384
O. Kriiger et al.
4
Inverse Modelling of Source Strengths
A reliable estimation of source strengths of gaseous mercury exclusively from
measurements of concentrations in the surrounding atmosphere near the sources
is impossible without modelling. The strong variability of the meteorological
parameters which determine the dispersion can only be taken into account using
a numerical model. On the other hand, results of a dispersion model based on
very uncertain and roughly estimated emission data most likely do not reflect the
real atmospheric conditions. Therefore, we decided to combine both methods.
The strategy was to use the measurements as constraints in fitting the model to
the emission source strengths. This procedure can be described as an inverse
modelling, because the model is applied oppositely to the usual way, in which the
transmission is calculated from known emission sources. Applying inverse
modelling, the problem can be formulated as follows:
In the surroundings of an ensemble of emission sources n measurements of the
concentrations dXi,Yi) of gaseous mercury at different points i exist. The location
of the measurements in a model grid, here in a downwind and cross-wind
direction of the plume, is given by Xi and Yi. Uncertainties of the measurements
are denoted by oci. The dispersion model calculates the concentrations Ci
(Xi,Yi) = f(xi,Yi,ex), where ex is the vector of free parameters being fitted. In this
case, ex represents the source strengths of the emission sources at the BSL Werk
Schkopau, which are treated variably in the procedure. In order to minimize the
differences between in-situ measurements and model results, a multiparameter
function is defined for a least-squares fit as a chi square:
2( ) = ~ [f(Xi) Yi) ex) - Ci]2
X ex
~
oc2
i=l
I
For the minimum, which is found at X2 = X2 (exmin), a statistical test for a X2distribution with [n - dim(exmin)] can be made to check the performance of the fit.
The parameter vector ex can usually be estimated by applying different methods
for minimizing X2. All investigations which are described in the following
sections are based on the simplex method (NeIder and Mead 1965). This genuine
multidimensional minimization procedure has the advantages that no accurate
start parameters have to be estimated, different ranges of the parameters can be
defined and it is even rather robust with respect to cross fluctuations in the
function value. Figure 5 shows the flow chart of the inverse modelling procedure.
5
Theoretical Tests with the Inverse Modelling Procedure
The applicability of the inverse modelling procedure for emission estimates at
the industrial site BSL Werk Schkopau was tested using characteristic vertical
O. Kriiger et al.
4
Inverse Modelling of Source Strengths
A reliable estimation of source strengths of gaseous mercury exclusively from
measurements of concentrations in the surrounding atmosphere near the sources
is impossible without modelling. The strong variability of the meteorological
parameters which determine the dispersion can only be taken into account using
a numerical model. On the other hand, results of a dispersion model based on
very uncertain and roughly estimated emission data most likely do not reflect the
real atmospheric conditions. Therefore, we decided to combine both methods.
The strategy was to use the measurements as constraints in fitting the model to
the emission source strengths. This procedure can be described as an inverse
modelling, because the model is applied oppositely to the usual way, in which the
transmission is calculated from known emission sources. Applying inverse
modelling, the problem can be formulated as follows:
In the surroundings of an ensemble of emission sources n measurements of the
concentrations dXi,Yi) of gaseous mercury at different points i exist. The location
of the measurements in a model grid, here in a downwind and cross-wind
direction of the plume, is given by Xi and Yi. Uncertainties of the measurements
are denoted by oci. The dispersion model calculates the concentrations Ci
(Xi,Yi) = f(xi,Yi,ex), where ex is the vector of free parameters being fitted. In this
case, ex represents the source strengths of the emission sources at the BSL Werk
Schkopau, which are treated variably in the procedure. In order to minimize the
differences between in-situ measurements and model results, a multiparameter
function is defined for a least-squares fit as a chi square:
2( ) = ~ [f(Xi) Yi) ex) - Ci]2
X ex
~
oc2
i=l
I
For the minimum, which is found at X2 = X2 (exmin), a statistical test for a X2distribution with [n - dim(exmin)] can be made to check the performance of the fit.
The parameter vector ex can usually be estimated by applying different methods
for minimizing X2. All investigations which are described in the following
sections are based on the simplex method (NeIder and Mead 1965). This genuine
multidimensional minimization procedure has the advantages that no accurate
start parameters have to be estimated, different ranges of the parameters can be
defined and it is even rather robust with respect to cross fluctuations in the
function value. Figure 5 shows the flow chart of the inverse modelling procedure.
5
Theoretical Tests with the Inverse Modelling Procedure
The applicability of the inverse modelling procedure for emission estimates at
the industrial site BSL Werk Schkopau was tested using characteristic vertical