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Chapter 6 - Principal Components Analysis Results

Principal Components Analysis

When the method of Principal Components Analysis (PCA, see Chapter 3) is applied to the Mars spectral cubes, the data are transformed into a new multi-dimensional space where the axes of that space represent orthonormal physical traits.  Mathematically there are as many dimensions in the new space (physical traits) as there are in the original space (wavelength); however, since the method orders the axes by the amount of data variance they represent, the space is usually reduced to only a few meaningful dimensions that can explain most of the data variation.  In the cases here the first three eigenvectors account for over 99% of the data variance. The 0th eigenvector accounts for, on average, over 97% of the variance with almost 2% of the variance accounted for by the 1st eigenvector.  The 2nd eigenvector then makes up most of the remaining 1% of the variance with the rest of the eigenvectors making up an insignificant part of the variance that appear to be dominated by the noise base on the incoherence of their spectra with wavelength.

The two methods used in determining the physical traits described by the various eigenvectors are plotting the eigenspectra and creating eigenimages.  The eigenspectra (Figures 6.1 to 6.7)  are a measure of how much each of the particular wavelengths contributes to that eigenvector.  An eigenimage (Figures 6.11 to 6.24) is a pictorial representation of a particular eigenvector's contribution to the pixels in an image.  This is a measure of how much of an individual pixel is explained by the particular eigenvector, where the brighter pixels indicate higher contribution from the eigenvector.

Plotted in the eigenspectra graphs are only the first three eigenvectors, due to the fact that these three account for over 99% of the spectral variance.  The 0th eigenvector looks like a typical average Mars spectrum.  Pixels having high values of this eigenvector lie within the bright regions of Mars and pixels having low values lie, not surprisingly, in the low albedo regions.  The pixels with the lowest values of 0th eigenvector lie within the sky.  Sky pixels on all these images do have a Martian spectral signature indicating that there is some amount of scattered Mars light in them which amounts to about 10% of the values of an average low albedo region.

The 1st eigenvector has an interesting shape that is similar to the spectrum of a fine water frost.  Figures 6.8 and 6.9 show the 1st eigenvector from each of the seven spectral scans plotted relative to various volatile spectra.  In these plots the Mars atmosphere spectrum is a dust-free radiative transfer model (Rothman et al. 1987, Pollack et al. 1993, Bell et al. 1994) assuming a surface pressure of 8mbar and a surface temperature of 275K and a 1 atmosphere pathlength. The second atmosphere model is the same but with a 2 atmosphere pathlength.  The CO2 frost from Fink & Sill (1982) is a laboratory grown fine frost reflectance spectrum.  The CO2 frost from Calvin (1990) is a Mariner 7 south polar cap spectrum and represents large, millimeter sized grains.  The water frosts are laboratory reflectance spectra from Roush et al. (1990).  The fine frost was grown at a temperature of 104K and sieved through a 90µm sieve.  The coarse frost was an ice cube at 82K.



As can be seen, the 1st eigenvector shows a strong correlation with the fine H2O frost. Many of the features in the eigenvector spectra, such as the 2.00µm feature, appear to match features in all the volatile spectra.  The 2.00µm absorption appears in the fine water frosts, both of the CO2 frosts, as well as the Mars atmosphere models.  Based on this feature alone there is not enough diagnostic information to decide what the eigenvector represents.  However, the spectral shapes of the eigenvectors on either side of the 2.00µm feature show a rise and a fall into features that only match the fine water frost.  Finally, the reflectance of the 3.00µm region relative to the reflectance in the absorption feature at 2.00µm also only matches the fine water frosts.

The steep drop in the eigenspectra from 3.75µm through 4.1µm is of some interest.  This steep of a drop is not at all present in the water ice spectra however there is a drop in the coarse CO2 frosts in this wavelength regime, however, the drop begins from a higher brightness level than the 3.00-3.75µm region and this is not the case with the 1st eigenvector.  It can also be noted that the low 3.00µm brightness of the eigenspectra is similar to this frost, although there is a lack of a feature at 3.33µm.  This would indicate that this eigenvector could be a representation of both coarse CO2 and fine H2O frosts.  The value of this eigenvector should thus be high at the poles and limbs where the band depth maps show evidence of clouds and ground frosts. However, this would indicate that wherever there are H2O frosts there are also CO2 frosts and it is easy to believe that this would not always be the case.

The 3.75µm downturn could also be interpreted as an inverse relation to surface temperature.  It is in this region that the thermal emission from the Martian surface begins to compete with the solar reflection brightness.  Figure 6.10 shows modeled emission spectra with Mars-Earth distance of 0.765AU and a Mars-Sun distance of 1.649AU, which are  equal to those at the time of the 14 JAN 95 observations.  The model was a simple blackbody emission model divided by a solar spectrum reflected from a grey surface using the albedo of a typical dark region. Surface temperature variations are taken from the Viking Lander 2 data at the season the same as the 14 JAN data. 

As can be seen, the thermal effect is negligible at wavelengths shorter than 3.5µm, being only 1.5% of the reflected spectrum at T = 240K.  The upturn at the warmest temperatures, corresponding to local noon regions, has a factor of 8 to 10 increase from 3.5µm to 4.1µm which is comparable to the factor of 4 to 6 downturn in the 1st eigenvector spectra.  This inverse relation would indicate that pixels with high values of 1st eigenvector would have colder temperature and thus, again, would be in the polar regions and limbs where it would be cold enough to condense clouds and ground frosts.



From the eigenimages it is easy to see that the 0th eigenvector represents albedo.  The eigenspectrum appears to be an average Mars spectrum and the 0th eigenimages correlate well with Mars continuum images.  The 1st eigenimages have high values in the polar regions and the limbs, correlating well with the various frost band depth maps and thus indicating a frost interpretation.  The dark, local noon areas have large, negative values which would be consistent with a temperature interpretation.  These eigenimages would also seem to indicate that cold temperatures and condensed volatiles are always correlated.  This will be investigated more in the linear mixture modeling.

Linear Mixture Modeling

The PCA by itself is quite useful in helping to interpret the spectral sets, however it can also be used as a tool for Linear Mixture Modeling (LMM).  Instead of using laboratory spectra of minerals and frosts or attempting to iteratively deduce spectral endmembers from the image set to do the LMM, the PCA technique can be used to find endmember spectra from within the data set that represent physical states on the planet.

The first step in finding these endmembers is to plot the spectral cubes in the PCA space with the 0th and 1st eigenvectors as axes (Figures 6.25 to 6.31).  The endmember pixels are then located by finding the extreme points in these plots.

The goal of endmember selection is to find all those pixels that have extreme values of the chosen eigenvectors and that surround the data cloud.  In order to do that with this data set, five to seven endmembers would be required, as seen by counting the number of extrema.  Based upon tests where the number, and order, of endmember groups are permuted, when that many endmembers are used to model a single pixel it becomes very difficult to ensure a unique model fit.  Based on the previously discussed difficulty in modeling limb pixels due to the Lambert falloff, as well as the non-linear process of atmospheric extinction, we will avoid modeling these areas as the technique obviously falls apart.  Since the areas of high negative 0th eigenvector values correspond to the limb and sky areas we will not choose endmember pixels that enclose this area.  That will allow the use of a smaller number of endmembers, thus allowing a more unique fit.  The endmember pixels are chosen from the most extreme values of the 1st eigenvector and the extreme positive values of the 0th eigenvector.  These three points will surround most of the data cloud leaving only the limb and sky pixels unmodeled.  The endmembers correspond to the north polar region and a dark, centrally located region for the 1st eigenvector and a bright, centrally located region for the 0th eigenvector.

Once the endmember pixels are chosen from the PCA plots, the spectra of those pixels are used to model the spectra of all the other pixels as a linear combination of those three endmembers.  Maps of these fractional abundances are then made and are presented in Figures 6.32 to 6.38 for the bright region endmember, Figures 6.39 to 6.45 for the dark region endmember, and Figures 6.46 to 6.52 for the north polar region endmember.


In the bright region fractional abundance maps there are no real surprises.  As expected the central bright regions have high values and the values lessen in the limbs and dark albedo regions.  In the 14 JAN images Syrtis Major is easily seen, in absence, as it rotates across the disk. Similarly in the 01 FEB images the absence of Acidalia is seen to rotate across the disk.  The images do show that the outlying areas of dark albedo regions have 10% to 20% bright endmember composition.  Singer et al. (1979) describe a simple additive model which finds that a typical dark region spectrum could be composed of at most 30% of a typical bright region endmember.  Bell (1992) preformed a similar two endmember spectral mixture and found that the dark regions contain 10% to 40% of the brightest image endmember spectrum.



The dark region fractional abundance maps show all the low albedo regions having values of 60% to 100% dark endmember composition, as expected.  As in the bright region fractional abundance maps, the features Syrtis Major and Acidalia can be seen rotating across the 14 JAN and 01 FEB images, respectively, only with high values instead of in absence.

The images also show typical bright regions as having 10% to 15% dark endmember composition implying that the many of the intermediate albedo regions are similar to the dark regions but covered with the a bright dust component.  Similarly, Bell (1992) in his simple two endmember spectral mixture analysis found that 5% to 10% of the bright region could be accounted for by a dark region endmember spectrum.



The north polar region fractional abundance maps show high values of this endmember not only in at the polar region, as expected, but also along the morning and evening hemispheres, with values of 50% and higher.  This would indicate that there are polar volatiles in these regions perhaps in the form of clouds or ground frosts.  In his visible to NIR wavelength studies, Bell (1992) added a frost spectrum to his mixture model and found that the morning limb and north polar cap had substantial fractions of this endmember.  It must be pointed out that the frost spectrum used in the Bell (1992) model was a CO2 frost and thus the south polar cap had the highest fractions of this endmember.  However, at the wavelengths used in his study it is not possible to discriminate between water and CO2 ices.

Now that we know how much the frost spectrum contributes to the limb pixels we can investigate the cold-wet correlation.  Eigenvector 1 seems to indicate a 1-1 correlation with cold regions and frost covered regions which would indicate that the atmosphere of the entire planet has enough water vapor in it that when it gets cold enough it will condense to form clouds of significant optical depth.  So, how true is this correlation?

It is already known that the water vapor in the Martian atmosphere undergoes large interannual variations.  The Viking Mars Atmospheric Water Detector (MAWD) measured values of 10 to 15pr.µm at mid-northern latitudes at seasons corresponding to those in the data presented here.  Rizk et al. (1991) measured water vapor abundances at four nights during the southern summer during the 1988-89 opposition and found that the abundance was up to a factor of two or more than the comparable season during the Viking era.  Clancy et al. (1992) measured water vapor abundances at the same season in 1990 and found it well below the Viking era measurements.  Converting the frost spectrum fractional abundances into a measurement comparable to the water vapor abundance would then allow a determination of how wet the Martian atmosphere is.  Such a conversion, if possible, has yet to be formulated and is something to be investigated in the future.

The question of the strength of the 1-1 correlation of wet and cold still remains. Presumably if there were significant cold areas that were frost free then the effects would be separate eigenvectors in the PCA.  This may be what the 2nd eigenvector is showing, as in the L band its spectrum is the inverse of the 1st eigenvector.  However, the 2nd eigenvector only accounts for less than 0.5% of the total data variance.  In comparing the north polar region fractional abundance maps to the 1st eigenimages the areas of high in eigenvector 1 are either high in north polar region abundance or they lie in regions too close to the limb that the LMM can not properly model them.  The answer to this question must then wait until the disk limb darkening effect can be accounted for in future work on these data sets.

The results of these frost fractional abundance maps correlate well with the results from the band depth maps but with two major advantages.  The first is that there was no a priori knowledge needed to search for the volatile signatures.  The north polar region comes out as an endmember based only on the PCA transformations.  The second advantage is that there is little to no correlation in these fractional abundance maps with albedo region.  The various band depth maps do show this correlation due to hydrated minerals and the fact that the map is dependant on only one absorption feature.  The eigenvector based fractional abundance maps are measuring the amount of a spectrum in each pixel, making it a more robust technique.

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