Mohinder S. Grewal - Global Navigation Satellite Systems, Inertial Navigation, and Integration

The DOP can be interpreted as the dilution of the precision from the measurement (i.e. range) domain to the solution (i.e. PVT) domain.

The observation measurement equations in three dimensions for each satellite with known coordinates ( x i, y i, z i) and unknown user coordinates ( X , Y , Z ) are given by

(2.21)

where is the pseudorange to the i th satellite and C bis the residual satellite clock bias and receiver clock bias, where an estimate of the satellite clock error from the GNSS ground control segment has been removed from C b.

These are nonlinear equations that can be linearized using the Taylor series (see, e.g., chapter 5 of Ref. [4]). The satellite positions may be converted to east–north–up (ENU) from Earth‐centered, Earth‐fixed (ECEF) coordinates (see Appendix B).

Let the vector of ranges be Z ρ= h( x), a nonlinear function h( x) of the four‐dimensional vector xrepresenting user solution for position and receiver clock bias and expand the left‐hand side of this equation in a Taylor series about some nominal solution x nomfor the unknown vector

(2.22)

of variables,

for which

(2.23)

where HOT stands for “higher‐order terms.”

These equations become

(2.24)

where H [1]is the first‐order term in the Taylor series expansion:

(2.25)

for v ρis the noise in receiver measurements. This vector equation can be written in scalar form where i is the satellite number as

(2.26a)

for i = 1, 2, 3, 4 (i.e. four satellites).

We can combine Eqs. (2.25)and (2.26a)into the matrix equation with measurements as

(2.26b)

which we can write in symbolic form as

(2.27)

(see table 5.3 in Ref. [4]).

To calculate H [1], one needs satellite positions and the nominal value of the user's position in ENU coordinate frames.

To calculate the geometric dilution of precision (GDOP) (approximately), we obtain

(2.28)

Known are δZ ρand H [1]from the pseudorange, satellite position, and nominal value of the user's position. The correction δ xis the unknown vector.

If we premultiply both sides of Eq. (2.28)by H [1]T, the result will be

(2.29)

Then, we premultiply Eq. (2.29)by ( H [1]T H [1]) −1:

(2.30)

If δ xand δZ ρare assumed random with zero mean, the error covariance ( E = expected value)

(2.31)

The pseudorange measurement covariance is assumed uncorrelated satellite to satellite with variance σ 2:

(2.32)

Substituting Eq. (2.32)into Eq. (2.31)gives

(2.33)

for

and

and the covariance matrix becomes

(2.34)

We are principally interested in the diagonal elements of

(2.35)

that represent the DOP of range measurement error to the user solution error (see Figure 2.4):

Hence, all DOPs represent the sensitivities of user solution error to pseudorange errors. Figure 2.4illustrates the relationship between the various DOP terms.

Figure 2.4DOP hierarchy.