Model for Transboundary Air Pollution

In our model, Finland, Estonia, and Russia negotiate over the amounts of sulfur and nitrogen emissions in 2010. We consider only those parts of Russia that are close to Finland and Estonia and thus affect depositions in those countries. Thus, four regions are considered in the model, namely, Finland, the Kola and Karelia region, the St. Petersburg region, and Estonia. In the sequel, vectorsx^S ∈IR⁴ and x^N ∈IR⁴denote the sulfur (SO2) and the nitrogen (NOx) emissions of the regions.

The decision vector isx:= ((x^S)^T (x^N)^T)^T ∈IR⁸, where x1 := the sulfur emission of Finland,

x₂ := the sulfur emission of the Kola and Karelia region, x₃ := the sulfur emission of the St. Petersburg region, x4 := the sulfur emission of Estonia,

x₅ := the nitrogen emission of Finland,

x₆ := the nitrogen emission of the Kola and Karelia region, x7 := the nitrogen emission of the St. Petersburg region, x₈ := the nitrogen emission of Estonia,

and all the emissions are given in kt/year.

We assume that if no negotiated agreement is reached, each country carries out the abatements required by current legislation. Thus a forecast for the emission levels in 2010 obtained using the current legislation scenario (clbunn scenario in Rains 7.2 software) can be considered as a status quo solution of the negotiation.

The numerical value for it isx¯= (155 473 136 175 152 86 170 73)^T. 5.2.1 The transportation model

The transportation model for air pollutants simulates the atmospheric processes and deposition rates of different pollutants. In RAINS, the transportation model is a linear mapping from the regions to a 150×150km² grid covering all of Eu-rope. The transfer matrices used in the model are provided by the EMEP (European Monitoring and Evaluation Programme) Meteorological Synthesizing Center-West (MSC-W) at the Norwegian Meteorological Institute. Letq_j^S, q_j^N denote the sulfur

96 Pirja Heiskanen and nitrogen depositions in the grid cell j, j = 1, . . . , n(n is the number of the grid cells). Then the total depositions in the countries,q¯^S ∈IR³andq¯^N ∈IR³, are obtained as a weighted sum of the depositions in the grid cells. Here the weight for a grid cell is the country’s area in that grid cell (for details see Heiskanen, 2000).

The transportation model for the emissions of the regions to the depositions in the countries is the following:

¯

q^S =G^Sx^S+b^S (5.1a)

¯

q^N =G^Nx^N +b^N . (5.1b)

HereG^S ∈ IR^3×4 andG^N ∈ IR^3×4 are the transportation matrices for sulfur and nitrogen, respectively. Vectors b^S ∈ IR³ and b^N ∈ IR³ contain depositions due to natural background depositions and emissions originating outside the re-gion of interest. The background depositions and the emissions for other countries than Finland, Russia, and Estonia are obtained using the current legislation sce-nario in the RAINS model. All the depositions are given in kt/year. The countries are enumerated so that the first components of the vectors denote the depositions in Finland, the second components denote the depositions in Estonia, and the last components denote the depositions in Russia. The numerical values for the trans-portation matrices and for the background depositions are as follows:

G^S =



 0.133 0.040 0.026 0.038 0.013 0.001 0.006 0.045 0.098 0.178 0.285 0.194





G^N =



 0.044 0.010 0.014 0.021 0.004 0.000 0.003 0.012 0.079 0.064 0.124 0.073





b^S = (40.99 16.29 943.22)^T b^N = (34.01 10.47 593.64)^T 5.2.2 Critical loads and exceedance functions

The effect of a pollutant on the environment depends not only on the amount of deposition, but also on the sensitivity of the ecosystem. For example, a sulfur level that is harmless to an ecosystem in central Europe may already pose a serious threat to a sensitive ecosystem in Lapland. Therefore, the concept of critical load (CL)

Efficient Alternatives in a Transboundary Air Pollution Negotiation 97 has been introduced. It is defined as “a quantitative estimate of an exposure to one or more pollutants below which significant harmful effects on specified sen-sitive elements of the environment do not occur according to present knowledge”

(Nilsson and Grennfelt, 1988). The critical loads are usually defined separately for every ecosystem, and, on their basis an average critical load for every grid cell is obtained.

The amount of the harmful effect on the environment is measured by ex-ceedance function (see, e.g., Nilsson and Grennfelt, 1988). If only one pollutant contributes to an effect, e.g., nitrogen to eutrophication, a unique critical load can be calculated and compared with deposition. Therefore, if the critical load of nutrient nitrogen for a grid celljis denoted byCL^nut_j , then the exceedance for eutrophica-tion for that cell is obtained as

e^nut_j =q_j^N −CL^nut_j . (5.2)

In the case of acidification, the definitions for critical load and exceedance are more complicated, since both sulfur and nitrogen contribute to acidification. How-ever, one equivalent of sulfur contributes more to excess acidity than one equivalent of nitrogen. As long as the deposition of nitrogen stays below the minimum critical load of nitrogenCL^{N min}, all deposited nitrogen is consumed by sinks. Then the critical load for sulfur is given byCL^Smax. The maximal critical load for nitrogen isCL^{N max}in the case of no sulfur deposition. Otherwise the critical load for nitro-gen is between the maximal and minimal value. Here we approximate the critical load function by a linear function. Then the exceedance for acidification for a grid celljgets the following form (for details see Heiskanen, 2000):

e^acid_j =q_j^S+ [ CL^Smax_j

CL^{N max}_j −CL^{N min}_j ][q_j^N−CL^{N max}_j ]. (5.3) The accumulated exceedance for a country is a weighted sum over the ex-ceedances of the grid cells. Here the weight of a grid cell is the fraction of area of the country in that cell. To obtain a measure that is independent of the area of the country, average accumulated exceedance is often used instead of the ac-cumulated exceedance. It is obtained by dividing the acac-cumulated exceedance by the total area of the country. The average accumulated exceedance has the same dimension (mg/m²/year) as deposition, and thus they can be directly compared.

Combining the previous results, we obtain a linear mapping from the emissions of the regions to the average accumulated exceedances of the countries as follows

98 Pirja Heiskanen

E^acid(x^S, x^N) =T¹x^S+T²x^N+v¹ (5.4a)

E^eut(x^N) =T³x^N +v², (5.4b)

where the numerical values for matricesT¹,T²,T³ and vectorsv¹ andv²are the following:

T¹ =



 0.398 0.119 0.079 0.114 0.286 0.014 0.131 0.979 0.028 0.051 0.082 0.056





T² =



 0.088 0.021 0.028 0.043 0.097 0.001 0.069 0.295 0.022 0.018 0.035 0.021





T³ =



 0.132 0.030 0.04 0.062 0.093 0.001 0.065 0.273 0.023 0.019 0.036 0.021





v¹ = (-228 -2566 -636)^T v² = (-93 -55 -135)^T