7.2.4 Basic Modelling Concepts

ADJUSTMENT MODEL CONSIDERATIONS

By focussing on the double-differenced phase solution, the following three issues will be discussed:

The mechanics of double-differencing.

The modification of the observation weight matrix under differencing.

The ambiguity modelling options.

 

(The former two are also applicable to triple-difference solutions.)

In Chapter 8 these and other issues relating to double-differenced solutions will be discussed further.

The Double-Differencing Process

Observation example: R = 2 receivers S = 4 satellites E = 60 epochs   Total of 480 one-way phase observations (R.S.E). For each epoch: there are R.S observations (here, 8). there are R(R-1)S(S-1)/4 possible double-differences that can be formed (here, 6). there are only (R-1)(S-1) independent double-differences (here, 3). There are therefore a total of 180 independent double-differences.

There are several ways to form, at an observation epoch, (R-1)(S-1) independent double-differences from the R.S one-way phase observations. Building differences between simultaneous one-way phase observables can be regarded as a pre-multiplication of the vector of one-way phase data by an appropriate matrix operator: the difference operator. Dropping super and subscripts as :

(7.2-6)

where D is the Double-Difference operator matrix. The dimension of the Double-Difference operator matrix is equal to [(R-1)(S-1),R.S].

Pre-multiplying both the functional model as well as the set of observations, the new system equations is:

(7.2-7)

where:

' =	is the misclose vector of double-differences,
v' =	is the residual vector of double-differences, and
A' =	is the design matrix of double-differences.

There are alternative forms for the Double-Difference operator. The two most common are the base satellite and sequential satellite options (see Figure 1 below).

Figure 1. Two between-satellite differencing strategies ((a) & (b)).

Although both approaches are mathematically equivalent (although the correlation matrices are different), in reality processing the same dataset using a different differencing scheme can lead to non-equivalent results because:

• of the presence of different characteristics of biases, errors and noise, and
• the different solution characteristics due to data dropouts, etc.

For example, if the satellite selected as the base satellite gives poor quality data, then this will contaminate all the double-differences. These alternative differencing schemes can be illustrated by way of an example. Assume that two receivers observe four satellites: PRN 3, 6, 9, 11.

Base satellite: In this option, the observable from one satellite, for example, the one with the lowest PRN number in this case, is held fixed as a reference satellite used to form the double-differences. The double-difference operator for a baseline observing the above four satellites has the form:

(7.2-8)

The resultant differenced observables are:

(7.2-9)

Note that the double-differenced observables _1-2^6-9, _1-2^6-11 and _1-2⁹¹¹ have not been constructed. These are dependent, and are not included in the observation set to be adjusted as the resultant Normal Equation matrix becomes singular (if the correlated nature of the double-differenced observables is reflected in the input observation weight matrix P). To illustrate this consider how they are formed as linear combinations of the independent double-differences:

(7.2-10)

Sequential satellite: In this approach the sequential double-difference operator is of the form:

(7.2-11)

The resultant differenced observables are:

(7.2-12)

Note that the double-differenced observables _1-2^3-9, _1-2^3-11 and _1-2^6-11 have not been constructed. They are linearly related to the three independent double-differences through similar relations to eqn (7.2-10).

The difference operator is the same for every epoch only if the same constellation of satellites is tracked for the entire observation session. The situation in which satellites rise and set during the session complicates things, and requires considerable "book-keeping".

The design matrices of the base satellite double-difference and sequential double-difference for a single epoch are illustrated in Figures 2 and 3 respectively. Figures 2a and 2b show the design matrices of double-differences in the case where the ambiguity parameters are represented by the undifferenced ambiguity model. Figures 3a and 3b illustrate the design matrices of double-differences when the ambiguity parameters are in terms of the double-differenced ambiguity model. (Ambiguity models are discussed later.)

Figure 2. Design matrices of base satellite double-differences for an epoch (2 sites, 4 satellites).

Figure 3. Design matrices of sequential satellite double-differences for an epoch (2 sites, 4 satellites).

Back To Chapter 7 Contents / Next Topic / Previous Topic

© Chris Rizos, SNAP-UNSW, 1999