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:


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