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|A PCA study to determine how features in meteorite reflectance spectra vary with the samples’ physical properties |
Mark Paton1, Karri Muinonen1,2, Lauri J. Pesonen1, Viljo Kuosmanen3, Tomas Kohout1, Jukka Laitinen3, and Martti Lehtinen4
1University of Helsinki, Dept. of Physics, P.O. Box 64, FI-00014, Helsinki, Finland
2Finnish Geodetic Institute, P.O. Box 15, FI-02431, Masala, Finland
3Geological Survey of Finland, P.O. Box 96, FI-02151, Espoo, Finland
4Geological Museum, University of Helsinki, P.O. Box 17, FI-00014, Finland
Corresponding author: Mark Paton
Meteorites have advanced our knowledge of processes in the Solar System with the application of high precision instruments here on Earth. The study of asteroids, the source of most meteorites, has in turn given us knowledge regarding the large scale evolution of the Solar System. Using the complementary information that asteroids and meteorites give us the story of our cosmic backyard can be more easily read. One efficient way to link meteorites to asteroids is by matching their respective reflectance spectra. There have been few convincing matches because of observational and scale differences as well as an incomplete knowledge of the light scattering physics involved. To better interpret the reflectance data we need to know the dependencies of the reflectance on physical properties and develop techniques for better comparisons of data sets. For these purposes we utilise our own measurements of 26 different meteorites together with spectra available on the NASA PDS.
We find that normalisation of reflectance at a wavelength between 1.1 and 1.3 µm gives the closest match of spectra from meteorites common to both data sets. The depth of the spectra bands deepens by similar amounts for different types of surface texture alterations i.e. rock to sawn surface, rock to polished surface and rock to powdered surface. Principal Component Analysis (PCA) is able to easily place carbonaceous chondrites, ordinary chondrites and achondrites into distinct groups using their reflectance spectra. We track the variation of spectral features in principal component space by using a set of meteorite spectra synthesised from mineral and elemental spectra. A spectral agent that reduces the reflectance at all wavelengths is required, in addition to olivine, pyroxene and carbon, to generate a set of synthesised spectra to match the distribution of measured spectra, in principal component space.
Keywords: Reflectance, spectra, meteorite, density, susceptibility, analysis
Meteorites contain a diverse range of over 300 minerals with about 40 not found in terrestrial rocks . Common mineral groups found in meteorites are silicates, oxides, carbonates, phosphides, phosphates and sulphides. Nickel-iron metal and carbon in the form of graphite or diamond are also relatively common in meteorites. Pyroxene, plagioclase and olivine are the most common and can be found in achondrites and chondrites. Table 1 shows the mineral and elemental (carbon) contents of meteorites. These reflect their formation processes e.g. see  or  for further details. It is generally accepted that chondrites are primitive while achondrites have experienced some differentiation.
Meteorites are graded in three main ways to describe petrology, shock and terrestrial weathering. Petrographical grades of chondrites, shown in Table 2, attempt to describe the thermochemical equilibration with 1 being the least affected by thermal metamorphism and 6 being the most effected by thermal metamorphism. A grade 6 chondrite has experienced a similar degree of melting as an achondrite but will not have experienced differentiation. The grades also describe the distribution and mixing of minerals like olivine and pyroxene in the meteorite. A meteorite with a lower grade will be more heterogeneous and a meteorite with a higher grade will be more homogeneous.
A similar petrographical grade is not available for achondrites as these are from differentiated bodies and have been at least partially melted, moving them beyond the petrographical grade 6 used to for chondrites. Achondrites are instead organised into subgroups with similar compositions and degrees of melting. They are sometimes organised into groups when it is suspected they share a common parent body. For example the subgroups Howardite, Eucrite and Digonite meteorites (see table 1) are thought to originate from asteroid Vesta and are known collectively as the Howardite-Eucrite-Diogonite (HED) group.
A grading system is employed to describe the degree of shock received by the meteorite during an impact event. The grade increases with increasing shock pressures. Lightly shocked meteorites may display dark shock veins while highly shocked impact melts will display a melted and deformed matrix. A grading system is also used to describe the amount of degradational terrestrial weathering affecting the meteorite sample. Terrestrial weathering first affects the metal grains, then the troilite and finally the silicates. The shock  and weathering grading systems are shown in Table 3 together with more details regarding to the petrographical grading system introduced in Table 2.
Fortunately, pyroxene, plagioclase and olivine have diagnostic absorption bands centred at wavelengths of ~1 μm and ~2 μm in the reflectance spectra. These are within the range of spectrometer instruments currently used for asteroid observations. The bands due to pyroxene and olivine have been found in reflectance spectra of a large number of asteroids providing evidence for a link with meteorites. Plagioclase has proved to be difficult to detect in reflectance spectra because olivine and pyroxene tend to dominate the spectrum in the near infrared.
It is desirable to further strengthen spectral links between meteorites and asteroids in order to relate results from high precision measurements of meteorites to their asteroidal parent bodies. This has been achieved for relatively few meteorite samples due to the difficulties associated with comparing reflectance spectra of asteroids with those of meteorites. While space weathering of asteroid surfaces is generally accepted to be a major cause of spectral mismatches between meteorites, models explaining weathering processes on asteroids have been difficult to test because multiple samples of asteroidal regolith representing the optical surface are required.
Space weathering tends to redden, darken and deplete the bands in the spectra of asteroid surfaces . These are thought to be due to the condensation of nano-phase iron on regolith grains due to vaporisation of material by solar wind particles and impacts from micro-meteorites . The number and relative importance of space weathering mechanisms, e.g. impact shocking, vaporisation-redeposition, solar wind implantation that act on the surface are not known for asteroids. Across the Solar System it is thought that solar irradiation is dominant . Also certain problems remain such as the timescales suggested by experiments and observations of asteroids e.g. see  and references therein. Asteroid surfaces are gardened, mixed over, by micrometeorite impacts which add to an already complicated picture and could account for results in different timescales of effectiveness for each process, depending on micrometeorite fluxes.
It is important to consider all possible mechanisms that contribute to asteroid surface alteration. It is conceivable that an asteroid could be mis-classified if for example it is covered in shock altered regolith . This hypothesis had been dismissed by previous workers due to dynamical considerations  and the low abundance of shocked material in our collections . Given the current state of meteorite-asteroid spectral links, it is necessary that we properly understand the light scattering physics from weathered and shocked material in order to properly account for spectral effects on asteroid reflectance spectra.
In this paper we investigate ways to improve the linking of meteorites and asteroids through use of reflectance spectra and physical property measurements. We use meteorite spectra that include our own measurements from meteorites kept at the Geological Museum in Helsinki, Finland and a larger data set of reflectance spectra kept on the NASA Planetary Data System (PDS)1. We use physical property measurements (bulk density and magnetic susceptibility obtained in the Department of Physics, University of Helsinki. Several issues regarding the linking of meteorites and asteroids are investigated.
First we introduce diagnostic features in meteorite reflectance spectra that we use and their relationship to the samples’ physical properties. Principal Component Analysis is then introduced as a mathematical tool to manage the large amount of information contained in reflectance spectra. A method to analyse the variation of mineral content in a data set of reflectance spectra is introduced using synthesised reflectance spectra. We describe a normalisation procedure for combining two data sets of reflectance spectra obtained under different observational circumstances.
In the results section we discuss the combination of the two data sets. We use PCA to explore the data sets for correlations between their physical properties (bulk density and magnetic susceptibility) and reflectance properties. PCA is used to investigate the effect of texture changes to the reflectance spectra. We track the composition, across a data set of meteorite spectra, using a set of synthesised spectra.
2. Diagnostic features in meteorite spectra
Shown in Fig. 1 are the reflectance spectra of important minerals in meteorites namely, pyroxene, olivine, plagioclase and an important element, carbon. Reflectance spectra can be used to identify important minerals and determine their relative abundances. See  for a review of reflectance spectroscopy applied to asteroids and meteorites. The bands in the pyroxene and olivine are due to the electronic absorption of photons within the crystal structure of the minerals causing a transition of Fe2+. Olivine has three bands that combine to form one broad band centred at around 1.0 μm. The pyroxenes have two separate bands one located at 0.9 μm and one at 1.9 μm. For both minerals the bands are superimposed on a continuum with a strong charge-transfer absorption band centered in the ultraviolet. The broad absorption band of plagioclase, centred at 1.25 μm, is also caused by the transition of Fe2+ but the feature will disappear if the material has been shocked above 25-30 GPa.
Opaques such as amorphous carbon will severely darken (reduce the reflectance of) the spectrum and provide little diagnostic information. Conversely free metal in the form of minerals (alloys) such as kamacite and taenite cause a smooth increase in reflectivity with wavelength .
One way to determine the abundance of materials in asteroids from reflectance spectra data is to use a linear combination of the reflectance spectra from the meteorite’s end member minerals. However, mixing is probably nonlinear to a degree that depends on the mineral distribution, the presence of opaques, the surface texture and the size of the mineral grains . An analysis of materials using a linear model may produce errors in derived abundances of the order of 10% or more. Therefore for quantative investigations a non-linear spectral mixing model is required . For large grains, where the grain size exceeds the optical depth, a linear mixing spectral model may be appropriate . Applying spectral mixing models to the reflectance spectra of meteorites makes the determination of mineral absolute abundances very difficult for several other reasons as well.
For example minerals may not be distributed homogeneously over the meteorite surface. A reflectance spectrum taken from one spot on the surface will then not be representative of the bulk mineralogy of the meteorite. If carbon is mixed intimately with the silicate material this will cause absorption over the whole continuum and greatly weaken the absorption band due to ferrous iron in the silicate materials like pyroxene and olivine. The scattering on a rough meteorite surface will result in increasing absorption of light and will thereby deepen the absorption bands. If the meteorite is ground into a powder then the absorption bands will deepen with increasing grain size .
There may be other factors that make interpretation difficult such as the influence of free metal on the overall slope of the spectrum and light scattering effects that may arise from different illumination and observation geometries when comparing reflectance spectra taken using different spectrometers.
Previous results have shown that the wavelength of absorption band minima varies systematically with relative composition . For example it is known that the olivine band minimum varies between 1.05 and 1.08 µm depending on the Fe-Mg composition. It has also been shown that the band area of the reflectance spectra of pyroxene and olivine varies systematically with relative abundance and can be predicted by measuring the Band Area Ratio (BAR), which is the area of Band II divided by Band I (see Fig. 2).
3. Principal Component Analysis
Principal Component Analysis (PCA) is a straightforward technique for the statistical analysis of the variation in large data sets, e.g. see . PCA can reduce the number of dimensions used to describe the variation in the data to a low level. In our case we have nearly 200 spectra that we want to analyse. PCA can reduce the pertinent information in the spectra to a small number of components that can be used to examine spectral features and grouping strategies (for classification purposes).
The data set is first rationalised so all the spectra are sampled over the same wavelength range using the same wavelength intervals. The spectra are placed in an array with each spectrum a column entry. The mean spectrum is calculated and then subtracted from the array. The resulting columns in the array represent vectors that describe m points in n-dimensional space, where m is the number of spectra and n is the number of spectral sample points.
Then the coordinate system is rotated, minimising the sum squared difference (i.e. minimising the error) of the points in a direction perpendicular to the axes. The axis with the greatest variance along its length is known as the first principal component. The axis with the second greatest variance along its length is known as the second principal component and so on with the higher dimensions. In this way the number of variables describing the structure of the data is minimised.
In PCA an eigenvector is a vector that describes the location of the principal component axis in n-dimensional space after the coordinate system has been rotated. It is a useful for interpreting the results of a principal component plot. The eigenvector will consist of n elements and each element maps back directly onto the original samples points which are, in our case, the wavelength. The first element in the eigenvector then directly corresponds to the first wavelength that the reflectance was sampled at. To illustrate further take, for example, an eigenvector that corresponds to the first principal component axis and has strong loadings (i.e. large values) in its middle elements. This will mean that the mid-range wavelengths of the spectrum are strongly correlated to the variation seen along the first principal component axis.
To illustrate the application of PCA a set of artificial meteorite spectra are synthesised from reflectance spectra of pure minerals and carbon. Here we use pyroxene and olivine in addition to carbon. Various types of meteorites are synthesised. First there are ordinary chondrites, with no carbon, with a more or less equal amount of olivine and pyroxene. Secondly there are high pyroxene achondrites with a high ratio of pyroxene to olivine. Thirdly there is a group of high olivine content meteorites with a high ratio of olivine to pyroxene. Last there is a group of carbonaceous chondrites that vary in combinations of olivine, pyroxene and carbon. Figure 3 shows some of the synthesised reflectance spectra for these meteorites.
We use a linear combination of olivine and pyroxene as the synthesis of ordinary chondrite and achondrite reflectance spectra is straightforward and we are interested in qualitative investigations. Carbonaceous chondrites are a little more complicated to synthesise and need some careful consideration. Carbon is an opaque that diminishes the 1 μm and 2 μm bands in the relatively abundant olivine and pyroxene. The large wing of the band centred in the visible is relatively unaffected as the absorption of light there is via a different mechanism. The generation of carbonaceous chondrite spectra uses the following equation:
where fi is the fraction of mineral i, r is the reflectance of mineral i (at wavelength λ), α is a factor introduced to reduce the depth of Band I and Band II. The term β is used to correct for the discontinuity in the spectrum caused by introducing the factor, α. The following spectra were used to make up the synthesised meteorite spectrum, olivine (i=1), pyroxene (i=2), carbon (i=3) and an artificial spectrum where all reflectance values equal to zero at all wavelengths (i=4).
The factor α in equation 1 is defined as follows,
The correction term β is defined as follows,
We note that 50% or more of carbon is required to flatten the band structure of pyroxene and olivine in these synthesised spectra while real carbonaceous chondrites only have a few percent carbon present. These synthesised meteorite spectra are only intended as a guide to interpreting real meteorite spectra and to identify trends.
The zero reflectance spectrum (i=4) is used to subject the synthesised meteorite spectrum of olivine, pyroxene and carbon to a hypothetical darkening mechanism, perhaps due to grain size effects or some other mechanism, across all wavelengths. This reduces the overall reflectance of the spectrum and therefore is different than the effect of carbon. As the contribution of the darkening agent increases the contribution of the synthesised spectra will be reduced in equal amounts. This differs in the behaviour that carbon will have in that it will also affect the visible part of the spectrum.
Figure 4 shows a result of PCA on the synthetic meteorite spectra plotted in principle component space. Variations in the amounts of the minerals manifest themselves in principal component space in easily identifiable clusters. To associate the clusters with the correct synthetic meteorite spectra each spectrum’s position in the array was used to trace the mineral spectra that were used. For example it is known the synthesised spectra that are combinations of only pyroxene and olivine reflectance spectra are clustered along a diagonal line located at the extreme negative end of the x-axis. Along this line there is increasing pyroxene up the positive y-axis and increasing olivine in the negative direction. The darkening agent has the effect of narrowing the spread of the spectra moving towards the extreme positive end of the x-axis. The carbonaceous chondrites are easily identifiable as the tight cluster located to the right on the chart.
The clustering of the spectra can be further understood by inspecting the eigenvectors in Fig. 5. Eigenvector one is relatively flat compared to the others, telling us that the spread of the spectra along principal component 1 is related to the average reflectivity of a spectrum across its whole wavelength range. Eigenvector 1 defines the first principal component axis because most of the variation in the data is due to flat spectrums (i.e. carbon and the darkening agent).
Eigenvector two shows more variation than eigenvector one with greatest values at 1.3 μm and 2 μm. The variation at 1.3 µm in eigenvector two tells us about the variation in Band I while the the variation at 2 µm tells us about the variation in Band II. Notice in Fig.1 that the greatest difference in reflectance between pyroxene and olivine occurs at 1.3 μm and 2 μm. So a linear combination of these two spectra would result in the greatest variations in reflectance at these points.
The band depths in the synthesised spectra change with changing ratio of olivine to pyroxene and hence the spectra with a deep Band I is at the extreme end of the positive y-axis and the spectra with deep Band I is at the extreme end of the negative y-axis. The darkening agent reduces the average reflectance together with the band depth and hence a convergence to zero is seen on the y-axis in the positive x direction. Even though the carbonaceous chondrites contain olivine and pyroxene, they are clustered tightly along the y-axis because the carbon reduces Band I and Band II relative to the visible part of the spectrum.
A common measure of interrelation between two quantities is the correlation coefficient. It is a measure of the linear interrelation and is often used in statistics. It is determined by dividing the covariance of the two data sets by the product of their standard deviations. A correlation matrix can easily be calculated during PCA and is useful for determining relationships between data sets. For example this would be useful for comparing the dependencies of features in the reflectance spectra such as the depths of Band I and Band II. The correlation coefficient is as follows:
where x and y are variables of the two data sets.