6.3.1 GPS Observation Modelling

INTRODUCTORY REMARKS

All GPS pseudo-range and carrier phase observations may be modelled as (section 6.1.1): Obsji = ji + bj + bi + bji + vji where ji is geometric range from stn.j to sat.i, bi are the satellite dependent biases, bj are the station dependent biases, bji are the observation dependent biases, and vji is the measurement noise. Only the geometric range contains the coordinate parameters of interest. All biases basically influence both the pseudo-range and integrated carrier phase observations by the same amount (there are some frequency-dependent effects), however only the integrated carrier phase observations contain the ambiguity bias, which is a constant for a satellite-receiver pair as long as the instrument remains locked onto the satellite. The measurement noise is about 2-3 orders of magnitude higher in pseudo-range data than for integrated carrier phase data.

A sample data file record for one epoch (date 26/7/92, time 6:52:30 UT) to the five satellites PRN2, 11, 18, 19, 28 (one record of five observation data types to each satellite) will help illustrate the impact of some of the measurement biases and errors:

Epoch marker--> 92 7 26 6 52 30.0000000 0 5 2 11 18 19 28

C/A pseudo-range (m)	L1 phase (L1 cycles)	L2 phase (L2 cycles)	L1 pseudo-range (m)	L2 pseudo-range (m)
PRN2 --> 22333042.96600	-12176065.17700	-9487835.16600	22333025.72200	22333027.39100
PRN11--> 22934353.22700	-7227959.70200	-5632169.61500	22934338.21900	22934340.52000
PRN18--> 20485466.29600	-23339757.04000	-18186808.56800	20485447.35300	20485447.55500
PRN19--> 20609091.79300	-20568272.40900	-16027208.73300	20609081.52200	20609081.81200
PRN28--> 23894196.98000	154816.73100	120640.79300	23894184.83300	23894185.36800

The following comments can be made concerning the various quantities:

• All five observation types are biased by the same amount (equivalent range) by the receiver and satellite clock errors, and the tropospheric delay.
  
• The phase observations have negligible noise. The P code pseudo-ranges have observation noise of a few decimetres, while the C/A code pseudo-ranges are the "noisiest". The multipath error (if present) is greatest for the C/A pseudo-ranges, and least for the phase measurements.
  
• The ionosphere accounts for most of the difference in the pseudo-range measurements on L1 and L2. This is equivalent to the difference in the L1 and L2 phase observations when they are converted to range (metric) equivalent values.
  
• The ionospheric delay on the C/A pseudo-range is equal to that on the L1 pseudo-range, and equal in value, though not in sign, to that of the L1 phase (when expressed in metric units).
  
• The ionospheric delay on pseudo-range measurements means that they measure range that is longer than "true", but that the phase observations are shorter than "true".
  
• The (unknown) ambiguity on the L1 phase is different to that of the L2 phase, and is different for each satellite.

Summary Remarks: Handling GPS Biases and Errors

Depending upon the level of accuracy sought, the various GPS biases and errors may be considered significant or not, and different options used in accounting for these effects. Below in Table below are summarised the options identified in section 6.2 for those applications requiring carrier phase data processing. (In section 7.2.2 the options appropriate for GPS surveying applications are discussed further.)

Options for handling GPS measurement biases and error

Bias or Error	A	B	C	D	E
Satellite Clock	*1	*1
Satellite Orbit	*2	*3
Receiver Clock	*1	*1
Fixed Reference Station	*4				*5
Ionospheric Delay	*6	*7	*8	*9	*10
Tropospheric Delay	*11	*12	*13	*14	*15
Phase Ambiguity 1b	*16	*17
Cycle Slips	*18	*19			*19
Antenna Phase Centre Movement					*20
Multipath					*21
Measurement Noise					*

A	parameter estimated	B	bias eliminated by differencing
C	bias measured	D	bias modelled

E	bias ignored, and assumed to be an error

Comments to the Table above:

1	these can be shown to be mathematically equivalent
1b	only for integrated carrier phase measurements
2	only practical for high precision GPS geodesy using specialised scientific software
3	orbit bias significantly reduced in baseline solutions -- "rule-of-thumb": baseline error is 1ppm per 20m orbit error.
4	completely free solution for all station coordinates not usually feasible
5	care should be taken to ensure that errors in reference station coordinates are not larger than orbit errors; if orbit is adjusted then this bias is absorbed into estimated orbit parameters
6	feasible if dual-frequency observations are not available
7	if single frequency observations are only available, short baseline results only mildly affected by residual ionosphere (generally highly correlated at both ends of the baseline), and effect tends to be at the 1-2ppm level or below
8	dual-frequency observations are a very satisfactory option for almost entirely eliminating ionospheric bias for many applications
9	broadcast ionospheric model is rather poor, accounting for 50% of effect
10	operational procedures to minimise residual ionospheric effect on baselines, for example, observe at night
11	estimated as a constant scale factor, or more elaborate stochastic parameter
12	as in the case of ionospheric bias, short to medium baseline results only mildly affected by residual wet troposphere component (generally highly correlated at both ends of the baseline), and effect tends to be at the few ppm level
13	can be measured directly by Water Vapour Radiometers, but not feasible for standard GPS surveys
14	models of tropospheric delay accounts for about 90% of bias (at a single station)
15	ignore residual (baseline) tropospheric bias
16	estimate ambiguity as real-valued parameter in an "ambiguity-free" solution
17	elimination of ambiguity by between-epoch observation differencing
18	carried out during data "pre-processing"
19	residual cycle slips can be ignored in triple-difference solutions
20	operational procedures can be used to eliminate effect on baseline solutions
21	avoid multipath environments

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© Chris Rizos, SNAP-UNSW, 1999