Agriculture Reference
In-Depth Information
to persistent differences in soil composition and do not necessarily result from differences in
'input' at the soil surface.
For the two limiting cases mentioned above, the 'diffuse' pollution case is called an ergodic
situation, because 'all' heterogeneity is sampled simultaneously. This leads to a distributed
answer, such as a spatially variable contaminant plume or breakthrough curve. If these
answers are measured or calculated at another place, they do not significantly change.
Hence uncertainty is limited.
However, the distributed answer is complicated and difficult to communicate to others who
cannot see the picture. Moreover, it is commonly too detailed for e.g. management
decisions. Therefore, it is appropriate to consider robust measures of the results, which focus
on the major issues. Useful examples of such robust measures are found in the theory of
moments. We will give the definition of these moments in terms of spatial moments
(representing the situation at one instant in time), but temporal moments (at one particular
location, plane or volume) are equally feasible. The spatial moments of a property P, where
the space coordinate is x, are given by
The zeroth' moment, or the mass of the distribution:
dP
(1)
M 0 f P

The first moment, or the mean of the distribution:
f P dP
1
M 0
M 1
P
(2)

The second central moment, or the variance of the distribution:
2
1
M 0
M c
P M 1
f P dP
(3)
For a contaminant plume as shown in Figure 3, the zeroth' moment is equal to the amount
of dissolved chemical in the contaminant plume, if P is equal to the position x, and the
function fx represents the spatial distribution of the contaminant concentration multiplied
with the water-filled porosity. The first moment represents the mean position (in direction
x) of the contaminant distribution and the second central moment represents the width of
the zone over which the concentration distribution occurs. Whereas we illustrated the
moments where P is equal to position, many other properties can be chosen. Which to
choose depends on the primary interest, but examples are the concentration or mass of
contaminant, the quantity or fraction of contaminant leached beyond a reference plane, such
as groundwater level and so on. In principle, the transport problem for de-icing chemicals is
a very complicated one, even in one direction such as depth. The reason for that is that a
complex of interactions is affecting this transport. Examples are infiltration conditions,
ad/desorption, microbial degradation according to different kinetics (1 st , Monod, 0 th order),
and a host of different redox-sensitive components that may affect the degradation rate
(Beltman et al., 1996; Keijzer et al., 1998) may occur, depending on ambient conditions as
well as the influx of degrading substrate (continuous, instantaneous). To give an impression

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