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David Parra-Guevara et al Int. Journal of Engineering Research and Applications ISSN : 2248-9622, Vol. 3, Issue 6, Nov-Dec 2013, pp.763-776 shown the usefulness of the Tikhonov regularization in the case when the functional to be minimized contains the derivative of the emission rate, and is subjected to integral constraints on the pollutant concentration which contains full information, in a compact form, on the dispersion phenomenon in the atmosphere and utilize the data on the anomalies of pollutant concentration. The adjoint solutions (or influence functions) used in the integral constraints exhibit the explicit relation between the emission rate and the data on the anomalies of pollutant concentration. The resulting variational problem is easily solved. In Section II, we formulate the general variational problem, while a dispersion model and its corresponding adjoint, which match the features of the general formulation, are presented in Section III. Finally, in Section IV the numerical approach of the variational problem is presented, and two examples of the solution of inverse problem are developed in detail using synthetic data.


Formulation of the problem

Let us suppose that a singular event, such as an explosion or release of a dangerous substance, happened completely during a time interval  0,T  in a known place r0 located in a region D , and the emission rate of new source Q(t ) is the control variable that must be determined. We also assume that 0 the distribution  (r ) of this pollutant in D at the initial moment t  0 , and the emission rates f (r, t ) of the other sources located in D are well known, and ) hence, the actual state  in region D for t  0 is


known (for example,  can be determined with a model). Taking into account the new source, the dispersion model for calculating the pollutant concentration  in time interval  0,T  can be written as

  A  f (r, t )  Q(t ) (r  r0 ) in D  0, T  t (1) (2)

 (r,0)   (r) in D 0

where A is the linear operator that describes the advection of pollutants by winds or currents, turbulent diffusion, deposition and decay of pollutants due to chemical reactions,  (r  r0 ) is the Dirac delta, and

r0 is the location of new source.


On the other hand, let us denote by  the solution of dispersion problem (1)-(2) only with forcing f (r, t ) ( Q(t )  0 ). Thus, the anomaly of the


pollutant concentration      dispersion problem

satisfies the

  A  Q(t ) (r  r0 ) in D  0, T  t  (r,0)  0 in D

(3) (4)

As it was noted before, the basic ) concentration of the pollutant  is supposed known as the solution of the dispersion model (3)-(4) with the initial distribution  0 (r ) and forcing f (r, t ) . Further, the concentration  can be partially evaluated by means of its monitoring in some places of region D . Therefore, the time series of the ) anomaly      of the pollutant can be obtained at the monitoring sites. It is reasonable to assume that the errors in the measurements of concentration  are small.


However, the errors in the basic function  are typically much larger than the errors of measurements, because this function is the result of solution of a computational model that does not describe exactly the processes of dispersion and transformations in the atmosphere. Therefore, we assume that the errors in ) the values of    , obtained in different monitoring sites, are of the same magnitude as the errors in ) function  . Such perturbations in the anomaly


     of concentration of pollutant generate the numerical instability in the process of inversion of data, and they are the fundamental reason for introducing one of the regularization methods. To this end, we first obtain integral equations that establish a direct relationship between the "cause" and the "effect" in the system. In order to find an explicit relationship between the anomaly  (state variable) and the emission rate Q(t ) (control variable), we introduce the influence function g as the solution of the following adjoint model:

g  A g  p(r, t ) in D  0, T  t g (r, T )  0 in D 

(5) (6)

The adjoint operator A is defined by means of the Lagrange identity [14]:

 A , g    , A g   h, g   D h g rd Hilbert space L2  D  . where

(7) is the inner product of the

Multiplying (3) by g and taking the integral over D  (0, T ) we get

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