International Journal of Aerospace Engineering

Volume 2017 (2017), Article ID 7854323, 11 pages

https://doi.org/10.1155/2017/7854323

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

Department of Geomatics Engineering, University of Calgary, Calgary, AB, Canada

Correspondence should be addressed to Fei Liu

Received 16 September 2016; Revised 6 January 2017; Accepted 17 January 2017; Published 21 February 2017

Academic Editor: Salvatore Gaglione

Copyright © 2017 Fei Liu and Yang Gao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 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 aswhere and represent the undifferenced dual-frequency ionosphere-free (IF) code and carrier-phase measurements (m), and represent the L1 and L2 frequencies, , and , are L1 and L2 code and phase measurements in unit of meter, respectively, is the geometric range between receiver and satellite, is the speed of light in vacuum, (satellite code clock error), (satellite phase clock error), (receiver code clock error), and (receiver phase clock error) are a function of the actual satellite clock error , receiver clock error , satellite dual-frequency IF code and phase biases, , , and receiver dual-frequency IF code and phase biases , . is the tropospheric delay, and are the noise including multipath of dual-frequency IF code and carrier-phase measurements, and and represent the wavelength and ambiguity of dual-frequency IF combination, which is given as where and are the integer ambiguities of L1 and L2 frequencies, and are the wide-lane and narrow-lane integer ambiguities formed by and , and and represent the wavelength of wide-lane and narrow-lane combination, respectively. In float PPP solution, the satellite code clock error can be provided by IGS precise clock products. The position, tropospheric delay, and receiver code clock error , 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 aswhere is the single-difference between-satellite operator. is the single-difference float ambiguity. and 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 aswhere is the single-difference MW combination, WL represents wide-lane combination while NL means narrow-lane combination, is the satellite MW bias, and , , , and represent the satellite phase and code biases of L1 and L2 frequencies. , , , and 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 bywhere 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 aswhere is the float ambiguity of . Similar to calculation of wide-lane FCB corrections, FCB corrections at a receiver in a network after convergence can be calculated byBy averaging the calculated FCB corrections from multiple receivers in a network, a precise value can be achieved. When users receive and apply the FCBs in the carrier-phase measurements, the integer property of ambiguities can be recovered. Then, 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 aswhere is the undifferenced triple-frequency IF carrier-phase measurement, is the wide-lane L1 and L2 carrier-phase measurement, is the wide-lane L2 and L5 carrier-phase measurement, (satellite phase clock error) and (receiver phase clock error) are a function of the actual satellite clock error , receiver clock error , and satellite and receiver triple-frequency IF phase biases , . and represent the wavelength and ambiguity of triple-frequency IF combination, which is given asSimilar to (3), after single-difference between satellites, the observation equation can be rewritten aswhere is the single-difference between-satellite operator. can be provided by IGS precise clock product and 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 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, . 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 has been proved, to obtain relative stable triple-frequency IF float ambiguities, one straightforward way is to recover after correcting the difference between and , namely, transforming to

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

As mentioned above, the IFCB is unstable, which is absorbed into float ambiguities, resulting in incorrect position estimation and unstable float ambiguity. In order to transform to , the difference between and 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 aswhere 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 needs to be eliminated, which can be achieved by calculating the difference between two consecutive epochs, given bywhere is the difference between the single-difference dual-frequency and triple-frequency IF phase measurements, is the at epoch , and is the difference of at two consecutive epochs. It can be seen from (12) that is the change of IFCB between two consecutive epochs. In order to improve the redundancy to calculate , datasets of a network are usually utilized to calculate the elevation-angle-dependent weighted average aswhere is obtained at station in the network, is the corresponding satellite elevation angle at station, represents the reference satellite elevation angle at station, and represents the weight of at station.

The IFCB at any epoch can be calculated by adding from the beginning and the IFCB at the initial epoch, expressed aswhere is the IFCB at epoch and is the IFCB at the initial epoch, which can be set as arbitrary value.

So far, can be obtained, which can be applied as another correction to triple-frequency IF carrier-phase measurements as . One thing needs to be mentioned is that there is a constant bias between the IFCB calculated and the actual value shown as belowThis 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 asBy 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 bywhere is the float WL ambiguity, is the IFCB obtained by (14), is the less than one cycle part of the bias in (15), and is the pseudo WL ambiguities after absorbing the majority part of bias in (15). Due to application of IFCB, the dual-frequency IF satellite phase bias 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 asThe 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 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 aswhere is the area of ambiguity search space, is the defined threshold, is the determinant of float ambiguity variance-covariance matrix, and and 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 Figure 1. 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.