Geochemical Fingerprinting - Application of Geochemical Tracers to Fluvial Sediment - page 29

variability of the fingerprint within the source materials, (2) the source material sam-
pling density, (3) analytical errors, and (4) changes in sediment characteristics during
particle entrainment, transport and deposition which may significantly influence the
chemical and physical nature of the sampled deposits, such as grain size and organic
matter content (Small et al. 2004 ).
Without the benefit of formalized numerical optimization techniques, initial
attempts at solving the constrained optimization problems (e.g., by Rowan et al.
2000 ; Jenkins et al. 2002 ; Phillips and Gregg 2003 ) involved pushing various com-
binations of that satisfied the non-negativity function and the unity

condition through the objective function to find the optimal solutions. All values of
the objective function were then recorded and compared to determine the optimum
value and those combinations of proportions which yielded this optimal value.
As an example, to solve the mixing problem using the efficiency function ( 2.10 ),
Rowan et al. ( 2000 ) created all possible combinations of proportions,

( x 0 , x 1 ,..., x n )

,
by generating all n -tuples (over 300,000 in all for 5 sources) differing by increments
of

( x 1 , x 2 ,..., x n )

02. They then substituted them into the efficiency function E . By plotting
E vs x i , Rowan et al. ( 2000 ) were able to determine for each source a range of
proportions that would generate an efficiency above a certain tolerance.
Jenkins et al. ( 2002 ) applied the same method to terrestrial and marine sediments
to determine sediment provenance usingmineral magnetic properties as a fingerprint.
The approach has also been used in disciplines other than geomorphology. Phillips
and Gregg ( 2003 ), for example, applied the method to determine the structure of
food-webs using stable isotopes as a fingerprint of food sources.
While, this technique allows one to generate numerous values of the objective
function, it is severely limited by the size of memory necessary to record all possible
values of f with

ʔ x = 0 .

( x 1 , x 2 ,..., x n ) for large numbers of sources beyond increments of

ʔ 01. Later attempts at solving the mixing model recognized the constrained

optimization problem as a standard quadratic programming problem and used avail-
able packages to solve it, as mention in Sect. 2.3.4.1 .

x ≤ 0 .

2.3.4.3 Modifications to the Mixing Model

Since the late 1990s, a number of modifications have been made to mixing models
to improve upon their overall effectiveness. One of the first modified the objec-
tive function in the mixing model ( 2.8 ) to account for differences in grain size and
organic matter content between the source area sediments and the river sediments
(Collins et al. 1997a , 2001 ). Later, Collins et al. ( 2010a ) made two additional mod-
ifications to the objective function. First, they added a 'within source variability'
weight. They found the use of this weighting parameter gave smaller ranges of pos-
sible source contributions when calculated using a Monte Carlo method (discussed
below). Second, they introduced a tracer discriminatory weight to 'reflect the tracer
discriminatory power.' This weighting factor takes into account the relative ability
of a specific fingerprinting parameter to differentiate between the various sediment
sources.