Forty-eight hour, 3-dimensional kinematic back-trajectories were calculated from the time and location of every aircraft spiral using the NOAA Air Resources Laboratory (ARL) HYbrid Single-Particle Lagrangian Integrated Trajectory (HY-SPLIT) model (Version 4) (R. R. Draxler and G. D. Rolph, 2003, http://www.arl.noaa.gov/ready/hysplit4.html) and 80 km Eta Data Assimilation System (EDAS) 3-hourly archive data. The latitude, longitude, and pressure were converted to Cartesian coordinates by treating the Earth as a sphere and calculating their position in 3-dimensional space. Vertical variability along the trajectory paths may have a large impact on transport and hence, pollution levels, but the spatial distances described by the variability in the vertical wind component are typically less than the horizontal spatial distances covered by the air parcels.  The spatial coordinates were normalized to give the vertical variability similar weighting to the horizontal variability.  The mean value for each coordinate at every time step was calculated.  The differences between the individual coordinates and the mean value were quantified and normalized by the standard deviation.  This process gives equal weighting to all three coordinates when performing the cluster analysis.  The Euclidean distance between each trajectory pair was then calculated for each hour of the 48 hour backtrajectory.  However, the first six time steps back from the receptor site were given zero weighting to account for the spatial heterogeneity of the aircraft spiral locations.  To further discount the spatial variability of the receptor locations and place the emphasis on the source regions, the trajectory time steps were weighted linearly back in time, increasing the weighting for each hour after the initial six zero-weighted time steps.

            After the distances between individual trajectories were calculated, the trajectories were clustered using an agglomerative, hierarchical clustering algorithm.  The algorithm used an average linkage function to determine the distance between the trajectories making up the clusters.  After reviewing the meteorology and pollutant profiles associated with each cluster, eight was determined to be the appropriate number of clusters. In the above link, the backtrajectory densities are plotted for each cluster. To determine these densites, a 1^o latitude by 1^o longitude grid map was made for each cluster, and the number of backtrajectory points crossing a grid box was divided by the total number of backtrajectory points in a cluster.  Densities between the 1^o by 1^o grid were interpolated using a linear weighting scheme.