Solving the Coefficient Matrix

The source-receptor coefficient matrix that was created in the previous section can be solved directly to determine the source term and location. The solution is the source term vector given a measured data vector and the coefficient matrix, which is composed of the dilution factors for each source-receptor pair created in the previous step.

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:
   • D_ij S_i = R_j,
   
   where the dilution factors are defined as the coefficient matrix. A plot of the product S_iD_ij can be presented as a map of the concentrations contributed by source i to all the receptors. A plot of S_iD_ij 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:
   • S_i = (D_ij)^-1 R_j.
   
   An example of using this approach to determine the CAPTEX source term follows from the previous tutorial. If those results are not available, go back and complete the previous section before proceeding.


2. Press the Display / Source-Receptor / Solve menu tab to open the matrix solution menu. The measured data vector file needs to be defined. An extract of the 3-hour sampling data should be used that only contains the measured values from the 26th from 0900-1200. The unit conversion factor of 1.0E-15 is the number of kg per pg (the measured data unit). Execute the iteration button to generate a solution, and then display the solution results which after pass #2 shows that an emission of 274 kg from 39.5 -84.5 would account for the measured data values. We know the correct answer is 201 kg. The results are written to a file called matrix.txt. Setting the verbose check box provides more detailed information on the solution process.

The solution of the coefficient matrix to determine source location or amount is an attractive simple and objective method as presented in this example. However, in reality there is an underlying subjective component which may require editing the measured data to reduce singularities. It may be difficult at times to obtain a solution because there will usually be many more source locations than measured data values. This section should be considered more experimental rather than a mature operational approach to source estimation.