Geoscience Reference
In-Depth Information
research and application projects. Such parallel systems are ideally suited to this type of calculation. If a
single run of the model fits in the memory of a single processor, then there is very little loss in efficiency
in the parallel system: essentially each processor runs at full calculation capacity during a run, except for
some short periods in which the results from a run are passed to the master processor or written to disk and
a new run is initiated. A recent alternative is the use of off-the-shelf graphics processing units (GPUs) for
scientific calculations using, for example, the CUDA extension of the C programming language. GPUs
consist of hundreds of parallel processors on a single board and can be used to greatly speed up run times
for certain types of modelling problem (e.g. Lamb
et al.
, 2009).
However, with large models, there are still some advantages in trying to make the Monte Carlo sampling
more efficient. In the GLUE methodology, for example, there is little advantage in sampling regions of
the parameter space with low likelihood measure values once those regions have been established. It
would be better to concentrate the sampling in the regions of high likelihoods. This is a subject that
has been studied in a variety of different fields, resulting in extensive literature on what is often called
importance sampling
. A number of methods have been developed to try to exploit the knowledge gained
of the response surface in refining an adaptive sampling strategy. These include the MC
2
and DREAM
techniques described in Box 7.3, which attempt to sample the response surface according to likelihood
density, so that regions of high likelihood are sampled more frequently. The hope is that considerable
savings in computer time will be made in defining the likelihood surface. Such methods may work well
when there is a well-defined surface but for surfaces with lots of local maxima or plateaux, the advantages
may not be so great. The efficiency of uniform sampling techniques can also be improved by running
the model only in areas of the parameter space where behavioural models are expected on the basis of
previous sampling. The nearest neighbour method of Beven and Binley (1992), the tree-structured search
of Spear
et al.
(1994), and the Guided Monte Carlo method of Shorter and Rabitz (1997) can all be used
in this way.
7.10.4 Deciding on a Likelihood Measure
There are many measures that can be used to evaluate the results of a model simulation. They, in part,
depend on what observational data are available to evaluate each model, but even if only one type of
data is available (such as observed discharges in evaluating a rainfall-runoff model) there are different
ways of calculating a model error and using those model errors to calculate a likelihood measure. What
is certain is that if we wish to rank a sample of models by performance, different likelihood measures
will give different rankings and the same measure calculated for different periods of observations will
also give different rankings.
The choice of a likelihood measure should clearly be determined by the nature of the prediction problem.
If the interest is in low flows, then a likelihood measure that gives more weight to the accurate prediction
of low flows should be used. If the interest is in water yields for reservoir design, then a likelihood
measure based on errors in the prediction of discharge volumes would perhaps be more appropriate. If
we are interested in predicting flood peaks, then a likelihood measure that emphasises accurate prediction
of measured peak flows should be chosen. In flood forecasting, a likelihood measure that takes account of
accuracy in predicting the timing of flood peaks might be chosen. If an evaluation of a distributed model
is being made then a likelihood measure that combines both performance on discharge prediction and
performance on prediction of an internal state variable, such as water table level, might be appropriate.
A summary of various likelihood measures is given in Box 7.1.
Most recently, applications of GLUE have been using the “limits of acceptability” approach suggested
by Beven (2006a). This is a way of trying to allow for epistemic errors in the modelling process by
estimating how accurately we might expect a model to make predictions given the multiple sources of
error in the modelling process. The limits should, ideally, be set prior to making any runs of the model
