Sect_9.1.2

9.1.2 Introduction to GPS Network Processing

SECONDARY NETWORK ADJUSTMENT CONCEPTS

From the remarks made above, for an average GPS campaign solution there are two types of solutions: a primary adjustment requiring the appropriate modelling of the GPS observables and the development of the processing strategies that permit the geodetic parameters of interest to be estimated (discussed at the single baseline level in Chapter 7 and Chapter 8), and a secondary adjustment that treats the outcome of the primary adjustment as an observation.

The basis for the following discussion is the theory of Least Squares adjustment. The mathematical models involved are much simpler than for the reduction of GPS phase data. In fact, the methodology is very similar to conventional network adjustments of geodetic observations such as distances, except for the fact that 3-D quantities are involved. The reader is referred to texts such as CROSS (1983) and HARVEY (1994) for background material on conventional geodetic adjustments.

The Input

The minimum input required for a Least Squares adjustment are the observations and their precision. In general these are output by the GPS (primary) phase reduction solution. The output is generally written to a "list" file which must be scanned to extract the relevant information (either automatically by the network adjustment program itself, or manually by the analyst using a text-editor program). Commercial GPS software now incorporates the network adjustment module within the main program, as in the case of the SKI^TM, GPSurvey^TM, etc.

The Nature of the Observations

3-D baseline components, as would be output by single or multi-baseline GPS phase solutions, either triple-differenced or double-differenced phase solutions (ambiguity-free or ambiguity-fixed).

Observation Weights and Re-scaling of VCVs

Each of the above "observations" has a corresponding VCV matrix Q_{l l} (the inverse of the weight matrix P). In the case of a baseline "observation", the VCV matrix is a 3x3 matrix:

(9.1-1)

This relates station i to station j, and does not involve any other GPS station. The total VCV matrix Q_{l l} formed from combining separate baseline solutions is likely to be block diagonal. Bx, By, Bz refer to the three baseline components. Using the variances (standard deviations squared) as the basis for assigning observation weight implies an assumption that the errors in the observations (the baseline components) are normally distributed. This may not be valid assumption! (However, there is little alternative to accepting this.)

In the case of multi-baseline "observations" the total VCV of a group of m stations (or m-1 independent baselines) is available. The VCV is of dimension 3(m-1)x3(m-1) and is generally full, with the following structure:

(9.1-2)

The weight matrix ensures that each observation contributes to the adjustment in accordance with its measurement accuracy. Very accurate observations should therefore have higher weights than less accurate observations. Hence, it should be noted that if the weight/VCV matrix is the output of a GPS phase reduction the precisions implied for the coordinate components are likely to be over-optimistic (that is, imply higher accuracies than are warranted). Invariably millimetre accuracies are claimed, particularly for the double-difference ambiguity-fixed solutions. The reason that the internal precision implied by the output VCV matrix (from the primary GPS phase reduction) is not a true measure of the external accuracy is that GPS phase observations are affected more by systematic errors (section 6.2.1 and section 6.3.1) than by random measurement errors. When such an unrealistic VCV is input into a network adjustment program the resulting solution will generally fail the Variance Factor Test (see below). However, the outcome of the adjustment (as far as the parameters is concerned) is dependent on the relative magnitudes of the weights, not just the absolute values.

If it were possible to determine the true VCV (including the effect of the unmodelled systematic errors), then this could be used in place of the VCV output by commercial GPS software. Unfortunately there is no (straightforward) means of obtaining such a VCV. In practice, analysts resort to various empirical techniques to modify the VCV matrix. This is usually done in an iterative manner, using the variance factor test (or some other suitable statistical test) to guide the analyst.

A scale factor w could be applied (say, 5), making all variances and covariances appear worse in eqns (9.1-1) and (9.1-2):

VCV^c = w . VCV (9.1-3)

However, a popular method is to describe the total error (that is, combined "internal" and "external" errors) in GPS session results through the use of relative weighting of the session observations according to the baseline lengths. The true errors in the observations are assumed to have constant ( a ) and length dependent ( b ) components, and a suitable population variance can be developed according to:

S² = a² + b² . L² (9.1-4)

This may then be used in place of the original diagonal values, or added to the ones already computed. The off-diagonal elements maybe left unaltered or scaled so that the correlations are preserved. Further discussion on this procedure is given in (section 9.4.1).

The Solution

The solution procedure is relatively straightforward. The sets of parametric equations (scaled by the appropriate weight matrices) are combined into the Normal Equations, the datum defect is accounted for in some way, and the Normal Equations inverted. The corrections to the apriori coordinates are obtained and the residuals are tested.

The Parameters

In the case of the secondary network adjustment there is only one class of parameters: the geodetic coordinates. Further, the "natural" coordinate parameter system is in the form of 3-D Cartesian components ( x_j, y_j, z_j ), nominally expressed in the WGS84 geodetic reference system, though some processing software carries out the adjustment in the ( , , h ) system. The standard Least Squares parametric equation has the form (CROSS, 1983; HARVEY, 1994):

(9.1-5)

The expression in brackets is the so-called "observed minus computed" term. l is the vector of the actual observations, are the approximate parameters, are the corrections to the approximate parameters and A is the design matrix containing the partial derivatives of the observations with respect to the parameters. It is possible to write the explicit form of this parametric equation for a baseline "observation" linking stations k and j as:

(9.1-6)

where the RHS contains the parameters and Bx_ij,, By_ij, Bz_ij are baseline component "observations". Note that there are in fact three "observations" per baseline, one for each Cartesian component (or , , h components if the ellipsoidal adjustment model is used). However the three "observations" are correlated (hence the weight matrix of the observations P is full).

The Solution Procedure

There are several characteristics of GPS secondary adjustments worth mentioning here:

As has been mentioned several times, apart from the unique nature of the observations, a distinguishing feature of a GPS secondary network solution is that it is a three-dimensional adjustment.
Furthermore the adjustment is strictly "geometric" in the sense that no gravity field influence is modelled, as is usually the case for conventional adjustments (through deflection of the vertical corrections, incorporation of zenith distance observations, levelling data, etc.).
The structure of the design matrix A of an adjustment of baselines or station coordinate "observations" is very simple, comprising +1 and -1 (eqn (9.1-6)).
The greater the redundancies (baseline connections, or multiple station occupations), the greater the usefulness of the secondary adjustment as a measure of the internal reliability and repeatability of the GPS survey. With no redundancy (that is, only one common station between sessions) the network solution degenerates to an algorithm for "concatenating" sessions.
A minimally constrained solution assumes that only one station is held "fixed" (that is, the coordinates are not permitted to adjust). Having one Datum Station is the minimum necessary to obtain a solution. (If no coordinate parameters are held fixed, then the resulting normal equation matrix A^TPA is singular and cannot be inverted.) It is possible to use alternate Least Squares algorithms that overcome this requirement, such as by specifying a suitable apriori VCV of the parameters, or by performing a pseudo-inverse, however, the majority of network adjustment software packages do not have such options.

The Output

As with any Least Squares adjustment the output consists of the estimated parameters (actually the corrections to the apriori values of the parameters ) and their precisions (inferred from the solution variance-covariance matrix Q). What is not output is the true accuracy of the parameters, hence the impact of unmodelled systematic errors is not obvious.

The output coordinates can be converted from Cartesian coordinates into ellipsoidal coordinates, or into map projection coordinates, and perhaps transformed to a local geodetic datum (Chapter 11). The solution VCV matrix contains the following information:

standard deviations of the estimated parameters,
correlations between the parameters,
absolute or point error ellipses (or ellipsoids), and
line or relative error ellipses (or ellipsoids).

In addition to the parameters and their precisions, other useful information that can be obtained from the solution is contained in the residuals and the VCV of the residuals. Least Squares theory does not require that the observation residuals be normally distributed. However, if the observation errors have a Gaussian Normal Distribution, normally distributed residuals may be expected. Several statistical tests can therefore be carried out on the residuals. The statistical tests may be applied to assess the quality of the observations and assist in outlier detection ("bad data") -- a useful summary of this topic can be found in HARVEY (1994). Tests also permit an assessment of the quality of the adjusted parameters and of the validity of the mathematical (functional) model to be made.

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