2.4.4 How Good is GPS?

RELATIVE POSITIONING ACCURACY PERFORMANCE

Correlated GPS Measurement Biases

Relative positioning is the most effective means of accounting for many of the troublesome GPS measurement biases, and hence is the basis for all high precision GPS positioning techniques. There are different implementations of relative or differential positioning.

The correlated nature of biases is best illustrated by an examination of the basic GPS pseudo-range measurement model (§6.2):

The subscript in brackets refers to the GPS station "i", and the superscript in brackets refers to the satellite "p". P is the measured pseudo-range, is the true geometric range from one receiver to one satellite, _rc is the receiver clock error, ^sc is the satellite clock error, ^orbit(i,p) is the satellite orbit error mapped into the range, _atmos(i,p) is the atmospheric refraction error, and _i^p are the remaining errors and biases not explicitly accounted for in the above observation model. Although the time argument has been dropped for the sake of clarity, all quantities in the above equation vary with time, and hence the equation represents a "snapshot" of a GPS pseudo-range measurement at a single epoch, or instant of time.

The spatially correlated nature of many GPS errors is obvious when an observation from another station "k" to the same satellite, at the same epoch, is modelled as in the above equation:

The following comments can be made with regard to these two equations:

• The receiver clock error _rc(i) systematically affects all measurements made at station "i" to all satellites by exactly the same amount. It is completely unrelated to the value of _rc(k) at the same measurement epoch. _rc(i) and _rc(k) across different epochs may be considered uncorrelated.
  
• The satellite clock error ^sc(p) systematically affects all measurements made to satellite "p" by any GPS receiver making a measurement at the same time-of-transmission. Hence ^sc(p) is spatially correlated (across different receivers) at an epoch, and this property can be exploited to overcome the effect of this error/bias.
  
• The satellite orbit error ^orbit(i,p), although explicitly associated with satellite "p", is mapped differently as a range error in the case of measurements made by station "i" compared to those made by station "k" (hence the use of the double index identifying both the receiver and satellite involved in the measurement). However, if the two stations are close together the residual bias (^orbit(i,p) - ^orbit(k,p)) will be very small. ^orbit(i,p) across different epochs will change comparatively slowly (hence there is a high temporal correlation).
  
• The atmospheric refraction error _atmos(i,p) expressly tags the measurement made to satellite "p" by station "i". However, if the two stations are close together the atmospheric conditions along the signal line-of-sight to the satellite can be expected to be very similar, and hence the residual bias (_atmos(i,p) - _atmos(k,p)) will be quite small in magnitude. The atmospheric refraction effect has an ionospheric and tropospheric component, each with their own spatial and temporal characteristics. _atmos(i,p) across different epochs will change comparatively slowly (hence a high temporal correlation).
  
• The residual error term _k^p contains the random effects of measurement noise (which will vary according to satellite, receiver and measurement epoch), disturbing biases which are not spatially correlated (their effects are too dissimilar at different stations), and any other biases not explicitly included in these two equations.

Characterising Accuracy of Differential Positioning

The accuracy of a relative position has two components, due to the two classes of errors contained within the term:

• The random measurement noise DOES NOT influence accuracy as a function of receiver separation. The magnitude of relative position error of two stations due to the random measurement noise is a function of:
  • The type of measurement, therefore carrier phase measurements are likely to contribute just a few millimetres, whereas in the case of pseudo-range measurements this may range from several decimetres to a few metres.
  • The degree of redundancy, influenced by such factors as the number of satellites tracked, the number of observation epochs, the degree of freedom of the solution (kinematic positioning has far less degrees of freedom than static positioning), etc.
  • The quality of the antenna centring over the physical station markers, including the stability of the electrical antenna centre.
  • The type of carrier phase solution, ranging from the millimetre level in the case of the best "ambiguity-fixed" solution, to several centimetres or a few decimetres where the carrier phase ambiguity is not fixed to an integer, or it is eliminated from the observation model through between-epoch differencing (see section 8.1.3).

     

• The residual biases that remain will largely influence accuracy as a function of receiver separation. These residual biases arise mainly because the satellite orbit errors and the atmospheric biases are not eliminated when observations from two receivers are combined. Their effect on relative position determination is greater for long baselines than for short baselines. This error signature is usually expressed as the ratio of the magnitude of the error to baseline length, as some many "parts per million" (ppm). The Figure below illustrates the relationship between relative error (ppm), relative error (centimetres), and baseline length. The dependency of relative error to baseline length has only been observed in GPS surveying results using carrier phase observations because the noise of the measurements is generally very much lower than the residual biases.

GPS relative accuracy as a function of baseline length.

Gross errors in the observations will largely propagate in a similar fashion to the random error effects. Independent testing of GPS surveying systems over a variety of baseline lengths has estimated that relative positioning error between static receivers typically is 1-10ppm, with a small constant term of 0.3-1cm (section 4.4.3).

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© Chris Rizos, SNAP-UNSW, 1999