# 12.7 Solving the Coefficient Matrix (CM) Previous Next

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 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. Reduce the particle release number from 50,000 to 20,000 to speed up the calculations. Then 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 the unique name tcm for Transfer Coefficient Matrix and later to be associated with the time of the release.

4. Now open the Special Runs Daily menu to configure a script to run the dispersion simulation once for each possible emission hour. 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 9 hours after the start time. Also check the radio-button to shorten the run duration of each new simulation by 1 hour. In this way each new simulation, starting one hour after the previous one, will have a duration of one hour 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 is for 23 UTC, only 8 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 tcm0925 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.

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. The results are rather disappointing. 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.

7. To illustrate the sensitivity of the results to small changes in model predictions or measurements, a hypothetical measurement file captex2_hypo.txt was created using the transfer coefficient matrix and the known source term. Comparing this to the actual measurements show the same major features with most of the differences in the concentration magnitudes. When the hypothetical measurement file is used, the solution captures the correct emission amounts at the correct times.

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.