Geoscience Reference
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When the above factors are incorporated into the objective function (
2.8
), it
exhibits the following form:
b
i
−
j
=
1
a
i
,
j
∗
2
m
ps
j
∗
om
j
∗
ws
i
,
j
∗
x
j
f
(
x
1
,...,
x
n
)
=
∗
W
i
(2.11)
b
i
i
=
1
where
ps
j
and
om
j
are the particle size and organic matter weights, respectively,
for the
j
th source;
ws
i
,
j
is the within source weight for the
i
th tracer within the
j
th
source; and
W
i
is the discriminatory weight for the
i
th tracer. Specific definitions
for the correction factors that have been added by Collins and his colleagues to the
model are provided Table
2.2
.
Another form of modification that has recently received considerable attention is
related toBayesian statistics.Walling andCollins (
2005
), andCollins et al. (
2013
), for
example, modified the approach to incorporate prior knowledge about river processes
into the model. More specifically, they limited the potential contribution from one of
the sources within the model by noting, in this case, that the contribution of sediment
to the channel by bank erosion could be no more than 50%. Thus, they imposed an
additional constraint on the model. That is, in addition to the unity constraint (
2.3
),
when
x
1
represents the proportion of the mixture due to channel bank erosion, it must
adhere to the following constraints: 0
≤
x
1
≤
0
.
5 and 0
≤
x
j
for
j
=
2
,
3
,...,
n
.
2.3.4.4 Handling Uncertainty of Tracer Data: The Monte Carlo Method
A significant source of uncertainty in the fingerprinting approach is the inherent
variability of a fingerprinting parameter within the source and river sediments. Take,
for example, a source area defined by its land-cover such as forest vegetation. Forested
areas may cover multiple geological units or soil types, each possessing a unique
mineralogical and geochemical set of characteristics. Thus, the variability inherent
in the collected geochemical data is likely to be relatively high. When combined with
Table 2.2
Definition of correction/weighting factors used in inverse models (Collins et al.
1997a
,
2001
,
2010a
)
Correction/weighting factor
Definition
Particle size correction factor
Ratio of specific surface area measured for a given source
to the average of the surface area from all the sources
Organic matter correction factor
Ratio of the organic carbon content within a given source
to the average organic carbon content of all the sources
Within source variability weight
The inverse of the standard deviation of a parameter for all
samples from the source
Tracer discriminatory weight
Calculated for each tracer as the ratio of the percentage of
source type samples classified correctly for that tracer to the
minimum of all such measurements determined during the
discriminant analysis used to identify the fingerprint
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