K. Le Bail – IGN/LAREG, Marne La Vallée, France
Email : firstname.lastname@example.org
We choose 16 stations in GPS-DORIS collocation distributed on the following map.
We study three time series:
- for DORIS (http://ids.cls.fr): the computing is made by LEGOS/CLS with GINS/DYNAMO software, the monthly solutions cover the period 1993 – end of 2001 in best. Toulouse station and EOP are fixed and the time series are given in the DORIS global system;
- for GPS:
o IGS series (ftp://macs.geod.nrcan.gc.ca): weekly solutions from the beginning of 1999 through the end of 2002;
o Michael Heflin series (ftp://sideshow.jpl.nasa.gov/pub/mbh/point): daily free-network solutions from the beginning of 1993 through the end of 2002, aligned with ITRF2000.
This is a statistical tool, very used in the time domain especially for the stability of the atomic clock.
The Allan Variance (or pair variance) is an estimator of the variance of a time series with regular steps. If are the measurements and t the sampling time, the mathematical expression of the Allan variance is:
, where <.> represents the average.
Let a temporal process which has a spectral density D proportional to f :
The a parameter allows to determine the noise :
- for white noise;
- for flicker noise;
- for random walk.
With the Allan variance, we can specify a noise in a time series thanks to the relation:
So we have (with a graph associated which looks like):
- m= -1 for white noise ;
- m= 0 for flicker noise;
- m= 1 for random walk.
In the graphs on the right papers, the Allan variance is calculated with a sampling time of one day for Michael Heflin series, one week for IGS series and one month for LEGOS/CLS series. That is why the first point hasn’t the same abscissa for the 3 series.
Let us note that the Allan variance isn’t the variance called repeatability. The Allan variance is dependant of the time, and this characteristic makes it more interesting in the understanding of some time series effects.
Effects that can influence the time behaviour of station coordinates:
- Network / system effects;
- Time series unification method;
- Geophysical signatures.
Analysis of GPS (JPL) series: Up
- The Allan variance analysis identifies various statistical behaviour of DORIS and GPS time series of coordinates.
- The horizontal and vertical motions have different spectral signatures:
o Up component has mainly flicker noise (slope = 0 in the Allan graphs) both in DORIS and GPS;
o North and East components have a variety of signatures : white noise (slope = -1), flicker noise (slope = 0), random walk (slope = 1).
- These results are preliminary, further investigations are under way. They can encourage to go on this study deeper, with longer series of more different analysis centres.
Allan, D. W. 1966, Proc IEEE, 54, 221