12.7 Solving the Coefficient Matrix (CM)


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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 secton we will try to determine the time variation of the source 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 captex_control.txt and captex_setup.txt settings into the GUI menu. We will run a series of 9 simulations, each representing an emission period of one hour on the 25th starting at 1500 and ending at 2300. Open the Concentration / Setup menu and set the start time to 83 09 25 15 and a run time of 21 hours. As in several of the previous exercises, we will only be using the 3 hour duration samples.

  3. Open the Pollutant setup menu and change the emission rate and duration each to 1.0. Then open the Grids menu and change the grid resolution from 0.25 to 0.05 to provide finer spatial resolution in an attempt to improve the fidelity of the short-range simulations. Also give the output file a unique name hysp015 associated with the time of the release. Save and exit all menus and Run Model. When finished, go back to the setup menus and increase the start time by one hour, decrease the run duration by one hour, and rename the output file to correspond with the start time. Run Model eight more times, each time changing the name of the output file, until all simulations for the 25th have been completed.

  4. 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 hysp0 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 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 measurements are the vector shown in the last column and the transfer coefficients are the model computations of the contribution of each release time to that measured value.

  5. 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. The results are rather dissapointing. Negative values 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. It is difficult to remove high values as they indicate good transport between the source and sampler location. In contrast, small values could be the result of the model plume edge being near the sampling location. There is greater uncertainty to these lower values.

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. 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. This section should be considered more experimental rather than a mature operational approach to source estimation.