Performance of Stand-Alone GPS - Understanding GPS: Principles and Applications - page 329

tively. The value for the PPS is reasonably consistent with observed performance,
but the value for the SPS is significantly pessimistic as compared with observed per-
formance (see Section 7.7). The reason for this discrepancy is that the dominant SPS
UERE component, residual ionospheric delay error, is highly correlated among sat-
ellites. This correlation invalidates one of the assumptions used in the derivation of
the DOPs. When the Klobuchar model's estimate of vertical ionospheric delay is too
high (low), the slant delay estimate for each visible satellite also tends to be too high
(low). A significant component of each residual ionospheric delay error is thus com-
mon to each satellite and drops out into the user clock solution.
With regard to horizontal position errors, (7.36) can be specialized to the hori-
zontal plane yielding

d RK = d (7.56)

2 × n

( dx , dy ) T is the vector component of the position error in the horizontal

where d R =

ρ n ) T is the pseudorange measurement errors, and n is the num-
ber of satellites being used in the position calculation. K 2× n is the upper 2

plane, d = ( d ρ 1 , ... , d

n
submatrix of K and consists of its first two rows. For the standard least square solu-
tion technique, K

×

( H T H ) −1 H T .
For a fixed satellite geometry, (7.56) expresses the horizontal position errors as
a linear function of the pseudorange measurement errors. If the pseudorange errors
are zero mean and jointly Gaussian, d R also has these properties. If the pseudorange
errors are also uncorrelated and identically distributed with variance

=

σ UERE 2 , the

covariance of the horizontal errors is given as

() (

)

− 1

T 2

cov d RHH = σ (7.57)

UERE

22 ×

2 submatrix of ( H T H ) −1 . The where the subscript notation denotes the upper left 2 ×

density function for d R is

1 1

2

[ ]

− 1

() ()

T

f xy , = exp − u cov d R u (7.58)

d R

1
2

[ ]

( )

()

2 π det cov d R

( x , y ) T and det represents the determinant of a matrix.
The density function defines a two-dimensional bell-shaped surface. Contours
of constant density are obtained by setting the exponent in parenthesis to a con-
stant. One obtains equations of the form

where u =

[ ]

()

− 1

T 2

u cov d R u = m (7.59)

where the parameter m ranges over positive values. The contour curves that result
form a collection of concentric ellipses when plotted in the plane. The ellipse
obtained when m equals 1 is termed the 1

ellipse and has the equation

[ ]

− 1

()

u

T

cov d R u = 1 (7.60)