Geochemical Fingerprinting - Application of Geochemical Tracers to Fluvial Sediment - page 27

Since the fingerprint will typically be comprised of m tracers, the mixing model
results in an m

×

n system of linear equations. Each equation represents the contri-
butions from each of the n sources determined on the basis of the measured amount
of a tracer in the sediment:

a 1 , 1 ∗ x 1 + a 1 , 2 ∗ x 2 + ··· + a 1 , j ∗ x j + ··· + a 1 , n ∗ x n = b 1

a 2 , 1 ∗ x 1 + a 2 , 2 ∗ x 2 + ··· + a 2 , j ∗ x j + ··· + a 2 , n ∗ x n = b 2

. . .

(2.5)

a i , 1 ∗ x 1 + a i , 2 ∗ x 2 + ··· + a i , j ∗ x j + ··· + a i , n ∗ x n = b i

. . .

a m , 1 ∗ x 1 + a m , 2 ∗ x 2 + ··· + a m , j ∗ x j + ··· + a m , n ∗ x n = b m

In theory, the mixture is exact, but in reality, there will exist some differences
(error) between the values of the m measured tracers in the source area, a i , j ( i =

1 , 2 ,..., m , j = 1 , 2 ,..., n ), and the downstream mixture (river sediment), b i ( i =

1 , 2 ,..., m ). The residual error corresponding to the i th tracer can be determined as

follows:

n

ʵ i = b i − a i , j ∗ x j (2.6)

j = 1

for i n ) are measurements

of the corresponding i th tracer within the j th source area and b i is the measurement
of the tracer of the i th tracer in the river sediment (mixture).
When the number of utilized tracers exceeds the number of source areas or types
within the catchment (as is often the case when using geochemical data), the system
of equation ( 2.5 ) is over-determined, and a 'solution' is typically obtained using a
computational method that optimizes an objective function. This function, subject
to the previously noted constraints, estimates a best fit solution to the entire data set
(Yu and Oldfield 1989 ).
There are several ways to obtain a best fit, but in previous studies, the objective
function, f , has taken the form of the sum of the relative errors where

= 1 , 2 ,..., m , where a i , j ( i = 1 , 2 ,..., m , j = 1 , 2 ,...,

m

ʵ i
b i

f ( x 1 ,..., x n ) = (2.7)

i = 1

(as defined by Yu and Oldfield 1989 ) or the sum of the squares of the errors (Collins
et al. 1997a ),

b i − j = 1 a i , j x j
b i

2

ʵ i
b i

2

m m

f ( x 1 ,..., x n ) = = (2.8)

i = i =

1 1

Note that the measurements of different tracers are often magnitudes apart; for
instance, Cu concentrations ranged from 2 . 78 to 823ppm, while Cd ranged from