Astro3310 – Planetary Image Processing Lab 4: Topographic Analysis

I. Photoclinometry

In you haven’t already, you’ll need to add the path of the matlab routines for this lab by typing:

“addpath(genpath(‘../.’))”

from the lab REPORT sub-directory in the Matlab window.

This program uses the I/F values in the image m0806185.lev1.cub. Some of the program lines are reproduced below:

res = 2.95;        %Size of a pixel in meters
sunaz = 320.65;        % Solar Azimuth
inc_ang = 48.18;        % Inclination Angle

These lines set up the variables specific to this image. The solar azimuth is important, as only tracks that cut across the image parallel to the illumination direction can be used for photoclinometry.

b = b-shadow;

b is the variable that stores the I/F values alone the sun's line of sight. Here we remove some estimate of the I/F value, which is due solely to the atmosphere (called shadow brightness because this is the only way shadows can be illuminated)

z = acos(b.*pi.*cos(inc_ang*(pi/180.0))./albedo);

These few lines are the guts of the program. The first calculates the incidence angle for each pixel based on its I/F value. The second then removes the incidence angle that a flat surface would have, leaving just the local slopes for each pixel. The third line converts the slope of each pixel to the change in height of that pixel. The for loop adds up all these height changes to find the actual height of each pixel.

Notice, for example, that we have to assume some value for the albedo and that the atmosphere can have an important role, due to its ability to scatter light.

You can start the program by typing 'pclin('m0806185.lev1.cub')' at the matlab prompt. The standard case will come up i.e. the program selects an interesting part of the image, guesses an albedo and assumes no atmospheric effects. You can force it to use some particular albedo by calling it like 'pclin('m0806185.lev1.cub',0.25)' for an albedo of 25%. You can use 'pclin('m0806185.lev1.cub',.25, 3000)'to select a starting line for the section of image you want to look at (line 3000, in this case). You can also make some correction for the atmosphere by assuming some amount of I/F is due to scattering, e.g. 'pclin('m0806185.lev1.cub',.25, 3000, .02)' will assume that 0.02 needs to be subtracted from each I/F value as a first order atmospheric correction. You can choose each of these three variables, although they must be in the order: “albedo, starting line, shadow brightness”. (If you want to specify the starting line but not the albedo, for instance, a nonsensical value like -1 will cause it to choose the default albedo value. 'pclin('m0806185.lev1.cub',-1, -1, -1)' will cause the program to use the defaults for all three.) Note that the elevation here is in arbitrary units. It's easy to get relative elevations with photoclinometry, but this technique (or more specifically, our rudimentary implementation of it) isn't reliable enough to actually measure heights in meters. So in discussing these heights, don't assume some unit, it's just an arbitrary scale.

Experiment with different albedos, letting the program guess first and then trying higher and lower values. Try this for a number of areas and try to figure out what goes wrong when the albedo is far away from the real answer. Explain it as best you can in words. Do the same for the shadow illumination parameter.

Export your Matlab plots and insert them into your lab writeup. 

II. Laser Altimetry

We'll look at one of the single tracks in particular that crosses over the image from part 1. Most MOC images have accompanying MOLA data through mission phase E07 or so. However, this data is only along the track of the spacecraft (the long axis of the image), and only along a small strip of the image. The whole volume of MOLA tracks has been gridded into one planet-wide map, but this map has slightly lower resolution in the low latitudes, and is not quite as reliable on small scales as the individual laser shots.

We have all the MOLA data on disks. The data structure of the MOLA files is long and complicated so to spare you the details and let you get straight to the data we have used a small program that will automatically go and get the MOLA track for any MOC image you ask for. This program returns 4 arrays, one of the longitude of each shot, one for the latitude, one for the topographic height and one for the time each shot occurred. All you need give it is the name of the MOC image you want data for. In the shell script you ran earlier, this was done by calling the program 'mocmola.pl' on line 5. We have written a program to get all this data we produced into Matlab. In MATLAB type:

“[time,y,x,z,d]=mocmolaplot('m0806185');”

Some plots will pop up, and some arrays will be spit out. In MATLAB you can find out information about arrays by typing 'whos ARRAY-NAME' if you leave off the array name and just type 'whos' you'll get information on all the variables MATLAB has stored in memory. Try things like 'y(1:10)' which will show you the first 10 elements in y (the projected y-coordinate, in meters, of the MOLA shots). Try 'max(x)' or 'min(time)' to see maximum and minimum values of some of the arrays. Note that the time is in awkward units of seconds after JD2000 (Julian date of 00:00 1/1/2000). Computers have no problem with numbers like this but humans generally do (at least I do).

The plot on the left hand side of the figure simply plots the MOC image we have been looking at, now projected, with the MOLA points on top. Note that the actual footprint of the laser beam is significantly larger than the points shown here, but I think this makes for easier viewing. But if a spot falls right on the edge of a cliff, the results may look a little funny. The plot on the right shows the elevation of the MOLA data vs. Latitude. You can now see the actual elevations of this image. Now plot the elevation on your own. For example, you can use the command:

“figure; plot(d,z);”

This will plot the topography recorded against the distance from the first point. Remember each point represents about 300 m along the ground, or 0.1 seconds. Notice in this dataset that we are dealing with absolute elevations, as opposed to unscaled and relative elevations in the photoclinometry section. This is very useful! So, what is the total relief along this track?  Because we now have absolute elevations, we can compare different parts of the image to each other. Are all of the high points in this image at the same elevation? What are the maximum slopes seen in this track, and are they consistent with loose sediment (as opposed to bedrock)? (The track isn't along the direction of maximum dip, so this will only be a lower limit on the actual maximum slopes. Too bad we don't have topographic data for the whole image...) Export your Matlab plots into your lab writeup when you're done.

The third topographic technique we will employ is stereo photogrammetry (see lecture notes). It just so happens that the image we have been looking at has been imaged twice by the MOC team. After MOLA was shut off MGS was oriented, for the most part, at 17 degrees off-nadir to save fuel. Images in this later phase of operation can be combined with earlier mapping phase images to extract topographic information, as in a 3-D picture. By measuring the parallax between various features, and with precise knowledge of the camera position, good topographic data can be obtained, at near the resolution (typically ~3X) of the original MOC images. Advantages to this method are the higher resolution, and the ability to extract absolute elevations. However, the extent of MOC stereo coverage is rather limited, and it does not have quite the vertical precision of the Laser Altimeter data. We will take a look at this data, and you can see the advantages and disadvantages for yourself.

To save you time, we've made a Digital Elevation Model (DEM) beforehand: dm270_1626.grd In matlab, type 'stereoplot' This will bring up two images. The first one is our friend M0806185. Again, this is the projected version of the image. The second plot is of the stereo DEM, with the image behind it. This is really different than the other two analysis methods! We have 2-D coverage of the image...sort of. Notice that there are lots of areas where the automatic correlator couldn't find matches between the images. This is usually because the images are too featureless in these areas. But you can see that where there are slope streaks on the image, the tracker is able to find matches between the images, because of their high contrast. (On the other hand, if you look closely, there are places where new slope streaks have formed in the time between the two images, and automatic correlation fails!) Also, you can see there is some noise to the data, much more than the MOLA data. We're going to use a function called proftool3, so you can take profiles of this plot in any direction you want. It can be used in two ways. Close the second (color) plot window, or just make the black and white image the active plot. Type:

“proftool3(z)”

Use the mouse to select the profile in the image window (right click when you're done to exit). This function lets you click on the image for reference, but extract the corresponding topography (from the elevation matrix, z). Follow the directions that come up in the matlab workspace. A plot should pop up showing the elevation profile. Alternatively, you can type:

“[xx,yy,zz,dd]=proftool3(z);”

This usage lets you save the x, y, z, and distance values of the profile in the four output arguments, in case you want to plot them again later. You have two tasks, using this tool: 1) Determine whether the layers exposed along the edge of this cliff are roughly horizontal or not. 2) Determine the angle of the slope streaks which can be seen throughout the image Is this what you would expect for a fine dust layer on the surface? Again, export your plots to support your conclusions for these questions. Finally, type 'h=reshape(z,1,1751*501);' and “figure; hist(h,20);” to view a histogram of elevations for this DEM. What are the maximum and minimum elevations?

Phew! Congratulations, you're done.

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

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)

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 what is really out there. Anyone wishing to go deeper into the (gruelling) mathematical modeling of reflection can come to either Jamie or Alex for references and/or help.

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