Geochemical Fingerprinting - Application of Geochemical Tracers to Fluvial Sediment - page 31

the fact that the number of samples collected to characterize the sediment sources
are generally limited because of financial and time constraints, the mean value of the
fingerprinting parameter calculated from the sample data from any source may not
necessarily represent the true mean of each tracer within the source. Thus, using the
average of the data points of all samples collected within the source for each tracer
elicits an error of unknown magnitude. This error is exacerbated by the fact that the
fingerprint is comprised of several tracers (Collins et al. 2010a ).
In order to reduce and quantify the uncertainty inmixing/unmixing models related
to this inherent variability in the source area data, recent studies have explored
the use of a Monte Carlo sampling framework (Small et al. 2004 ; Collins et al.
2010a , 2012 ). Validation of these approaches using constructed laboratory mixtures
of source materials and synthetic data show the methods hold considerable promise
(Small et al. 2004 ).
As an example, Small et al. ( 2002 , 2004 ) used the sample data from each source
to create a probability distribution to estimate the mean value for each tracer within
the source. The estimated means of all tracers within all sources are then used as
parameters in the objective function which is minimized to solve for the proportions.
This process is repeated numerous times (on order of several thousands) until there
are sufficient results to estimate confidence intervals (e.g., 95%). As described in
more detail below, Rowan et al. ( 2012 ) used this method to determine the effects
land use management practices have on algae blooms.
Collins et al. ( 2010a ) used a similar approach in which a goodness of fit function,
which they refer to as a relative mean error (RME), was assumed to measure the
robustness of the optimized solutions of the mixing model:

⊡ ⊤

b i − j = 1 a i , j ∗ 2

m

ps j ∗ om j ∗ ws i , j ∗

x j

1
m

⊣ ⊦

RME = 1 − ∗ W i (2.12)

b i

i = 1

Subsequent studies by Collins et al. ( 2012 , 2013 ) modified the approach by using
the median values instead of mean values for the tracer parameter within each source.
Themedian values were estimated on the basis of a probability density function. They
also use an estimated frequency-weighted average median, initially introduced by
Collins et al. ( 2012 ), in which R = i = 1 v i Fr i where n is the number of intervals

for the predicted deviate relative contribution, scaled between 0 and 1, and v i and Fr i
are the mid-value and the frequency for the i th interval, respectively.
A question that arises in the use of the fingerprinting approach is how results
obtained from using mean source and river sediment values compare to the out-
put generated using the Monte Carlo approach. While additional investigations are
needed to answer this question, Fig. 2.6 compares the result generated using only
mean values (arrows) to the results obtained using the objective function (2-norm
error squared) and confidence interval of 95% using the RME defined above ( 2.12 )
for six samples collected from a wetland core within the Mkabela catchment of
South Africa. Grain size and organic matter correction factors were not included
in the analysis. While differences exist, the results for these specific samples are