International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
One can see that the horizontal accuracy of a single handheld
receiver is 10.2 m, and accuracy of differential measurements
would be 3.1 m. The table is based on 1 6 corresponding to a
68% confidence interval. Note that the user does not have any
assurance to obtain this accuracy in practice. Hence, the need
for an empirical quality measure computed during GPS
measurements at a particular time and place. In the following
we provide the error analysis needed to compute the positional
error of a code receiver on the basis of the coordinates of
known reference points.
Error source Bias Random Total DGPS
Ephemeris data 2.1 0.0 2.1 0.0
Satellite clock 2.0» 0.7 2.1 0.0
Ionosphere 4.0 0.5 4.0 0.4
Troposphere 0.5 0.5 0.7 0.2
Multipath 1:0 21.0 1.4 1.4
Receiver noise 0.5 0.2 0.5 0.5
UERE, rms* , 5.1. 1.4 5.3 1.6
Filtered UERE, rms 621:51:0.4. 5105.1 1.5
VDOP- 2.5 12.8 3.9
HDOP- 2.0 10.2 34.
Table 1 Standard GPS error model based on 1 ©
(corresponding to a 68% confidence interval)- LI C/A (no
selected availability)
Where DGPS is Differential GPS, UERE, is the user equivalent
range error that sums the various error components, VDOP is
vertical dilution of precision, and HDOP is horizontal dilution
of precision. The table shows errors of Pseudorange
measurements (Ephemeris data, Satellite clock . error,
lonospheric error, Tropospheric error, Multipath, and the
Receiver noise).
1.ASSESSING THE POSTIONAL ERROR USING
REFERENCE POINTS
The accuracy of GPS observation can be estimated in various
ways. One method is to define confidence regions are ellipsoids
(in 3D space), or ellipses (in 2D space), such that their volume
or area contains the true value within a preselected level of
probability. A useful scalar for assessing the error often
displayed on handheld units is the dilution of precision (DOP),
being a factor of reduction of precision (in function of satellite
constellation at the moment of observation). In the following
we will show how DOP and confidence regions can be derived
from the covariance matrix of the unknown vector of the three
geocentric coordinates and travel time (e.g. x, y, Z, dt) to be
determined from the GPS observations (Wells 1986)
A pseudo-range observation equation, using code, for one
satellite can be written as:
[4 e c t S t
edi esed E eti ud +d a (1)
Sat _
Rec Ion Trop
P
Where c — 299792458 m/sec (the speed of light in a vacuum),
: sat :
dT,ec the receiver clock error, dt the satellite clock error, dion
ionospheric error, dirop tropospheric error and
a. 3: 2 Sat 2 s 2
pego —XRec) *(y a = Von) Te — ZRec Q)
the true range receiver-satellite, ignoring atmospheric effects
For more than one satellite, we obtain a set of equations that is
over determined:
I ] - 1 1
P'-edt =p +c.(dTr ee - dt dj a trop
2 2. 2
P =edt sp e (dT e -dt^)ed? +r
eprreoovemocsnaeceseccopooteoso oo smb oras oo esoopnacseeeesoosemooo soo moooseoo
These GPS observation equations can be written in matrix
notation as:
Ax XY = B +
E (4)
Where X is the unknown vector matrix containing (z, y, z, dt),
A is the coefficient matrix of the unknowns (number of columns
= number of unknown parameters, number of rows = number of
visible satellites), B is the matrix of the known values from
reference points, and V is the error matrix. Applying
generalized least squares; an estimate of the unknown vector
matrix is obtained as:
7 -l 7
X- (A WA) A WB 6
Where W is the weight coefficient matrix. In addition estimated
observation errors are obtained as:
T -i T
y-4((4 WA) A4 WB-P (6)
And a-posteriori variance factor is defined as:
sns pe Twy
70 = (7)
n-—k
where O, is the mean square error of unit weight, n is the
number of observations, k is the minimum number of
observation required, and 1 — k is the number of redundant
observations.
The variance-covariance matrix of the unknowns is defined as
C.202 (d WA) ! (8)
X 0
628
Intei
—
Wh
vari
eler
can
The
HI
and
VI
The
eige
198
The
WG
diffi
the
