13.6 Solving the Coefficient Matrix (CM)




The source-receptor matrix created in the previous section was compared to the measurement data to determine the source location. Other previous examples used the measurement data to determine the source strength. In this section we will try to determine if there is any time variation of the source strength by creating a transfer coefficient matrix representing the contribution of each time period to the measurement data.

  1. Assume that the concentration at receptor R is the linear sum of all the contributing sources S times the dilution factor D between S and R:
    • Dij Si = Rj,

    where the dilution factors are defined as the coefficient matrix. A plot of the product SiDij can be presented as a map of the concentrations contributed by source i to all the receptors. A plot of SiDij for receptor j would show a map of the concentration contributed by each source to that receptor. In the case where measurements are available at receptor R and source S is the unknown quantity, the linear relationship between sources and receptors can be expressed by the inverse of the coefficient matrix:
    • Si = (Dij)-1 Rj.

    The dilution factors can be computed from various source locations, the same location but with releases at different times, or a combination of the two. For the example in this section, we assume we know the source location but not the time variation of its strength or even its start and stop time.

  2. First we will need to run a series of dispersion simulations for each of the potential release times. Start by retrieving the previously saved geol_control.txt and geol_setup.txt settings into the GUI menu. We will run a series of 5 simulations, each representing an emission period of 12 hours. The first starting on the 1st at 0000 UTC and ending 60 hours later on the 3rd at 1200 UTC. The next 12 h emission simulation starts at 1200 UTC on the 1st and ends 48 h later on the 3rd at 1200 UTC. Open the Concentration / Setup menu and set the starting location to 43.0 -75.0. Open the Pollutant setup menu and change the emission rate to 1.0 and the duration to 12 hours. Set the emission starting times to 00 00 00 00 00 because for each simulation, the emission start time will match the simulation start time. Finally open the Grids menu and change the grid resolution from 0.25 to 0.10 to provide finer spatial resolution in an attempt to improve the fidelity of the simulations. Also give the output file the unique name tcm for Transfer Coefficient Matrix which will later be appended with the time of the release. Save to close the setup menus.

  3. Because of the finer concentration grid, we should release more particles. Open the Advanced / Configuration / Concentration / menu #4 and increase the particle release number from 5000 to 10,000. Save to close the menus. The GUI variables are now configured to run multiple simulations.

  4. Now open the Special Runs Daily menu to configure a GUI script to run the dispersion simulation once for each emission period. The menu shows the start time of the simulation as previously configured in the SETUP menu and the only additional information required is the period over which new simulations will be started. In this case, up to 48 hours after the start time. Also check the radio-button to shorten the run duration of each new simulation by 12 hours. In this way each new simulation, starting 12 hours after the previous one, will have a duration of 12 hours less than the previous one, insuring that all simulations end at the same time. Once configured, Execute Script and as each new run is started the output file name is shown on the run log. Output files are named with the base plus the start time of the simulation. Note that the last run starts on the 3rd at 0000 UTC, 48 hours after the start time.

  5. When all the runs have been completed, go to the Utilities / Transfer Coefficient tab and open the SVD solution menu. In step 1 replace the concentration file name wildcard with tcm and then press the Create to generate the INFILE of filenames. In step 3 define the units conversion factor and any other simulation specific requirements. Then in step 4, press Create to generate the transfer coefficient matrix in a comma delimited format. This can be opened in Excel. Each row represents a measurement. Each column represents the dilution factor for the release time (1st row of that column) and measured data value, which is given in the rightmost column for that row. The transfer coefficients are the model computations of the contribution of each release time (column) to that measured value (row).

  6. In step 5, pressing Solve generates the default solution using all matrix elements. Dates are given in the raw format (days since the year 1900) and the results are grams. Negative values should be treated as zero and indicate that the model transfer coefficient is too large for the measured value. The solution to these equations is driven by the model transport errors. High values indicate good transport between the source and sampler location. In contrast, small TCM values could be the result of the model plume edge being near the sampling location and result in higher emission estimates. Although the release rate is overpredicted, ranging from 2000 to 9000 g/h, the results suggest a continuous source.

Although in this example the source-term vector varied only in time, solutions using a spatial variation, or both space and time, are also possible. The solution of the coefficient matrix to determine source location or amount appears to be a simple and objective method. However, in reality there is an underlying subjective component which may require editing the measured and model data to reduce singularities due to uncertainties in the data or model predictions. It may be difficult at times to obtain a solution because there will usually be many more source locations or release times than measured data values. The next section will review a method that considers the errors in the measurements and model predictions in determining the source.

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