Preamble
Rocks and sediments
inevitably contain mixtures of magnetic minerals, grain sizes, and
weathering states. Most rock magnetic interpretation techniques rely on
a set of value parameters, such as susceptibility and
isothermal/anhysteretic remanent magnetisation (ARM or IRM). These
parameters are usually interpreted in terms of mineralogy and domain
state of the magnetic particles. In some cases, such interpretation of
natural samples may be misleading or inconclusive.
A less constrained approach to magnetic mineralogy models is based on
the analysis of magnetisation curves, which are decomposed into a set
of elementary contributions. Each contribution is called a magnetic
component, and characterises a specific set of magnetic grains with a
unimodal distribution of physical and chemical properties. Magnetic
components are related to specific biogeochemical signatures rather
than representing traditional categories, such as single domain
magnetite. This unconventional approach can be regarded as a kind of
principal component analysis (PCA) that gives a direct link to the
interpretation of natural processes on a multidisciplinary level. Since
magnetic components rarely occur alone in natural samples, unmixing
techniques and rock magnetic models are interdependent.
Valuable information
for rock magnetic and environmental studies can be obtained directly
from the coercivity distribution of the sample, which provides a
richness of details hidden in the measurement curve (see examples).
What
is MAGMIX?
The software
package MAGMIX provides computer programmes for the analysis of
isothermal magnetisation curves* and coercivity
distributions and was developed by Ramon Egli. This
first release includes the programmes CODICA and GECA. It
consists of two packages: CODICA (COercivity
DIstribution CAlculator)
and GECA (GEneralised Coercivity Analyser).
*The
term magnetisation curve has here and in the following the meaning of
an acquisition curve, alternating field demagnetisation curve or
backfield curve (latter also called direct current demagnetisation).
What
does CODICA do?
CODICA is
a programme that calculates the coercivity distribution of a
magnetisation curve and estimates the measurement errors. CODICA
eliminates efficiently measurement noise from
acquisition or demagnetisation curves of remanent magnetisation (based
on rescaling procedures)
and calculates then the derivative (also called coercivity
distribution or coercivity spectrum).
CODICA proceeds essentially on three steps that are shown in the
figures below, considering a simple example. First, a set of scale
transformations is applied to the field and the magnetisation (b). After field and magnetisation scales have been
changed, the magnetisation curve becomes close to straight line, and is
said to be linearised (c). A socalled residual curve
is obtained after removing the linear trend from the scaled curve by
subtraction of a polynomial (d). The residual curve
has the characteristics of a stochastic signal, because it oscillates
more or less randomly around a mean value of zero. Wiggles may arise
from small asymmetries of the original magnetisation curve, and as well
as from measurement errors. Measurement errors are easy to
recognise in the residual curve, since they are enormously amplified.
This gives the possibility to optimise and to correct experiments in
order to obtain optimal results. The residual curve is then fitted
using a method called leastsquares collocation, which is a
particularly effective model for stochastic (nonperiodic) signals red
line in (d). The interpolated residual curve 
supposed to be free of measurement errors  is transformed back
into a magnetisation curve red line in (e) and its
first derivative, called coercivity distribution (f),
is calculated. The leastsquares collocation method provides also a way
to estimate the error associated to the operations described above and
thus provides confidence limits for the results it produces.
The following figures (a) to (f) illustrate the working principle of
CODICA on a
simple example (sample kindly provided by Christoph Geiss).



(a) Original AF demagnetisation
curve showing a characteristic multidomain (MD) shape.


(b) The field axis is rescaled in
order to get a sigmoidalshaped curve. To do so, the scale is expanded
at small fields.







(c) The magnetisation is now
rescaled in order to linearise the magnetisation curve. To obtain this
result, CODICA expands the magnetisation scale near the beginning and
the end of the magnetisation curve. The red line is the linear bestfit
to the data.


(d) The linear trend of the
rescaled curve (red line in c) is subtracted to obtain the socalled
residuals. As it can be clearly seen, the measurement errors are quite
evident in this plot. The red curve is a bestfit of the residuals that
CODICA obtains from an autocorrelation model.







(e) A model for the
errorfree magnetisation curve (red) is obtained from the fitted
residuals (red curve in d), by inverting the mathematical functions
used to transform the original measurement (a) into the residuals (d).


(f) A coercivity distribution is
calculated as the analytical derivative of (e). The thickness of the
line corresponds to the estimated confidence limits of the coercivity
distribution. CODICA calculates the coercivity distribution on a
logarithmic field scale, as in (f), as well as on a linear scale.

Examples
Some
applications of CODICA to rock magnetic and environmental studies. The
left plots show the original measurements, the right plots are
coercivity distributions calculated with CODICA. The thickness
of the curves corresponds to the error estimate of CODICA.



(a) Alternating field
demagnetisation curves of a Tiva Canyon Tuff that contains
acicular magnetite in the SP/SSD grain size range. Measurements have
been started 8.5 and 160 hours after an ARM was imparted (see the IRM
Quarterly, 14(3), 2004, for more details about the Tiva Canyon
Tuff). 

(b) Coercivity distributions
calculated from (a). Notice the bimodal
character of the 8.5 h curve, showing the magnetisation of viscous and
of more stable particles. The difference between both curves (inset) shows
the coercivity distribution of the viscous particles with relaxation
times between 8.5 and 160 h. 






(c) Normalised alternating field
demagnetisation curves of ARM from samples of particulate matter
collected from the atmosphere at three places in the city of
Zürich,
Switzerland (see S. Spassov et al., 2004). 

(d) The normalised coercivity
distributions show the
increasing relative contribution of a high coercivity component in more
polluted
areas (GMA: forest near Zürich, WDK: center of Zürich, GUB:
motorway
tunnel).

What
does Geca do?
GECA is a
programme for modelling a coercivity distribution as a linear
combination
of special model functions, that are supposed to represent the
coercivity distribution of specific groups of magnetic particles,
called components (= unmixing). A Pearson's X
^{2}
goodness of fit test evaluates the number of functions required for the
bestfit and an error estimation allows to calculate the confidence
limit for each model parameter.
The figure below displays the modelling of a coercivity distribution
(sample WDK from above). The grey band
corresponds to the coercivity distribution (in absolute) and its thickness
is the error estimate of CODICA. The coercivity distribution is fitted
with a linear combination (blue) of two
model functions (red
and green).
Τhe difference between the measured (grey) and the modelled (blue) coercivity
distribution (= residuals) is plotted in the lower diagram (blue). The
residuals fit well within the error of the coercivity distribution (grey). Pearson's X
^{2}
goodness of fit test gives a value of 1.5 for the solution obtained,
which is inside the interval of confidence [0.59;1.7], hence the test
is passed  model and measurement are not significantly different.
Click on the figure to zoom in. Another example of unmixing (lake
sediment) is given below.
Example



(a) alternating field
demagnetisation of an anhysteretic remanent magnetisation imparted to
an anoxic sediment from lake
Baldeggersee, Switzerland (see Egli, 2004, for a
detailed discussion about magnetic
measurements on this lake).


(b)
Coercivity distribution calculated from (a), and results of a component
analysis (colored areas). Three components can be clearly
distinguished: the lowest coercivity component has been attributed to
detrital magnetite, the middle coercivity component to magnetosomes
that survived reductive dissolution, and the high coercivity component
to a high coercivity mineral such as haematite.

Requirements
Both
programmes are written in Mathematica,
a programming language similar to other traditional computer languages
such as C, Pascal or FORTRAN. The usage of CODICA implies that Mathematica is installed on
your computer. MAGMIX has been developed for Mathematica
5.0 under Windows, but with small code modifications it is also
compatible with Mathematica for
Linux/Unix. At least 128 MB RAM
and a 1 GHz CPU are recommended.
Installation
1.
Install Mathematica on your computer.
2. Download the
zipfile from below, store it in a local directory and unpack it.
3. Install
CODICA. To install CODICA 5.0 copy the source code file MAG_MIX_1/CODICA/Install/Codica.m
into the following directory:
C:/.../WolframResearch/Mathematica/5.0/AddOns/StandardPackages/Utilities
whereby C:/.../ depends on the
installation of Mathematica on your computer.
4. To install GECA,
copy the source code file MAG_MIX_1/GECA/Install/Geca.m
and the file MAG_MIX_1/GECA/Install/components.txt
into the
following directory:
C:/.../Wolfram
Research/Mathematica/5.0/AddOns/StandardPackages/Utilities
whereby C:/.../
depends
on the installation of Mathematica on your computer.
Download
Both
releases can be
downloaded following the links below.
MAGMIX
Release 1 (2005), Mathematica v.5.0 for Windows must be
installed in your computer.
The zip file
contains: installation instructions for CODICA 5.0 and
GECA
2.1, the source code (files Codica.m,
Geca.m),
the reference manual (pdfformat) and examples (including data).
Manual
only
Here you can
download the reference manual, including theoretical aspects.
Notice
Your feedback is
important to improve the MAGMIX. If you encounter problems in
using the programmes, or if you have suggestions, please do not hesitate to
contact the author by
email: eglix007@umn.edu.
References
Egli, R.,
Analysis of the field
dependence of remanent magnetization curves, Journal
of Geophysical
Research, 108 (B2),
2081, doi 10.1029/2002JB002023, 2003.
Egli, R., Characterization of individual rock
magnetic components by analysis of remanence curves, 2. fundamental
properties of coercivity distributions, Physics and Chemistry of the Earth,
29, 851867, 2004.
Heslop,
D., Dekkers, M.J., Kruiver, P.P. and I. H.
M. van Oorschot, Analysis of isothermal remanent magnetization
acquisition curves using the expectationmaximization algorithm, Geophysical Journal International,148, 5864, 2002.
Kruiver
P.P., Dekkers M.J. and D. Heslop, Quantification of
magnetic coercivity components by the analysis of acquisition curves of
isothermal remanent magnetisation, Earth and Planetary Science Letters, 189,
269276,
2001.
Spassov,
S., Egli, R., Heller, F., Nourgaliev, D.K. and J. Hannam, Magnetic
quantification of urban pollution sources in atmospheric particulate
matter, Geophysical Journal
International, 159,
555564, 2004.