8.1 Summary
This thesis presents the development of an inverse model that may be used to estimate the parameters of the source term for a pollutant gas released into the at mosphere from a point source above the ground. We categorise the analysis process for the accidental release of gases from a single point source into the following:
1. The instantaneous release from a known location,
2. The instantaneous release from an unknown location,
3. The extended release over a period of time from a known location,
4. The extended release over a period of time from an unknown location.
The developments of inverse models for these cases are presented in Chapters 4, 5 and 6. The models use measured pollutant concentrations at a minimum of three observation sites on the ground, as well as meteorological data such as wind speed and percentage cloud cover. The problem where the pollutants are released instantaneously is well posed and the source term parameters are estimated with reasonable degree of accuracy. The methodology is based on nonlinear least squares,
186 Final Summary and Conclusions
and multiple linear regressions coupled with the solution of an advection-diffusion equation for an instantaneous point source.
But the problem where the pollutants release from the non-steady extended source is ill posed. The main difficulty with the ill-posed problem is that the mea sured data always have errors and consequently the solution is extremely sensitive to errors in the measurement data. Tikhonov's regularisation algorithm, which sta bilises the solution process, is used to overcome the ill posedness of this problem. Here the approach taken is to develop the inverse model as a least squares minimi sation problem coupled with the solution of an advection-diffusion equation for an extended point source.
For the case where the location of the source is known is a linear ill posed prob lem. We used the properties of L-curve and generalised cross-validation criterion from linear inverse theory to estimate the optimal value of the regularisation pa rameter. For the case where the location of the source is not known is a non-linear ill-posed problem. This problem is different from the linear problem in several ways. First, we can't use only linear algebra to determine the minima of the objective function. Second, the non-linear objective function may have more than one mini mum for each value of the regularisation parameter. To deal with these problems, we developed a number of different algorithms, each of which is applied to many test cases. The implementation of all the algorithms has a number of features in common. First, we often have to determine local minima for a given value of reg ularisation parameter. In all our algorithms exploiting the fact that some of the parameters appear in the problem linearly and hence determined using simple lin ear algebra for given values of nonlinear parameters and regularisation parameter. The nonlinear parameters. are computed in directly in MATLAB as the least-squares solution of the objective function using the optimisation routine lsqnonlin. This is speeded up enormously because, after the elimination of linear parameters using linear algebra, only few non-linear variables remain.
To deal with the multiple minima, we found all or most of the local minimum of the objective function for a sequence regularisation parameter values and then we selected the minimum solution which has the lowest function value. We employ basically two approaches to find the optimal value of the regularisation parameter. In the first approach we solved the problem for a sequence of regularisation pa rameter values and then we used three different methods described in section 6.3
to pick the optimal value the regularisation parameter. In the second approach we solved the problem for a sequence of regularisation parameter values but here we start from smaller value and then slowly increase its value in conjunction with the L-curve from linear inverse theory until its steady state. In our experience, both approaches are equally useful. If we consider only the amount of computation time, then approach 2 is a better choice than approach 1.
8 . 2 Conclusions
The research presented in this thesis addresses the problem of finding estimates of the release rate of pollutant gas from a single point source. The main contributions towards resolving this problem are:
(i) data from at least three spatial locations are needed to estimate reliably the release rate of atmospheric pollutants where the source of the pollutant location is not known,
(ii) the model is capable of finding the release rate of atmospheric pollutants by using the measured pollutant gas concentration data at just one location on the ground where the location of the pollutant source is known,
(iii) reliable estimates are obtained even with the partial capture of the data where the pollutants are released instantaneously but accuracy of the estimates are higher for the complete sampling data,
188 Final Summary and Conclusions (iv) for the cases where the pollutant is released from extended point source, the
developed method is able to recover the release rate with reasonable degree of accuracy. But, if the data sampling are incomplete, then the data are unable to provide correct information about the release rate at earlier times. This is because the plume is more dispersed, and information about the plume released at the earlier times is lost. Furthermore, accuracy of the solution does not decrease much on the increasing amount of noise in the data for the known location problem (linear problem), however, if the location of the source is not known (nonlinear problem), accuracy of the solution decreases roughly at the same rate with the increasing amount of noise in the data,
(v) accuracy of the reconstructed solution depends on the following
Instantaneous release
(i) size and randomness of noise in the data
(ii) distance between the source
and the receptor
(iii) distance and angle between
the receptors
(iv) number of measurement data
Extended release
(i) size and randomness of noise
in the data
(ii) distance between the source
and the receptor
(iii) distance and angle between
the receptors
(iv) number of measurement data
(v) regularisation
(a) order of regularisation
(b) regularisation parameter
selection method
(vi) discretisation size of
the source function
Table 8.1 summarises the above in short form. Problems like the post-accident management plan, leak detection in pipelines and other installations that are used