Geoscience Reference
In-Depth Information
2.3.4 Inverse/Unmixing Models
2.3.4.1 Derivation of the Inverse/Unmixing Models
Early work by Yu and Oldfield (
1989
,
1993
) and Collins et al. (
1997a
,
b
) was partic-
ularly instrumental in defining the general fingerprinting approach most often used
today. It can be viewed as a process in which the composite fingerprint created for
the sediment sources is compared to the river sediments using an inverse/unmixing
model to unravel the relative amount of sediment from each source that comprises
the river sediment of interest.
Mathematically, constraints on the mixing model require that (1) each source
type contributes some sediment to the mixture, and thus the proportions (
x
j
,
j
=
1
,
2
,...,
n
), derived from
n
individual source areas must be non-negative (0
≤
x
j
)
,
and (2) the contributions from all source areas must equal unity, i.e.:
n
x
j
=
x
1
+
x
2
+···+
x
n
=
1
.
(2.3)
j
=
1
Three significant factors may lead to situations where this latter assumption of
linear additivity in property values is not fully achieved. First, analytical errors may
be associated with the characterization of the measured geochemical parameters.
These errors are typically on the order of
5%, and in most instances do not pose
a significant issue. Second, an important sediment source may not have been recog-
nized or sampled. The failure to characterize a significant source primarily occurs
when dealing with large basins composed of a large number of sediment sources
(geological units, soils types, or land-use/land-cover categories). Third, the tracer(s)
may have exhibited non-conservative behavior either during transport, or as a result of
diagenetic alterations following deposition (Walden et al.
1997
; Rowan et al.
2012
).
Assuming that a particular tracer has been established as comprising part of a
fingerprint for the
n
sources, the downstream mixture of this particular tracer is
represented by
±
n
a
j
x
j
=
a
1
x
1
+
a
2
x
2
+···+
a
n
x
n
(2.4)
j
=
1
where
a
j
represents the measurement of the tracer within the
j
th source area.
Initial studies used the mean or median of the data points from each source in the
fingerprint. Collins et al. (
2010a
), for example, noted that the “use of the mean con-
centration value to represent a particular source can be justified as being physically
realistic since the sediment collected from the catchment outlet inevitably represents
a mixture of material mobilized and delivered from numerous locations upstream.
As a result, the collection of representative source material samples from a range of
locations throughout the catchment and the use of the sample to derive the mean fin-
gerprint property concentrations can be assumed to be analogous to natural sediment
mixing during the sediment mobilization and delivery process.”
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