Geoscience Reference
In-Depth Information
tAble 9.3
differencing Modes and their error Characteristics
error Source
Single difference
double difference
Ionosphere
Reduced, depending on the baseline length
Reduced, depending on the baseline length
Troposphere
Reduced, depending on the baseline length
Reduced, depending on the baseline length
Satellite clock
Eliminated
Eliminated
Receiver clock
Present
Eliminated
Broadcast ephemeris
Reduced, depending on the baseline length
Reduced, depending on the baseline length
Ambiguity term
Present
Present
Noise level w.r.t. one-way
observable
Increased by
2
Increased by 2
stochastic model describing the accuracy of the measurements and its statistical properties. A fun-
damental part of the stochastic model is a covariance matrix of observations, Σ. The primary least-
squares formula representing a general matrix form of a solution of a system of GPS observation
equation is shown in Equation (9.4). Note that Equation (9.1) through Equation (9.3) (and in general,
any GPS observation equation) are nonlinear; thus, they are linearized before processing through
Equation (9.4):
ξ=
(
−
−
−
1
T
1
T
1
AAAy
Σ
Σ
(9.4)
where
A
is a design matrix containing the partial derivatives of the observable with respect to the
unknown parameters, ξ; and
y
is the vector of observations minus “calculated observation” which
is computed based on the approximated values of the parameters. For more details on the least-
squares solution, see, for example, Strang and Borre (1997).
9.7.2 dgPs s
e R v i c e s
: a
n
o
v e R v i e w
As explained earlier, the GPS error sources are spatially and temporally correlated for short to
medium base-user separation. Thus, if the reference station and satellite coordinates are known
(satellite location is known from broadcast ephemeris), the errors in the GPS measurements can
be estimated at specified time intervals and made available to the nearby users through a wireless
communication as differential corrections. These corrections can be used to remove the errors from
the observables collected at the user's (unknown) location (Yunck et al., 1996). This mode of posi-
tioning uses DGPS services to mitigate the effects of the measurement errors at the user's location,
leading to the increased positioning accuracy in real time. DGPS services are commonly provided
by the government, industry, and professional organizations, and enable the users to use only one
GPS receiver collecting pseudorange data, while still achieving superior accuracy as compared to
the point-positioning mode. Naturally, in order to use a DGPS service, the user must be equipped
with additional hardware capable of receiving and processing the differential corrections.
DGPS services normally involve some type of wireless transmission system. They may employ
VHF or UHF systems for short ranges, low-frequency transmitters for medium ranges (beacons),
or L-band or C-band geostationary satellites for coverage of entire continents, which is called Wide
Area DGPS (WADGPS) or Global Satellite Based Augmentation System (GSBAS). WADGPS
involves multiple GPS base stations with precisely known locations that track all GPS satellites in
view. These data are sent to the master control station (see Figure 9.12) that estimates the errors in
the GPS pseudoranges and forms a satellite uplink message that is transmitted to a geo-stationary
satellite. This satellite, in turn, broadcasts the information to all users with specialized GPS
receivers; the average correction latency is about 5 sec (www.aiub.unibe.ch/download/igsws2004/
