13.2 Backward versus Forward Dispersion




In the previous section we computed the dilution factor from the source to various sampling locations and then used the measured concentration value to estimate the emission rate. In situations where the source location is not known, a frequently employed option is to compute the dilution factor from the sampler to all possible source locations using an inverse integration. If there are multiple samplers, it may be possible to combine the information from each sampler to estimate the source location as well as the release rate. Here we will determine if the "backward dispersion" calculation from the sampler gives the same dilution factor as the forward calculation from the source.

  1. Start by retrieving the previously saved srcfwrd_control.txt and srcfwrd_setup.txt settings into the GUI menu from the last section. In that calculation we compared results to the 3-hr sampling data. Here the run should start at 83 09 25 17 corresponding with the real start time for the emissions, leave the run duration at 21 h. Open the Setup Run / Pollutant Emission menu and simplify the emissions even further, set the rate to 1.0 unit for 1 hour. Then open the concentration grid menu and change the averaging time from 03 hours to 01 hour and the start of the sampling back to the run (and emission) start time 83 09 25 17. Save the changes and run the model.

  2. When the run has completed, open the Utilities / Convert to Station menu and pick a site, such as S316, Oberlin OH, which we know is within the calculated and measured plume. Set the location to 41.30 -82.22. Also to keep enough significant digits in the fixed format output, increase the multiplier from 1.0E+12 to 1.0E+15. This last step is necessary because we reduced the emission rate from its actual value 67000.0 to a unit value. The resulting hourly output file at station 316 shows a value of 30.772 near the peak time (0700-0800) of the plume passage.

  3. To configure the reverse calculation, open the setup run menu and set the start time to 83 09 26 08 and the run duration to 15 hours; set the back radio-button to change the sign to negative. Open the starting location menu and change the starting location to the position of sampler 316 (41.30 -82.22). Then open the concentration grid menu and reverse the sampling start and stop times. Set the start time of sampling to 83 09 26 08 00 and the stop time to 83 09 25 17 00. Save the changes and run the simulation.

  4. When the run completes, open the Convert to Station menu and set the sampling site to the original release location 39.90 -84.22. The resulting time series at the source location shows a value of 9.679 between 1800-1700, the same transport time as the original calculation to 0700-0800. There is about a factor of three difference in the calculated dilution factors between the two approaches, although differences in the peak dilution factor are only 28% if we ignore the travel time.

The results shown here indicate that the dilution factors between the forward and backward calculations are not identical, although they may be similar and for certain applications, these differences are sometimes ignored due to the convenience of the backward calculation. Some of these issues were explored in previous sections, for instance when a particle trajectory intersects the ground the transport calculation is no longer reversible. Also the particle dispersion calculation depends upon the turbulence the previous time step:

  • U'(t+Δt) = R(Δt) U'(t) + ...

through the auto-correlation, which means that the current dispersion rate depends upon its previous history. If the turbulence was the same (stationary) during the computational period, then the forward and backward integration would be the same. If for example, the forward calculation period went from day to night and the corresponding backward calculation from night to day, the turbulence history of the two calculations would be quite different. The use of backwards dispersion was addressed in more detail by Eslinger and Schrom (2019), Utility of atmospheric transport runs done backwards in time for source term estimation, J. Environ. Radioact., 203, 98-106.

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