12.8 Cost Function Minimization of the CM


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In the previous section the transfer coefficient matrix (TCM) and air concentration measurement data were explicitly solved for the unknown source term using matrix inversion methods. In this section, with the same TCM, we use a cost function approach to minimize the difference between the observations and model predictions by varying the source term. The nine output files from the previous section are required to continue.

  1. In this example, the cost function F is minimized, qij are the emissions over M time periods and N source locations (in this case one location), qbij is the first-guess emission estimate, σ2ij is the emission error variance. In the second term, ahm and aom are the HYSPLIT air concentration predictions and observations, respectively and ε2m is the variance of the observations.

    Note that ahm is defined as the product of qij and the TCM for each release time at that observation location. Further technical details regarding the computational approach used to solve the TCM can be found in Source term estimation using air concentration measurements and a Lagrangian dispersion model–Experiments with pseudo and real cesium-137 observations from the Fukushima nuclear accident, T. Chai, R. Draxler, A. Stein, Atmospheric Environment, 106, 241-251.

  2. Using the simulations from the last section for each of the nine potential release times (hysp015 - hysp023 go to the Utilities / Transfer Coefficient tab and open the cost function 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.

  3. Step 5 is used to create the PARAMETER_IN_000 input file for the inverse modeling executable lbfgsb. Detailed information is required that is not always be well known and several solution iterations may be required before the optimal input parameters have been properly defined. In this CAPTEX example, change the first-guess source term to 30000 g which will be assumed for all simulation hours (1500-2300) although we know the source was only active from 1700-2000. Leave the other parameters with their default values. If further changes are required, then consider the following:

    • The first guess value only represents an estimate of the source term.
    • The scaling factor is used to reduce the numeric range of the source and predictions; a smaller range improves the solution convergence.
    • The first-guess uncertainty needs to be defined as the sum of a fraction and constant value. The emission inversion results are less sensitive to the first guess when large uncertainties are prescribed.
    • The measurement uncertainty should also be defined as a fraction and sum.
    • The source term solution may be bounded or unbounded. A lower bound can be used to eliminate negative solutions.
    • A logarithmic transformation can be applied to the source or the TCM results prior to computing a solution. The log transformation of the source also ensures non-negative solutions. In general, log transformation of the air concentrations perform better due to the large range of air concentrations.

  4. In the final Step 6, run the inverse modeling executable by pressing the Solve button. The solution is displayed and the results are grams. Overall the results are a considerable improvement over the SVD solution with the maximum emissions occurring during the known emission period (1700-2000). The results are also shown graphically where the green dots are the previous SVD solution while the red squares show the cost function solution. The solid black line shows the actual source magnitude and duration. Negative solutions are not shown. Different solutions can be tested by sequentially repeating steps 5 and 6. The PARAMETER_IN_000 can also be edited manually to set other parameters not defined in the GUI. In this case only repeating step 6 is required.

This section demonstrates a more stable alternative approach to solving the transfer coefficient matrix by varying the source term to minimize the differences between observations and model predictions. However, unlike the direct SVD solution, multiple input parameters need to be defined to achieve a reasonable solution.