Trajectory regression analysis is a way to determine a simple relationship between a measured air quality parameter at a specific receptor location and the amount of time that air is flowing across potential source regions on their way to the receptor location. The dependent variable in the multiple-linear regression is the air quality parameter and the independent variables are the estimates of time that air spends over each of a group of specific source regions. Back-trajectory analysis is used to estimate the amount of time an air parcel spends over each source region.

Implicit in the trajectory regression attribution analysis approach is the concept that the amount of time air spends over a region determines that regions contribution to pollutants measured at the receptor site. Obviously this approach is too simplistic to capture the effects of varying atmospheric factors that are known to be influential modulating concentration (e.g. washout by precipitation, enhanced chemistry in clouds, etc.). Another source of uncertainty results from errors in the trajectory estimated position and movement of air over source prior to arriving at the receptor sites. In spite of these sources of error, long-term average source region contributions determined by trajectory regression attribution analysis have been found to be reasonable and useful. [add references to papers by Gebhart & others]

The trajectory regression analysis for the Causes of Haze Assessment is done two ways, with an additive intercept term and with no intercept. The intercept term is thought to account for both contributions from beyond the selected source regions (like a global background value) and statistical noise from imprecise parameters and an imperfect model. Regression without the intercept forces the sources to account for some of the background contributions, so will overestimate some of the source region contributions, while regression with an intercept may underestimate some source region contributions by incorporating statistical noise in the intercept term. The most reasonable attribution results by the trajectory regression method are likely within a range set by regression with and without an intercept.

A feature of the regression analysis methodology is that an uncertainty level and statistical significance are estimated for each regression coefficient. The uncertainty is the standard error for the regression coefficient, meaning that there is a 67% probability that the “true” coefficient is within a range of plus or minus the uncertainty around the regression coefficient value. The magnitude of the attribution uncertainties are shown in the attribution bar plots as a vertical line extending above and below the top edge of each attribution bars for the source regions. If the uncertainty line is of a comparable magnitude to the attribution bar height (e.g. ½ or greater than the bar), then the attribution should be considered uncertain. Another way to assess the question of statistical uncertainty is with the significance or p value for each coefficient. It estimates the probability that the regression coefficient value is statistically different than zero. A p value of 0.0000 indicates that the regression coefficient has less than 0.01% chance of being zero. Larger p values correspond to higher probability of being insignificantly different from zero (e.g. p = 0.0010 → 0.1% probability; p = 0.0100 → 1% probability; and p = 0.1000 → 10% probability of being zero), so the smaller the p value the more reliable is the regression attribution method from a statistical prospective.

Selection of source regions for trajectory regression can be done in any number of ways. Some approaches are more likely to be successful than others in terms of producing a statistically meaningful result. In general you want to have relatively larger source regions at a greater distance to the site than those near the site. It’s generally good to avoid source region boundaries that bisect high emissions density regions, but it is also good to chose source regions that are of interest to the ultimate user, in our case the WRAP. The approach chosen to define source regions for the trajectory regression for the Causes of Haze Assessment is outlined below.