10 Applications of the Inverse Problem of Pollution Propagation
347
case when the length of the optimum fairway does not play a role and the basic goal
is to minimize the average probability of coastal hit or to maximize the particle age
along the fairway or to share costs equally.
If the intention is to divide costs equally, an inherent approximation to the optimum solution is the equiprobability line (Soomere et al. 2010). Sailing along this
line naturally results in dividing the potential costs equally between the opposite
coasts: a deviation from it to either side increases the projected costs of consequences for this side. It is heuristically clear that the equiprobability line generally
does not provide a global minimum for environmental risks. This solution is only
suitable for elongated basins and accounting for the presence of islands may be
complicated.
For a sea area of elongated shape the maps of probability of coastal pollution and
of particle age, ideally, provide an elongated ‘trough’ of low values for probabilities
or, equivalently, a similar ‘crest’ of particle age, and a more or less gently sloping
pattern of the measure −1 ≤ ˆ
p ≤ 1 between the opposite coasts. Figures 10.7–10.9
suggest that the resulting fields generally have one local extremum of the probability
and age (and also one zero-crossing point of the quantity ˆ
p) for all cross-sections of
the Gulf of Finland except for the entrance and for the easternmost part of the gulf.
It is natural to assume that the optimum fairway goes roughly along this trough or
crest, or follows the zero-crossing line of ˆ
p.
In practical applications the underlying data for Figs. 10.7–10.8 were first corrected manually in order to adjust a few clearly erroneous values (zeros of the probability or very large values of the particle age) for several bays deeply cut into mainland. In the model resolution, currents in these areas were very small and almost no
particle propagation occurred in the RCO model that worked without local spreading. The latitude for the points of the optimum fairway was found from the analysis
of the cross-section of the relevant field along each longitude (Fig. 10.9). It was set
at the centre of the grid cell corresponding to either the minimum probability p ij or
the maximum particle age p ij (Soomere et al. 2011c; Lu et al. 2012). The obtained
estimates for the optimum fairway were confined to discrete values of the centres of
grid cells and sometimes contained abrupt shifts by 2–3 cells to the south or north.
Optionally the curves were smoothed over five neighbouring values in the east–west
direction (Fig. 10.10).
Another estimate of the optimum fairway roughly following the equiprobability line was constructed from the zero-crossing points of the measure ˆ
p (Soomere
et al. 2010). These points were well defined in the relatively narrow parts of the gulf
that hosted comparatively large north–south gradients of ˆ
p. The small-scale fluctuations in the distribution of ˆ
p in some parts of the bay (especially in the eastern
part where its north–south slope was small) were smoothed by averaging the crosssection over five neighbouring grid points. An estimate for the zero-crossing of ˆ
p
for each longitude was found using a linear approximation between its smoothed
values at adjacent grid points. Doing so led to a single clearly defined zero-crossing
point for each longitude. The curves consisting of the zero-crossing points required
no additional smoothing in the east–west direction (Soomere et al. 2011c).
347
case when the length of the optimum fairway does not play a role and the basic goal
is to minimize the average probability of coastal hit or to maximize the particle age
along the fairway or to share costs equally.
If the intention is to divide costs equally, an inherent approximation to the optimum solution is the equiprobability line (Soomere et al. 2010). Sailing along this
line naturally results in dividing the potential costs equally between the opposite
coasts: a deviation from it to either side increases the projected costs of consequences for this side. It is heuristically clear that the equiprobability line generally
does not provide a global minimum for environmental risks. This solution is only
suitable for elongated basins and accounting for the presence of islands may be
complicated.
For a sea area of elongated shape the maps of probability of coastal pollution and
of particle age, ideally, provide an elongated ‘trough’ of low values for probabilities
or, equivalently, a similar ‘crest’ of particle age, and a more or less gently sloping
pattern of the measure −1 ≤ ˆ
p ≤ 1 between the opposite coasts. Figures 10.7–10.9
suggest that the resulting fields generally have one local extremum of the probability
and age (and also one zero-crossing point of the quantity ˆ
p) for all cross-sections of
the Gulf of Finland except for the entrance and for the easternmost part of the gulf.
It is natural to assume that the optimum fairway goes roughly along this trough or
crest, or follows the zero-crossing line of ˆ
p.
In practical applications the underlying data for Figs. 10.7–10.8 were first corrected manually in order to adjust a few clearly erroneous values (zeros of the probability or very large values of the particle age) for several bays deeply cut into mainland. In the model resolution, currents in these areas were very small and almost no
particle propagation occurred in the RCO model that worked without local spreading. The latitude for the points of the optimum fairway was found from the analysis
of the cross-section of the relevant field along each longitude (Fig. 10.9). It was set
at the centre of the grid cell corresponding to either the minimum probability p ij or
the maximum particle age p ij (Soomere et al. 2011c; Lu et al. 2012). The obtained
estimates for the optimum fairway were confined to discrete values of the centres of
grid cells and sometimes contained abrupt shifts by 2–3 cells to the south or north.
Optionally the curves were smoothed over five neighbouring values in the east–west
direction (Fig. 10.10).
Another estimate of the optimum fairway roughly following the equiprobability line was constructed from the zero-crossing points of the measure ˆ
p (Soomere
et al. 2010). These points were well defined in the relatively narrow parts of the gulf
that hosted comparatively large north–south gradients of ˆ
p. The small-scale fluctuations in the distribution of ˆ
p in some parts of the bay (especially in the eastern
part where its north–south slope was small) were smoothed by averaging the crosssection over five neighbouring grid points. An estimate for the zero-crossing of ˆ
p
for each longitude was found using a linear approximation between its smoothed
values at adjacent grid points. Doing so led to a single clearly defined zero-crossing
point for each longitude. The curves consisting of the zero-crossing points required
no additional smoothing in the east–west direction (Soomere et al. 2011c).