GPS CORRELATION |
Corr
elations express the inter-dependence between variables. For two
variables
x and y in a linear relationship, the correlation between them is
defined
as:
 |
(7.2-13) |
where:
x
is the standard deviation of x, is the standard deviation of y, and is the covariance between x and y.
The value of correlation lies in the range -1 and +1. When the correlation of two variables approaches the maximum of ±1, the two variables are said to be highly correlated. It is worth noting that high correlation does not mean that the variations of one are caused by the variations of the others, although it may be the case. In many cases, external influences may be affecting both variables in a similar fashion. There are two types of correlation encountered in GPS measurement and data processing: physical correlation and mathematical correlation.
Physical correlation refers to the correlations between the actual field observations. It arises from the nature of the observations as well as their method of collection. If different observations or sets of observation are affected by common external influences, they are said to be physically correlated. Hence all observations made at the same time at a site may be considered physically correlated because similar atmospheric conditions and clock errors influence the measurements.
Mathematical correlation is related to the parameters in the mathematical model. It can therefore be partitioned into two further classes which correspond to the two components of the mathematical adjustment model:
The accuracy of an absolute position is a function of the accuracies of the observed quantities in the functional model. Due to uncertainties in the satellite's position, satellite and receiver clock errors and propagation delays, the absolute point position using pseudo-range data will be accurate to no better than several dekametres (section 2.4.3). However, the error sources in the GPS system (satellite orbits, satellite and receiver clocks, atmospheric propagation, etc.) will exhibit some physical correlation among the signals received at several stations that are simultaneously tracking the same set of satellites. This is why for survey applications it is necessary to use GPS in a "relative mode", in which two or more receivers must observe the same satellites simultaneously. These physical correlations can then be physically modelled as common bias terms in the function model. Using mathematical operations such as data differencing, these effects can be eliminated or greatly reduced, leading to centimetre level accuracy. As the inter-station distances become larger, a decorrelation of the physical effects of orbit error and atmospheric delay will tend to occur.
The fundamental problem which is encountered when performing differencing operations is to accurately and completely determine to what extent various elements are correlated and how well they can be modelled. Physical correlations may be both spatial and temporal in nature, or a combination of both. An example of this is the refraction delay which a satellite's microwave signal experiences as it propagates through the ionosphere and troposphere. This delay is a significant source of both spatial and temporal physical correlation which is difficult to handle.
On the basis of experimental data, EL-RABBANY (1994) has derived an exponential covariance function that describes the temporal physical correlation:
 |
(7.2-14) |
where 
(
) is the correlation coefficient
for a time lag of (in secs), and T
is the correlation length (in secs). A typical value of T is
about 250-350 seconds, implying that data collected more than 350 seconds apart,
from the same receiver to the same satellite, can be considered completely uncorrelated
(correlation coefficient is zero). The values of () for different data rates (between
epoch measurement rate) quoted by EL-RABBANY (1994) are: 0.996, 0.981,
0.944, 0.891, 0.794, for equal to 1, 5, 15, 30, 60 seconds
respectively. Hence, assuming a constant data rate, the correlation coefficient
between any two epochs can be expressed as:
 |
(7.2-15) |
For example, for a 15 second data rate (1)=0.944, and the correlation
coefficient for data between consecutive epochs is 0.944, but it is 0.9442
for two sets of data 30 seconds apart, and 0.9443
for two sets of data 45 seconds apart, and so on. Hence the temporal correlation
decreases significantly as the time interval between data increases. This empirically
derived correlation could be introduced into the VCV matrix of one-way observations,
however no commercial software incorporates this and hence this is one of
the reasons why the accuracy of baseline results will be optimistic. This
problem is discussed further in sections 9.2, 9.3 and 9.4. In the following
development, no between-epoch correlation will be assumed to be present.
When differencing occurs, the original carrier phase observations and their one-way phase mathematical model are modified, hence leading to mathematical correlation of the resulting observables. This means that both the functional and stochastic models change. The new functional model has already been given in eqn (7.2-4). Generally it is assumed that at a particular epoch all (one-way) carrier phase observations are independent and have the same variance. (However, this is not entirely true because the magnitude of residual atmospheric biases are likely to be a function of satellite elevation angle, and hence the observation weights should vary as a function of the elevation angle.) Of interest is the error propagation rather than the absolute values, hence the variance can be set to unity. Therefore:
| QI = I | (7.2-16) |
where I is the Identity matrix. Applying the Law of Propagation of Variances, the VCV matrix of the double-differenced observations, and hence the new stochastic model, becomes:
| QI' = DQIDT | (7.2-17) |
The covariance matrix of the double-differenced observations is dependent on the double-difference operator used to construct the differenced observables. Using the double-difference operators (defined by eqns (7.2-8) and (7.2-11)) in eqn (7.2-17) leads to the VCV matrix of the base satellite double-differences and the sequential double-differences respectively:
 |
(7.2-18a) |
 |
(7.2-18b) |
Both VCV matrices illustrate stochastic correlation (that is, non-zero off-diagonal terms). However, in the case of between-receiver differencing (section 6.3.4), the resultant single-differences are uncorrelated (see HOFMANN-W ELLENHOF et al, 1998).
The double-differencing process therefore introduces mathematical correlations into the resultant observables. As a consequence, to ensure the mathematical equivalence between processing using the differenced and undifferenced phase data approach (section 6.1.2), it is necessary to construct a variance-covariance matrix of the double-differenced observables which expresses this correlation between observables. In practice it is not necessary to compute the VCV matrix QI' at every epoch, because the covariance matrix only changes with a change in the satellite-receiver combinations used in the construction of the double-differences. Triple-differenced data is also mathematically correlated (see, for example, HOFMANN-WE LLENHOF et al, 1998).
In most commercial GPS software packages the mathematical correlation arising from data differencing is usually neglected, and a diagonal VCV matrix for the double-differenced and triple-differenced data is assumed. The impact of this simplification on data adjustment is generally small for short baselines, perhaps of the order of 1 or 2ppm. Not taking into account the mathematical correlations between double-differences formed when more than two receivers are operating simultaneously (two or more independent baselines) is an important distinction between single-baseline and multi-baseline processing.
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