The Integer Recovery Clocks (IRC) method

The Integer Recovery Clocks (IRC) method is presented in Laurichesse and Mercier (2007), Laurichesse et al. (2008, 2009a, 2010), Laurichesse (2011), Collins (2008) and Collins et al.

(2008). In Collins (2008) and Collins et al. (2008), the IRC method is referred to as the De-coupled clock model. The general principle of the IRC method is to provide separate code-phase and carrier-code-phase satellite clock corrections. This eliminates FCBs from carrier-code-phase measurements and enables ambiguity resolution.

The ionosphere-free measurement combination is calculated as shown in equations (3.11) and (3.12) for code-phase and carrier-phase measurements in metres, respectively. When the ambiguity term (𝑏₃^𝑖) in the ionosphere-free measurement combination is solved in equations (4.19) and (4.20), the ambiguity term includes carrier-phase (𝐵_𝐿3,𝑟 − 𝐵_𝐿3,𝑠^𝑖 ) and

79 code-phase (𝐵_𝑃3,𝑠^𝑖 − 𝐵_𝑃3,𝑟 ) biases (Collins, 2008). Therefore, the ambiguity term (𝑏₃^𝑖) is not an integer even when all measurement errors excluding carrier and code-phase biases are corrected perfectly and the distance between the receiver and satellite is known exactly (Collins, 2008). The symbols in the equations are following: 𝑑^𝑖 is the distance between the receiver and satellite, 𝛿𝑑𝑇₃ the receiver clock error, 𝛿𝑠^𝑖 the satellite clock error, 𝐷_{𝑡𝑟𝑜𝑝}^𝑖 the tropospheric error, 𝑀₃ ^𝑖 the code-phase multipath error, 𝑚₃^𝑖 the carrier-phase multipath error, 𝑄₃^𝑖 the code-phase measurement noise, 𝑞₃^𝑖 the carrier-phase measurement noise, 𝐵_𝑃3,𝑟 the receiver code-phase bias term, 𝐵_𝑃3,𝑠^𝑖 the satellite code-phase bias term, 𝐵_𝐿3,𝑟 the receiver carrier-phase bias term, 𝐵_𝐿3,𝑠^𝑖 the satellite carrier-phase bias term and 𝑖 the satellite.

{ 𝑃₃^𝑖 = 𝑑^𝑖+ 𝛿𝑑𝑇_𝑃3− 𝛿𝑠_𝑃3^𝑖 + 𝐷_{𝑡𝑟𝑜𝑝}^𝑖+ 𝑀₃ ^𝑖 + 𝑄₃^𝑖 𝐿^𝑖₃ = 𝑑^𝑖 + 𝛿𝑑𝑇_𝑃3− 𝛿𝑠_𝑃3^𝑖 + 𝐷_{𝑡𝑟𝑜𝑝}^𝑖 + 𝑚₃^𝑖 + 𝑞₃^𝑖 + 𝑏₃^𝑖

(4.19)

{

𝛿𝑑𝑇_𝑃3= 𝛿𝑑𝑇 + 𝐵_𝑃3,𝑟 𝛿𝑠_𝑃3^𝑖 = 𝛿𝑠^𝑖 + 𝐵_𝑃3,𝑠^𝑖

𝑏₃^𝑖 = 𝜆₃(𝑁₃^𝑖 + 𝐵_𝐿3,𝑟 − 𝐵_𝐿3,𝑠^𝑖 ) + 𝐵_𝑃3^𝑖 − 𝐵_𝑃3,𝑟

(4.20)

The motivation to develop the IRC method was to prevent ambiguities from being affected by code biases when employing the traditional float solution PPP model, which is similar to the model used in Section 4.2 (Collins, 2008). Satellite and receiver clock errors are estimated separately for the carrier-phase and code-phase measurements when employing the IRC method. The measurement model for the IRC method is shown in equations (4.21) and (4.22), where 𝛿𝑑𝑇_𝑃3 is the receiver code-phase clock error, 𝛿𝑠_𝑃3^𝑖 the satellite code-phase clock error, 𝛿𝑑𝑇_𝐿3 the receiver carrier-phase clock error, 𝛿𝑠_𝐿3^𝑖 the satellite carrier-phase clock error and 𝜆₃ the wavelength of the ionosphere-free combination (Collins, 2008).

The benefit of estimating the separate code-phase and carrier-phase clock errors, compared to the traditional model in equation (4.19), is that ambiguities have an integer nature. This enables carrier-phase ambiguity resolution.

{ 𝑃₃^𝑖 = 𝑑^𝑖+ 𝛿𝑑𝑇_𝑃3− 𝛿𝑠_𝑃3^𝑖 + 𝐷_{𝑡𝑟𝑜𝑝}^𝑖+ 𝑀₃ ^𝑖 + 𝑄₃^𝑖 𝐿^𝑖₃ = 𝑑^𝑖+ 𝛿𝑑𝑇_𝐿3− 𝛿𝑠_𝐿3^𝑖 + 𝐷_{𝑡𝑟𝑜𝑝}^𝑖 + 𝑚₃^𝑖 + 𝑞₃^𝑖+𝜆₃𝑁₃^𝑖

(4.21)

80 {

𝛿𝑑𝑇^𝑃3 = 𝛿𝑑𝑇 + 𝐵_𝑃3,𝑟 𝛿𝑠_𝑃3^𝑖 = 𝛿𝑠^𝑖 + 𝐵_𝑃3,𝑠^𝑖 𝛿𝑑𝑇_𝐿3 = 𝛿𝑑𝑇 + 𝜆₃𝐵_𝐿3,𝑟

𝛿𝑠_𝐿3^𝑖 = 𝛿𝑠^𝑖 + 𝜆₃(𝐵_𝐿3,𝑠^𝑖 )

(4.22)

Expressing the ionosphere-free fixed ambiguity term (𝑁₃^𝑖) in equation (4.21) as multiples of 6 mm, which is the wavelength of the ionosphere-free combination, is possible in a theory (Dach et al., 2007). However, it is not practical because the wavelength is too short compared to the typical total magnitude of the errors. Therefore, it is not possible to resolve the ambiguity term (𝑁₃^𝑖) directly in practice. To overcome the issue, the ionosphere-free ambiguity term (𝑁₃^𝑖) is decomposed into the wide-lane (𝑁_𝑤𝑙^𝑖 ) and narrow-lane (𝑁_𝑛𝑙^𝑖 ) terms as in equation (4.23) (Collins, 2008). The wavelength of the wide-lane combinations is 86 cm and the wavelength of the narrow-lane combination is 10.7 cm (Collins, 2008).

𝑁₃^𝑖 = 𝑓₁

𝑓₁+ 𝑓₂(𝑁_𝑛𝑙^𝑖 ) + 𝑓₁𝑓₂

𝑓₁²− 𝑓₂²𝑁_𝑤𝑙^𝑖 (4.23)

Wide-lane ambiguities are estimated using the Melbourne-Wubbena combination (Melbourne, 1985, Wubbena, 1985). The combination is calculated using equation (4.11).

The magnitude of noise and multipath in the wide-lane estimates depends on the receiver, antenna and environment around the receiver (Collins, 2008).

Narrow-lane float ambiguities are calculated based on the float ionosphere-free ambiguities and fixed wide-lane ambiguities as shown in equation (4.18) (Collins, 2008). The fixed ionosphere-free ambiguity (𝑁₃^𝑖) can be obtained after fixing both the wide-lane (𝑁_𝑤𝑙^𝑖 ) and narrow-lane (𝑁_𝑛𝑙^𝑖 ) ambiguities for the satellite as described in equation (4.23) (Collins, 2008).

Data from a global GNSS reference network is required for IRC correction estimation. For example, data from 32 reference stations (Collins, 2008) or from 50 stations (Laurichesse et al., 2010) are used to estimate IRC corrections. It is beneficial to use as many geographically diverse located stations as possible in the calculation in order to improve the accuracy of the corrections (Ge et al., 2012). In general, the larger number of stations adds redundancy to the estimation and geographical diversity ensures that as many satellites as possible are tracked by the stations. Nevertheless, increasing the number of stations also increases the

81 amount of computational power required, which could make real-time processing impossible (Ge et al., 2012). Real-time IRC correction generation is demonstrated, for example, in Laurichesse (2010).

In the network processing, float wide-lane ambiguity estimates, obtained using the Melbourne-Wubbena combination, can be filtered using a sliding window to reduce measurement noise. Equation (4.11), which is used to estimate wide-lane ambiguities, is singular, because the wide-lane ambiguity (𝑁_𝑤𝑙^𝑖 ), wide-lane receiver FCB (𝐵_{𝑤𝑙,𝑟} ) and wide-lane satellite FCB (𝐵_{𝑤𝑙,𝑠}^𝑖 ) terms cannot be separated (Mercier and Laurichesse, 2008). Having a singular equation means that an infinite number of solutions is possible.

A method to perform un-differenced wide-lane estimation using data from a GNSS receiver network is shown in Laurichesse et al. (2009c). To address the singularity issue, the FCB is first estimated for a station with a stable receiver FCB. At the station, the receiver FCB (𝐵𝑤𝑙,𝑟 ) is set to zero. Thereafter, wide-lane ambiguities are fixed by rounding the ambiguities to the nearest integers and the satellite FCBs are estimated as shown in equation (4.24). The satellite wide-lane FCBs are then applied in the other stations. In these stations, the receiver FCB can be estimated using the least-squares method and the initial value for the receiver FCB can be obtained based on equation (4.25), where 𝑁_𝑤𝑙^𝑖 is the closest integer. The ambiguity term must be rounded to the closest integer to separate ambiguity from receiver FCB. Finally, the batch least-squares method is employed all over the network to obtain new estimates of the receiver and satellite FCBs (Laurichesse et al., 2009c). The previous FCB estimates are used as initial values in the estimation.

𝐵_{𝑤𝑙,𝑠}^𝑖 = 𝑏_𝑤𝑙^𝑖 − 𝑁_𝑤𝑙^𝑖 (4.24)

𝐵_{𝑤𝑙,𝑟} = 𝑏_𝑤𝑙^𝑖 − 𝑁_𝑤𝑙^𝑖 − 𝐵_{𝑤𝑙,𝑠}^𝑖 (4.25)

IRC corrections can be estimated using the Kalman filter method as in Laurichesse et al.

(2010) or the least-squares method as in Collins (2008). The number of Kalman filter states estimated for 34 satellites using 50 stations is shown in Table 4.2 (Laurichesse et al., 2010).

82

Kalman filter state type Quantity Total number

Satellite carrier-phase clock error One per satellite 1×34 Station carrier-phase clock error One per station 1×50 Satellite code-phase clock error One per satellite 1×34 Station code -phase clock error One per station 1×50

Tropospheric wet delay One per station 1×50

Station coordinate corrections Three per station 3×50 Satellite orbit corrections Three per satellite 3×34 Narrow-lane carrier-phase ambiguities 12 per station 12×50

Total number of states 1070

Table 4.2 Kalman filter states for IRC corrections estimation (Laurichesse et al., 2010) For the IRC correction estimation, initial satellite orbit estimates can be calculated based on the IGS Ultra-rapid orbit predictions (Laurichesse et al., 2010). However, the accuracy of these predictions is not always sufficient (with errors larger than 10.7 cm) (Laurichesse et al., 2010). Therefore, additional orbit corrections which are relative to the IGS Ultra-rapid corrections are estimated using EKF when estimating satellite clock corrections (Laurichesse et al., 2010).

Station coordinates can be fixed to the known values in the estimation (Laurichesse et al., 2010). Tropospheric wet delay is estimated as a Kalman filter state at each station. The IRC correction estimation begins with wide-lane and narrow-lane float ambiguity estimation.

The wide-lane ambiguities must be fixed first, for example, after 30 minutes in the case of the implementation presented in Laurichesse et al. (2010). The reason for not fixing wide-lane ambiguities immediately is to let ambiguities to converge to ensure stability of the float ambiguity solution.

The system used to estimate IRC corrections is singular, because of the singularity between the receiver carrier-phase clock error, satellite carrier-phase clock error and narrow-lane ambiguity and singularity between the receiver and satellite code-phase clock errors (Collins, 2008). Therefore, one station with a stable receiver clock error is selected and the receiver clock errors are fixed to zero for this station. To remove the singularity between the narrow-lane ambiguities and carrier-phase clock error, one narrow-narrow-lane ambiguity in each station is fixed to an arbitrary value.

IRC corrections with, for example 5 s data-rate can be estimated using a Kalman filter (Laurichesse et al., 2010). Selecting the data-rate is a compromise between the

83 computational power requirement and performance of the corrections. The corrections delivered to rovers are wide-lane FCB and code-phase and carrier-phase clocks for each satellite. The satellite wide-lane FCBs can be delivered as daily files while code-phase and carrier-phase clock corrections are updated, for example, every five seconds (Laurichesse et al., 2010).