THE NATURE OF GPS OBSERVATION MODEL BIASES |
The basic GPS observables of carrier beat phase and pseudo-range are essentially
biased ranges (section 1.3.1).
The ideal quantity required for GPS data processing is the true
range 
ji. The true range contains all the geometric
information necessary to determine the receiver coordinates and/or the satellite
position. One of the challenges in GPS processing is therefore to develop
data analysis strategies that best handle the measurement
biases. These biases arise from a number of sources (section
6.2.1):
rcj) and
satellite clock (sci),
atmos), and
Upon adopting eqn (6.1-13) as the basic
measurement model, and proceeding with the normal linearisation process
required for a Least Squares adjustment, a further class of biases is introduced
into the residual quantity: observed range (phase or code) minus modelled
or calculated (biased) range. This "O-C" quantity now also includes
errors in the apriori information contained in the modelled range, primarily
the errors in the assumed known receiver coordinates and satellite ephemeris
information (used to derive the "calculated" value for the quantity
ji).
There are two options, these additional biases can be:
A significant challenge in GPS processing is therefore to develop strategies
that either reliably estimate the additional model biases introduced into
a Least Squares adjustment, or at the very least, minimise the effects of
neglected residual systematic biases and random errors.
Dealing with the various biases in GPS solutions based on the mathematical
model according to eqn (6.1-13) is a considerable
problem. In fact much of the development effort in GPS software has gone
into investigating ways of best reducing the computational burden imposed
by having to account for the biases present in GPS range/phase measurements.
Several options are available:
They can be estimated as explicit
(additional) parameters.
Those biases linearly correlated across different datasets can
be eliminated by differencing.
Those biases which are a function of frequency can be eliminated
by constructing linear combinations of dual-frequency data.
The biases can be directly measured,
for example using Water Vapour Radiometer (WVR) observations in the case
of the tropospheric delay.
The biases can be considered known, being adequately modelled,
for example as is sometimes attempted for the tropospheric delay.
The biases can be simply ignored.
These strategies may completely account for some of the biases, or reasonably
handle other biases (the unaccounted for part being subsumed into "residual
biases" which are usually then lumped together with the measurement
"errors"). The Table in section 6.3.1
summarises the options for handling GPS biases, affecting carrier phase
as well as pseudo-range data.
The basic observable in GPS data processing contains the following information:
In particular, attention will be focussed on the parameters that are explicitly modelled in GPS surveying applications: the geodetic parameters, and the remaining "nuisance" parameters (cycle ambiguity, clock errors, atmosphere). Within this latter group of nuisance parameters, the bias subset comprising the cycle ambiguity and clock phase error terms are often referred to collectively as the "clock biases".
The two most frequently used options applicable in the case of handling
the "clock biases" associated with GPS measurements: (1) explicit
parameter estimation, and (2) elimination by observation differencing. The
questions that then immediately spring to mind are:
The Fundamental Differencing Theorem is useful in this
regard:
| "Linear biases can be accounted for either by reducing the number of observations so that the biases cancel, or by adding an equal number of unknowns to model the biases. Both approaches give identical results." (LINDLOHR & WELLS, 1985) |
Thus, the effect of the clock phase errors ( rcj and sci
) and cycle ambiguity ( nji ) may be eliminated by
differencing the basic eqn (6.1-13) between
satellites and between receiver sites, observed simultaneously, and perhaps
between observation epochs as well; or by estimating them explicitly, and
the result would be equivalent.
However, although they may be mathematically equivalent, there are significant differences in how the algorithms are implemented for GPS phase reduction. On the one hand, differencing leads to a smaller parameter set having to be estimated, but on the other hand there is more data management, or "book-keeping", required if satellites rise and set during an observation session. The undifferenced observation model option is more "intuitive", being more like a range observable, however the need to estimate many more parameters weighs against it heavily. Consequently only a few software packages employ the principles of undifferenced phase reduction, and these are invariably "scientific" software for high precision geodetic applications. All commercial GPS software use the double-differenced approach.
The observation modelled by eqn (6.1-13) is sometimes referred to as "one-way phase" or "undifferenced phase" to distinguish it from the more commonly used single-, double- or triple-differenced phase observations described in section 6.3. The term "undifferenced phase" is, strictly speaking, a misnomer because eqn (6.1-6) is actually formed by differencing the incoming and local (receiver) oscillator phases. Nevertheless it is useful to refer to the two approaches to dealing with the clock biases as being the "undifferenced" or "nuisance parameter" approach on the one hand, and the "differenced" approach on the other.
However, a further comment on the equivalence of the "undifferenced"
and the "differenced" approach is worth making. The double-differencing
approach (difference between satellites to eliminate rcj,
and between receivers to eliminate sci) does introduce mathematical correlations in the resultant double-difference
observables (section 7.2). Only if this correlation is reflected in the
variance-covariance (VCV) matrix of the double-differenced observables will
mathematical equivalence between the "differenced" and the "undifferenced"
approach be assured. (The same holds true for triple-differenced data.)
As a consequence, to ensure this equivalence, it is necessary to explicitly
construct a new VCV matrix at each observation epoch (of course in practice,
advantage is taken of the fact that the weight matrix would only change
with a change in the satellite/receiver combinations used to construct the
double- or triple-differences, and that happens only if satellites rise
or set during the session).
In the "undifferenced" approach, there is more flexibility in the clock modelling options. For example, the clock phase errors in eqn (6.1-13) are a function of time. If the satellite clock, or receiver clock, or both, had a stable, predictable behaviour (as when external atomic frequency standards such as cesium or rubidium oscillators are used), a bias parameter model based on a clock polynomial could be used, and the coefficients estimated, together with the cycle ambiguity, as session parameters. In general however, the clock phase errors are assumed to possess the characteristics of "white noise" and are explicitly modelled by independent bias terms on an epoch-by-epoch basis. The total number of clock bias parameters is (R+S)T+RS, where R is the number of receivers, S the number of satellites and T is the number of epochs in a session of data. There is therefore a potentially large number of these bias parameters to be estimated. In the case of four receivers, four satellites and 60 (one minute) measurement epochs, the total number of estimable clock bias parameters would be 496!
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© Chris Rizos, SNAP-UNSW, 1999