1.4.9 Positioning Strategies

THE IMPACT OF SATELLITE GEOMETRY

The accuracy with which positions can be determined is not just a function of the measurement precision, and the appropriate modelling of biases. It is also a function of the satellite(s) - receiver(s) geometry. Hence, although the systems of eqns (1.4-2), (1.4-6), (1.4-8), (1.4-9), (1.4-11), (1.4-12) are all theoretically valid solutions to the positioning problem, geometric considerations may make a certain solution strategy better than another. The simplest case to consider is point positioning using receiver-biased range measurements -- the GPS navigation mode referred to above.

The co-factor matrix Q from the Least Squares solution contains the contribution to position error of both the geometry and the random measurement error. While in the surveying discipline the components of the co-factor matrix of parameters are transformed into components of an "error ellipsoid" (orientation and length of the three axes), in the case of the navigation applications the effect of satellite configuration geometry is usually expressed by the Dilution of Precision (DOP) factor. DOP is the ratio of the positioning accuracy to the measurement accuracy:

= DOP . o

(1.4-13)

where

o	is the measurement accuracy, and
	is the position accuracy.

DOP is always a number greater than unity when there are no redundant observations.

There are a number of different definitions of DOP factors, depending on the coordinate component, or combination of coordinate components, being considered:

(1.4-14)

where:

E2 , N2 , H2	are the variances of the east, north and height components,
X2 , Y2 , Z2	are the variances of the X, Y and Z components, and
T2	is the variance of the estimated receiver clock error parameter.

are all obtained from the diagonal elements of the co-factor matrix of the Least Squares position solution Q. (All elements have been divided by the variance of unit weight.) The range solution is likely to be in the form of Cartesian coordinate components (X, Y, Z) -- eqn (1.4-2). The corresponding co-factor matrix for the local geographic components (E, N, H) is obtained as follows:

Q = R . Q . RT

(1.4-15)

or

(1.4-16)

In the case of GPS point positioning, which requires the estimation of four parameters: 3-D position and receiver clock error, the most appropriate DOP factor is the Geometric Dilution of Precision (GDOP):

(1.4-17)

GDOP can be interpreted as the reciprocal of the volume of a tetrahedron that is formed from the four satellites and receiver position, hence the best geometric situation for point positioning is when the volume is a maximum, which therefore requires GDOP to be a minimum. Figure 1 below illustrates the situation of good and poor GDOP.

Figure 1. The relationship between satellite configuration geometry and GDOP.

The following comments can be made regarding DOPs:

• The smaller the value of DOP, the higher the precision of the position results -- the measurement errors are not as strongly amplified.
• DOP is usually greater than unity, however, if many satellites are observed (say >8) the value of DOP can be less than unity.
• DOPs can be used as the basis of selecting satellites for solution -- if GPS receiver cannot track all satellites that are in view.
• A high DOP (say >10) defines an "outage" -- a situation where the position solution is too unreliable.
• DOP varies with time of day and geographic location -- but the pattern of DOP at a location repeats itself each day because the constellation is unchanged from day-to-day (except it rising approximately four minutes earlier each day), hence it is highly predictable.
• DOP varies with number of satellites considered-- due to such factors as elevation cutoff angle used, number of satellites used by receiver to give "fix", etc.
• DOP can be computed without the need for any measurements -- only the satellite positions are required (from an appropriate ephemeris) and an approximate receiver position.
• DOP has only a limited role in differential positioning -- it may be useful for certain types of GPS survey planning, as well as for quality assurance.

An example of the variation in PDOP over 24 hours at Sydney, Australia, calculated for all visible satellites above an elevation cutoff angle of 5, is shown in the Figure below.

Figure 2. Variation of PDOP at Sydney, Australia (elevation cutoff 5 degrees).

Further details may be found in LANGLEY (1999).

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© Chris Rizos, SNAP-UNSW, 1999