Environmental Engineering Reference
In-Depth Information
heterogeneity (King, 1991; Rastetter et al ., 1992). Proba-
bility density functions that describe the heterogeneity at
subgrid scales in the input data have to be determined to
estimate the expected model output value over the region
of interest. The equation used may be expressed as:
local model in time and space, which of course provides
not only an accurate, but even an 'exact estimate':
F
=
f ( x , y , t ) dt dy dx
(5.11)
x
y
t
where f ( x , y , t ) is the local model in time t and space ( x , y ).
This method is typically not feasible because closed-form
analytical integrals cannot be found for most environ-
mental models (Bugmann et al ., 2000).
+∞
F
=
E [ f ( X )]
=
f ( x ) p ( x ) dx
(5.9)
−∞
where f ( X ) is a local model, E [ f ( X )] is the expectation
value of f ( X ), X is a random variable (or vector of
random variables), and p ( x ) is the probability density
function (or the joint frequency distribution) describing
the subgrid scale variance in X . In an environmental
model, parameters always include attributes derived
from factors such as climate, landscape and soil. Clearly,
the join frequency distribution of such parameters is
rarely known if there are a big number of parameters
for a model. Nevertheless, using such an approach to
account for one or two of the most important nonlinear
model input parameters can substantially improve model
results (Avissar, 1992; Friedl, 1997).
To solve this equation simply, Bresler and Dagan
(1988) and Band (1991) use a Taylor-series expansion
about the vector mean of model parameters, where
the distribution information required is reduced to the
variance-covariance matrix of the key model variables.
Taylor-series expansions can transformmodels from fine
to coarse scales. The equation becomes:
5.5.2.8 Parameterizing interaction
Fluxes in the natural world always not only interfere
but also have feedbacks with each other (Harvey, 2000).
It is necessary either to parameterize the interaction
effects between grid cells directly, or to create a whole
new model that incorporates these interactions and their
effects. The parameterization approach of representing
inter-grid interactions can be used when the small-scale
effects do not significantly affect the process at large scales.
This approach is commonly used in climate modelling
(see Chapter 9 for further details).
5.5.3 Approachesofdownscaling
climatemodels
General circulation models (GCMs) produce climate sce-
narios for assessing the impacts of global climate changes
on ecological, physical, and cultural systems. However,
GCM outputs are very coarse with grid cells ranging from
1 to 5 and temporal resolution of a month, which is
insufficient for detailed assessment of land-surface pro-
cesses and climate-change impacts at local and regional
scales using ecosystem models, biological models, soil-
erosion models, and hydrological models (Zhang, 2005;
Tabor andWilliams, 2010). Tomeet the needs of local and
regional modelling, it is necessary to downscale GCM cli-
mate data. Downscaling of climate data is concerned with
deducing the changes from global change models such
as GCM to finer scales. This approach has subsequently
emerged as a means of interpolating global/regional-scale
atmospheric predictor variables (such as a mean sea-
level pressure or vorticity) to station-scale meteorological
series or agricultural production (Karl et al ., 1990; Wigley
et al ., 1990; Hay et al ., 1992; Wilby and Wigley, 1997).
The fundamental assumption is that relationships can be
established between atmospheric processes occurring at
disparate temporal and/or spatial scales.Wilby andWigley
(1997) reviewed four categories of downscaling methods
in GCM. First, regression methods involve establishing
linear or nonlinear relationships between fine-resolution
f ( i ) ( µ x )
i !
E ( X
µ x ) i
F
=
(5.10)
µ x )isthe i th derivative of
f ( X ) evaluated at µ x with respect to X ,and E ( X µ x ) i
is the expected value of the i th moment of X about
µ x is the mean of X , f ( i ) (
where
µ x .
To implement this type of solution, the model must
either have higher order derivatives all equal to zero,
or the expansions must be estimated from sample data.
In either case, this approach is typically limited to an
approximation of the exact solution provided by par-
tial transformations (Friedl, 1997). Using this method,
it is only possible to derive areal average values of envi-
ronmental variables rather than their spatial distribution.
Such a result for a large area - such as estimating the aver-
age global carbon storage - is very useful, whereas it would
be meaningless for looking at land-surface properties.
5.5.2.7 Analytical integration
Themethod of analytical integration is used to extrapolate
a model employing explicit, analytical integration of the
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