Geoscience Reference
In-Depth Information
4.1.2
Spatial variations of temporal distribution
If we are to consider more than one point in space, the temporal distributions at different
points in space can be different. For example in case of a catchment, the temporal
distribution of precipitation over subbasins can be different. Denoting by
m
the number of
subbasins, Equations (4.1)-(4.4) can be rewritten as
(4.7)
w
i,j
=
W
i
b
i,j
(4.8)
(4.9)
y
=
f
(
w
i,j
;i
=1,…,
m; j
=1,…,
n
)
=
f
[(
w
1,1
,…,
w
1,
n
),…,(
w
m,
1
,…,
w
m,n
)]
(4.10)
where
W
i
is the accumulated value of the input for subbasin
i
and
w
i,j
are the
disaggregated signals for subbasin
j
and subperiod
j
. Whereas varying the coefficient
b
i,j
over subperiods allows the possibility of different temporal distributions, varying the co-
efficient over subbasins allows for different spatial distributions. Thus by generating dif-
ferent values of
b
i,j
,
as many temporal and spatial patterns as are needed can be generated.
4.1.3
Uncertainty in the input quantity
The principles described so far deal with the uncertainty due to the unknown temporal dis-
tribution of a variable over the subperiods and subbasins. There can be uncertainty also in
the quantity of the input (accumulated sum) measured or forecasted over the period. This
uncertainty is normally represented by a probability density function (PDF). Such a PDF
is either derived from the available data or assumed in the absence of a sufficient number
of data values. In case of the fuzzy set theory-based approach, the uncertainty is repre-
sented by a membership function (MF). Arbitrary representations of the uncertainty in the
magnitude of the input (accumulated sum) by a PDF and a MF are shown in Figure 4.2.
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