Sect_8.2.4

8.2.4 Introduction to Ambiguity Resolution

SELECTING THE BEST SET OF INTEGER AMBIGUITIES

The st andard criteria for successful ambiguity resolution is if the identified ambiguity set ( n* ) clearly fits the double-differenced phase data better than any other ambiguity set:

(8.2-3)

The testing criteria is generally the lowest weighted root-sum-of-squares (RSS) of the double-differenced data residuals:

(8.2-4)

where v is the vector of residuals, P is the observation weight matrix.

This procedure requires:

The computation of residuals for each ambiguity-fixed solution being tested -- estimate a new baseline for each ambiguity set.
Assumes unbiased (for example, no residual atmospheric refraction), clean (that is, no cycle slips), low noise (for example, no multipath) data.
Observation data series long enough for reliable residual testing.

Such a strategy seeks to find the "best" set of ambiguities, the one that is clearly better than the "second best" set by some rejection criteria (which can be varied). In many software packages the "ratio" value is examined: the ratio of the RSS obtained using the second best ambiguity set to the RSS for the best set. This value should be as large as possible. Ambiguity resolution is obviously least reliable when there is no obvious "best" set of ambiguities (that is, the "ratio" value is small, perhaps less than 2 or 3). This is generally implemented as an "all or nothing" process.

An alternative approach is to resolve only some ambiguities. Using the round-off strategy, only those ambiguities that are near integers, and which have low standard errors (Figure 1 below), are assumed to have been reliably resolved. Another approach is a sequential ambiguity resolution search procedure, also based on minimising the estimated "weighted RSS". However, in this procedure the best determined (the one with the lowest standard error) ambiguity parameter is used as the test value about which the search window is defined. Once that ambiguity has been resolved, new ambiguity values are derived for the remaining estimable parameters, and the search continues using the best determined of the remaining ambiguities, and so on. This process is more flexible and can be halted when no further ambiguities can be resolved reliably.

Figure 1. Scenarios in ambiguity resolution: ambiguity well determined
and close to an integer.

There are several factors that make ambiguity resolution difficult, including:

The degree to which the geodetic parameters are reliably separated from the ambiguity parameters.
The magnitude of any unmodelled biases still present in the double-differenced phase data.
The length of the baseline.
The quality of the receiver-satellite geometry, and how much it has changed during the observation session.
The data quality.
Sub-optimal algorithms.

When the baseline is comparatively long (>20 km), or there are some unmodelled biases present (high ionospheric activity, etc.), some ambiguities may be far from an integer value but still have small standard deviations. Or, alternatively, the ambiguity values may be close to integers but the standard deviations may be too large (arising from poor receiver-satellite geometry, data outages, etc.). These two situations are illustrated in the two figures below.

Figure 2. Scenarios in ambiguity resolution: Session not long enough? Poor geometry?

Figure 3. Scenarios in ambiguity resolution: Biased data? Baseline too long?

The situations illustrated in figures 2 & 3 mean that the rejection criteria cannot discriminate between more than one candidate ambiguity set, and hence ambiguity resolution cannot proceed reliably. Hence, because an ambiguity-fixed solution is not possible, the ambiguity-free solution may be considered to be the "optimal" one.

Another useful way of visualising this ambiguity search procedure is from a geometric viewpoint, as described in the next section.

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