Concentration field method

In this method, a mean or logarithmic mean concentration is calculated for each grid cell $(i,j)$. The equation for the logarithmic mean is

$$\log \overline{c}_{ij} = \frac{1}{\displaystyle \sum_{l=1}^M {\tau^l_{ij}}} \sum_{l=1}^M {\log (c_l) \tau^l_{ij}},$$ (2)

where $l$ is the index of the trajectory, $M$ is the total number of trajectories, $c_l$ is the concentration observed on arrival of trajectory $l$, and $\tau^l_{ij}$ is the time spent in $(i,j)$ by trajectory $l$ (Seibert et al., 1994; Stohl, 1998). A high value of $\overline{c}_{ij}$ implies that air parcels travelling over $(i,j)$ will have, on average, high concentrations at the receptor.

If the sampling time and/or the frequency of available trajectories are not constant, we make, after reindexing, the following adjustment to the weights in Eq. 2,

$$\log \overline{c}_{ij} = \frac{1}{{\displaystyle \sum_{s=1}^... ...{ij}}{M_s}} \sum_{s=1}^N {\log (c_s) \frac{\tau^s_{ij}}{M_s}},$$ (3)

where $s$ is the index of the sample, $M_s$ is the number of trajectories arriving at the receptor site during the sampling time of sample $s$, $N$ is the total number of samples, and $\tau^s_{ij}$ is the total time spent in $(i,j)$ by the $M_s$ trajectories. If trajectory data endpoints are given at equal time intervals, $\tau^s_{ij}$ can be replaced by the corresponding number of trajectory segment endpoints, $n_{ij}^s$. When the sampling time is the same for all samples and the frequency of trajectories is constant, Eq. 3 reduces to Eq. 2.