Mathematical models of pseudorange and carrier phase observables

Pseudorange is a geometric range between the transmitter and the receiver, distorted by the propagation media and the lack of synchronization between the satellite and the receiver clocks. It is recovered from the measured time difference between the epoch of the signal transmission and the epoch of its reception by the receiver. The actual time measurement is performed with the use of the PRN code. In principle, the receiver and the satellite generate the same PRN sequence. The arriving signal is delayed with respect to the replica generated by the receiver, as it travels ~20,000 km. In order to find how much the satellite's signal is delayed, the receiver-replicated signal is delayed until it falls into synchronization with the incoming signal (it is achieved at the point of maximum correlation between the incoming PRN code and the receiver-generated replica). The amount by which the receiver's version of the signal is delayed is equal to the travel time of the satellite's version (Figure 9.8). The travel time, At (~0.06 s), is converted to a range measurement by multiplying it by the speed of light, c.

FIGURE 9.8 Principles of pseudorange measurement based on time observation.

There are two types of pseudoranges: C/A-code pseudorange and P-code pseudorange. The precision of the pseudorange measurement is partly determined by the wavelength of the chip in the PRN code. Thus, the shorter the wavelength, the more precise the range measurement would be. Consequently, the P-code range measurement precision (noise) of 10 to 30 cm is about ten times higher than that of the C/A code. Under the Anti-Spoofing policy, the P-code is encrypted to Y-code, as already explained, resulting in more complicated signal recovery on L2 frequency. Because there is no C/A code on L2, signal correlation, as explained above, does not work anymore, as the receiver cannot generate a replica of the unknown Y-code. Consequently, more sophisticated signal tracking techniques must be used (see, for example, Ashjaee, 1993).

The pseudorange observation can be expressed as a function of the unknown receiver coordinates, satellite and receiver clock errors, and the signal propagation errors (Equation (9.1)):

PSi = pS + 4 + T + c(dtr - dts) + Mi, + esr,i f p2 = PS + 4 + T + c(dtr - dts) + M,2 + &

where pr = sqrt[(Xs - Xr)2 +(YS - Yr)2 + (( - Zr)2 (9.2)

Is f2

Pseudoranges measured between receiver r and satellite s on L1 and L2 Geometric distance between satellite s and receiver r

Range error caused by ionospheric signal delay on L1 and L2 The r-th receiver clock error (unknown)

The s-th satellite clock error (known from the navigation message) The vacuum speed of light

Multipath on pseudorange observables on L1 and L2

Range error caused by tropospheric delay between satellite s and receiver r (estimated from a model)

Measurement noise for pseudorange on L1 and L2 Coordinates of satellite s (known from the navigation message) Coordinates of receiver r (unknown) Carrier frequencies of L1 and L2

Carrier phase is defined as a difference between the phase of the incoming carrier signal and the phase of the reference signal generated by the receiver. At the initial epoch of the signal acquisition, the receiver can measure only the fractional phase, so the carrier phase observable contains the initial unknown integer ambiguity, N. Integer ambiguity is a number of full phase cycles between the receiver and the satellite at the starting epoch, which remains constant as long as the signal tracking is continuous. After the initial epoch, the receiver can count the number of integer cycles being tracked. Thus, the carrier phase observable can be expressed as a sum of the fractional part, 9 (in cycles), measured with millimeter-level precision, and the integer number of cycles counted since the starting epoch, t0. The integer ambiguity can be determined using special techniques referred to as

0 1 ^ FIGURE 9.9 Carrier phase range measurement.

ambiguity resolution algorithms. Once the integer ambiguity is resolved, the ambiguous carrier phase observable can be converted to unambiguous range measurement R = (N + 9)X by multiplying the sum of the measured phase (in cycles) and the initial integer ambiguity (in cycles) by the corresponding wavelength, X (see Figure 9.9). It should be noted that starting from epoch t0, the carrier phase measurement 9 will include not only the fractional part of a cycle, but also the number of full cycles since the initial epoch t0. The phase-range observable, O (in meters; Equation (9.3)), equals the sum of R and all the error sources affecting the measurement. This observable is used in the applications where the highest accuracy is required.

= pr -4 + Ts + XiNS,i + c(dtr - dts) + mS,i + er.1 f

$r,2 = pr - 4 + T + X2NS,2 + c(dtr - dts) + mS,2 + <2 f where

&S.2- Phase-ranges (in meters) measured between station r and satellite 5 on L1 and L2 NS.b NS2: Initial integer ambiguities on L1 and L2, corresponding to receiver r and satellite 5 X « 19 cm and X2 » 24 cm are wavelengths of L1 and L2 msrM msr22: Multipath error on carrier phase observables on L1 and L2 e5,i,£5,2: Measurement noise for carrier phase observables on L1 and L2

Another observation sometimes provided by GPS receivers, and primarily used in kinematic applications for velocity estimation, is instantaneous Doppler frequency. It is defined as a time change of the phase-range, and thus, if available, it is measured on the code phase (Lachapelle, 1990).

Equation (9.2), which is a nonlinear part of Equation (9.1) and Equation (9.3), requires Taylor series expansion to enable the estimation of the three unknown user coordinates (X, Y, Z)r. Secondary (nuisance) parameters in the above equations are satellite and user receiver clock errors,

tropospheric and ionospheric errors, multipath, and integer ambiguities. These are usually removed by differencing mode of GPS data processing (see Section, by empirical modeling (troposphere), or by the processing of dual-frequency signals (ionosphere). As already mentioned, ambiguities must be resolved prior to users' position estimation with carrier phase measurements.

Equation (9.1) through Equation (9.3) are parameterized in terms of Cartesian geocentric coordinates X, Y, Z; however, after the positioning solution is obtained, Cartesian coordinates can be converted to geodetic latitude, longitude, and height, which represent an equivalent triplet of coordinates. Because GPS is a geometric system, its coordinates are related to a reference ellipsoid (WGS84 ellipsoid), whose semimajor axis and flattening are needed to convert Cartesian to and from geodetic coordinates (Torge, 1980). Since GPS provides heights above the WGS84 ellipsoid, in order to convert these heights to topographic (orthometric) heights, a geoid undulation (geoid-ellip-soid separation) must be used. National Geospatial-Intelligence Agency (NGA) provides an online service that calculates geoid undulation for a given location using the latest NGA geoid model ( For more information on geoid and height conversion, the reader is referred to Torge (1980).

The main principle behind positioning with GPS is triangulation in space, based on the measurement of a range (pseudorange or phase-range) between the receiver and the satellites (Figure 9.10). Essentially, the problem can be specified as follows: given the position vectors of GPS satellites (such as ps of satellite 5 in Figure 9.10) tracked by a receiver r, and given a set of range measurements (such as, Prs ) to these satellites, determine a position vector of the user, pr. A single range measurement to a satellite places the user somewhere on a sphere with a radius equal to the measured range. Three simultaneously measured ranges to three different satellites place the user on the intersection of three spheres, which corresponds to two points in space. One is usually an impossible solution that can be discarded by the receiver. Even though there are three fundamental unknowns (coordinates of the user's receiver), the minimum of four satellites must be simultaneously observed to provide a unique solution in space (see Figure 9.1 in Section 9.2), as explained next.

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