The filter was first developed by Wdowinski et al. (1997) for data analysing of five sites of Permanent GPS Geodetic Array (PGGA) in the southern California to re-evaluate the far-filed displacements induced by Landers earthquake, 1992, in the post-processing step as anew spatial filtering technique. They aimed to distinguish between coseismic and short-term post seismic displacements of the PGGA sites affected by the earthquake. The Signal to Noise Ratio (SNR) of the estimated site positions was significantly improved by the filter,and consequently they could successfully resolve the total surface displacement into its coseismic and post-seismic components site-by-site, rather than by cumbersome analysis of relative displacements between pairs of sites. Concept of the filter has been elaborated in Wdowinski et al. (1997) as it was initially used by them for analysis of GPS data for the Landers earthquake, and a concise summary of the filter is given here from the same source with respect to GNSS data analysis.

When an earthquake displacement is studied, it is important to identify and remove features in the time series that are clearly not resulted from the earthquake deformation cycle. As a first step, data outliers from all three components (north, east and up) are eliminated. Two types of outliers are identified: outliers points (1) with error bars larger than three times the Root Mean Square (RMS) scatter in any of the components or (2) that deviate from the mean of the series by more than three times the RMS scatter. These operations are performed independently on the data segments before and after the earthquake, and in an iterative manner until all outliers are eliminated. Then the time series are cleaned of outliers by this way.

Tectonic signals are considered to be either abrupt changes in position (coseismic displacements) or longer-term continuous changes (e.g., inter-seismic, pre-seismic, and post-seismic displacements).Therefore, the day-to-day position changes can be considered as a daily noise super imposed on the longer term tectonic signals. This signature is obvious at all sites when the various components are aligned (or “stacked”), particularly in the east and vertical components. It is assumed that this signature is a common bias for all sites and is most likely non-tectonic because of its large regional extent. Therefore, it is justified to eliminate this bias from the data series in the following three-step procedure:

**De-trending:**The component intercept that best fits the data is determined by simplified linear regression. Then the residual (i.e., the difference in position between the observed and predicted values) is calculated for each day t and site s:

The determination of the common-mode is improved if the number of sites increased. If S=1, however, the data point from the one site is deleted from the corresponding time series.**Stacking:**The common-mode bias is calculated on a day-by-day basis by averaging the values from all sites:

In the case of an abrupt change (such as a step function) in the time series, for example coseismic displacements, the algorithm is applied separately to the time series segments before and after the day of the change.**Filtering:**For each site, the common-mode error is subtracted from the observed position Os(t) to obtain the filtered position:

Obviously, for post-seismic displacement a special consideration should be taken into account. Sites which are supposed to have experienced the post-seismic displacement do not participate in the stacking for the affected period. Considerably less RMS scatter is seen in the “filtered” time series compared to the cleaned time series. The filtered time series for each site will generally have lower scatter than individual baseline series (relative position), assuming that the common signal dominates the errors at each site. Baseline components remain unchanged because the filtering algorithm removes the same value from all sites.

**Bibliography**

Wdowinski,S. et al., 1997. Southern California Permanent GPS Geodetic Array:Spatial filtering of daily positions for estimating coseismic and postseismic displacements induced by the 1992 Landers earthquake.Journal of Geophysical Research, 102(B8), p.18057-18070.

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