Caution: This page is very long and contains explicit trigonometry and graphic ...well, ... graphic graphs! It is not recommended for younger or more sensitve net surfers; only inveterate anomaly hunters and dedictaed skeptics with plenty of time on their hands should venture beyond this point!
Whether the Blair Cuspids are anomalously tall objects or mere boulders depends on how their shadows are interpreted, and that in turns depends on knowing the angle of the slopes on which the objects' shadows are cast in the Lunar Orbiter photographs. The method described previously that exploited the size of the apparent penumbra of Cuspid 5 supports, I think, the possibility that at lest the largest of the Cuspids, #5, is anomalously tall, but it is a non-standard method that is fraught with uncertainties, not the least of which is determining the boundary of a partial shadow that by definition has no sharp boundaries. The standard and preferred method for deciding such questions is photometric analysis. Previously I said that the information necessary to do a photometric analysis had been lost, but I have learned since that this is not the case.
This page presents the results of such an analysis on both of the Cuspid photos (Lunar Orbiter 2 Frame 61H3 and 62H3). Before describing those results, however, both the mathematical basis of the method and the procedure for extracting photometric information from Lunar Orbiter photographs have to be understood.
When a person looks at a two-dimensional black and white photograph of an uneven three-dimensional terrain, the brain interprets the differences in shades of gray as differences in the slopes of the terrain. Bright areas are interpreted as surfaces sloping upward in the direction away from the sun and darker areas are interpreted as surfaces sloping downward in the same direction. This information is then synthesized into a mental image of hills and valleys, craters and rocks. This unconscious process is based on a real mathematical relationship well-known to imaging professionals. The relationship, which holds over most surfaces for very low sun angles such as that in the Lunar Orbiter 2 photographs of the Blair Cuspid region, is:
A = arcsin(KE)
where A represents the elevation angle of the light source (i.e., the sun) above the plane of the surface and E represents the exposure, or total light reflected from the surface onto the camera's film during the time the camera shutter is open. This means that the illumination angle, A, at any given point on a photograph can be easily computed if the exposure E at that point and the constant K are both known and the height of objects casting shadows on the photographed surface can be computed with a high degree of confidence - something still lacking in the previous estimates. The problem with the Lunar Orbiter photographs is that neither of the necessary numbers is immediately available.
As noted previously, the Lunar Orbiter photographic negatives from NSSDC have been enhanced. The enhancement usually involves making dark areas darker and light areas lighter than in the original photograph, increasing the overall contrast of the enhanced image. This makes it easier for the eye to perceive small detalis in the photograph but it creates difficulties for photometric work because there is no longer a direct relationship between film density (the extent to which the negative is darkened) and the amount of light exposure that was received by the camera. The original magnetic tapes with the unenhanced photographs have apparently been lost. But all is not lost!
Each Lunar Orbiter high-resolution photograph is composed of 30+ framelets. On one edge of every framelet is a row of nine blocks. The block on one end of the row is almost black. Moving from that end of the row to the other, the blocks are progressively lighter shades of gray until the last block is almost completely white. The row of blocks on Framelet 386 from LO2-61H3 is shown in Figure 1 below.
FIGURE 1
Gray scale blocks on Framelet 386 from LO2-61H3. This framelet is near
the one that contains the Blair Cuspids. Blocks 0 and 1, on the far left
are somewhat difficult to make out because they are nearly the same shade
of gray as asthe background.Part of the lunar surface can be seen at the
bottom of the image.The series of vertical, horizontal, and slanted lines
are used to determine the photograph's resolution.
NOTE: If you cannot discern the first three blocks from each other or from the background, then your web browser is not showing the image properly. It and other images on this page will appear much too dark. Netscape and other browsers usually allow you to click on an image to bring up a menu with the option to save the image to a file. If you do that, you can then use an imaging application to view the image and adjust the brightness and contrast until the individual blocks are distinguishable. The same procedure can be used on any of the other images that appear too dark.
While puzzling over the problem of how to apply photometric analysis to photographs that have undergone unknown modifications, it occurred to me that perhaps these blocks were used to preserve exactly the information needed. I surmised that prior to loading the film onto the Orbiter, the blocks in each row had been exposed to a constant light source but for different lengths of time so that the darkest was exposed for 0 time units, the block next to it for 1 time unit, the block after that for 2 time units, and so on to the final block, exposed for 8 time units. The time and light intensity were, I supposed, selected so that the film in the final block was darkened to the maximum extent possible. If all that were true, then a graph could be constructed of film density versus exposure for the full range of photographic densities for that type of film. I checked with the National Space Science Data Center, and they verified that this was correct.
Because the gray-scale blocks were subjected to the same enhancements that the rest of the photograph was, a plot of the blocks' brightnesses as measured with the Photoshop imaging application versus their sequence numbers should be equivalent to a plot of photographic density versus the original exposure - just what is needed for photometric analysis. The value of E, the exposure, for the slope of the terrain near any of the Cuspids could be determined! Figure 2 shows an example of how this kind of graph is constructed.
FIGURE 2
Illustration of how a graph of film density vs exposure is constructed. "Film
density" is considered to be the same property as the brighness of the digital
image. Exposure is considered to be the block number on the scale of 9 values
from 0 to 8 with Block 0 representing no exposure. To construct the curve,
a point is marked where a horizontal line from the vertical axis at the value
measured for each block's digital brightness intersects a vertical line drawn
from horizontal axis at the block's sequence number on the framelet.(The
corresonding horizontal and vertical lines are in blue). The curve is
then drawn between the points. (The curve and the points determined from
the measurements are in red). The maximum possible digital brightness value
is 255 but the maximum density/digital brightness is generally somewhat less
than that..
The remaining unknown was the factor K that relates density to the sine of the sun's elevation angle. Two points on the graph are needed to determine K. One point is easy to get. The value of the measured image brightness of Block #0 represents the sine of a zero-degree sun elevation angle. The measured brightness on the digital image of this block is NOT zero, however, but it is very small: an average gray-scale value of about 10 out of a possible 255 values on the digital image was found for one of the photos. The value of the photographic density corresponding to the bright end of the scale, however, is not so easy to get. That would be exposure value corresponding to the sine of a 90 degree sun elevation, or 1.0. The film is very likely saturated, or maxed out, by illuminations much less than that, so simply assuming that the lightest gray-scale block represents a 90 degree illumination angle is not valid.
But there is another point that can be used: the digital image gray-scale value corresponding to the 10.9 degree sun elevation above the lunar horizontal from the NASA support data for the two photographs. While it cannot be known that any given point on the photograph is precisely horizontal, the average brightness of the photograph should represent the 10.9 degree angle assuming that the terrain, on average, is horizontal. This should be a conservative assumption, since the terrain in that region of the moon tends to slope upward away from the sun, which may mean that the average sun elevation is greater than 10.9 degrees, possibly as much as 12 degrees. That would mean the computed sun elevation for any given point in the photograph is too low and that objects casting shadows are taller (and therefore more anomalous) than what was computed.
To find the average overall brightness, I first made two graphs, one from the gray block series of a framelet in each photograph, of photographic density (digital image brightness) versus exposure (block sequence number). I then scanned twenty 30mm square areas at 2700 DPI from the 8X10-inch negatives of both photographs. For each 30mm sample area, I found the corresponding exposure value on the graph for the same photogrpah. Finally, I took the average of the forty exposure values as my estimate of the brightness of the surface at the 10.9 degree sun angle. The value I computed was 4.14. I used a statistical technique involving what is called a "t" distribution to determine that this value was within 0.07 of the true exposure value on the scale from 0 to 8 at a confidence level of 95%.That means there is a 5% chance that the actual value is less than 4.07 and a 5% chance that it is greater than 4.21. This result gives an average exposure value right in the middle of the film's range where it should be, which shows the Lunar Orbiter imaging staff did a good job at selecting their exposures. A higher average exposure would tend to wash out lighter features because their brightness values would be all scrunched together at the high end of the density range and a lower average exposure would make darker features indistinguishable from black shadows. An average value of 4 optimizes the amount of information captured by the camera.
Assuming that this average is the exposure value, E, for a 10.9 degree sun angle, the estimated value of K is:
K = sin(10.9) / 4.1 = .046
This is the value of K that I used to determine the sun elevation for all of the objects in LO2-61H3 and the additional object I found interesting in LO2-62H3 - the "Crater Cuspid".
With all these lengthy (but necessary) preliminaries out of the way, we are ready to consider the measurements of the slopes of the terrain in the specific areas of interest - the slopes on which the shadows of the Blair Cuspids appear. Again using Photoshop, I measured the average brightness of 10-pixel wide strips along the lengths of the objects' shadows as close to the shadows as I could get without including them. I did these measurments on 2700DPI images scanned from the negatives The results for Cuspid 5 for Frame 61H3 are shown in Figure 3. The results for 62H3 are in Figure 4.
FIGURE 3
The measured density and corresponding exposure from Frame 61H3
for the slope on which the shadow of Cuspid 5 appears. The horizontal and
vertical blue lines indicate the photographic density and original estimated
exposure respectively of the slope. The curve was plotted from the gray-scale
blocks of a framelet near the Cuspid.The framelet in which the Cuspid appears
was not used because Block 0 was lighter than the shadowed region of the
same framelet, indicating that unevenness in either the Orbiter telemetricy
signal, the electronic enhancement , or chemical film processing caused the
blocks to be brighter than they should be.
FIGURE 4
The measured density and corresponding exposure from Frame 62H3
for the slope on which the shadow of Cuspid 5 appears. The horizontal and
vertical blue lines indicate the photographic density and original estimated
exposure respectively of the slope. The curve was plotted from the gray-scale
blocks of a framelet near the Cuspid. In this case, the curve for the framelet
in which the Cuspid actually appears is almost identical to the one used
for this graph.
Despite the fact that the density vs exposure curves are significantly different for the two photographs, the slope exposures for Cuspid 5 found by this procedure are very similar, which adds considerable confidence in the validity of the results. Figure 3 shows that the measured digital brightness (32 on the 255 value digital gray scale) in 61H3 corresponds to an exposure value of 2.8 while Figure 4 shows that a much lower brightness (12) in 62H3 corresponds to an exposure of 2.6 on the 8-valued exposure scales. Ignoring the somewhat higher exposure of the former in favor of the more conservative result for the latter, the estimated sun elevation, A, on the slope beneath Cuspid 5 is:
A = arcsin(2.6 K) = arcsin(2.6 *.046) = 6.9 degrees
Note that this angle computed by the "preferred" photometric method is almost exactly within the middle of the 6 to 8 degree range previously proposed as the reasonable range of probably sun elevation angles. From the previously described relationship between shadow length, L, and object height, H:
H = L tan(A)
It can therefore be estimated that the height of Cuspid 5 is 0.12 times the length of its shadow. This is the factor by which the shadow's length was compressed to give the profile of Cuspid 5 presented previously. While there is certainly some "wiggle room" with these numbers, the validity of this profile is still, in my opinion, strongly supported by two completely independent types of analyses, the original one based on the apparent size of the penumbral shadow and the photometric analysis, which produced very similar results for two different photographs with very different density vs brightness curves.
For anyone who would like to verify the brightness measurements, the unenhanced 2700DPI gifs for Cuspid 5 can be downloaded for 61H3 here (65K) and for 62H3 here (58K).You will need software that will display histograms of selected areas of the images.
In hopes of luring the unwary reader into reading this long account of the photometric analysis of Cuspid 5, I deferred showing the profile for the object I referred to previously as the "Crater Cuspid", or Cuspid 6. I'll describe it now. This was a small object on the rim of a crater near the main group of Cuspids casting a long needle-like shadow. Due to the curving shadow cast by the crater rim itself, there is some uncertainty as to where the base of the "Cuspid" shadow should be marked, but the base-line indicated in Figure 5 below shows what seems the point at which the shadow is unambigously that of the Cuspid and not the crater rim.
FIGURE 5.
Section of LO2-62H3 showing the "Crater Cuspid" and the shadow it ccasts
across the wall of the crater on whose rim the object rests. The arrows
on the left indicate the point selected as the base of the shadow.
FIGURE 6.
Profile of Cuspid 6 (the "Crater Cuspid") for the 9.9 degree sun elevation
derived from photometric measurements. The profile was produced by compressing
the shadow of the object from its tip down to the point shown in Figure 5
where it intersects the shadow of the crater. (It is not valid to use the
full length of the shadow from its tip to the object itself in this case
because the height of the object above the crater floor must be substracted).
While this object is only half the height of Cuspid 5, the resemblance of
the shape and proportions of the two objects is obvious. Outlines have been
added to the version on the right to make the object's apparent shape more
visible.
The estimated illumination angle used to produce the profile in Figure 6 was 9.9 degrees. This angle was determined using the exact same photometric procedure by which the angle for Cuspid 5 was determined as already described. The 2700DPI image used for the measurements can be downloaded here (46K).
Despite the fact that this object (Cuspid 6, for short) is casting its shadow down into a crater, the computed illumination angle is only one degree less than the 10.9 degree angle for a horizontal surface under the same lighting conditions. This small difference is perfectly reasonable because the shadow does not extend into the deepest part of the crater. Walking along the length of this shadow, you would descend down an almost imperceptible 1-degree slope. As you should be able to measure yourself from Figure 5, the length , L, of the shadow is about 7.14 times its width, W, at most points, which means that the object 's height, H, is 25% greater than its width, assuming the sun angle of 9.9 degrees: H /W = 7.14 tan(9.9) = 1.25. This is the same ratio that was found for Cuspid 5, the largest object and the one casting the most dramatic-looking shadow in the region, so the much smaller "crater cuspid" has very similar proportions to those of its big brother.
To produce the original profiles I showed on the first Cuspid page, I assumed that the objects were sitting on horizontal surfaces for all but Cuspid 5. With the exception of Cuspid 4, none of the other objects fare too well under photometric analysis in terms of having anomalous height, except for Cuspid 4. My initial tentative assumption that they were on horizontal surfaces was not very accurate. Cuspids 1 through 3 appear to be on downward slopes similar to that of Cuspid 5. Cuspid 4, while also on a downward slope, still is illuminated at a sun elevation of about 9.8 degrees, indicating there is a small slope at that position of 1.1 degrees below the horizontal. This does not appreciably change the object's profile, and I won't reproduce it here. The other objects still may be unusual due to their apparently conical shapes, but they are not very tall cones. Under what I hope will be considered a serious and methodical examination, the claim that some of the objects known as the Blair Cuspids are true anomalies has been weakened, but the case for a few of them has in my opinion been substantially strengthened. As to what these objects actually ARE, I will not hazard a guess except to say that it seems very unlikely that they are all the type of common boulders favored by the NASA experts who dismissed them without any apparent scientific evaluation.
