# Triple-Frequency GPS Precise Point Positioning Ambiguity Resolution Using Dual-Frequency Based IGS Precise Clock Products

Authors: Fei Liu; Yang Gao
Journal: International Journal of Aerospace Engineering
 (2017)

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## Abstract

With the availability of the third civil signal in the Global Positioning System, triple-frequency Precise Point Positioning ambiguity resolution methods have drawn increasing attention due to significantly reduced convergence time. However, the corresponding triple-frequency based precise clock products are not widely available and adopted by applications. Currently, most precise products are generated based on ionosphere-free combination of dual-frequency L1/L2 signals, which however are not consistent with the triple-frequency ionosphere-free carrier-phase measurements, resulting in inaccurate positioning and unstable float ambiguities. In this study, a GPS triple-frequency PPP ambiguity resolution method is developed using the widely used dual-frequency based clock products. In this method, the interfrequency clock biases between the triple-frequency and dual-frequency ionosphere-free carrier-phase measurements are first estimated and then applied to triple-frequency ionosphere-free carrier-phase measurements to obtain stable float ambiguities. After this, the wide-lane L2/L5 and wide-lane L1/L2 integer property of ambiguities are recovered by estimating the satellite fractional cycle biases. A test using a sparse network is conducted to verify the effectiveness of the method. The results show that the ambiguity resolution can be achieved in minutes even tens of seconds and the positioning accuracy is in decimeter level.


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## Body

## 1. Introduction

With precise satellite orbit and clock products, Precise Point Positioning (PPP) using ionosphere-free (IF) code and carrier-phase observations can achieve centimeter-level accuracy if advanced error calibration models are applied [1]. The main disadvantage of PPP is that it needs significant time to reach convergence. Fast ambiguity resolution (AR) is requested to reduce this convergence time. Ambiguity fixed solutions can also further improve the PPP accuracy. Several PPP integer ambiguity resolution methods have been explored and developed in recent years [2–4]. However, it still takes a few tens of minutes to obtain reliable ambiguity resolution (AR) if only with dual-frequency observations. This is because large noise of code measurements leads to long time smoothing with Melbourne-Wübbena (MW) measurement combination [5, 6]. Moreover, the narrow-lane (NL) ambiguities need more than ten minutes to be fixed due to short wavelength [7].At present, the third Global Positioning System (GPS) civil signal L5 is available with the launch of the latest Block IIF satellites, which enables more flexible ambiguity resolution strategies. Triple-frequency PPP AR can be achieved faster with longer wavelength, which has been studied by researchers [4, 8–10]. Gu et al. [9] verified the effectiveness of triple-frequency PPP using BeiDou datasets. For GPS triple-frequency PPP, Geng and Bock [8] fixed the ambiguities in each frequency and proved higher efficiency using simulated GPS datasets; Wang [10] only fixed the extra-wide-lane (EWL) and wide-lane (WL) ambiguities with a windowed phase smoothing phase technique to reduce the large noise of triple-frequency IF measurements. However, both GPS triple-frequency PPP AR researches mentioned above used simulated datasets, which would not suffer the inconsistency between the dual-frequency based precise clock products and the triple-frequency IF carrier-phase measurements. Very few researchers implemented the GPS triple-frequency PPP with measured datasets. Laurichesse [11] implemented the GPS triple-frequency PPP AR with integer clock products applicable for each single frequency carrier-phase measurements, which is however not widely adopted by applications yet. If widely applied dual-frequency based precise clock products are used in GPS triple-frequency PPP with observed data, there is obvious inconsistency between the dual-frequency based precise clock products and the triple-frequency IF measurements. This is because unlike the interfrequency clock biases (IFCBs) between dual-frequency IF code and phase measurement, which is stable over certain time [3], the IFCB between dual-frequency IF code and triple-frequency IF phase measurement varies up to meters over time [12, 13]. In this paper, GPS triple-frequency PPP AR method using widely used dual-frequency based precise clock products and measured datasets is proposed. In this method, the IFCBs between the dual-frequency and triple-frequency IF carrier-phase measurements are first estimated and then applied to triple-frequency IF carrier-phase measurements before using triple-frequency IF carrier-phase measurements in PPP. In this way, the widely applied dual-frequency precise clock products can be used in triple-frequency PPP. After applying the estimated IFCBs, the satellite fractional cycle biases (FCBs) can be estimated and used to recover the integer property of ambiguities. In other words, compared to dual-frequency PPP, in addition to FCBs, IFCBs between dual-frequency and triple-frequency IF carrier-phase measurements are also broadcasted to users to implement the triple-frequency GPS PPP AR.Montenbruck et al. [13] and Li et al. [12] provided solutions for estimating L1/L2 and L1/L5 IFCBs, which can be also applied for estimating L1/L2 and triple-frequency IF carrier-phase IFCBs. Then, the triple-frequency GPS PPP AR using dual-frequency based IGS precise clock products and observed datasets can be conducted. Specifically, single-difference between-satellite PPP AR fixing only EWL and WL ambiguities is implemented. Due to limited number of GPS satellites with triple-frequency observables, dual-frequency and triple-frequency IF carrier-phase measurements are applied together to obtain PPP solutions. Datasets of one sparse network with eight reference stations in Europe are used to first estimate the IFCBs. The extra-wide-lane (EWL) and wide-lane (WL) FCBs can be generated by the reference stations in the networks after applying the estimated IFCBs. Both IFCB and FCB products have to be sent to test user stations. At the test user stations, the IFCBs are used to obtain stable float solutions. Then, after applying the estimated EWL and WL FCBs, the EWL ambiguities can be fixed instantaneously with the Melbourne-Wübbena combination while the WL ambiguities can be determined by the LAMBDA method [14]. Different from dual-frequency PPP AR which has to resolve the narrow-lane (NL) ambiguity, L1/L2 wide-lane ambiguity with valid wavelength of 3.4 m needs to be fixed in triple-frequency PPP AR, which means that fast AR can be achieved with longer wavelength. The paper is organized as follows. In Section 2, the procedures of dual-frequency single-difference PPP ambiguity resolution using dual-frequency based IGS precise products are presented. Section 3 introduces the problem of implementing triple-frequency PPP ambiguity resolution using the method in Section 2 and provides the solution. The test results and conclusions are shown in Sections 4 and 5, respectively.


## 2. Dual-Frequency PPP AR Using Dual-Frequency Based Precise Products

The dual-frequency undifferenced ionosphere-free (IF) combination of code and phase measurements after applying each error model (e.g., Sagnac effect, relative effect) can be expressed as(1)Pdual=f12f12-f22P1-f22f12-f22P2=ρ+cdtr+bPdualr-cdts+bPduals+T+εPdual=ρ+cdtPdualr-cdtPduals+T+εPdualΦdual=f12f12-f22Φ1-f22f12-f22Φ2=ρ+cdtr+bΦdualr-cdts+bΦduals+T+λdualNdual+εΦdual=ρ+cdtΦdualr-cdtΦduals+T+λdualNdual+εΦdualcdtPdualr=cdtr+bPdualrcdtPduals=cdts+bPdualscdtΦdualr=cdtr+bΦdualrcdtΦduals=cdts+bΦduals,where Pdual and Φdual represent the undifferenced dual-frequency ionosphere-free (IF) code and carrier-phase measurements (m), f1 and f2 represent the L1 and L2 frequencies, P1, P2 and Φ1, Φ2 are L1 and L2 code and phase measurements in unit of meter, respectively, ρ is the geometric range between receiver and satellite, c is the speed of light in vacuum, cdtPduals (satellite code clock error), cdtΦduals (satellite phase clock error), cdtPdualr (receiver code clock error), and cdtΦdualr (receiver phase clock error) are a function of the actual satellite clock error dts, receiver clock error dtr, satellite dual-frequency IF code and phase biases, bPduals, bΦduals, and receiver dual-frequency IF code and phase biases bPdualr, bΦdualr. T is the tropospheric delay, ε(Pdual) and ε(Φdual) are the noise including multipath of dual-frequency IF code and carrier-phase measurements, and λdual and Ndual represent the wavelength and ambiguity of dual-frequency IF combination, which is given as (2)λdualNdual=f12λ1N1-f22λ2N2f12-f22=λNLN1+f2f1+f2λWLNWL,where N1 and N2 are the integer ambiguities of L1 and L2 frequencies, NWL and NNL are the wide-lane and narrow-lane integer ambiguities formed by N1 and N2, and λWL and λNL represent the wavelength of wide-lane and narrow-lane combination, respectively. In float PPP solution, the satellite code clock error cdtPduals can be provided by IGS precise clock products. The position, tropospheric delay, and receiver code clock error cdtPdualr, together with the float ambiguities are going to be estimated.In this study, PPP AR is implemented using the strategy proposed by Ge et al. [3] where single-difference (SD) FCBs are applied to recover the single-difference ambiguity integer property. The single-difference eliminates the receiver clock and bias. Here the dual-frequency PPP AR is introduced first to better illustrate the triple-frequency PPP AR. The single-difference dual-frequency observation model after applying various error correction models can be rewritten as(3)∇Pdual=f12f12-f22∇P1-f22f12-f22∇P2=∇ρ-∇cdtPduals+∇T+ε∇Pdual∇Φdual=f12f12-f22∇Φ1-f22f12-f22∇Φ2=∇ρ-c∇dtΦduals+∇T+λdual∇Ndual+ε∇Φdual=∇ρ-c∇dtPduals+∇T+∇Adual∇Adual=c∇dtPduals-c∇dtΦduals+λdual∇Ndual+ε∇Φdual=∇bPduals-∇bΦduals+λNL∇N1+f2f1+f2λWL∇NWL+ε∇Φdual,where ∇ is the single-difference between-satellite operator. ∇Adual is the single-difference float ambiguity. ∇NWL and ∇NNL are the single-difference wide-lane and narrow-lane integer ambiguities. In this section, because signals in only two frequencies are involved, the wide-lane and narrow-lane represent the corresponding combination of L1 and L2. In the next section, the wide-lane combinations will be specified. It can be seen that the IF ambiguities can be resolved by fixing the wide-lane and L1 ambiguities sequentially, which is illustrated as follows.The single-difference between-satellite Melbourne-Wübbena (MW) combination can be formed as(4)∇AMW=∇ΦWL-∇PNL=λWL∇NWL-∇bMWs+ε∇AMW∇bMWs=αWL∇bΦ1s+βWL∇bΦ2s-αNL∇bP1s+βNL∇bP2sαWL=f1f1-f2βWL=-f2f1-f2αNL=f1f1+f2βNL=f2f1+f2,where ∇AMW is the single-difference MW combination, WL represents wide-lane combination while NL means narrow-lane combination, bMWs is the satellite MW bias, and bΦ1s, bP1s, bΦ2s, and bP2s represent the satellite phase and code biases of L1 and L2 frequencies. αWL, βWL, αNL, and βNL are the corresponding wide-lane and narrow-lane coefficients. The single-difference satellite wide-lane FCB at a receiver in unit of meter can be achieved by(5)∇NWL=∇AMWλWL∇bMWs=∇AMW-λWL∇NWL,where ∗ denotes rounding of the real value to the nearest integer value. The wide-lane FCB correction with high precision can be calculated by averaging the corrections obtained in a network. For users, after applying the averaged wide-lane FCBs, the wide-lane ambiguities can be calculated by rounding of the WL ambiguity real value to the nearest integer value.After solving the wide-lane ambiguities, if the receiver coordinate is already known, (3) can be rewritten with known parameters on the left side and unknowns on the right side as(6)∇Φdual-f2f1+f2λWL∇NWL-∇ρ+c∇dtPduals=∇T+λNL∇N1+∇bPduals-∇bΦduals+ε∇Φdual=∇T+∇A1∇A1=λNL∇N1+∇bPduals-∇bΦduals+ε∇Φdual,where ∇A1 is the float ambiguity of N1. Similar to calculation of wide-lane FCB corrections, N1 FCB corrections at a receiver in a network after convergence can be calculated by(7)∇N1=∇A1λNL∇bN1s=∇bPduals-∇bΦduals=∇A1-λNL∇N1.By averaging the calculated N1 FCB corrections from multiple receivers in a network, a precise value can be achieved. When users receive and apply the N1 FCBs in the carrier-phase measurements, the integer property of N1 ambiguities can be recovered. Then, N1 can be resolved by applying LAMBDA method.


## 3. Triple-Frequency PPP AR Using Dual-Frequency Based Precise Products

According to Wang [10] and Feng [15], undifferenced triple-frequency IF carrier-phase observation model after applying error models can be written as(8)Φtriple=f1f1-f5ΦWLL1L2-f5f1-f5ΦWLL2L5=ρ+cdtr+bΦtripler-cdts+bΦtriples+T+λtripleNtriple+εΦtriple=ρ+cdtΦtripler-cdtΦtriples+T+λtripleNtriple+εΦtriple,where Φtriple is the undifferenced triple-frequency IF carrier-phase measurement, ΦWLL1L2 is the wide-lane L1 and L2 carrier-phase measurement, ΦWLL2L5 is the wide-lane L2 and L5 carrier-phase measurement, cdtΦtriples (satellite phase clock error) and cdtΦtripler (receiver phase clock error) are a function of the actual satellite clock error dts, receiver clock error dtr, and satellite and receiver triple-frequency IF phase biases bΦtriples, bΦtripler. λtriple and Ntriple represent the wavelength and ambiguity of triple-frequency IF combination, which is given as(9)λtripleNtriple=f1f1-f5λWLL1L2NWLL1L2-f5f1-f5λWLL2L5NWLL2L5.Similar to (3), after single-difference between satellites, the observation equation can be rewritten as(10)∇Φtriple=f1f1-f5∇ΦWLL1L2-f5f1-f5∇ΦWLL2L5=∇ρ-c∇dtΦtriples+∇T+λtriple∇Ntriple+ε∇Φtriple=∇ρ-c∇dtPduals+c∇dtPduals-c∇dtΦtriples+∇T+λtriple∇Ntriple+ε∇Φtriple=∇ρ-c∇dtPduals+∇T+∇Atriple∇Atriple=c∇dtPduals-c∇dtΦtriples+λtriple∇Ntriple+ε∇Φtriple=∇bPduals-∇bΦtriples+f1f1-f5λWLL1L2NWLL1L2-f5f1-f5λWLL2L5NWLL2L5+ε∇Φtriple,where ∇ is the single-difference between-satellite operator. cdtPduals can be provided by IGS precise clock product and ∇Atriple is the single-difference triple-frequency float ambiguity.One issue in triple-frequency PPP AR is that, unlike the dual-frequency PPP AR in which∇bPduals-∇bΦduals is relatively stable over certain time [3], triple-frequency PPP suffers a totally different situation where the variation of the single-difference ambiguities using dual-frequency based precise products can be up to meters [12, 13]. This is caused by the instability between satellite dual-frequency IF code bias and satellite triple-frequency IF phase bias, namely, ∇bPduals-∇bΦtriples. To implement PPP AR, one prerequisite is to obtain relatively stable float ambiguities. Then, after recovering the integer property of ambiguities, integer ambiguity resolution can be achieved. Since the stability of ∇bPduals-∇bΦduals has been proved, to obtain relative stable triple-frequency IF float ambiguities, one straightforward way is to recover ∇bΦduals after correcting the difference between ∇bΦduals and ∇bΦtriples, namely, transforming ∇bPduals-∇bΦtriples to ∇bPduals-∇bΦduals


### 3.1. Interfrequency Clock Bias (IFCB) Estimation

As mentioned above, the IFCB∇bPduals-∇bΦtriples is unstable, which is absorbed into float ambiguities, resulting in incorrect position estimation and unstable float ambiguity. In order to transform ∇bPduals-∇bΦtriples to ∇bPduals-∇bΦduals, the difference between ∇bΦtriples and ∇bΦduals has to be estimated. According to Montenbruck et al. [13] and Li et al. [12], the IFCBs can be obtained based on the difference between the two ionosphere-free phase combinations (L1/L2/L5-minus-L1/L2) since the ionosphere impact is greatly reduced and other nondispersive errors contained in the observations can be eliminated. One thing needs to be mentioned is that the L2 phase center offset (PCO) and phase center variation (PCV) is used to correct for L5 due to similar frequency and unavailability of L5 PCO and PCV. To improve the computation efficiency, Li et al. [12] adopted two strategies namely epoch-differenced (ED), and satellite-differenced and epoch-differenced (SDED) method to calculate the IFCB. Difference between epochs removes the ambiguities while difference between satellites eliminates the contribution of receiver IFCB. The SDED method is applied to calculate the IFCB between dual-frequency and triple-frequency IF phase measurements in this work. IFCB in this section refers to the single-difference between-satellite IFCB. In SDED method, IFCB can be obtained by adding all the IFCB difference between two consecutive epochs from the beginning and the IFCB value at the initial epoch. The specific procedures are presented as follows.To calculate the IFCB, the difference between the single-difference between-satellite dual-frequency and triple-frequency IF carrier-phase measurements using (3) and (10) can be formed as(11)∇Φdual-∇Φtriple=∇Adual-∇Atriple=∇bΦtriples-∇bΦduals+λdual∇Ndual-λtriple∇Ntriple,where ∇bΦtriples-∇bΦduals is the IFCB between triple-frequency and dual-frequency IF phase measurements. As can be seen from (11), the variation of IFCB is reflected on the variation of difference between triple-frequency and dual-frequency IF phase measurements if no cycle slip occurs. To calculate the IFCB, the constant λdual∇Ndual-λtriple∇Ntriple needs to be eliminated, which can be achieved by calculating the difference between two consecutive epochs, given by(12)d=∇Φdual-∇ΦtripleΔd=dk-dk-1=∇bΦtriples-∇bΦdualsk-∇bΦtriples-∇bΦdualsk-1,where d is the difference between the single-difference dual-frequency and triple-frequency IF phase measurements, dk is the d at epoch k, and Δd is the difference of d at two consecutive epochs. It can be seen from (12) that Δd is the change of IFCB between two consecutive epochs. In order to improve the redundancy to calculate Δd, datasets of a network are usually utilized to calculate the elevation-angle-dependent weighted average Δd as(13)Δd=∑piΔdi∑pi,pi=sin⁡θisin⁡θrisin2⁡θi+sin2⁡θri,where Δdi is Δd obtained at i station in the network, θi is the corresponding satellite elevation angle at i station, θri represents the reference satellite elevation angle at i station, and pi represents the weight of Δd at i station.The IFCB at any epoch can be calculated by addingΔd from the beginning and the IFCB at the initial epoch, expressed as(14)δk=δ0+∑Δd,where δk is the IFCB at epoch k and δ0 is the IFCB at the initial epoch, which can be set as arbitrary value.So far,∇bΦtriples-∇bΦduals can be obtained, which can be applied as another correction to triple-frequency IF carrier-phase measurements as ∇Φtriple+δk. One thing needs to be mentioned is that there is a constant bias between the IFCB calculated and the actual value shown as below(15)a=∇bΦtriples-∇bΦdualsk-δk.This is because the IFCB at the initial epoch is assumed to be arbitrary value. However, this would not affect integer ambiguity resolution because the majority part of the constant bias only changes the integer values of WL ambiguities while the less than one cycle part will be treated as part of the FCB corrections.


### 3.2. Ambiguity Resolution

After obtaining the IFCBs, relative stable float ambiguities can be achieved. The next step is to recover the integer property of ambiguities, which is discussed in this part. From (10), it can be seen that the triple-frequency IF ambiguity resolution can be achieved by resolving the wide-lane L2/L5 and wide-lane L1/L2 integer ambiguities sequentially. Since the wavelength of WL L2/L5 is as long as 5.86 m, it will be called extra-wide-lane (EWL) and WL refers to wide-lane of L1 and L2 from now on. Similar to (4) and (5), EWL FCBs at a receiver can be determined by forming MW combination of L2 and L5 measurements as(16)∇AMWL2L5=∇ΦWLL2L5-∇PNLL2L5=λWLL2L5∇NWLL2L5-∇bMWL2L5s+ε∇AMWL2L5∇bMWL2L5s=αWL∇bΦ2s+βWL∇bΦ5s-αNL∇bP2s+βNL∇bP5sαWL=f1f1-f2βWL=-f2f1-f2αNL=f1f1+f2βNL=f2f1+f2∇NMWL2L5=∇AMWL2L5λWLL2L5∇bMWL2L5s=∇AMWL2L5-λWLL2L5∇NWLL2L5.By averaging the satellite EWL FCBs obtained by the receivers in the network, precise EWL FCBs can be achieved and broadcasted to users. After applying the EWL FCBs, the EWL integer ambiguity can be obtained by rounding the real-valued EWL ambiguity to its nearest integer value.After solving EWL ambiguities, similar to (6), the WL FCBs can be obtained by(17)∇Φtriple-∇ρ+f5f1-f5λWLL2L5NWLL2L5+c∇dtPduals+δ=∇T+∇bPduals-∇bΦduals+ba+f1f1-f5λWLL1L2N~WLL1L2+ε∇Φdual=∇T+∇AWL∇AWL=∇bPduals-∇bΦduals+ba+f1f1-f5λWLL1L2N~WLL1L2+ε∇Φdual,where ∇AWL is the float WL ambiguity, δ is the IFCB obtained by (14), ba is the less than one cycle part of the bias a in (15), and N~WLL1L2 is the pseudo WL ambiguities after absorbing the majority part of bias a in (15). Due to application of IFCB, the dual-frequency IF satellite phase bias ∇bΦduals is on the right side of the equation above instead of the triple-frequency one in (10), which means stable float ambiguities can be expected. After convergence, the float WL ambiguities can be obtained. Similar to (8), the WL FCB corrections at one receiver can be obtained as(18)N~WLL1L2=∇AWLf1/f1-f5λWLL1L2∇bNWLs=∇bPduals-∇bΦduals+ba=∇AWL-f1f1-f5λWLL1L2N~WLL1L2.The WL FCBs broadcasted to users is the average of WL FCBs in a network. With the WL FCB corrections, the integer property of WL ambiguities can be recovered. The integer WL ambiguity will be searched by LAMBDA method. It can be seen that the valid wavelength of the WL ambiguity f1/f1–f5λWLL1L2 is 3.40 m, which makes it much easier to obtain the WL ambiguities.According to Geng and Bock [8] and Teunissen [16], the search area of ambiguity search space can be represented as(19)S=πχ2Q≈πχ2σLδPkλ1λ2,where S is the area of ambiguity search space, χ2 is the defined threshold, |Q| is the determinant of float ambiguity variance-covariance matrix, and λ1 and λ2 are the carrier-phase wavelength. It indicates that, with longer wavelength and smaller measurement noise, the ambiguities would be easier to be fixed. Although, for the triple-frequency IF carrier-phase, the noise is enlarged around 100 times, the valid wavelength for LAMBDA to be fixed is 3.40 m, which makes it much easier to be fixed, compared to the situation in dual-frequency PPP where NL ambiguities need to be fixed.


## 3.1. Interfrequency Clock Bias (IFCB) Estimation

As mentioned above, the IFCB∇bPduals-∇bΦtriples is unstable, which is absorbed into float ambiguities, resulting in incorrect position estimation and unstable float ambiguity. In order to transform ∇bPduals-∇bΦtriples to ∇bPduals-∇bΦduals, the difference between ∇bΦtriples and ∇bΦduals has to be estimated. According to Montenbruck et al. [13] and Li et al. [12], the IFCBs can be obtained based on the difference between the two ionosphere-free phase combinations (L1/L2/L5-minus-L1/L2) since the ionosphere impact is greatly reduced and other nondispersive errors contained in the observations can be eliminated. One thing needs to be mentioned is that the L2 phase center offset (PCO) and phase center variation (PCV) is used to correct for L5 due to similar frequency and unavailability of L5 PCO and PCV. To improve the computation efficiency, Li et al. [12] adopted two strategies namely epoch-differenced (ED), and satellite-differenced and epoch-differenced (SDED) method to calculate the IFCB. Difference between epochs removes the ambiguities while difference between satellites eliminates the contribution of receiver IFCB. The SDED method is applied to calculate the IFCB between dual-frequency and triple-frequency IF phase measurements in this work. IFCB in this section refers to the single-difference between-satellite IFCB. In SDED method, IFCB can be obtained by adding all the IFCB difference between two consecutive epochs from the beginning and the IFCB value at the initial epoch. The specific procedures are presented as follows.To calculate the IFCB, the difference between the single-difference between-satellite dual-frequency and triple-frequency IF carrier-phase measurements using (3) and (10) can be formed as(11)∇Φdual-∇Φtriple=∇Adual-∇Atriple=∇bΦtriples-∇bΦduals+λdual∇Ndual-λtriple∇Ntriple,where ∇bΦtriples-∇bΦduals is the IFCB between triple-frequency and dual-frequency IF phase measurements. As can be seen from (11), the variation of IFCB is reflected on the variation of difference between triple-frequency and dual-frequency IF phase measurements if no cycle slip occurs. To calculate the IFCB, the constant λdual∇Ndual-λtriple∇Ntriple needs to be eliminated, which can be achieved by calculating the difference between two consecutive epochs, given by(12)d=∇Φdual-∇ΦtripleΔd=dk-dk-1=∇bΦtriples-∇bΦdualsk-∇bΦtriples-∇bΦdualsk-1,where d is the difference between the single-difference dual-frequency and triple-frequency IF phase measurements, dk is the d at epoch k, and Δd is the difference of d at two consecutive epochs. It can be seen from (12) that Δd is the change of IFCB between two consecutive epochs. In order to improve the redundancy to calculate Δd, datasets of a network are usually utilized to calculate the elevation-angle-dependent weighted average Δd as(13)Δd=∑piΔdi∑pi,pi=sin⁡θisin⁡θrisin2⁡θi+sin2⁡θri,where Δdi is Δd obtained at i station in the network, θi is the corresponding satellite elevation angle at i station, θri represents the reference satellite elevation angle at i station, and pi represents the weight of Δd at i station.The IFCB at any epoch can be calculated by addingΔd from the beginning and the IFCB at the initial epoch, expressed as(14)δk=δ0+∑Δd,where δk is the IFCB at epoch k and δ0 is the IFCB at the initial epoch, which can be set as arbitrary value.So far,∇bΦtriples-∇bΦduals can be obtained, which can be applied as another correction to triple-frequency IF carrier-phase measurements as ∇Φtriple+δk. One thing needs to be mentioned is that there is a constant bias between the IFCB calculated and the actual value shown as below(15)a=∇bΦtriples-∇bΦdualsk-δk.This is because the IFCB at the initial epoch is assumed to be arbitrary value. However, this would not affect integer ambiguity resolution because the majority part of the constant bias only changes the integer values of WL ambiguities while the less than one cycle part will be treated as part of the FCB corrections.


## 3.2. Ambiguity Resolution

After obtaining the IFCBs, relative stable float ambiguities can be achieved. The next step is to recover the integer property of ambiguities, which is discussed in this part. From (10), it can be seen that the triple-frequency IF ambiguity resolution can be achieved by resolving the wide-lane L2/L5 and wide-lane L1/L2 integer ambiguities sequentially. Since the wavelength of WL L2/L5 is as long as 5.86 m, it will be called extra-wide-lane (EWL) and WL refers to wide-lane of L1 and L2 from now on. Similar to (4) and (5), EWL FCBs at a receiver can be determined by forming MW combination of L2 and L5 measurements as(16)∇AMWL2L5=∇ΦWLL2L5-∇PNLL2L5=λWLL2L5∇NWLL2L5-∇bMWL2L5s+ε∇AMWL2L5∇bMWL2L5s=αWL∇bΦ2s+βWL∇bΦ5s-αNL∇bP2s+βNL∇bP5sαWL=f1f1-f2βWL=-f2f1-f2αNL=f1f1+f2βNL=f2f1+f2∇NMWL2L5=∇AMWL2L5λWLL2L5∇bMWL2L5s=∇AMWL2L5-λWLL2L5∇NWLL2L5.By averaging the satellite EWL FCBs obtained by the receivers in the network, precise EWL FCBs can be achieved and broadcasted to users. After applying the EWL FCBs, the EWL integer ambiguity can be obtained by rounding the real-valued EWL ambiguity to its nearest integer value.After solving EWL ambiguities, similar to (6), the WL FCBs can be obtained by(17)∇Φtriple-∇ρ+f5f1-f5λWLL2L5NWLL2L5+c∇dtPduals+δ=∇T+∇bPduals-∇bΦduals+ba+f1f1-f5λWLL1L2N~WLL1L2+ε∇Φdual=∇T+∇AWL∇AWL=∇bPduals-∇bΦduals+ba+f1f1-f5λWLL1L2N~WLL1L2+ε∇Φdual,where ∇AWL is the float WL ambiguity, δ is the IFCB obtained by (14), ba is the less than one cycle part of the bias a in (15), and N~WLL1L2 is the pseudo WL ambiguities after absorbing the majority part of bias a in (15). Due to application of IFCB, the dual-frequency IF satellite phase bias ∇bΦduals is on the right side of the equation above instead of the triple-frequency one in (10), which means stable float ambiguities can be expected. After convergence, the float WL ambiguities can be obtained. Similar to (8), the WL FCB corrections at one receiver can be obtained as(18)N~WLL1L2=∇AWLf1/f1-f5λWLL1L2∇bNWLs=∇bPduals-∇bΦduals+ba=∇AWL-f1f1-f5λWLL1L2N~WLL1L2.The WL FCBs broadcasted to users is the average of WL FCBs in a network. With the WL FCB corrections, the integer property of WL ambiguities can be recovered. The integer WL ambiguity will be searched by LAMBDA method. It can be seen that the valid wavelength of the WL ambiguity f1/f1–f5λWLL1L2 is 3.40 m, which makes it much easier to obtain the WL ambiguities.According to Geng and Bock [8] and Teunissen [16], the search area of ambiguity search space can be represented as(19)S=πχ2Q≈πχ2σLδPkλ1λ2,where S is the area of ambiguity search space, χ2 is the defined threshold, |Q| is the determinant of float ambiguity variance-covariance matrix, and λ1 and λ2 are the carrier-phase wavelength. It indicates that, with longer wavelength and smaller measurement noise, the ambiguities would be easier to be fixed. Although, for the triple-frequency IF carrier-phase, the noise is enlarged around 100 times, the valid wavelength for LAMBDA to be fixed is 3.40 m, which makes it much easier to be fixed, compared to the situation in dual-frequency PPP where NL ambiguities need to be fixed.


## 4. Tests and Results

To test the validity of the proposed triple-frequency ambiguity resolution method, one sparse network with 10 stations in Europe where more than four GPS Block IIF satellites (transmitting the L5 signal) can be observed for more than 60 minutes is used. Among the 10 stations, 8 stations are used as reference stations to generate the single-difference IFCBs and FCBs while the other two stations are used as test user stations. PPP AR is implemented at the test user stations. All the datasets used in this work can be downloaded from the IGS Multi-GNSS Experiment (MGEX) website. Trimble R9 receiver is used in every station to receive triple-frequency signals. The distribution of 10 stations used is shown in Figure1. The eight stations to generate IFCBs and FCBs are cebr, vill, dlf1, gop7, metg, kiru, dyng, and mas1, which are denoted in blue. The red stations redu and tlse are used to implement PPP AR. Dual-frequency PPP AR is first implemented in Section 4.1 using the same datasets as triple-frequency PPP AR in Section 4.2 to make a comparison. The same datasets are applied to fix the ambiguities of the same Block IIF satellites in both dual-frequency and triple-frequency PPP AR tests.Figure 1
Distribution of observation stations.The observation span for redu and tlse is chosen on UTC May 9, 2015, from 02:35 am to 03:35 am when four Block IIF satellites (PRN 01, 03, 06, and 09) are in view simultaneously for all the stations mentioned above. The further details of the network can be found in Table1. In both dual-frequency and triple-frequency processing, IGS final precise orbit and clock products are applied by both reference stations and test user stations. Single-difference is applied to eliminate the receiver contribution to biases in estimating the IFCBs, FCBs, and PPP AR implementation. In the tests, PRN 03 is set as the reference satellite due to high elevation. The datasets of reference stations are used to generate the FCBs to recover the integer property of ambiguities in dual-frequency processing. In triple-frequency processing, apart from the WL FCBs, IFCBs are first estimated to compensate the inconsistency in triple-frequency IF carrier-phase observations.Table 1
Details of test (unit for distance is kilometer).
Name Interval (s) Average distance Min distance Max distance Number of Ref stations Inter-Ref station distance redu 1 1538 227 3006 8 35-4494 tlse 1 1620 568 2913 8 35-4494


### 4.1. Dual-Frequency PPP AR

As illustrated in previous researches [2–4], the dual-frequency ionosphere-free ambiguities can be decomposed as WL and NL ambiguities. To recover the integer property of ambiguities, the WL and NL FCBs are first estimated by the reference stations. The WL ambiguities can be determined by MW combinations easily with a wavelength of 0.86 m. With known coordinates of reference stations, the float ambiguities can be converged quickly. Therefore, the NL FCBs can be determined quickly as well. Shown in Figure 2 is the histogram of NL ambiguities after applying estimated NL FCBs. It can be seen that over 95% fraction parts of NL ambiguities fall into the −0.1 to 0.1 cycle after applying the estimated NL FCBs.Figure 2
Histogram of the fractions of SD NL ambiguities after applying FCBs.The one-hour observation at user test stations redu and tlse is divided into 2 30-minute sessions at each station. Due to short wavelength of narrow-lane combination, it takes tens of minutes for the test stations to fix the ambiguities. The positioning results for the four sessions at the test user stations are shown in Figure2. It can be seen that, after successfully fixing the correct integer ambiguities, the positioning results are improved immediately. The time to first fix (TTFF) in these four sessions is 15, 21, 19, and 29 minutes, respectively. The average TTFF is 21 minutes. For the two sessions at station redu, the RMS of horizontal positioning errors is reduced from 2.30 cm and 8.12 cm to 1.45 cm and 0.36 cm after fixing the ambiguities. The vertical RMS is reduced from 2.06 cm and 12.04 cm to 0.59 cm and 6.83 cm. The positioning results for the two sessions at tlse station are improved as well. The horizontal errors are reduced from 5.47 cm and 3.05 cm to 0.76 cm and 0.53 cm. As for the vertical direction, the error is reduced from 13.57 cm to 4.63 cm in the first session while the second session suffered slight degradation from 3.79 cm to 3.98 cm.The results of the dual-frequency PPP AR of GPS Block IIF satellites show that the dual-frequency PPP AR can achieve centimeter-level accuracy, but it takes long time to fix the ambiguities.


### 4.2. Triple-Frequency PPP AR

In triple-frequency PPP processing, dual-frequency IF code measurements of Block IIF satellites are applied instead of triple-frequency IF code measurements. This is because the triple-frequency IF combination would enlarge the noise about 100 times, which degrades the positioning results. Moreover, there is inconsistency between the triple-frequency IF code measurements and the dual-frequency based precise products. The bias between the dual-frequency and triple-frequency IF code measurements is shown in Figure4.It can be seen from Figure3 that the biases between dual-frequency and triple-frequency IF code measurements fluctuate greatly. Therefore, it is more appropriate to apply dual-frequency IF code measurements. Unlike dual-frequency PPP, the IFCBs between triple-frequency and dual-frequency IF phase measurements need to be estimated to compensate the inconsistency between the triple-frequency IF carrier-phase measurements and the IGS dual-frequency based precise orbit and clock precise products. The weighted IFCBs based on (12)–(14) are first calculated with the datasets of reference stations in the network. Shown in Figure 5 are the estimated weighted IFCBs with reference station datasets for PRN 01, PRN 06, and PRN 09 with PRN 03 as reference satellite and the first common epoch of each satellite as reference epoch. The initial single-difference IFCB (δ0) is set as 0. From Figure 5, we can see that the variation of IFCB can be up to almost 3 meters over 2 hours which accounts for more than half a valid cycle (3.4 m) of WL L1/L2 in triple-frequency PPP AR. The aim of IFCB estimation is to obtain converged stable float ambiguities, which is the prerequisite to fix the ambiguities. Otherwise, the large variation IFCBs would contaminate the ambiguities, resulting in unstable float ambiguities which cannot be fixed.Figure 3
Dual-frequency positioning results of 4 30-minute sessions.Figure 4
Bias between dual-frequency and triple-frequency IF code measurements.Figure 5
Reference epoch based SD IFCB.As introduced before, the triple-frequency IF ambiguities can be decomposed as EWL ambiguities and WL ambiguities. After applying the correction of single-difference IFCBs and achieving stable single-difference float ambiguities, WL FCBs can be obtained after fixing the EWL ambiguities at reference stations with known coordinates. Single-difference EWL FCBs at reference station can be obtained by rounding the WL L2/L5 SD MW combinations directly due to large valid wavelength (5.86 m). Similarly, after the float ambiguities get converged and EWL ambiguities are fixed at reference stations, the WL FCBs can be obtained by rounding the converged WL ambiguities. Figure6 shows that over 95% fractional parts of WL ambiguities are within 0.1 cycles after applying the WL FCBs.Figure 6
Histogram of the fractions of SD WL ambiguities after applying FCBs.With estimated IFCBs, EWL FCBs, and WL FCBs, the triple-frequency PPP AR can be implemented at the test user stations redu and tlse. The cut-off angle is set as 15° and an elevation-dependent weighted strategy was applied to measurements to reduce the impact of poor quality of measurements in low elevation. For EWL ambiguities, bias rounding is directly applied due to the large wavelength of EWL. As for WL, LAMBDA method is applied to search for integer ambiguities and the ratio of the second minimum to the minimum quadratic form of residuals (a threshold of 3) is calculated to validate an ambiguity resolution. Because of the large noise of triple-frequency IF measurements, the weight of triple-frequency IF measurements is lower, compared to dual-frequency IF measurements. The weight ratio of dual-frequency IF phase measurements to triple-frequency IF phase measurements is about 33. Because of large noise of triple-frequency IF carrier-phase measurements and limited number of Block IIF satellites, the ambiguity resolution results can be expected that it could improve the results, but in limited extent.As noted in the previous sections, the valid WL wavelength is about 3.4 meters, so ambiguity resolution is expected to be accomplished faster than conventional dual-frequency GPS PPP. With collected datasets at test user stations, one-hour observation with four Block IIF satellites is utilized for triple-frequency PPP AR. SD EWL and SD WL FCBs can be obtained at reference stations to recover the integer property of ambiguities. In the first test, the one-hour observation at each station is divided into 10 sessions with 6 minutes in one session. In order to ensure the success of fixing ambiguities, ambiguities are attempted to be resolved after 3 minutes. Figures7 and 8 are the positioning errors in horizontal and vertical directions for the 10 sessions at redu and tlse, respectively. Each color represents positioning error of one session. Incorrect ambiguities are fixed for the last session at test user station tlse. It can be seen that, in most cases, the ambiguity resolution would have immediate impact on the positioning results. However, the positioning results still keep at decimeter level with slight improvement for some sessions. This is because only WL ambiguities are fixed in this work and the large noise of triple-frequency IF carrier-phase measurements and limited number of Block IIF satellites would contribute little to improving positioning results. To see the improvement of positioning results, the RMS of positioning errors after fixing ambiguities is listed in Table 2. The fixed solution results are compared with the float solution in the same period. It can be seen that correct triple-frequency WL ambiguities can be fixed in 19 sessions out of the 20 test sessions, except for the last session at station tlse. For the WL ambiguities fixed sessions at tlse, the positioning accuracy is slightly improved. The positioning accuracy after ambiguities fixed still keeps at decimeter level. For station redu, positioning accuracy degrades after WL ambiguity resolution in 4 sessions. The positioning accuracy keeps at decimeter level as well after WL ambiguity resolution. To further see the fast ambiguity resolution using triple-frequency measurements, the one-hour dataset is divided into 30 sessions with 2-minute observation in each session. The ambiguities are attempted to be fixed after 30 seconds of processing. The positioning results are shown in Table 3. Similarly, the positioning errors of fixed solution and float solution in the same period are compared. This time all the WL ambiguities can be fixed for all the 60 sessions at two test user stations. For station redu, the positioning accuracy degrades slightly after fixing the WL ambiguities in 40% sessions (12/30), while 2 sessions suffer positioning accuracy degradation at tlse. The fixed ambiguities of all sessions are correctly fixed, regardless of improvement of positioning or not. Therefore, the reason that leads to positioning degrading is not incorrect fixed ambiguities. There is great chance to obtain worse positioning results with limited number of WL ambiguities fixed partly due to the large noise of triple-frequency IF carrier-phase measurements. Besides, the quality of SD IFCBs and SD FCBs also has great impact on the positioning results.Table 2
RMS of positioning results with fixed ambiguities and float ambiguities during the same period (after fixing ambiguities) for 10 6-minute sessions at redu and tlse. In the table, “float” means RMS of positioning error with float ambiguities, “fixed” means RMS of positioning error with fixed WL ambiguities, “—” means there is no corresponding float solution RMS for comparison, and “fail” means failure of fixing correct WL ambiguities during the session. The unit in this table is centimeter.
Station 1 2 3 4 5 6 7 8 9 10 redu Float 33.5 41.4 43.1 43.1 56.9 49.1 62.1 69.8 60.9 54.5 Fixed 30.8 41.1 50.4 38.6 72.2 49.3 54.3 56.7 70.9 49.1  tlse Float 45.8 63.1 97.6 81.3 89.0 125.8 106.0 61.8 98.0 — Fixed 44.1 69.5 66.3 67.3 85.6 84.9 64.0 57.5 45.8 FailTable 3
RMS of positioning results with fixed ambiguities and float ambiguities during the same period (after fixing ambiguities) for 30 2-minute sessions at redu and tlse. In the table, “float” means RMS of positioning error with float ambiguities and “fixed” means RMS of positioning error with fixed WL ambiguities.
Station 1 2 3 4 5 6 7 8 9 10 redu Float 58.2 68.6 68.6 59.2 96.7 86.7 94.3 67.6 76.2 51.6 Fixed 58.1 41.3 61.9 48.3 53.5 68.7 76.8 70.3 51.9 44.8  tlse Float 132.8 54.1 88.8 132.7 91.6 80.1 118.8 173.0 146.7 97.1 Fixed 89.1 44.1 80.7 123.8 88.3 59.5 55.2 113.9 92.2 73.5  Station 11 12 13 14 15 16 17 18 19 20  redu Float 92.6 84.0 48.3 65.2 92.7 66.8 63.8 67.8 80.5 41.6 Fixed 41.7 44.3 55.1 104.7 100.6 47.3 51.1 80.6 61.2 46.8  tlse Float 86.4 142.0 141.0 85.1 86.9 186.6 142.3 122.1 140.6 121.0 Fixed 53.7 134.3 99.2 96.3 74.9 113.5 93.5 67.7 72.2 68.5  Station 21 22 23 24 25 26 27 28 29 30  redu Float 60.8 78.5 87.9 55.5 64.9 57.9 76.7 58.3 120.6 51.8 Fixed 74.8 75.9 96.9 59.4 62.0 80.5 103.4 61.9 62.2 85.8  tlse Float 102.2 80.4 75.0 159.9 123.9 161.0 100.1 163.8 160.9 162.4 Fixed 57.8 78.2 39.3 99.6 44.7 51.2 52.8 59.8 249.8 108.8Figure 7
Positioning results of 10 6-minute sessions at redu.Figure 8
Positioning results of 10 6-minute sessions at tlse.


## 4.1. Dual-Frequency PPP AR

As illustrated in previous researches [2–4], the dual-frequency ionosphere-free ambiguities can be decomposed as WL and NL ambiguities. To recover the integer property of ambiguities, the WL and NL FCBs are first estimated by the reference stations. The WL ambiguities can be determined by MW combinations easily with a wavelength of 0.86 m. With known coordinates of reference stations, the float ambiguities can be converged quickly. Therefore, the NL FCBs can be determined quickly as well. Shown in Figure 2 is the histogram of NL ambiguities after applying estimated NL FCBs. It can be seen that over 95% fraction parts of NL ambiguities fall into the −0.1 to 0.1 cycle after applying the estimated NL FCBs.Figure 2
Histogram of the fractions of SD NL ambiguities after applying FCBs.The one-hour observation at user test stations redu and tlse is divided into 2 30-minute sessions at each station. Due to short wavelength of narrow-lane combination, it takes tens of minutes for the test stations to fix the ambiguities. The positioning results for the four sessions at the test user stations are shown in Figure2. It can be seen that, after successfully fixing the correct integer ambiguities, the positioning results are improved immediately. The time to first fix (TTFF) in these four sessions is 15, 21, 19, and 29 minutes, respectively. The average TTFF is 21 minutes. For the two sessions at station redu, the RMS of horizontal positioning errors is reduced from 2.30 cm and 8.12 cm to 1.45 cm and 0.36 cm after fixing the ambiguities. The vertical RMS is reduced from 2.06 cm and 12.04 cm to 0.59 cm and 6.83 cm. The positioning results for the two sessions at tlse station are improved as well. The horizontal errors are reduced from 5.47 cm and 3.05 cm to 0.76 cm and 0.53 cm. As for the vertical direction, the error is reduced from 13.57 cm to 4.63 cm in the first session while the second session suffered slight degradation from 3.79 cm to 3.98 cm.The results of the dual-frequency PPP AR of GPS Block IIF satellites show that the dual-frequency PPP AR can achieve centimeter-level accuracy, but it takes long time to fix the ambiguities.


## 4.2. Triple-Frequency PPP AR

In triple-frequency PPP processing, dual-frequency IF code measurements of Block IIF satellites are applied instead of triple-frequency IF code measurements. This is because the triple-frequency IF combination would enlarge the noise about 100 times, which degrades the positioning results. Moreover, there is inconsistency between the triple-frequency IF code measurements and the dual-frequency based precise products. The bias between the dual-frequency and triple-frequency IF code measurements is shown in Figure4.It can be seen from Figure3 that the biases between dual-frequency and triple-frequency IF code measurements fluctuate greatly. Therefore, it is more appropriate to apply dual-frequency IF code measurements. Unlike dual-frequency PPP, the IFCBs between triple-frequency and dual-frequency IF phase measurements need to be estimated to compensate the inconsistency between the triple-frequency IF carrier-phase measurements and the IGS dual-frequency based precise orbit and clock precise products. The weighted IFCBs based on (12)–(14) are first calculated with the datasets of reference stations in the network. Shown in Figure 5 are the estimated weighted IFCBs with reference station datasets for PRN 01, PRN 06, and PRN 09 with PRN 03 as reference satellite and the first common epoch of each satellite as reference epoch. The initial single-difference IFCB (δ0) is set as 0. From Figure 5, we can see that the variation of IFCB can be up to almost 3 meters over 2 hours which accounts for more than half a valid cycle (3.4 m) of WL L1/L2 in triple-frequency PPP AR. The aim of IFCB estimation is to obtain converged stable float ambiguities, which is the prerequisite to fix the ambiguities. Otherwise, the large variation IFCBs would contaminate the ambiguities, resulting in unstable float ambiguities which cannot be fixed.Figure 3
Dual-frequency positioning results of 4 30-minute sessions.Figure 4
Bias between dual-frequency and triple-frequency IF code measurements.Figure 5
Reference epoch based SD IFCB.As introduced before, the triple-frequency IF ambiguities can be decomposed as EWL ambiguities and WL ambiguities. After applying the correction of single-difference IFCBs and achieving stable single-difference float ambiguities, WL FCBs can be obtained after fixing the EWL ambiguities at reference stations with known coordinates. Single-difference EWL FCBs at reference station can be obtained by rounding the WL L2/L5 SD MW combinations directly due to large valid wavelength (5.86 m). Similarly, after the float ambiguities get converged and EWL ambiguities are fixed at reference stations, the WL FCBs can be obtained by rounding the converged WL ambiguities. Figure6 shows that over 95% fractional parts of WL ambiguities are within 0.1 cycles after applying the WL FCBs.Figure 6
Histogram of the fractions of SD WL ambiguities after applying FCBs.With estimated IFCBs, EWL FCBs, and WL FCBs, the triple-frequency PPP AR can be implemented at the test user stations redu and tlse. The cut-off angle is set as 15° and an elevation-dependent weighted strategy was applied to measurements to reduce the impact of poor quality of measurements in low elevation. For EWL ambiguities, bias rounding is directly applied due to the large wavelength of EWL. As for WL, LAMBDA method is applied to search for integer ambiguities and the ratio of the second minimum to the minimum quadratic form of residuals (a threshold of 3) is calculated to validate an ambiguity resolution. Because of the large noise of triple-frequency IF measurements, the weight of triple-frequency IF measurements is lower, compared to dual-frequency IF measurements. The weight ratio of dual-frequency IF phase measurements to triple-frequency IF phase measurements is about 33. Because of large noise of triple-frequency IF carrier-phase measurements and limited number of Block IIF satellites, the ambiguity resolution results can be expected that it could improve the results, but in limited extent.As noted in the previous sections, the valid WL wavelength is about 3.4 meters, so ambiguity resolution is expected to be accomplished faster than conventional dual-frequency GPS PPP. With collected datasets at test user stations, one-hour observation with four Block IIF satellites is utilized for triple-frequency PPP AR. SD EWL and SD WL FCBs can be obtained at reference stations to recover the integer property of ambiguities. In the first test, the one-hour observation at each station is divided into 10 sessions with 6 minutes in one session. In order to ensure the success of fixing ambiguities, ambiguities are attempted to be resolved after 3 minutes. Figures7 and 8 are the positioning errors in horizontal and vertical directions for the 10 sessions at redu and tlse, respectively. Each color represents positioning error of one session. Incorrect ambiguities are fixed for the last session at test user station tlse. It can be seen that, in most cases, the ambiguity resolution would have immediate impact on the positioning results. However, the positioning results still keep at decimeter level with slight improvement for some sessions. This is because only WL ambiguities are fixed in this work and the large noise of triple-frequency IF carrier-phase measurements and limited number of Block IIF satellites would contribute little to improving positioning results. To see the improvement of positioning results, the RMS of positioning errors after fixing ambiguities is listed in Table 2. The fixed solution results are compared with the float solution in the same period. It can be seen that correct triple-frequency WL ambiguities can be fixed in 19 sessions out of the 20 test sessions, except for the last session at station tlse. For the WL ambiguities fixed sessions at tlse, the positioning accuracy is slightly improved. The positioning accuracy after ambiguities fixed still keeps at decimeter level. For station redu, positioning accuracy degrades after WL ambiguity resolution in 4 sessions. The positioning accuracy keeps at decimeter level as well after WL ambiguity resolution. To further see the fast ambiguity resolution using triple-frequency measurements, the one-hour dataset is divided into 30 sessions with 2-minute observation in each session. The ambiguities are attempted to be fixed after 30 seconds of processing. The positioning results are shown in Table 3. Similarly, the positioning errors of fixed solution and float solution in the same period are compared. This time all the WL ambiguities can be fixed for all the 60 sessions at two test user stations. For station redu, the positioning accuracy degrades slightly after fixing the WL ambiguities in 40% sessions (12/30), while 2 sessions suffer positioning accuracy degradation at tlse. The fixed ambiguities of all sessions are correctly fixed, regardless of improvement of positioning or not. Therefore, the reason that leads to positioning degrading is not incorrect fixed ambiguities. There is great chance to obtain worse positioning results with limited number of WL ambiguities fixed partly due to the large noise of triple-frequency IF carrier-phase measurements. Besides, the quality of SD IFCBs and SD FCBs also has great impact on the positioning results.Table 2
RMS of positioning results with fixed ambiguities and float ambiguities during the same period (after fixing ambiguities) for 10 6-minute sessions at redu and tlse. In the table, “float” means RMS of positioning error with float ambiguities, “fixed” means RMS of positioning error with fixed WL ambiguities, “—” means there is no corresponding float solution RMS for comparison, and “fail” means failure of fixing correct WL ambiguities during the session. The unit in this table is centimeter.
Station 1 2 3 4 5 6 7 8 9 10 redu Float 33.5 41.4 43.1 43.1 56.9 49.1 62.1 69.8 60.9 54.5 Fixed 30.8 41.1 50.4 38.6 72.2 49.3 54.3 56.7 70.9 49.1  tlse Float 45.8 63.1 97.6 81.3 89.0 125.8 106.0 61.8 98.0 — Fixed 44.1 69.5 66.3 67.3 85.6 84.9 64.0 57.5 45.8 FailTable 3
RMS of positioning results with fixed ambiguities and float ambiguities during the same period (after fixing ambiguities) for 30 2-minute sessions at redu and tlse. In the table, “float” means RMS of positioning error with float ambiguities and “fixed” means RMS of positioning error with fixed WL ambiguities.
Station 1 2 3 4 5 6 7 8 9 10 redu Float 58.2 68.6 68.6 59.2 96.7 86.7 94.3 67.6 76.2 51.6 Fixed 58.1 41.3 61.9 48.3 53.5 68.7 76.8 70.3 51.9 44.8  tlse Float 132.8 54.1 88.8 132.7 91.6 80.1 118.8 173.0 146.7 97.1 Fixed 89.1 44.1 80.7 123.8 88.3 59.5 55.2 113.9 92.2 73.5  Station 11 12 13 14 15 16 17 18 19 20  redu Float 92.6 84.0 48.3 65.2 92.7 66.8 63.8 67.8 80.5 41.6 Fixed 41.7 44.3 55.1 104.7 100.6 47.3 51.1 80.6 61.2 46.8  tlse Float 86.4 142.0 141.0 85.1 86.9 186.6 142.3 122.1 140.6 121.0 Fixed 53.7 134.3 99.2 96.3 74.9 113.5 93.5 67.7 72.2 68.5  Station 21 22 23 24 25 26 27 28 29 30  redu Float 60.8 78.5 87.9 55.5 64.9 57.9 76.7 58.3 120.6 51.8 Fixed 74.8 75.9 96.9 59.4 62.0 80.5 103.4 61.9 62.2 85.8  tlse Float 102.2 80.4 75.0 159.9 123.9 161.0 100.1 163.8 160.9 162.4 Fixed 57.8 78.2 39.3 99.6 44.7 51.2 52.8 59.8 249.8 108.8Figure 7
Positioning results of 10 6-minute sessions at redu.Figure 8
Positioning results of 10 6-minute sessions at tlse.


## 5. Conclusions

In this paper, triple-frequency GPS PPP AR is investigated based on observed datasets. A general data processing method is proposed and verified. The collected datasets of triple-frequency IF carrier-phase measurements indicate an inconsistency with the IGS dual-frequency based precise clock product obtained by L1/L2 ionosphere-free combination. The interfrequency biases are estimated and applied to triple-frequency IF carrier-phase measurements in order to obtain relatively stable float ambiguities. Limited by the available triple-frequency datasets, one network with ten stations is utilized. In the network, the PPP AR is implemented at the two test user stations with the IFCBs and FCBs generated by eight reference stations. Compared with dual-frequency PPP AR, triple-frequency ambiguities can be fixed much faster. However, the fixed triple-frequency solution still keeps at decimeter level. In some cases, the accuracy even degrades after correctly fixing the ambiguities, because only a limited number of WL ambiguities are fixed and the triple-frequency IF combination enlarges the measurement noise. The test results show that most positioning results become better after fixing ambiguities, which confirms the effectiveness of the proposed fast triple-frequency PPP AR method. In this study, the satellite L5 PCO and PCV are assumed as the same value of satellite L1 and L2. The receiver L5 PCO and PCV are assumed to be the same value of L2 due to similar frequency. Actually, due to the large coefficients in triple-frequency IF combination, the effect of PCO and PCV could have a decimeter-level impact on the measurements in some situations, which may cause inaccurate estimation of IFCBs. The estimation of IFCBs can be omitted if triple-frequency precise clock product is available in the future, and the results can be better if precise information (e.g., satellite and receiver PCO and PCV) for the third frequency signal can be available.



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*Source: 7854323-2017-02-21.xml*