6.4.11 Dual Frequency Relations

USING CARRIER PHASE DATA TO SMOOTH PSEUDO-RANGE DATA

One way of overcoming the two main problems associated with pseudo-range data, that is: (a) the high measurement noise, and (b) the greater multipath disturbance, in comparison to carrier phase data is to create a pseudo-range / carrier phase combination that, in effect, "smooths" the pseudo-range data. The basis of all data smoothing techniques is to derive the rate of change of range from the carrier phase data (the Doppler data can also be used for this purpose), and to combine this with the absolute measurement of range provided by the pseudo-range data (though it is a biased range).

A first example of a GPS data smoothing technique was described by HATCH (1982), and makes use of dual-frequency phase and pseudo-range data. The basis of the technique is eqn (6.4-26), which is both "geometry-free" and "ionosphere-free". The difference between the wide-lane phase combination and the narrow-lane pseudo-range combination is a constant: the wide-lane ambiguity. Differencing between epochs eliminates this ambiguity term, and hence the rate of change of the (narrow-lane) pseudo-range is identical to the rate of change of the (wide-lane) integrated carrier phase.

The steps involved are (HOFMANN-WELLENHOF et al, 1998):

• At the first observation epoch compute the quantities:

	(6.4-34a)
(L5) = (L1) - (L2)	(6.4-34b)

Set P_(L6)(t1) = P_(ex)(t1) = P_(sm)(t1) .

• For all observation epochs ti after t1, extrapolated values of the pseudo-range are calculated:

(6.4-34c)

• The smoothed value is obtained as the arithmetic mean:

(6.4-34d)

• The above formulae can be generalised for an arbitrary epoch ti following ti-1, by modifying eqn (6.4-34c) slightly:

(6.4-34e)

The above algorithm cannot handle cycle slips in the carrier phase data, therefore a variation to eqns (6.4-34d) and (6.4-34e) is (HOFMANN-WELLENHOF et al, 1998):

(6.4-34e)

where is a time dependent weight factor. For the first epoch = 1, thus giving full weight to the current measured pseudo-range. With the accumulation of further epochs the weight is continuously reduced, and hence increasing the influence of the carrier phase data. A reduction rate of 0.01 in the weight from epoch to epoch has been recommended for 1 second kinematic data. In such a case, after 100 epochs (100 seconds), only the smoothed value from the previous epoch (augmented by the carrier phase rate of change) is taken into account. Again, the occurrence of cycle slips would cause the algorithm to fail. (Cycle slips are detected by comparing the rate of change of the carrier phase data across two epochs with the measured Doppler value times the time interval between epochs.) If a cycle slip is detected, the weight is reset to = 1 and the smoother is restarted (but the cycle slip does not have to be repaired ).

Alternative smoothing algorithms have been developed which use Doppler data in place of carrier phase data. Furthermore, all smoothing algorithms are also applicable to single frequency data, in which case the P_(L6)(t_i) and _(L5)(t_i) are replaced by the observations P_(L1)(t_i) and _(L1)(t_i). Of course such techniques are inferior to the dual-frequency techniques because they cannot account for the ionospheric bias.

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© Chris Rizos, SNAP-UNSW, 1999