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Reduced-Order Modelling of Dispersion 251
value for a white noise for angles between 0 and π. Onced is calculated by this
procedure one needs to define d ⊥ E which is unknown as B⊥ is unknown. We
make the approximation d ⊥ E ∼ d E(B,B ∗ )whereB ∗ is the projection of B over
the vector u, the averaged velocity along the polynomial characteristic. This
approach gives satisfactory results for smooth atmospheric flow fields which
is our domain of interest as no phyto treatments is, in principle, applied when
the wind is too strong or if the temperature is too high (e.g., for winds stronger
than 20 km/h and air temperature more than 30 ◦ C). This also makes that the
polynomial construction above gives satisfaction with low order polynomials.
Ground Variations
At this point one accounts for the topography or ground variations ((x, y) →
ψ(x, y)) in the prediction model above. These are available from digital terrain
models (DTM) [Arc06]. Despite this plays an important role in the dispersion
process, it is obviously hopeless to target direct simulation based on a detailed
ground description. One should mention that ground variations effects are implicitly
present in observation data for wind and transport as mentioned in
Section 2.2. However, as we said, observations are quite incomplete and to
improve the predictive capacity of the model one needs to model the dependency
between ground variations and migration time. Therefore, in addition
to the mentioned assimilation problem, one scales the migration times used
for transport over large distances by a positive monotonic decreasing function
f(φ) with f(0) = 1 where
φ =(∇ x,y ψ · u H )/‖u H ‖.
Here u H is the ‘close to ground’ constructed flow field based on the assimilated
observations.
3 Parameter and Source Identification
Two types of inverse problems have been treated. The first inverse problem
is for parameter identification in the model above assimilating either local
experimental data (as described in Section 2.1) or partial data available on
wind u obs and transported species c obs measured by localized apparatus. In
particular, the unknown parameters in our global transport model comes from
the solution of a minimization problem for:
J(p, u obs ,c obs )=‖c(p, u(p, u obs )) − c obs ‖, (6)
where p gathers all unknown independent parameters in Section 2.2. ‖·‖ is
a discrete L 2 -norm over the measurement points. u(p) is the completion of
available wind measurements (u obs ) over the domain described in Section 2.2.