INTRODUCTION |
To carry out a reliable multi-session GPS network adjustment, the input VCV matrix of the baseline component "observations" must be correct. There are a number of reasons why the input VCV matrix may need to be modified, including:
This chapter discusses each of these topics.
As has been mentioned several times, the output VCV matrices of phase data
processing software are unlikely to be the appropriate ones to input into
a secondary network adjustment. What practical methods can be employed
to produce a VCV matrix that is representative of the external factors influencing
the GPS results, as well as the internal "noise only" contribution?
One method, known as Variance Component Estimation, is described in CASPARY (1987, pp.97-110). It has been used to combine GPS and terrestrial networks, and can be applied to both homogeneous and heterogeneous sets of observables in order to determine the relative uncertainties of the various components that make up the total stochastic model. In GPS session observations, the components could be a constant error plus an error that is distance dependent. In the case of heterogeneous observations, each observational group may be assigned its own error characteristics.
The VCE method uses an iterative numerical procedure starting with reasonable apriori estimates of the variance components. It relies on the assumption that all significant errors are randomly represented in the data set, and that a single scale factor is sufficient to account for the errors in one observation type, particularly where observations may seem consistent and yet still be affected by undetected systematic errors. This is usually true when dealing with the propagation of errors into new observation types such as is the case in GPS. Therefore it is recommended that methods such as VCE be accompanied by attempts to evaluate the effects of systematic errors.
An alternative, empirical method of modifying VCVs, uses the knowledge that session errors are likely to have both a constant and length dependent component (section 2.4.4 and section 10.2.2), and to apply suitable factors to the VCV matrix of the baseline vectors. Each GPS network (single session, or even individual baseline) is then combined, after altering their VCV, so as to ensure that the aposteriori variance factor of the combined multi-session adjustment passes the Variance Factor Test (section 9.1.4).
In modifying the VCV of a GPS phase data reduction solution, it is necessary to take into account both the "internal" errors as well as the "external" (largely undetectable) influences. Because GPS is a new surveying technology, there is still comparatively little experience to use as a reference for the expected precision or accuracy of each type of receiver or software reduction package (that provided by the manufacturer is likely to be very sensitive to many factors such as length of observation session, satellites tracked, etc.). The internal precision of a GPS relative positioning solution could in fact be assessed by its aposteriori variance factor, as it is derived directly from the residuals and will reflect the effects of receiver errors such as measurement noise, and multipath. The VCV matrix could then be scaled through multiplication by the aposteriori variance factor, the solution re-run and the new VF tested. However, scaling the VCV matrix by the aposteriori variance factor will produce estimates of the VCV that only reflect these "internal" error sources.
The above two approaches: (a) modification of individual elements of the VCV matrix, and (b) scaling by the aposteriori variance factor; are common options implemented in commercial network adjustment software. These are discussed further, in this chapter.
A popular approach to describing the total error (that is, combined "internal" and "external" errors) in GPS session adjustment results is by constant ( a ) and length dependent ( b ) terms, and to develop a suitable population variance such as (eqn (9.1-4)):
| S2 = ( a + b.L )2 | (9.4-1) |
where L is the interstation distance in kilometres, a is in millimetres and b is in parts per million (ppm).
The constant term can be justified, for example, as a centring (horizontal effect) or height-of-antenna measurement (vertical effect) uncertainty. On the other hand, residual ionospheric and tropospheric errors in the carrier phase observations, as well as the effect of satellite orbit and fixed station errors, map into the positioning results with magnitudes that grow with increasing baseline length.
The effect of external systematic biases on GPS phase data reductions can only be detected through their consistency as compared to other data. This is sometimes rather difficult, as comparing GPS results to existing terrestrial networks will show differences that may be due to errors in the terrestrial network. A more appropriate method could be to compare repeated session "measurements" of GPS baselines at different times of the day, different times of the year for consistency. However, care must be taken that other effects on baseline/session accuracy are isolated from these analyses (such as radically different baseline lengths). Note, systematic errors that effect all session results with a similar magnitude will not be detected using this technique.
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© Chris Rizos, SNAP-UNSW, 1999