Positioning With

FIGURE 9.10 Range from satellite s to ground receiver r.

FIGURE 9.10 Range from satellite s to ground receiver r.

As already mentioned, the fundamental GPS observable is the signal travel time between the satellite and the receiver. However, the receiver clock that measures the time is not perfect and may introduce an error to the measured pseudorange (even though we limit our discussion here to pseudoranges, the same applies to the carrier phase measurement that is indirectly related to the signal transit time, as the phase of the received signal can be related to the phase at the epoch of transmission in terms of the signal transit time). Thus, in order to determine the most accurate range, the receiver clock correction must be estimated to bring the receiver clock to synchronization with the satellite clock, and its effect must be removed from the observed range. The synchronization error between satellite and receiver clocks is particularly important given that an error of only 0.1 microsecond in the satellite or receiver clocks results in distance error on the order of 30 m. Hence, the fourth pseudorange measurement is needed, because the total number of unknowns, including the receiver clock, is now four. If more than four satellites are observed, a least-squares solution is employed to derive the optimal solution.

9.7.1 Point versus Relative Positioning

There are two primary GPS positioning modes: point positioning (or absolute positioning) and relative positioning. However, there are several different strategies for GPS data collection and processing, relevant to both positioning modes. In general, GPS can be used in static and kinematic modes, using both pseudorange and carrier phase data. GPS data can be collected and then postprocessed at a later time, or processed in real time, depending on the application and the accuracy requirements. In general, postprocessing in relative mode provides the best accuracy.

9.7.1.1 point (Absolute) positioning

In point, or absolute positioning, a single receiver observes pseudoranges to multiple satellites to determine the user's location. For the positioning of the moving receiver, the number of unknowns per epoch equals three receiver coordinates plus a receiver clock correction term. In the static mode with multiple epochs of observations, there are three receiver coordinates and n receiver clock error terms, each corresponding to a separate epoch of observation 1 to n. The satellite geometry and any unmodeled errors will directly affect the accuracy of the absolute positioning.

9.7.1.2 Relative positioning

The relative positioning technique (also referred to as differencing mode or differential GPS, DGPS) employs at least two receivers: a reference (base) receiver, whose coordinates must be known, and the user's receiver, whose coordinates can be determined relative to the reference receiver. Thus, the major objective of relative positioning is to estimate the 3D baseline vector between the reference receiver and the unknown location. Using the known coordinates of the reference receiver and the estimated AX, AY, and AZ baseline components, the user's receiver coordinates in WGS84 can be readily computed. Naturally, the user's WGS84 coordinates can be further transformed to any selected reference system.

An observable in differencing mode is obtained by differencing the simultaneous measurements to the same satellites observed by the reference and the user receivers (between receiver differencing), or through "between satellite differencing" and "between epoch differencing." The most important advantage of relative positioning is the removal of the systematic error sources (common to the base station and the user or both satellites and epochs of observation) from the observable, leading to the increased positioning accuracy. Because for short to medium baselines (up to ~40 to 60 km) the systematic errors in GPS observables due to troposphere, satellite clock, and broadcast ephemeris errors are of similar magnitude (i.e., they are spatially and temporally correlated), the relative positioning allows for a removal or at least a significant mitigation of these error sources, when the observables are differenced. In addition, for baselines longer than 10 km, the ionosphere-free linear combination must be used (if dual-frequency data are available) to mitigate the effects of the ionosphere (Hofman-Wellenhof et al., 2001). Single-frequency users are limited to short baselines, unless differential corrections are provided via DGPS services, as explained in the next section. In summary, the following are the primary consequences of GPS data differencing:

• Elimination or reduction of several bias errors

• Reduction of data quantity

• Introduction of mathematical correlation among data

• Increase of the noise level of the differenced data

The primary differential modes are (1) single differencing mode, (2) double differencing mode, and (3) triple differencing mode. The differencing can be performed between receivers, between satellites, and between epochs of observations as already mentioned. The single-differenced (between-receiver) measurement, <&ktJ, is obtained by differencing two observables to the satellite k, tracked simultaneously by two receivers i (reference) andj (user): <&ktJ = — (see Figure 9.11). By differencing observables from two receivers, i and j, observing two satellites, k and l, or simply by differencing two single differences to satellites k and l, one arrives at the double-differenced (between-receiver/between-satellite differencing) measurement: = - $ j - $t + $j = $kuj - $'lj . Double difference is the most commonly used differential observable. Furthermore, differencing two double differences, separated by the time interval dt = t2 - tx, renders triple-differenced measurement, dt) = §k¡j(tz) — $tj(ti), which in case of carrier phase observables effectively cancels the initial ambiguity term. Differencing can be applied to both pseudorange and carrier phase. However, for the best positioning accuracy with carrier phase double differences, the initial ambiguity term should be first resolved and fixed to the integer value. Relative positioning may be performed in static and kinematic modes, in real time (see the next section), or for the highest accuracy, in postprocessing. Table 9.3 shows the error characteristics for between-receiver single and between-receiver/between-satellite double differenced data.

Basic pseudorange and carrier phase Equation (9.1) through Equation (9.3) represent the functional relationships between the true observations and the underlying parameters. However, the observations (and models involved) are not perfect; thus, the functional models require a respective

FIGURE 9.11 Between-receiver between satellite phase-range double differencing.
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