Geometric Dilution of Precision (GPS)

This section derives dilution of precision (DOP) factors that related the URE to the expected position estimation accuracy. In the discussion of this section, it is assumed that H includes a rotation matrix so that the vector p, which contains the first three states of x, represents horizontal position and altitude (i.e., north, east, and down).

After convergence of the GPS update of eqn. (8.13), we expect that dx = 0 which implies that

Also, from eqns. (8.6) and (8.11), we have that

where x is defined in eqn. (8.7). Combining these equations yields

where the number of measurements m is greater than three and the inverse matrix exists.

Assuming that the error vector x is zero mean with covariancewe can determine the expected value and covariance of the position error. The expected value ofis

therefore, the estimate of the receiver antenna position and receiver clock bias is expected to be unbiased. This statement should be interpreted with caution. The ensemble average has an expected value of zero. In fact, if a temporal sequence of positions was averaged long enough, the position error should converge to zero. However, the temporalaverage has significant time correlation due to the range errorsall being time correlated. Therefore short-term averages can contain significantly biased errors.

DefineFrom eqn. (8.80) the covariance matrix P can be computed as shown in eqn. (8.81):

This P matrix could be used with the methods of Section 4.9.1 to define probability ellipses. Define the matrixBased on eqn. (8.81),which shows that the variance of the x estimation error is determined by the product of the URE variance a2 and the matrix G which is determined by the geometry of the satellites relative to the user. Due to the structure of the covariance matrix, as discussed in eqn. (4.24), the variance of the position estimates and various useful error metrics can be computed, for example:

andThe matrix G defines various dilution of precision (DOP) factors that are convenient for specifying the amplification of the URE in the estimation of specific portions of x. Typically used DOP factors are

where VDOP quantifies the magnification of URE in estimating altitude, HDOP quantifies the magnification of URE in estimating the horizontal positionetc.

Most GPS receivers calculate and can be configured to output the various DOP factors and URE. This allows the user (or real-time software) to monitor the expected instantaneous estimation accuracy. GPS planning software is also available which allows users to analyze the DOP factors as a function of time and location to determine either satisfactory or optimal times for performing GPS related activities.

The matrix H is a function of the number of satellites in view and the geometry of the line-of-sight vectors h* from the satellites to the user. The effect of satellite-to-user geometry on position estimation accuracy is illustrated in Figure 8.10 using two dimensions.

Figure 8.10: Two dimensional illustration of the relationship between satellite-to-user geometry and the resulting position accuracy.

This figure shows two possible user-satellite configurations. In each configuration, a receiver measures the range to two satellites at the indicated positions. Due to measurement errors, the range measurement is not known exactly, but the receiver position is expected to lie between the two indicated concentric circles. In the left half of the figure the receiver vectors h1 and h2 are nearly collinear; the intersection of the two concentric circles for the two satellites results in a long thin region of possible positions with the largest uncertainty direction orthogonal to the satellite vectors. In this case, the DOP factor would be large. In the right half of the figure the vectors h1 and h2 are nearly orthogonal; the intersection of the concentric circles for each satellite results in a much more equally proportioned uncertainty region.

The measurement matrix H is not constant in time because of the GPS satellite positions change as they orbit Earth. Large DOP values result when the rows of H are nearly linearly dependent. This could be measured by the condition of the matrix HTH, but this approach is not often used or necessary. There may be times when for a given set of four satellites the H matrix approaches singularity and GDOP approaches infinity. This condition is called a GDOP chimney, see Figure 8.11. This was a difficulty in the early days of GPS, when satellites had fewer tracking channels or a limited number of satellites were in view. Fortunately, there are usually more than four satellites in view and modern receivers track more than the minimum number of satellites; hence, either all or an optimal set of the tracked satellites can be selected to minimize a desired DOP URE amplification factor.

Example 8.4 Consider the following case for which six satellites are in view of the receiver antenna. The rows of the H matrix for each satellite are:

Figure 8.11: GDOP chimney for a set of four satellites.

SV 2	-0.557466	+0.829830	-0.024781	+1.000000
SV 24	+0.469913	+0.860463	+0.196942	+1.000000
SV 4	+0.086117	+0.936539	-0.339823	+1.000000
SV 5	+0.661510	-0.318625	-0.678884	+1.000000
SV 7	-0.337536	+0.461389	-0.820482	+1.000000
SV 9	+0.762094	+0.267539	-0.589606	+1.000000

For this set of six satellites, there are fifteen different combinations of four satellites, as shown in Table 8.6. The GDOP figures for the various sets of four satellites range from 3.58 to 16.03. Instead of selecting the best four satellites, an all-in-view approach using all six satellites would result in a GDOP of 2.96. A

A comparison of the above discussion with the discussion of observability in Section 3.6.1 shows that the various DOP factors are measures the degree of observability of a selected set of position states are at a given time instant (and location) for a given set of satellites. If observability is temporarily lost (e.g., due to satellite occlusion), some of the DOP factors become infinite and a full position solution is not possible by stand-alone GPS. In an aided navigation system, the integration of the high rate sensors through the vehicle kinematic equations will maintain an estimate of the vehicle state which includes the position. In addition, the navigation system will maintain an estimate of the state error covariance matrix denoted as P. During the time period when PDOP is infinite, some linear combination of the states will be unobservable from the available GPS measurements. The Kalman filter will use the available measurements in an optimal fashion to maintain the estimate accuracy to the extent possible.

SV Combination	GDOP	Satellite Combination	GDOP
24, 2, 4, 5	3.72	24, 2, 4, 7	6.25
24, 2, 4, 9	4.86	24, 2, 7, 5	4.48
24, 2, 9, 5	5.09	2, 7, 4, 5	3.58
2, 9, 4, 5	16.03	24, 7, 4, 5	8.63
24, 9, 4, 5	5.45	7, 9, 4, 5	8.59
24, 9, 4, 7	6.96	7, 9, 4, 2	4.96
24, 9, 5, 7	4.89	7, 9, 5, 2	3.92
24, 9, 2, 7	13.10

Table 8.6: Example satellite combinations and corresponding GDOP values.

The unobservable portions of the state vector will be indicated by growth in the corresponding directions of the P matrix. This could be analyzed via a SVD or eigen-decomposition of P.

This section has worked entirely with covariance of position error. Other performance metrics (e.g., CEP, R95, 2drms, etc.) are discussed in Section 4.9.1.