Tutorial 1: Reflectance from Planetary Surfaces

Ge151: Spring 2011

Reflected visible and near infrared light from planetary surfaces is arguably the most important remote sensing window available to us. Not only are most planetary images taken in the visible (where the sun provides a convenient abundant illumination source) but reflected light in the near-infrared provides important spectral information capable of distinguishing many minerals. This tutorial will cover a few basic things to make sure everybody is starting from the same level.

When dealing with reflection the first thing to consider is the spectrum of the source (the sun for planetary work). The solar luminosity is roughly 4x10²⁶ Watts, emitted isotropically over 4pi ster-radians. The solar constant for each planet is the solar flux at that planets position (for Earth its around 1300 Watts). The solar flux dies off as an inverse square law, Jupiter at ~5 AU receives about 1/25^th (~4 %) of the Earth's solar flux. The shape of the solar spectrum follows that of a 6000 K blackbody. It peaks in the visible region (not surprisingly since our eyes have evolved to take advantage of that) at about half a micron. To convert the Plank function to specific flux (flux per hertz) you have only to multiply it by the solid angle subtended by the sun. The solar constant is then just the integral of the specific flux over all frequencies. More will be said in tutorial two about these quantities and the relationships between them.

Most surface images of planetary bodies are taken with a broadband filter which covers most of the visible range. Several things affect the brightness received by each pixel in the CCD array or each element in a vidicon image (believe it or not for some bodies this is still the best we have). The flux falling on the surface is the primary factor and is the same for all pixels unless the image covers such a large fraction of the planet that the solar incidence angle varies considerably from one part of the image to another. The albedo of the surface making up that particular pixel is a measure of what fraction of the incident light is reflected, there are many types of albedo and we'll discuss some of them later. Albedo varies from pixel to pixel, e.g. sand is darker than dust (for reasons we'll also discuss later). Local slopes can also effect the incidence angle and so effect the brightness of that pixels, e.g. the sunlight side of a hill is brighter than the shadowed side. Before we go any further lets define our observational geometry to prevent any confusion.

The above figure is taken from Hapke, 'Theory of Reflectance and Emittance Spectroscopy' (Cambridge University Press, 1993). The incidence angle is denoted by i and is the angle the incident solar radiation makes with the local vertical. The emission angle is denoted by e and is the angle made by the emitted radiation toward the detector with the local vertical. Nadir pointing spacecraft observe only whats directly below them, so the emission angle is usually very small. The phase angle is denoted by g and is the angle between the incident and emitted rays. For nadir observations (where e is zero) the phase angle is equivalent to the incidence angle. When g is zero the sun is directly behind the observer this can lead to a surge in brightness known as the opposition effect, which we'll talk about later.

For a spherical planet the incidence angle on the sun is given by the following formula, where L is the latitude, H is the hour angle and D is the solar declination.

cos(i)=sin(L)·sin(D) + cos(L)·cos(D)·cos(H)

Taking into account the position of the sun, planet and spacecraft it is possible to derive an I/F value for each pixel. This is the fraction of light which hit the surface contained within the pixel that was reflected. It depends both on the albedo of the surface at that point and on the local slopes which may have effected the incidence angle. That's fine for bodies like the Moon, Mercury and the Galilean satellites but not for bodies like the Earth or Mars which have atmospheres. The atmosphere can scatter light out of the incoming solar radiation which reaches the surface providing diffuse illumination rather than collimated and it can also scatter radiation out of the outgoing beam of radiation. In short the presence of an atmosphere is bad news when trying to interpret I/F values. We'll restrict ourselves to how the surface effects the I/F values but keep in mind that any atmosphere is playing a major role.

Albedos

Before describing the different kinds of albedo it is necessary to describe the concept of a Lambert Surface. A Lambert surface is one which appears equally bright from whatever angle you view it (any value of e). The Moon is a very good approximation of a Lambert surface, it appears equally bright at its edges and at its center making it appear almost like a two dimensional disk rather than a sphere. The flux from 1 m² of a lambert surface is proportional to cos(e) due to the geometrical effect of foreshortening, however the same angular size includes more surface area at grazing angles and so these effects cancel out leaving the brightness independent of e. There are many kinds of albedo, all of which were defined for use with the Moon and then generalised to cover other bodies. Some of the more common ones you may come across are:

Normal Albedo: Defined as the ratio of the brightness of a surface element observed at g=0 to the brightness of a Lambert surface at the same position but illuminated and observed perpendicularly i.e. i=e=g=0.

Geometrical (Physical) Albedo: It is the weighted average of the normal albedo over the illuminated area of the body. Defined as the ratio of the brightness of a body at g=0 to to the brightness of a perfect Lambert disk (not sphere) of the same size and distance as the body illuminated and observed perpendicularly i.e. i=e=g=0.

Bond (Spherical) Albedo: Defined as the total fraction of incident irradiance scattered by a body in all directions. This quantity will be important when considering the total amount of energy absorbed by a surface for things like thermal balance etc...

Slopes

We've already mentioned that local slope can serve to increase or decrease the apparent brightness of a patch of ground. If we assume an albedo for each pixel then we can figure out what the solar incidence angle is for each pixel based on the I/F value. We can subtract the incidence angle for a flat piece of ground at that time,latitude and season, we are left with the slope of each pixel but only in one direction! This procedure is known as photoclinometry, it works best when you can remove the albedo easily e.g. if the surface is covered with some uniform albedo material like frost or dust. This can only tell you what the slopes toward and away from the sun are, if you want to reconstruct the entire topographic surface then you need at least two observations illuminated from different directions. Again the atmosphere wreaks havoc with this sort of technique and make it difficult to get quantitative results however if you could constrain the answer with other methods such as a laser altimeter in the case of Mars then you can generate very high resolution topography maps for small areas.

Reflection Phenomena

Photometric and Phase Functions: Surfaces behave differently when viewed under different geometries. The phase function gives the brightness of the surface viewed under some arbitrary phase angle divided by the brightness expected if the surface were viewed at zero phase. The photometric function is the ratio of surface brightness at fixed e but varying i and g to its value at g=0. For a Lambert surface the photometric function is given by cos(e).

Grainsize Effects: Scattering from a particulate medium in the visible and near infrared tends to be dominated by multiply scattered light i.e. photons that have been scattered within the surface more than once. Photons pass through grains getting absorbed along the way, they get scattered by imperfections within the grains and by grain surfaces. The more surface area there is the more scattering there is and the bigger the grains (more volume) the more absorption there is. The surface area to volume ratio of the grains therefore has an effect on the amount of light scattered by the surface. The surface area to volume ratio scales linearly with the grain size so for the same material surfaces with larger grain sizes will in general be darker. The most familiar example of this is probably sand on the beach, dry sand is bright but when sand get wet it clumps into larger grains and turns darker. In reflection spectra larger grains produce broader, deeper absorption bands.

Opposition Effect: When observed at zero phase surfaces experience a sharp increase in brightness known as the opposition effect (also known as: opposition surge, heiligenschein, hot spot and bright shadow). The pictures below show the opposition surge in the case of the moon, look at the edges of the astronauts shadow (which is at zero phase angle), can you see the extra brightness compared to the surrounding regolith. The plot shows the same effect but in a more quantitative way.

The physical explanation offered for the opposition effect is that of shadow hiding. If the illumination source is directly behind you then you cannot see any of shadows cast by surface grains (since they're all hinding behind the grains themselves), whereas if you were looking toward the illumination source then you can see all the shadows cast by the surface making it appear darker. Coherent backscatter is another mechanism offered as an explanation and it is likely that both mechanisms play some role although from recent work on Clementine data it looks like shadow hiding is the dominant mechanism at least in the case of the Moon.

Reflection from particulate mediums is an area of research in its own right, which has been largely pioneered by Bruce Hapke in recent years. This is only meant as a taste of whats really out there. Anyone wishing to go deeper into the (gruelling) mathematical modeling of reflection can come to me for references and/or help.

Questions

(1) What is the flux at Venus for a filter 0.05 microns wide centered on 0.7 microns?

(2) At equinox on Mars the MOC camera aboard MGS is making observations at 60 N at 2pm local time. The pixel size is 1.4 meters across, how much detectable power is falling on the surface within the area of each pixel ? The narrow angle band is centered on 0.7 microns and has a bandwidth of 0.4 microns.

- Shane Byrne -

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