows the standard deviations for each receiver combination estimated from the hourly bias estimates themselves presented in Figure 5, not to be confused with posteriori standard deviations from the solution presented as error bars. The receiver combinations T1-J1 and T1-S1, which consist of receivers of different types, clearly show larger variations than the receiver combinations of the same types, which is also obvious from Figure 5 itself. The columns 6 and 8 shows a theoretical estimate of the expected standard deviation for the receiver combinations T1-J1 and T1-S1 if there would not exist any correlation between receivers of the same type in columns 2-4. As can be observed, most of the actual sizes of the receiver combinations T1-J1 and T1-S1 in columns 5 and 7 are several times larger than the theoretical estimate. Some of the theoretical values for the combination T1-J1 shows large values due to individual outliers for the receiver combination J1-J2. As the theoretical values differ greatly from the actual values for the receiver combinations T1-J1 and T1-S1, it can be concluded that there exists a satellite-dependency of receiver phase biases and that these are characteristic for a specific receiver architecture. The satellitedependency of receiver phase biases is thereby only revealed when SDs are formed between different receiver types. 3. Relative between-satellites phase bias values for SD of different receiver types seem to differ between satellite passes, i.e. the satellite-dependent phase bias varia-

8 Satellite dependency of GNSS phase biases between receivers and between signals 137 tions seem to vary over time. This might for instance be observed for the differences between satellite PRN 10 and 28 for the hours 06:00-07:00 and 18:00-19:00 for the combination T1-J1. This property will be further examined in the next section where the temporal characteristics of the biases are investigated. day solution between DOY 2-4 for the same receiver combinations as in Figure 5. This diagram shows that the variations of relative phase biases with combinations of mixed receiver types have variations that are significantly larger than those of the same-type-combinations. It is however impossible to distinguish individual satellites from this diagram as all satellites use the same colors. Figure 7: Temporal variations for all GPS satellites for various receiver combinations Figure 6: Residuals over time DOY 2-4 for phase biases T1-J1 As was mentioned earlier, even though the receivers were connected to the same external oscillator a drift of the relative clock error between the receivers was observed. This drift becomes more obvious over longer time periods. Over a 3-day solution, where the estimation method of constant-over-time phase biases was employed, this drift revealed itself as a trend in the residuals of the solution. Figure 6 shows the residuals of the 3-day solution with the receiver combination T1-J1 between DOY The drift of the residuals here is obvious; the red lines are regression lines for each one of the satellites. They show that the drift is similar for all satellites, which indicates that the drift is related to the relative clock error. It is thus clear that at least one of the receivers deviates from its external frequency input. It is however, from the setup of the experiment with relative estimates, impossible to determine which one of the receivers that would be Temporal characteristics of relative receiver phase biases To characterize temporal variations as those shown in Figure 5 and 6 for relative phase biases and clock errors respectively, a Kalman filter was designed according to the description given earlier in this paper. Figure 7 shows a 3- Figure 8: Temporal variations for 3 representative GPS satellites for various receiver combinations To get a clearer picture of the temporal variations for individual satellites, the relative phase biases of individual satellites are shown in Figure 8. Again, the same receiver combinations are shown as in Figure 5 and 7. For the sake of brevity only the diagrams for three satellites were included in this paper. They were chosen as they show temporal variations that are representative for all of the GPS satellites observed during the three days. Two observations can be made from Figure 8:

9 138 M. Håkansson Table 2: Standard deviations in millimeters of receiver phase biases for various receiver combinations Time (UTC) σ T1 T2 σ J1 J2 σ S1 S2 σ T1 J1 σ T1 T2 2 + σ J1 J2 2 σ T1 S1 σt1 T2 2 + σ S1 S2 2 00:00-01: :00-07: :00-13: :00-19: The temporal variations for mixed receiver combinations seems to repeat themselves every sidereal day. It is likely that this is somehow connected to fact that the GPS satellite geometry repeats every sidereal day. 2. The temporal variations show similar patterns for all 3 satellites. As a result of how satellites emerging above the elevation mask are handled, all phase bias estimates start at zero. It then takes the filter a short duration before the estimated biases adjust to their actual values. The estimated phase biases for the receiver combinations T1-J1 and T1-S1 in most cases appear as monotonically decreasing functions over time. The opposite will of course be true, with monotonically increasing functions, if the single differences are opposite, i.e. the combinations were J1-T1 and S1-T1 instead. To ensure that the results in Figure 5, 7, 8 and Table 2 were the results from phase biases alone, and not affected too much by other error sources and especially local multipath, some further investigations were performed. When forming SDs of the receivers in a zero-baseline setup most of the error sources will cancel out as they will be common to the receivers. However, the handling and impact of multipath errors differs between different receiver architectures both as a result of differing multipath mitigation techniques but also as a result of the discriminator design. It has furthermore been shown that various multipath mitigation techniques might amplify the effect that signal distortions originating from the satellite payload induce on the code error (Simsky, et al. 2004). In this study, all multipath mitigation options were disabled in the receivers to eliminate potential multipath mitigation side effects. Multipath is a phenomenon that repeats itself along with the satellite geometry if the environment remains unchanged, and in the case of GPS, the repetition occurs every sidereal day (Ragheb, et al. 2007). This corresponds well with the observations made in Figure 8, and it could therefore be a potential cause for the phase bias variations. We thus investigated further whether different environments affected the estimation of SD receiver phase biases in different way, as the multipath effect would be different in different environments. The environments for the antenna sites in Gävle and Mårtsbo are shown in Figure 3 and 4 respectively. Figure 9 shows the SDs phase bias time series estimates for three representative satellites with the receiver combination Trimble NetR9- Javad TRE_G3T Sigma for three receiver pairs located in Gävle and Mårtsbo. The estimates were calculated from observations collected between DOY 2-4, As can be observed in these diagrams, similar results are shown for all three receiver pairs, and it is also obvious that the deviations between the receiver pairs in Gävle and Mårtsbo are not bigger than between the two receivers located in Gävle. This indicates that multipath does not have a dominating effect on the phase bias estimates. 3.2 Handling of multipath Figure 9: Temporal variations for 3 GPS satellites for the receiver combination Trimble-Javad at Gävle and Mårtsbo

10 Satellite dependency of GNSS phase biases between receivers and between signals Satellite dependent phase biases between different phase observables In the second part of the experiment, phase biases between different phase observables were estimated in accordance with Eq. (18) and (19). Figure 10 shows two diagrams with phase bias estimates between the signals L1C (carrier phase of C/A-code on L1) and L1W (carrier phase of P-code on L1), and between the signals L2C (carrier phase of C-code on L2) and L2W (carrier phase of P-code on L2). The first diagram shows the relative phase biases for L1 of one of the Javad TRE_G3T Sigma receivers in Gävle and in Mårtsbo. This estimate was unfortunately not possible to calculate for the Trimble receivers on L1, as these do not track P-code for this carrier frequency. The second diagram shows the corresponding L2 relative biases both for Javad TRE_G3T Sigma and Trimble NetR9 on the Gävle and Mårsbo sites. It is clear from these diagrams that all the receivers show similar values for the estimated biases, regardless of which site they are located on and regardless of the receiver type. It can thereby be concluded that local effects from the environment does not have a dominating effect on the results. It is also apparent that these biases show larger values than for the first part of the experiment, with variations as large as 2 mm and 3.5 mm between the greatest and smallest values on the L1 and L2 frequencies. For the SD biases estimated in the first part, the size of variations was about 0.8 mm between the greatest and the smallest values. Figure 10: Javad and Trimble phase biases between signals on L1 and L2 on DOY 2 in Gävle and Mårtsbo It is also clear in Figure 10 that both receiver types show very similar biases on the L2 frequency. This indicates that these biases to a large extent are a result of delays occurring at transmission from the satellites as both receiver types experience the same bias variations between the satellites. From Eq. (16) we can thus consider both the receiver terms φ S1 S 2,r and to be close to zero. Contrary to the phase biases investigated in the first part of the paper, which were related to the relative receiver bias, the Figure 11: Sigma phase biases between signals L2C and L2W on DOY 2 and 3 in Gävle biases in this part mainly originate from the satellite hardware captured in the initial phase φ s S 1 S 2 and the bias b s S 1 S 2 terms. To assess the bias stability, biases of the L2C-L2W difference of one of the Javad TRE_G3T Sigmas in Gävle were estimated for both DOY 2 and DOY 3, The results are depicted in Figure 11 and show similar values for both days. This indicates that the biases are stable over a time period of 24 hours. 4 Conclusions In this study the satellite dependency of relative phase biases between receivers and between signals were investigated. In the first part, it was shown that a satellite dependency on receiver phase biases exists. Although for the receiver combinations investigated, this bias was small enough to be ignored in most applications, as the maximum variations were below 0.8 mm. The satellite dependency only appeared between receivers of different types which indicates that this is a bias that is characteristic for the receiver architecture. It was furthermore shown that this bias was varying periodically over the day. Multipath could furthermore be excluded as a source for the periodic phase bias variations as the two sites Gävle and Mårtsbo showed similar results. This however does not exclude that the phase bias variations somehow are related to the variations of the GPS satellite geometry, which also has a period of one sidereal day. The results from this study alone is not enough to determine the cause for these variations. One possibility discussed in the paper is signal distortions, which have already shown the ability to affect code mea-

11 140 M. Håkansson surements in other studies. The temporal variations would in this case be possible to explain if the distortions also had a nadir dependency. Another plausible explanation would be that the phase bias variations are somehow related to the topocentric range rate and the Doppler tracking in the receivers. This corresponds well with the results in Fig. 8 which show the relative phase biases to be monotonically decreasing or increasing functions during the time the satellites are visible and tracked by the receiver, which is also the case with the topocentric range rate. Other causes are of course also possible. In the second part, the satellite dependency of relative phase biases between signals was investigated. On Javad TRE_G3T Sigma the satellite dependency of relative biases between carrier phase from C/A-code and P-code tracking for L1, and civil code and P-code on L2 were investigated. For Trimble NetR9 relative biases between civil code and P-code on L2 were investigated. These biases showed intersatellite variations with maximum of 2 mm and 3.5 mm between the greatest and smallest values for L1 and L2, respectively. For L2 tracking, the biases of the Javad TRE_G3T and Trimble NetR9 turned out to be similar which indicates that these biases are caused by delays occurring at the satellite at transmission. This bias will for instance influence the accuracy in relative positions if different tracking modes are employed in the reference station and user receivers. However, this bias was shown to be stable over time which opens up the opportunity to calibrate for this in advance. Acknowledgement: This study is included as a part of the author s Ph. D. education, which is financed by Lantmäteriet - The Swedish Mapping, Cadastral and Land Registration Authority. Further acknowledgments are given to Anna B. O. 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