Operational Oil Spill Modelling: From Science to Engineering Applications in the Presence of Uncertainty - Mathematical Modelling and Numerical Simulation of Oil Pollution Problems - page 119

to require extensive hindcast analyses that are customized to the driving models and
the regional/local conditions [ 39 ].
An ensemble approach requires setting perturbation values for the parameters and
initial conditions for the oil spill model. These can be set as part of a Monte Carlo
approach over the expected range. Lagrangian particle from all simulations are then
combined into a single data set that can be analyzed for any particular time point
using the following steps:

(i) Define a bounding box covering all the particles at a given forecast time.
(ii) Subdivide the bounding box in a grid.
(iii) Compute the probability density based on particle counts in the grid cells.
(iv) Accumulate probability starting from high values and working downwards so
that the integrated probability over the area is 100%.
(v) Construct and visualize probability contours.

A demonstration of the probability approach is illustrated in Fig. 5.11 . The grid
size within the bounding box is chosen to ensure statistically significant grid cells
over majority of the forecast spill area, which will depend on the number of simulated
particles and their effective diffusion. The probability density can be computed by
any number of kernels, (e.g. [ 28 , Chap. 9]), but a simple Gaussian kernel is arguably
appropriate for an oil spill [ 1 , 39 ]. Accumulating probability contours from high to
low values allows development of probability distributions with multiple local max-
ima and disconnected contours. The probability approach, whether adopted within
a multi-model forecast system or simply within the oil spill model itself, provides
the end user with a better understanding of the results of an oil spill simulation than
does the traditional point cloud or particle tracks.

Fig. 5.11 The simulation from Fig. 5.7 using a probability approach for the particle distribution
instead of the particle cloud. Colour in left and centre panels represents probability of finding a
Lagrangian particle in each subgrid cell of the bounding box. Right panel compares the particle
cloud ( red )to50 , 70 and 90% contours of accumulated probability