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1
A Method for Searching for Artificial Objects on Planetary
Surfaces
†
Mark J. Carlotto and Michael C. Stein
The Analytic Sciences Corporation
55 Walkers Brook Drive
Reading, MA 01867 USA
Abstract
The focus of the search for extra-terrestrial intelligence (SETI) has been to look outside our solar
system at radio frequencies for signs of intelligent life. Such a strategy is consistent with current
information which suggests that it is unlikely that intelligent life could have evolved on the other
planets in our solar system. Our knowledge to date cannot, however, rule out the possibility that
extra-terrestrials or their probes may have reached this solar system. If so, they may have altered
planetary surfaces in ways that are detectable through remote sensing. An alternative strategy is
proposed which involves extending SETI to include a systematic search for anomalous, i.e.,
possible non-natural objects on planetary surfaces. An approach for detecting anomalous objects in
planetary imagery based on the fractal modeling of terrain is described. Fractals have been
successfully used to model a wide range of physical and biological phenomena, including natural
terrain. This success is due, in part, to a property of fractals known as self-similarity, i.e., fractals
look the same, in some sense, across a range of scales or resolutions. The anomaly detection
technique described in this paper is based on identifying areas in terrain that lack self-similarity.
Preliminary results of applying the technique to Viking orbiter imagery suggest that certain objects
on the Martian surface currently under investigation may not be natural.
Key Words: Search for Extraterrestrial Intelligence (SETI), Mars, Viking orbiter, fractals, object
detection, image processing.
1. Introduction
The search for extra-terrestrial intelligence at radio frequencies began with project Ozma [1] and
has, as of yet, produced no evidence for their existence. The success of such an endeavor requires
that: 1) there are advanced technological civilizations in the galaxy that are either communicating
with one another or trying to communicate with us, 2) we are able to detect their signals, and 3) we
will be able to decipher the message when we receive it.
The Drake equation [2] has been a starting point for speculating on the number of technological
civilizations in the galaxy. At one extreme, some say that we are the only advanced civilization in
the galaxy since if there were others we would know about them. This is known as the Fermi
Paradox due to a question asked by Enrico Fermi in the 1940s. At the other extreme, others have
estimated upwards of billions of advanced technological civilizations in the galaxy. Evidence in
favor of either extreme is scarce. Estimates derived from the Drake equation depend on parameter
values which are obviously based on our very limited experience in contacting extra-terrestrials
(ETs). Freitas [3] argues that the Fermi Paradox cannot be proven. In any case, the consensus is
that the number of advanced civilizations is somewhere in between the two extremes and is
sufficient to warrant some kind of search.
†
Corrected and revised version of original paper published in the Journal of the British Interplanetary Society, Vol.
43, pp 209-216, 1990.
2
Many believe that the best channel for ET communications is at 1420 MHz, the frequency at which
atomic hydrogen radiates [4]. Even with this as a starting point, as the bandwidth of spectrum
analyzers decreases and the angular resolution of radiotelescopes increases, the number of
frequencies and directions that we must search will continue to increase. Most feel that the lack of
success is due to the very limited portion of the "search space" covered to date. Others believe that
we do not yet have the technology to detect and decipher ET radio transmissions or may be looking
for a signal that is simply not there [5].
An alternative strategy has been put forth more recently that involves searching for ET probes or
artifacts. Freitas [6] has proposed that we search for messenger probes in the vicinity of the earth.
Based on the argument that intelligent life has the natural tendency to expand into and occupy space,
and requires raw materials to support such expansion, e.g., for building space colonies,
Papagiannis [7] argues that if ETs are in our solar system, they may be in the asteroid belt. Using
arguments similar to those based on Drake equation, Foster [8] estimates frequencies of visitations
by ETs or their messenger probes and goes on to suggest the possibility that past encounters may
have left behind artifacts or indirect evidence (e.g., deranged planetary terrain).
This last idea is the point of departure for this paper, that there may be artifacts that have already
been imaged by probes such as the Viking Orbiter. For example, there has been considerable
interest recently in a collection of unusual surface features in the Cydonia region of Mars [9-12]. Is
there a way to objectively assess these and other surface features so that future missions such as the
Mars Observer can gather more data?
In order to carry out an objective search for artificial features, detection criteria must be established.
Current SETI strategies have been criticized on epistemological grounds, i.e., we are looking for
what we think is out there [5]. Since we do not really know what we are looking for, it is not
possible to define in a direct way what is "artificial". However, one can attempt to define what is
natural and proceed to measure the "naturalness" of unknown features. Artificial features are then
identified as those farthest from "natural" in some sense.
The contributions of this paper are: 1) to attempt to define objective criteria for identifying possible
artificial objects, 2) to describe a technique based on fractal geometry for detecting such objects, 3)
to present some preliminary results of applying the technique to Viking orbiter imagery that suggest
that certain unusual Martian surface features may not be natural, and 4) to outline a strategy for
continuing the search as part of on-going planetary exploration programs.
2. Characterizing Natural Landscapes Using Fractal Geometry
Mandelbrot [13] has defined a fractal as an object whose Hausdorff-Besicovitch dimension is
strictly greater than its topological dimension. For example, a fractal surface in 3-d space whose
topological dimension is 2 would have a fractal dimension between 2 and 3. Fractals have been
shown to be good models for diverse physical and biological phenomena such as the length of
coastlines, stream flow volume, terrain surface area, volume of blood vessels and many more [13].
A reason is that natural phenomena often possess the property of being self-similar at different
scales or resolutions. Self-similarity is defined by the relation
M(rX)= r
f ( D)
M(X) (1)
where M(X) represents any measurable property of the fractal (e.g., surface area),
X
represents a
scale of measurement of the metric property,
r
is a scaling factor between zero and one, and f (D)
is a simple function of the fractal dimension that depends on the metric property. This equation
states that the metric property computed at a reduced scale of measurement is equivalent to scaling
3
the metric property of the fractal at the original scale. This implies that the metric property obeys a
power law function of the scale of measurement
M( X) = KX
f (D )
(2)
where
K
is a constant.
For the purpose of modeling landscapes, a class of statistical fractals known as fractional Brownian
motion [14] has been used extensively. Fractional Brownian surfaces may be described by their
second order difference statistics,
Var B
H
(t)− B
H
(t + T) ∝ T
2 H
(3)
their surface area,
A(r) ∝r
2− D
(4)
or their power spectral density,
S( f )∝1/ f
(1−2 H )
(5)
where
D = 3−H
. Eq. 3 states that the variance of the difference between any two points on the
fractional Brownian surface a distance
T
apart increases at a constant power of the distance (since
0 < H <1
; Eq. 4 states that the surface area decreases as a constant power of the scale (since
2 < D < 3
); Eq. 5 states that the power spectral density (power per unit frequency) varies as a
power of the spatial frequency.
Mark and Aronson [15] analyzed 17 topographic data sets in the United States using variograms
which plot variance versus distance as defined in Eq. 3. They found that most of the data sets could
not adequately be characterized by a single fractal dimension, i.e., the logarithm of the variance was
not linearly related to the logarithm of the distance over all distances or scales. Rather, the behavior
of topography tended to be divided into scale ranges. Over small scales (< 0.6 km) many of the
surfaces could be modeled as fractional Brownian surfaces with D around 2.2 - 2.3. Over larger
scales, higher dimensions around 2.75 were noted while at still larger scales many surfaces
exhibited periodicities. A similar result was noted by Clarke [16] who suggests that a combination
of fractals, to model local behavior, and Fourier methods, to model longer-term variations, be used
to describe topography. Locally, at least, fractals seem to be good models for topography. Clarke
goes on to suggest that on planets like Mars where the types of processes that shape terrain over
larger scales on Earth are for the most part absent, fractals may be adequate by themselves. For
example, Woronow [17]
shows that fractals can be used for classifying certain kinds of large scale
Martian impact craters.
The above suggests a reason why the generation of realistic synthetic terrain using fractional
Brownian motion has been so successful as demonstrated by Voss [18] and by Fournier et al [19].
Fig. 1 is a 3-d plot of a synthetic terrain surface (D = 2.1) produced by Voss' method for
generating fractional Brownian surfaces using discrete Fourier synthesis. The method is based on
passing the discrete Fourier transform of a 2-d white Gaussian noise field through a linear filter
with transfer function proportional to 1 (k
2
+l
2
)
4 − D
where
k
and
l
are the spatial frequencies in the
x
and
y
directions, and inverse transforming the result. Fig. 2 shows several views of this
synthetic terrain surface. A Lambertian reflectance function was used to first create a shaded
rendition of the terrain surface (left). The sun is south (solar azimuth θ
0
=180°) at a zenith angle
4
φ
0
=45°. An oblique view was then generated by viewing the shaded rendition mapped onto the
elevation surface via an oblique parallel projection (right). The image was computed from a view at
an azimuth angle θ
1
=180° and a zenith angle φ
1
= 60°. (Rendering techniques are described
further in Appendix B.)
3. Detecting Manmade Objects
A major focus in artificial intelligence and machine vision has been on the problem of recognizing
instances of known objects in imagery (see, for example Ballard and Brown [20].) The problem of
recognizing an unknown object as an instance of a known object or class of known objects involves
comparing features of the unknown object that can be computed from the available data to those of
known objects and selecting the object with the best match.
The problem of recognizing unknown or unexpected objects, and the related problem of detecting
man-made objects embedded in natural terrain are fundamentally different. One possible approach
might be to try to determine the characteristics that are common to all man-made objects. For
example, one thinks of man-made objects as having flat surfaces, sharp boundaries, and different
brightness from the background. Unfortunately, due to lighting conditions, imaging geometry, and
obscuration, the strong linear features that one might expect are often not there.
An alternative approach based on modeling the background has recently been proposed by Stein
[21]. His approach does not rely on an explicit model for man-made objects (e.g., that they are
rectangular in shape or brighter than the background); rather, it is based on the observation that
man-made objects tend not to be self-similar in structure and so fractals should be poor models for
man-made objects. The method is based on estimating the fractal dimension of the image intensity
surface within a rectangular window that is about the size of the objects one would like to detect,
along with the error that results from assuming fractal or self-similar behavior. The technique used
to estimate the fractal dimension involves computing the surface area A(r) of the image intensity
surface as a function of scale
r
(see Appendix A). The metric properties of self-similar sets scale
according to a power law as noted in Eq. 4. The fractal dimension of the image intensity surface is
estimated by performing a linear regression of log A(r) vs. logr . A measure of the degree to which
the image intensity surface lacks self-similarity
ε
is estimated by summing the residuals of the linear
regression over scale, i.e.,
ε =
1
R
log A(r)− (2 − D)log r
[ ]
2
r=1
R
∑
(6)
Surfaces that are not self-similar will not follow a power law relationship, hence the residuals in Eq.
6 will be large and so
ε
will be large.
An entire image is processed by repeating the above process on a pixel-by-pixel basis within a
"sliding window". Two images are produced: one is the local fractal dimension D(x,y), i.e., the
fractal dimension of the portion of the image intensity surface within the rectangular window
centered at (x,y); the other is the local fractal model-fit image ε(x,y). Fig. 3 shows an image of
military vehicles embedded in natural terrain (a) along with the computed fractal dimension image
(b) and fractal model fit error image (c) for
R =10
scales and a 21 by 21 window.
For natural textures on earth, typical ranges can be used for thresholding the fractal dimension
image in order to generate detections. At the upper end, terrestrial observations by Mark and
Aronson [15] indicate that fractal dimensions over short scales are less than about 2.5. At the low
5
end, it has been observed that discontinuities in the image intensity surface (e.g., due to shadows,
object boundaries, and obscuration) produce fractal dimension estimates that are below the
topological dimension. Thus, for detecting man-made objects, regions whose fractal dimension is
not greater than 2.0 and less than 2.5 are considered anomalous.
The fractal model fit is another independent measure of anomalous behavior. Unfortunately, since
the fractal model fit error is a relative measure, absolute thresholds do not exist. If the relative
frequency of occurrence of man-made objects is small however, the model fit image can be
thresholded at a given false alarm rate (the probability that a man-made object may be detected when
one is not actually present). The detection result (d) in Fig. 3 indicates possible man-made objects
where the fractal dimension is not between 2.0 and 2.5, and where the fractal model fit error is
greater than the 90th percentile. Three of four vehicles have been detected with two "false alarms".
Lack of data like that compiled by Mark and Aronson limits the use of the fractal dimension for
anomaly detection on Mars. For the imagery processed in the next section, the fractal dimension is
used only to remove some object and shadow boundary effects by eliminating regions whose fractal
dimension is less than 2.0. The unthresholded fractal model fit error image is used by itself in the
remaining areas to indicate the degree to which the data lacks the self-similar behavior of terrain on a
local basis.
4. Preliminary Mars/Viking Orbiter Results
Fig. 4 is a mosaic of parts of three Viking frames: 35A72, 35A73, and 35A74. A 1280 by 1024
pixel area is shown. This is the area in Cydonia that is currently under investigation by a number of
individuals [9-12]. The resolution is about 50 meters per pixel and the total area shown is
approximately 3000 sq. km.
The result obtained by applying the anomaly detection technique to the imagery over this area is
shown in Fig. 5. The image was produced by combining the fractal dimension and model fit images
as described in the previous section and shows the top four detections.
R =10
scales and a 21 by
21 pixel analysis window were used. The analysis window thus covers an area about 1 sq. km and
is near the upper scale limit for self-similarity based on Mark and Aronson's results for terrestrial
landscapes. The "face" [9] was found to have the largest fractal model fit error which implies that it
is the least natural object in this area. Close-ups of the face are shown in Fig. 6. A number of
objects in the "city" [11] also have large fractal model fit error. Close ups of one of those objects,
the "fortress" [12] are shown in Fig. 7.
†
As was seen earlier in the example in Fig. 3 the object detection technique may indicate the presence
of a man-made object when there is no such object (false alarms) and may fail to detect a man-made
object when one is there (missed detections). In Figs. 4 and 5, several features which appear to be
natural seem to exhibit a certain degree of non-fractal behavior. On the other hand, several other
unusual objects (e.g., the "cliff" [11] and the "D&M pyramid" [9]) do not appear to be anomalous
by this technique.
As for the false alarms, it is certainly possible for nature to conspire to produce a structure that is
not self-similar over short scales. In fact, this has been said of the face and the other nearby objects
in Cydonia. It is also noted that there are many features on Mars as well as on the earth that exhibit a
periodic structure over short scales and are therefore not locally fractal.
†
Descriptive terms such as the "city" and "fortress" have been used by various investigators for the sake of
convenience and consistency. They do not imply an endorsement of particular interpretations of these objects.
6
In regard to the missed detections, it was observed earlier that images of fractal surfaces are also
fractal. The converse is not necessary true however. If a man-made object is illuminated so that its
3-d structure does not induce significant shading and shadowing effects, structural information will
be lost in the image formation process. The image of the object will look smoother than it really is
and will thus appear less anomalous. In other words, the ability to discriminate anomalous objects
from the background may be reduced at certain sun and viewing angles. This is illustrated in Fig. 8
which shows two images of the face (Viking frames 35A72 and 70A13) and surrounding terrain. In
35A72 where the sun is more than 15° lower than in 70A13, the ability to discriminate the face from
the background is much greater.
To verify that the technique does not detect a plethora of natural objects, it was applied to the full
Viking frame containing the above objects (35A72) as well as three other nearby frames: 35A70,
35A71, and 35A73. The strength of the strongest detection in 35A72, the "face", was 1.75, 1.88,
and 4.31 times greater than the strongest detections in these three other frames, respectively. It is
noted that the size of these objects (1-2 km) precludes a comparative analysis of terrestrial analogs
(e.g., the Great Pyramid) using imagery such as Landsat or SPOT. It is also worth noting that it
would be difficult, if not impossible to obtain images of facial profiles such as those carved on Mt.
Rushmore since they were not meant to be viewed from above.
Finally, a result from Viking frame 70A10 is presented that suggests that there may be other objects
on Mars worth investigating. Fig. 9 shows a 512 by 512 pixel region from 70A10. It is over 100
km from the Cydonia region in Fig. 4. Also shown in Fig. 9 is the fractal model fit error image
obtained with the same parameters used earlier. Ignoring the bright areas caused by periodic noise
in the data, a strong anomaly is present over an unusual rectangular structure having a circular
depression and a tapered "access ramp". Nearby are a pyramidal object and sharp angular features
etched into the surrounding terrain.
5. Summary
This paper has presented criteria for identifying objects of possible artificial origin on planetary
surfaces, described a technique based on fractal geometry for detecting such objects in planetary
imagery, and presented some preliminary results of applying the technique to Viking orbiter
imagery. The results presented in this paper suggest there are a number of surface features on Mars
that may not be natural. Although it is beyond the scope of this paper to speculate on their origin,
the results warrant further investigation. In particular, higher resolution imagery must be collected
by the planned U.S. Mars Observer. More importantly, current thinking in the SETI community
[22-24] needs to be broadened to include a systematic search for ET artifacts in our solar system.
The criteria and techniques discussed in this paper are a starting point for beginning an objective and
systematic search for ET artifacts. It is well within the state-of-the-art to process the current Viking
orbiter imagery archive (about 60,000 images). Estimated costs of the order of ten million dollars to
process existing planetary imagery and create an automated system for screening future imagery is
many orders of magnitude less than some of the more ambitious radio search projects that have been
proposed.
To paraphrase Van Leeuwenhoek and Denton, perhaps they are not only "dancing in our lenses" but
were, or are, at our very doorstep.
7
Acknowledgments
The authors wish to thank Keith Hartt for the 3-d surface reconstructions used in the paper.
Appendix A - Object Detection Algorithm
The object detection technique [21] is briefly presented here so that others can independently verify
the results presented in the paper. Given an imagex(i, j) a series of grayscale erosions and dilations
are computed
t
r+1
(i, j) = max t
r
(i, j) +1, t
r
(
′
i ,
′
j )
{ }
b
r+1
(i, j) = min b
r
(i,j)−1, b
r
( ′ i , ′ j )
{ }
(A1)
where b
0
(i, j) = t
0
(i,j) = x(i,j), ′ i ≤1 and ′ j ≤1. These are used to compute the volume of a
covering of the image intensity surface as a function of the scale parameter
v
r
(i, j) = t
r
(i, j) −b
r
(i, j) (A2)
To obtain an estimate of the surface area for an
m x n
rectangular patch centered at (i, j) the
covering volume at each pixel over a window of the same size is summed and divided by twice the
scale parameter:
A
r
(i, j) =
1
2r
v
r
′ i < m /2, ′ j <n /2
∑
(i− ′ i , j − ′ j ) (A3)
The fractal model is formally a linear regression model relating the logarithm of the surface area
estimates A
r
(i, j) to the logarithm of the scale parameter
r
log A
r
(i, j) =[2− D(i, j)]logr +E
r
(i, j) (A4)
where E
r
(i, j) is the residual of the linear fit at scale
r
(i.e. the difference between the actual value
of the logarithm of the surface area and the value predicted by the linear model). D(i, j) is the fractal
dimension and is related to the slope of the linear regression of log A
r
(i, j) onto logr . The fractal
model fit error is the average of the squared residuals
ε(i, j) =
1
R
E
r
2
r
∑
(i,j) (A5)
where
R
is the number of scales.
8
Appendix B - Image Formation Model
The anomaly detection technique tacitly assumes that the self-similar structure of terrain is preserved
through the imaging process. Pentland [22] and Kube and Pentland [23] have analyzed the
properties of images of fractal surfaces and show that under certain conditions the image of a fractal
Brownian surface is also fractal Brownian, and hence, self-similar.
The simplified model of the image formation process shown in Fig. B1 was used to verify
experimentally that images of fractal surfaces are also fractal. The model ignores atmospheric effects
and sensor degradation. It assumes a single point source (the sun), a Lambertian surface reflectance
function, constant albedo, and an imaging sensor that is far enough away for a parallel projection to
hold. These assumptions appear to be reasonable over small areas on Mars under limited
illumination and viewing conditions [12]. A computational model of the image formation process
can be divided into two parts: 1) computing a shaded rendition of the surface, and, 2) projecting the
shaded surface onto the focal plane of the imaging sensor.
The shaded rendition i(x,y) is related to the elevation surface z(x,y) by the reflectance map R(p,q)
where the gradients p = ∂z / ∂x and q = ∂z / ∂y are the partial derivatives of the elevation in the x
and y directions. The reflectance map depends on the reflectance properties of the surface and on
the position of the sun. The use of the reflectance map in computing shaded renditions of terrain is
discussed by Horn [24]. The location of the sun in gradient space is (p
0
,q
0
) where
p
0
= tanφ
0
cosθ
0
, q
0
= tanφ
0
sinθ
0
and θ
0
and φ
0
are the azimuth and zenith angles of the sun.
The reflectance map for a Lambertian reflectance function is
R(p,q) =
(pp
0
+qq
0
+1)
(p
2
+q
2
+1)
1/2
(p
0
2
+q
0
2
+1)
1/2
(B1)
Areas that face away from the sun, i.e., where the numerator is negative are assigned zero
reflectance.
It is assumed that the imaging sensor is far enough away for a parallel projection to hold [28]. For
oblique viewing along the x axis, the point (x,y) is mapped to the point ( ′ x , ′ y ) where
′ x = xcosφ
1
+ z(x,y)sinφ
1
′
y = y
(B2)
and φ
1
is the zenith angle of the sensor. That is, the surface is foreshortened along the viewing
direction (in this case along the x direction). For viewing from another azimuth, the surface is
simply rotated so that the line-of-sight is along the x axis. Since for φ
1
> 0, obscuration can occur,
the hidden surfaces must first be removed. Then, the shaded rendition i(x,y) is mapped onto the
image plane via Eq. B2 and intermediate pixel locations are interpolated to obtain the projection
i( ′ x , ′ y ).
Fig. B2 summarizes the fractal properties of images of the fractional Brownian surface shown in
Fig. 1. The stability of the fractal dimension and model fit error over a wide range of illumination
and viewing conditions implies that images of natural terrain are also fractal. It does not imply
however that images containing man-made objects will not also be self-similar. For example, at low
solar zenith angles where shading effects are reduced it may be difficult to discriminate man-made
objects from the natural background. This phenomenon was noted in Section 4.
9
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28. J. D. Foley and A. Van Dam, Fundamentals of Interactive Computer Graphics, Addison-
Wesley, Reading, MA, 1983.
11
Fig. 1 3-d plot of synthetic terrain produced by Voss' method for generating fractional Brownian
surfaces (fractal dimension = 2.1)
Fig. 2 Orthographic view of shaded rendition of synthetic terrain (left) and an oblique parallel
projection (right).
12
Fig. 3 Detection of man-made objects. (a) Original image (upper left) has four military vehicles
embedded in a complex natural background. (b) Fractal dimension (upper right) and (c) fractal
model fit (lower left) images indicate that man-made objects are different than the background. (d)
Areas whose fractal dimension is outside of range 2 < D < 2.5 and whose model fit is greater than
90%-ile are shown (lower right).
13
Fig. 4 Mosaic of Viking orbiter frames 35A72, 73, and 74. Landmarks shown are the "face", the
"city", the "D&M pyramid", and the "cliff".
14
Fig. 5 Detection results show four largest anomalies over the "face" and within the "city".
15
Fig. 6 Close ups of the "face": down-looking (left) and an oblique view near ground level on the
sunlit side (right).
Fig. 7 Close ups of the "fortress" within the "city": down-looking (left) and an oblique view
looking into the inner space (right).
16
Fig. 8 Effect of variable illumination on the detection of anomalous features. Part of 35A72 and
fractal model fit error image (top left and right). Part of 70A13 and fractal model fit error image
(bottom left and right). Note the difference in the number of false alarms between the right top and
bottom images. Square "rings" are caused by noise in imagery.
17
Fig. 9 Analysis of part of 70A10 frame (top). Fractal model fit error image indicates strong anomaly
over rectangular feature with central depression and v-shaped opening. Note pyramidal object and
linear features above and to the right.
z(x,y)
(φ ,θ )
1 1
(φ, θ)
I
0
x'
x
y
y'
(φ ,θ )
0 0
i(x,y)
i'(x,y)
18
Fig. B1 Image formation model. Shaded rendition i(x,y) is a function of the reflectance properties
of the surface and the position of the light source. An oblique view i(x',y') is generated by a parallel
projection.
0 30 60 90
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
0.0e+0
1.0e-3
2.0e-3
3.0e-3
4.0e-3
5.0e-3
6.0e-3
7.0e-3
8.0e-3
9.0e-3
1.0e-2
Solar Zenith Angle (degrees)
0 15 30 45 60 75
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
0.0e+0
1.0e-3
2.0e-3
3.0e-3
4.0e-3
5.0e-3
6.0e-3
7.0e-3
8.0e-3
9.0e-3
1.0e-2
View Zenith Angle (degrees)
Fig. B2 Stability of fractal dimension and model fit as a function of sun zenith (top) and view zenith
(bottom) angle.
