Sect_9.4.4

9.4.4 Modifying the Stochastic Model

USING EMPIRICALLY DERIVED VARIANCES
TO ALTER THE VCV MATRICES

The use of empirically derived variance information to modify the GPS VCV matrices is now an accepted practice when combining individual baseline solutions, single session solutions or multi-session solutions.

Generally the values of a range from about 5-40mm and the values of b range from 2-10ppm, depending on the type quality of the initial GPS reduction. Some additional remarks regarding the way in which variances are derived from the values of a and b:

The value of s² adopted for a certain baseline is the same for all three baseline components: Bx, By, Bz.
The same a quantity may be assumed to apply to all three baseline components, or a different value used for each component ( a_X, a_Y, a_Z ), or the VCV_XYZ matrix may be transformed into the topocentric form VCV_ENH (section 11.1.8) and a different value assumed for the horizontal components ( a_E, a_N ) (arising from centring error) than for the vertical component (arising from height-of-antenna error) ( a_H ).
The value of b may be assumed to relate to the baseline length and is to be distributed according to the baseline component lengths, in which case the variances are approximated by:

s_x² = ( a_x² + b.Bx )²
s_y² = ( a_y² + b.By )²
s_z² = ( a_z² + b.Bz )² (9.4-3)
The value of b may be assumed to relate to the baseline length and is to be distributed according to the estimated baseline component standard deviations, in which case the variances are approximated by:

(9.4-4)

where _Bx, _By, _Bz are the component standard deviations and is the baseline length standard deviation.
The VCV_XYZ matrix may be transformed into the topocentric form VCV_ENH and a different value of b assumed for the horizontal and vertical components (for example, to account for the fact that the height determination in GPS is weaker than the determination of the horizontal components):

s_E² = ( a_E² + b_E.L )²
s_N² = ( a_N² + b_N.L )²
s_H² = ( a_H² + b_H.L )² (9.4-5)

Variations based on eqns (9.4-3) and (9.4-4) are also possible.

Note that for short baselines, the influence of the constant component a is the strongest, while in the case of long baselines the length dependent component b has a greater influence than the constant part.

Although there are several strategies for computing the variances (see above), there are also several options for how they are used to modify the original VCV matrix:

(1) The computed VCV matrix of the initial baseline phase data reduction is ignored, and replaced by a diagonal VCV matrix containing only the empirically derived variance s². There are no covariance terms, hence the correlations between baseline components have been changed.

(2) Preserve the computed VCV matrix, and add the appropriate variance quantity s² to the diagonal elements of the matrix. Maintaining the original covariance terms means that the correlations between baseline components have been changed.

Construct a VCV matrix using the new variance information s² and the original correlation information. The procedure would be:

Compute the coefficients of correlation using the original VCV information:

(9.4-6)

Determine new values for the variances s_X², s_Y², s_Z².
Construct the new VCV matrix using eqn (9.1-7):

(9.4-7)

The latter is the preferred option.

Example of a Multi-Session Network Adjustment

The Molong GPS survey referred to earlier may be used as an example. Seven observation sessions, in which four receivers were used, were reduced independently (an example was given in section 9.2.5). Although only Trimble GPS receivers were used, the observations were made over a two day period by the same field parties and all ambiguities were resolved, there were a number of factors that made deriving a multi-session network solution far from a straightforward process:

Different single session solutions:
- Sessions A, B, C, and F were reduced as single baselines using the commercial software, and all the baselines (trivial and independent) were included in the secondary network adjustment.
- Sessions D and E were reduced together in a rigorous multi-station, multi-session solution using GAMIT, a sophisticated scientific program designed for GPS geodesy applications.
- Session G was reduced in multi-baseline mode using the early TRIMMBP software.
Varying length of baselines. The baselines varied in length from less than one kilometre to over 30 kilometres. The precisions derived from the various VCVs did not reflect any length dependence.
Antenna station setup varied from being tripod mounted to using geodetic pillars. The precision of centring and/or antenna height measurement is expected to be uneven.

Using the various single baseline, multi-baseline and multi-station/session solution outputs, with their original computed VCVs, in a secondary multi-session network adjustment caused the Variance Factor Test to fail at the 95% confidence level (with 31 degrees of freedom). After some experimentation the stochastic information associated with the baseline observations was modified according to the following:

The observations were divided into two groups: (1) the results of session A, B, C, F, G processing, and (2) session D and E processing.
Stations occupied in sessions D and E were almost all existing geodetic pillars.
The stochastic model for the observations was altered by adding to the diagonal elements of the computed VCV matrices the variance developed from eqn (9.4-5), in which a=10mm and b=5ppm for group (1) baselines; a=5mm and b=3ppm for group (2) baselines.

The Variance Factor Test was then successfully passed. The relative 2-D error ellipses are almost circular, and range in size from about 5 to 10mm. It should be pointed out that the magnitude of the semi-major axes of the error ellipses is dependent on the output VCV of the multi-session adjustment, and that this aposteriori VCV then is empirically altered leading to a new input VCV matrix for an iterated solution. However, the amount by which the VCV is changed is determined by the variance factor, which itself is influenced by the amount of redundancy in the overall network! Which brings up an important issue: should trivial baselines be included in a network adjustment? This is discussed in the next section.

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9.4.4 Modifying the Stochastic Model

USING EMPIRICALLY DERIVED VARIANCES TO ALTER THE VCV MATRICES

Example of a Multi-Session Network Adjustment

USING EMPIRICALLY DERIVED VARIANCES
TO ALTER THE VCV MATRICES