Parallax can be thought of as the apparent difference in the distance separating two points in three-dimensional space in a pair of photographs that were taken along different lines of sight. This is the basis of stereographic measurement of the shapes and sizes of three-dimensional objects from aerial or satellite photographs. This article describes the estimates of the elevations of the Mars Face and its surroundings I derived using stereography and the implications of these results.

For these measurements, two photographs are required. The angle of the line of sight relative to the vertical direction of the imaged region (the "emission angle") must differ significantly between the two images of the pair. I wrote a program that computes the elevation of a feature given the X-Y coordinates of its position in a pair of images and various kinds of ancillary data for the two images . The ancillary data is obtainable from the Planetary Data System archive. For those interested in the details of the measurement method, a description is provided on a another web page.

For brevity, I refer to the first image acquired by the Mars Global Surveyor in April 1998 (MOC SP122003) as image "98", the image acquired in June 2000 (MOC M1600184), as "00", and the image from April 2001 (MOC E0300824) as "01". The three stereo pairs used for height measurements are referred to as 98:00, 98:01, and 01:00. The height measurements for the three pairs of images serve as a cross-check on the validity of the measurements. The June 2000 image, however, was a partial image covering only the western side of the Face, so the cross-checking is necessarily limited to that region of the landform..

The results, as shown in Figure 1, are fairly consistent for those features that appeared in all three photos . The 98:00 heights (in green) are always lower than the heights measured on the other two pairs and the 01:00 measurements (in yellow) are always lower than the 98:01 measurements (in red). This is almost certainly because of differences in the elevations of the "ground" reference points used for the three image pairs. If the measurements were exact, the difference in measured height between any two image pairs for any given point would always be the same as the difference in the elevation of the ground reference points used for the respective image pairs.

Figure 1. Section of Mars Global Surveyor image E0300824 showing measured elevations in meters for three pairs of MGS images. Large arrows point to ground reference points used for each of the three image pairs.

The differences in computed elevation, as can be seen in Figure 1, range from 35 to 65 meters higher for elevations measured for the 98:00 pair than for the 00:01 pair and 65 to 95 meters higher for 98:00 pair than the 98:01. The range of +/- 15 meters in the differences in elevation relative to two ground references implies that the error margin is about +/- 7.5 meters for any single elevation measurement.

(If the elevation of a point as measured for ground reference 1 is h1 and the elevation of the same point is h2 as measured for a ground reference point 2, and the uncertainty for both measurements is e, then the error in the value of h1 -h2 will range from (h1 + e) - (h2 - e) to (h1 - e) - (h2 + e), or from 2e to - 2e.)
 
 

Primary Results of Interest

The peak height of the Face appears to be just above the circular, "nostril"-like feature. This point was measured to be 380 meters above the ground reference point for 98:00. The blue arrow shows the direction of the parallax displacement of this feature from its true position, which is significantly closer to the vertical ("head to chin") centerline of the landform than it appears to be in the 01 image. The "nose" ridge is therefore positioned symmetrically halfway between the left and right edges of the landform. Because the "hare-lip" ridge beneath the "nostril" is also at nearly the same elevation, this feature, too, would be closer to the centerline than it appears in the MSSS enhancements, which do not take into account the parallax displacements of three-dimensional objects. This makes a subtle but important difference in the impression of the symmetry of the landform.

The parallax displacements such as the one represented by the blue arrow in Figure 1 were used to compute the heights of various points on the landform. But they could in principle be used as well to create a true orthorectified image of the Face that takes parallax into account. However, I don't have the "image-warping" software for such a task at present. This could be an interesting exercise, but much of what might be seen in such a rectified image has already been provided directly by the 18-meter resolution image of the Face taken by the Odyssey spacecraft's Themis camera in April 2002. Figure 2 shows a comparison the 01 and 98 MGS images with the Odyssey image. Unlike any MGS image taken to date the Themis image is very close to a true orthophotograph (i.e., a photograph looking "straight down at the surface from above). The parallax distortion in the 98 image, which had an emission angle 20 degrees greater than that of the 01 image, is particularly large. With no mention of the presence and severity of these distortions in the NASA press statements accompanyng the release of the partially orthorectified MGS images, they were highly misleading to the public. But it is the measurement of these distortions that was the basis for my stereographic estimates of the height of the landform.
 
 

Figure 2. Comparison of the Face in incompletely orthorectified MGS images of the Face taken in 2001 (left) and 1998 (right) with the true orthophoto taken by the Odyssey Themis camera in April 2002 (center). The vertical lines in all three images indicate the left and right vertical boundaries of the landform and the vertical centerline. The parallax distortions introduced by the partial orthorectification of the MGS images can clearly be seen.

NOTE:  I never found the software to do the necessary tranformation of the 2001 MGS image into an orthophotograph, but I have since written the software myself. The resultant "synthetic orthophoto" is shown at this link.  -- Lan Fleming 10/12/02

As Mark Carlotto noted in a New Frontiers in Science paper, the "nostril" and two other circular features above and below it are closely aligned with each other along the vertical centerline.

Recently, one person who has been a vehement skeptic about the Mars Face somewhat grudgingly admitted that he could see why some people are interested in it after seeing the Themis image, although he didn't say why. I believe that it's these repeated indications of alignments on the vertical centerline that are not apparent in the orthorectifications of MGS images that do not take into account the parallax effect. In any case, most SPSR members and others interested in this subject were well aware of the parallax distortions of the MGS images, and the Themis image was rather anticlimactic.
 
 

General Trends in Terrain Elevation

As can be seen from the elevation numbers in Figure 3, If one were to walk around the visible base of the Face landform the peak height would vary from 295 meters to the southwest to 335 meters to the northeast. There is a slight decrease in elevation in every direction away from visible perimeter of the Face in the 2001 image. It is clear that the elevations to the north and east are about 80 meters lower than elevations to the south and west. Points on the eastern side of the Face are also as much as 100 meters lower than corresponding points on the western side of the landform's vertical centerline.

Figure 3. Section of Mars Global Surveyor image E0300824 at 19% full size. Numbered points indicate elevations in meters measured from the ground reference point at upper right. The 1998 and 2001 image pair was used for these measurements.The hand-drawn ovals indicate the regions that probably contain the ground reference points used in a previous estimate of the height of the landform from MGS MOLA data.

This conformance of elevations on the landform with differences in the elevation of the surrounding terrain is consistent with the interpretation that its eastern side was artificially "built up" and then experienced a collapse at some later time, perhaps due in part to the unevenness of the terrain. Of course, it would also be consistent with intentional asymmetry or with random variations in the height of a natural mesa. These terrain differences are worth noting but don't really favor one theory over another.
 

Comparison with Prior MOLA Results

A previous article described the results of the Mars Orbiter Laser Altimeter (MOLA) measurements for the height of the Face and compared them to Carlotto's shape-from-shading (SFS) estimates. The maximum height of the landform measured by the MOLA was 330 meters relative to two points 1500 meters to either side of the maximum. More precisely, the measured height was 345 meters relative to the ground point 1500 meters to the northeast of the maximum and 321 meters to a point on the opposite side of the profile maximum, 1500 meters to the southwest. The elevation of the northeastern point was therefore 24 meters lower than the elevation of the southwestern point.

The ovals in Figure 3 indicate the areas that probably contain the corresponding positions of my MOLA reference points. Their exact positions relative to the MOC images is unknown, at least by me. But the agreement between the MOLA results and my stereographic measurements is very good. The elevation of the martian plains in the northeastern area containing or close to the northeastern MOLA reference is also about 25 meters lower than the elevation of the corresponding area to the southwest according to the stereographic measurements. The peak heights of the Face measured using stereography are about 320 and 345 meters respectively, again in close agreement with the MOLA profile.

The MOLA then, probably passed very close to the peak as I had previously concluded, but it is still unclear whether it measured the maximum height. Given the error margin of +/- 7.5 meters that I estimate for my stereographic measurements, the true peak height could easily be 7 meters and as much as 14 meters greater.
 

Comparison with Prior Shape-From-Shading Results

In a 1988 paper published in the journal Applied Optics, Mark Carlotto gave estimates of the Face's dimensions based on his shape-from-shading (SFS) analysis of Viking Frames 35A72 and 70A13. He estimated a height of 430 meters from 70A13 and 395 meters from 35A72. In contrast, I estimated a maximum height of 380 meters relative to a point 1 km to the northeast of the Face on lower-elevation terrain. However, this difference cannot be attributed to flaws in Carlotto's SFS methodology. His estimates were based on NASA's data on the resolutions of the two Viking images as of 1988: 51.73 meters for 35A72 and 48.13 meters for 70A13. Those estimates have since been revised upwards to higher resolutions, as shown on Michael Malin's "Face" page. The most recent resolution estimates are 46 meters for 35A72 and 43 meters for 70A13. Scaling Carlotto's original height estimates by the ratio of the new resolutions to the old gives a range of heights for the SFS method of 380 to 350 meters. This is within the range of heights relative to most points within 1 km from the Face I found using stereography. One thing that seems remarkable about the SFS measurements is that Carlotto appears to have derived a height estimate from Viking images with a resolution of 10 to 20 times less than the resolutions of the MGS images I used for my stereographic measurements but did so with an error margin that was not much more than twice as great.

Comparison with "3D Perpsective Image" Released by NASA

Displayed in the NASA article accompanying the release of the April, 2001 MGS Face image was a picture that was called a "3D perspecitve" enhancement of the MOC image. It has already been pointed out that it could not have been based on the MOLA data for the Face as was claimed because there were only two closely-spaced MOLA tracks over the landform. The stereographic measurements of the landform, confirmed by the more recent Odyssey Themis orthophoto, show that this "enhancement" has no basis in reality. Figure 4 shows that the orientation of the "nose ridge" and "harelip" structures as presented in the article (indicated by a red line) are grossly misaligned with their true position along the centerline of the landform (indicated by the green line). Further, it is impossilbe for the edges of the landform's base to diverge with increasing distance from the observer as is shown in NASA's "perpsective" image. Since the eastern and western edges as viewed from above are essentially parallel, they should instead tend to converge, in the same manner and for the same reason that parallel railroad tracks appear to converge with increasing distance from the observer due to perspective.
 
 

Figure 4. Grossly distorted image released by NASA that was claimed to show a perspective view of the Face. Red lines indicate erroneous orientations of the eastern and western edges and the central "nose" ridge. Green lines indicate the (approximately) correct orientations of those same features.

The Odyssey team's article on the Themis image employed the same misleading tactic previously employed for the 2001 MGS image (see article on Middle Butte) of comparing the Face to a landform that was not shown in the article and that obviously bears no resemblance to the Mars Face. (See Figure 5).

However, the Odyssey image was something of a milestone since it was the first image of the Face released by NASA in recent years that they did not distort in such a way as to make the Face appear more "natural" than it really does.

            ---- Lan Fleming