A Masochist's Guide to the Modified Gaussian Model of Spectroscopy

On a lazy summer day, you may have, at times, found yourself incapable of getting off the couch. You roll your head to the side and notice a small bit of food lying just out of reach. What is it? The most certain way of determining this would be to actually get up, grasp the object, and taste it. That, however, would require getting up. This common conundrum is paralleled by planetary scientists who seek to know the identity of planetary bodies, but find themselves incapable of physically transporting their bodies to collect samples.

Missions such as NASA's New Horizons have reached to the ends of the Solar System and, quite literally, shined light upon the nature of exotic worlds. How can the instruments on board these spacecraft tell us about the composition of planets not only while zipping through space at 36,373 mph, but also being millions of miles away from their targets?

New Horizons is equipped with instrumentation that allows us to interpret the composition of  bodies like Pluto... but how?!

Judging Books by their Covers

The minerals that compose the rocky materials of planets have specific effects on how light reflects from their surfaces. Visible light is part of a continuous spectrum of radiation, and each color is produced by different wavelengths of light.

Most of the light that comes from the Sun is a combination of many wavelengths, which our brains collectively interpret as white light. However, once light interacts with a material some of those wavelengths are absorbed, while others are reflected. The cumulative effect of this preferential absorption is the production of different colors. For example, Jay's eyes appear as a beautiful blue color because they are composed of molecules that reflect blue wavelengths of light while absorbing others. Consequently, if you know how different materials interact with light, you can interpret the composition of something remotely by examining the absorption of specific wavelengths.

Spectroscopy would allow us to interpret the composition of Jay's eyes without having to poke them first.

Okay, light is absorbed differently by different things. How do we use that to interpret planetary surfaces?

The spectrum of light extends beyond the range that can be visibly seen, and instruments called spectrometers can see far beyond the limitations of the human eye. Absorption features within the near-infrared (wavelengths a little longer than our eyes can detect) can be diagnostic of the specific minerals of which rocks are composed. The interpretations of these absorption features, however, were not well-constrained during the infancy of spectroscopy.

Laboratory-based spectral measurements of individual minerals have been used to determine mineral-specific absorption features, and those features can be overlain on top of remotely-collected (extraterrestrial) spectra in order to model and interpret composition. Historically, each absorption feature was modeled as a Gaussian distribution, which assumed the slope of the absorption feature is symmetrical on either side. This is due to a statistical feature that arises when examining large amounts of data (like all the photons that are reflected from a planetary surface).

While this approach was useful in providing constraints for possible mineral assemblages, it also yielded multiple solutions for complex absorption features. In the case of pyroxene, a mineral that is dominant in many different kinds of extraterrestrial materials, these simple Gaussian models could not produce spectra that reflected real, known compositions.

Traditional Gaussian modeling of orthopyroxene (a variety of the mineral  pyroxene) requires multiple absorption features to explain measured results, which is inconsistent with physical models (the three solid curves, from bottom to top, represent the measured absorption, and two independent modeled absorptions, respectively). The curves are offset on the y-axis to display their relative shapes.

So Statistics has lied to us again! What can we do??

Instead of assuming that these absorption features are Gaussian in their shape, what if we assume that the slope of the absorption feature on either side can be asymmetrical, and is dependent upon some physical property of the material in question?

Enter: the Modified Gaussian model.

This model makes the assumption that the slope of the absorption feature corresponds to the distance between atoms where absorption occurs.

Where:

s = strength of absorption (amplitude)

µ = center of absorption (mean)

σ = band width (standard deviation)

n = a value that can be solved for empirically to match measured spectra

x = energy (random variable)

e = an exponential function

Wow, that's a lot of weird math, so let's break it down in simple terms:

The strength of the absorption just means how much of the that wavelength is absorbed relative to other wavelengths: a high strength = greater absorption.

The center of absorption is the mean (average) of the wavelengths absorbed.

Band width measures how much that absorption varies from the mean value.

The x value corresponds to the energy (photons) absorbed by the material, to the power of n. That value is proportional to the distance between atoms where absorption occurs.

By adjusting the value of n, we can alter the slope of the absorption feature on either side of its center. In doing so, we can produce a modeled spectrum that fits to measured values without incorporating multiple absorption features. This results in much more accurate spectral modeling that reflect real-world mineral assemblages, and produces errors that are equivalent to the errors that are inherent in the actual measured observations.

In the Modified Gaussian model, single absorption features can explain observed spectra. The bottom curve is measured spectra, and the curve just above that is produced by the model.

These models are still limited by the quality of their data. For example, not all minerals absorb light in such a way to produce characteristic spectra. Some assemblages may also contain detectable minerals in such low abundances that they're effectively drowned out by their surroundings. Additionally the data can be further complicated by thermal vibrations (the release of heat) and variations in mineral structures.

While these methods may not give a us an exact representation of extraterrestrial mineralogies, they do give us a reliable way narrowing down the compositions of planetary bodies that we may be incapable of visiting ourselves.