This paper shows how local sensitivity analysis, in respect of the parameters which specify the boundary conditions, can be used for relating the total load of non-conservative pollutants to their distributions within a water body. The method is applied to the estimation of the Maximum Permissible Load of inorganic nitrogen in the lagoon of Venice, that is of the maximum load of nitrogen which keeps its average yearly concentration below a prescribed threshold. The use of the spatial distributions of sensitivity coefficients in order to rank the sources of pollution and to forecast the effect of a reduction in the pollution is also discussed.
Since 1999, in Italy the pollutant loads which enter the water-bodies have been disciplined by means of the so-called maximum-permissible-loads (MPLs) policy. Within this framework, Local Authorities should make an inventory of the sources of pollution and then fix the level of emission of each activity, in order to maintain the concentrations of potentially dangerous substances below prescribed thresholds, called ‘Quality Targets’ (QTs). The implementation of this policy may clearly benefit from the use of mathematical models, which can be used as tools for both estimating the MPLs, by solving the so-called’ inverse problem’, and exploring the consequences of different input scenarios.
In fact, in mathematical terms, the loads are specified by a set of boundary conditions: numerical models can then be used for determining a functional relationship between the set of parameters which specify the boundary conditions and the output variables which one decides to compare with the QTs. Once this task has been accomplished, one can invert this function in order to estimate the MPLs which are compatible with the targets.
These problems are investigated in this paper using the lagoon of Venice as a case-study and the sensitivity analysis in respect of each source of pollution as a tool. In fact, because of its peculiarity, this system was thoroughly investigated and a 3D reaction–diffusion water quality model is already available [1]. Advective transport is not included in the model, as numerical simulations suggested that the residual currents in the lagoon of Venice are very small [2], [3] and [4]. However, diffusivities embody information about the contribution of the tide to the dispersion, as they are computed at each grid point on the basis of a statistical analysis of the results of a Lagrangian particle dispersion model [5].
The reaction–diffusion Eq. (1) is solved using a finite-difference scheme
equation(1)
In Eq. (1), c is the state vector; K, the tensor of eddy diffusivities; f, the reaction term; β is the set of site-specific parameters. The model simulates the dynamic of the ecosystem up to the second trophic level by using 12 state variables. The state vector includes the concentrations of the two main forms of inorganic nitrogen, ammonium and nitrate, as well as the inorganic reactive phosphorous one. These chemicals are considered to be the main cause of the eutrophication and, therefore, the current legislation fixes their QT for the lagoon of Venice. In this paper, we have applied the method outlined below to the estimation of the MPLs of ammonium and nitrate: the sum of their concentration gives the dissolved inorganic nitrogen (DIN), as the concentrations of nitrite is very low. At present, the concentration of DIN is above the target, while the concentration of reactive phosphorous is very close to it, as its use in detergents was prohibited in 1989.
Ammonium, NH4, and nitrate, NO3, are carried into the lagoon by the rivers, and are directly released from the Industrial area of Porto Marghera, on the edge of the lagoon, and from the city of Venice and the nearby islands. The yearly evolutions of these inputs were modelled using Von Neumann-type time-dependent boundary conditions: the fluxes Φi were specified using a set of trigonometric polinomia
equation(2)
equation(3b)
where ΦiNH4(t) and ΦiNO3(t) are the daily fluxes of ammonium of nitrate released by the ith source at time t, expressed in days.
The parameters αi,0NH4,…,αi,6NH4,…,αi,0NO3,…,αi,6NO3 were estimated for each source by means of a least squares regression of the monthly data reported in Ref. [6] and on total loads data presented in Ref. [7]. The exchanges with the Adriatic sea at the three inlets were described by means of Dirichlet-type boundary conditions: the concentrations of ammonia and nitrate at the boundaries were taken from Ref. [8]. On the basis of the available data, it was possible to define 16 sources, which are shown in Fig. 1together with the sampling stations which were monitored from 1995 to 1999. Therefore, 2×7×19 parameters had to be considered as potential input factors, if one wishes to take into consideration also the fluxes through the three inlets.
Full-size image (14 K)
Fig. 1.
Sources of DIN, white circles, and the monitoring network, filled circles.
Local sensitivity analysis, LSA, with respect to the parameters which characterise the sources of pollution in waterbodies appears to be a flexible and effective tool for dealing with problems which are posed by environmental policies which are based on the respect of water quality thresholds. Such approach seems particularly suitable for complex transition systems, as the simulation of their dynamics requires the use of time-consuming reaction transport models. However, it must be kept in mind that LSA relies on the linearization of the state-equation and, therefore, the results may not always be accurate. In this paper, LSA was used for obtaining a quick first estimation of the Maximum Permissible Load, which was based on the nominal simulation: this finding was then checked by taking as input the MPL which were determined and re-running the model. Such a check, which in this case-study gave satisfactory results, should always be made.
The possibility of computing very quickly the distributions of the concentration of the pollutants from the results of a single run, even though with some degree of inaccuracy, is extremely interesting in cases when the remediation policies must be agreed upon in a public contexts, when the local authorities, the population and the stakeholders concerned are encouraged to express their views. In such occasions, the post-processing module could be included in a GIS system and used interactively to show the effects of alternative remediation strategies.
