Distribution of polygon characteristic scale in Martian patterned ground terrain in the northern hemisphere using the Fourier transform

Authors

  • T. Orloff,

    Corresponding author
    1. Department of Earth and Planetary Sciences, University of California, Santa Cruz, California, USA
    • Corresponding author: T. Orloff, Department of Earth and Planetary Sciences, University of California, 1156 High St., Santa Cruz, CA 95064, USA. (travis.orloff@gmail.com)

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  • M. Kreslavsky,

    1. Department of Earth and Planetary Sciences, University of California, Santa Cruz, California, USA
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  • E. Asphaug

    1. Department of Earth and Planetary Sciences, University of California, Santa Cruz, California, USA
    2. School of Earth and Space Exploration, Arizona State University, Tempe, Arizona, USA
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Abstract

[1] Permafrost patterned ground forms on the surface of nearly all landscapes between 50°N and 70°N on Mars. This landform appears in satellite imagery as an interconnected network of polygonal shapes. Previous studies used geomorphologic observations to characterize patterned ground terrains on Mars. These classification systems prove useful but suffer from somewhat subjective methods. We find objective analysis of the polygons comprising patterned ground terrains on the surface of Mars feasible using High Resolution Imaging Science Experiment imagery acquired from orbit. We perform a two-dimensional Fourier transform on 124 images of patterned surfaces to characterize the spatial scale pertinent to scenes and analyze the distribution of these properties on the surface of Mars between 50°N and 70°N. We find two distinct sets of polygons: large polygons which form below 60°N and small polygons primarily above 60°N with exception for sites within Acidalia Planitia. Our findings show similar trends to those found in previous studies but also fundamental differences in the populations of polygons above and below 60°N. The polygons within 60°N–70°N share roughly the same polygonal scale, implying that seasonal temperature change is not the only factor forcing polygon development as previous models found. In some cases, our method indicates the presence of multiple scales of polygons, although it cannot quantify the larger scale. This method in conjunction with observational analysis improves our ability to characterize surfaces and examine patterned ground terrains on Mars.

1 Introduction

[2] The surface of Mars displays a wide variety of permafrost patterned ground across a wide range of latitudes covering tens of millions of square kilometers of the surface [Costard and Kargel, 1995; Mangold, 2005; Balme and Gallagher, 2008; Levy et al., 2009; Orloff et al., 2011]. The morphologic variations suggest that high-latitude (>50°) permafrost patterned ground forms due to a variety of environmental drivers dependent on local climate and setting. Formation of the patterns evidently involves some combination of physical processes such as thermal contraction of ground ice or ice-rich ground, frost heave, and ice sublimation [Mellon, 1997; Seibert and Kargel, 2001; Marchant and Head, 2007; Morgenstern et al., 2007; Balme et al., 2008; Mellon et al., 2008; Lefort et al., 2009].

[3] Individual polygons in the different patterned ground terrains have distinct visual appearances from satellite imagery. Two such examples are shown in Figure 1. Figure 1a shows classic high-latitude (>65°N) small polygons, and Figure 1b shows significantly larger polygons found at lower latitude (50°N–55°N). Note the differences in the polygonal edges and spacing of these polygonal networks. Additional examples are in Figures 4 and 10. These examples do not represent the whole range of polygon morphologies but illustrate that polygons form at different length scales in addition to different visual appearances.

Figure 1.

Images of two types of patterned ground. (a) Smaller polygons at higher latitude, 69.2°N, from HiRISE image PSP_001418_2495. (b) Larger polygons at lower latitude, from ESP_018294_2365. Note the differences in the polygonal edges and spacing of these two types of polygons. These two examples do not represent the whole range of polygon morphologies (see, e.g., Figures 4 and 10). They illustrate that in addition to distinct visual appearances, polygons also form at different scales. Illumination of both images is from the right and images are ~125 m across. Image: NASA/JPL/University of Arizona.

[4] Patterned ground has been classified, described, and mapped using primarily subjective methods based on geomorphological observations (polygon size, shape, slopes, and presence of boulders) in satellite imagery from the Mars Orbiter Camera (MOC) [Malin and Edgett, 2001; Mangold et al., 2004; Mangold, 2005; Kostama et al., 2006] and the High Resolution Imaging Science Experiment (HiRISE) [McEwen et al., 2007; Levy et al., 2009; Korteniemi and Kreslavsky, 2013]. These studies showed significant latitudinal trends in the types of patterned ground further supporting the idea that climate directly influences polygon formation.

[5] Although these qualitative photogeological studies provided useful and important observations, they did not provide an objective method to consistently analyze these terrains. Several studies have shown quantitative ways to characterize patterned ground terrains. For example, Yoshikawa [2003] characterized a specific type of Martian polygons in Utopia Planitia with MOC narrow angle images using measurements of polygon area, perimeter, and crack orientations. He then calculated the mean nearest neighbor distance between the intersections of polygon-forming troughs and divided it by the mean expected distance in a random distribution. This gave a statistic that quantifies the degree to which an observed distribution departs from a random distribution [Clark and Evans, 1954]. This study found similar values of this statistic for Martian and terrestrial polygons, but terrestrial polygons varied to a higher degree.

[6] Dutilleul et al. [2009] and Haltigin et al. [2010] also calculated the nearest neighbor distance between the intersections of troughs outlining polygons, using a method called spatial point pattern analysis. However, they argued that the mean distance is insufficient to characterize the trough network. Instead, they considered the whole distribution of nearest neighbor distance values to classify the polygonal network as aggregated, random, or clustered. Haltigin et al. [2011, 2012] also analyze the trough networks based on major and minor trough junctions to determine the evolution of polygonal networks in time.

[7] Ulrich et al. [2011] used the methods of multivariate analysis to describe polygon properties. First, all polygons were manually digitized based on images, digital elevation models, and slope models. Then for each polygon, topographical properties and geomorphometric parameters were extracted. After collecting these data, Ulrich et al. [2011] used principal component analysis to assess the relationships between these parameters and redundancy analyses including environmental parameters as explanatory variables to identify factors that could significantly explain variations in polygon shape and dimension.

[8] Each of these quantitative methods requires human input and manual outlining of the polygons; thus, these techniques could only be applied to a limited geographic area. Pina et al. [2006, 2008] applied an automatic image processing method known as “watershed segmentation” [Beucher and Meyer, 1993; Bleau and Leon, 2000] to outline polygons in an automated way using both MOC and HiRISE images. Then they used the polygon outlines in vector data formats to measure polygon sizes and count the number of polygon neighbors. They use several statistical measures to characterize their trough network. Their automated segmentation procedure depends strongly on manual tuning filters for each individual image, which again introduces a subjective element into processing.

[9] Here we develop a method that quantitatively characterizes the polygonal pattern in a given image, in a manner that is free of any subjective input. We use a two-dimensional Fourier transform (2-D FT) to objectively analyze patterned ground terrains across Mars in northern high latitudes and determine a spatial characteristic which allows us to quantify patterned ground in a geologically meaningful way. Several studies have used the Fourier transform to analyze periodic structures in topographic and bathymetric data [Rayner, 1972; Hanley, 1977; Stromberg and Farr, 1986; Ricard et al., 1987; Mulla, 1988; Gallant, 1997; Perron et al., 2008]. We intend to determine if a similar objective technique finds regional or latitudinal spacing variations that are in agreement with those found in previous geomorphological studies. Such variations would indicate regional- and global-scale differences in patterned ground formation and hence signatures of variations in the driving geology and climate that can be “read” across wide swaths of Mars.

2 Method

[10] The goal of our study is to characterize permafrost patterned ground terrain using an objective measure and to analyze the distribution of this objective measure on the surface of Mars between 50° and 70° latitudes. We perform a Fourier transform on HiRISE images between 50°N and 70°N on Mars and determine the wavelength most representative of the terrain, and in this paper, use that wavelength as our objective measure.

[11] We start with a catalog of HiRISE images (Table 1) that we aim to analyze. We choose the same images as those used for a geomorphological study in Korteniemi and Krelavsky [2013]. The images chosen do not represent a random population, nor do they form a uniform distribution. Instead, images were chosen such that no large craters or other large primary features would influence our results. Only full-resolution HiRISE images (map projected with 25 cm sampling) were chosen. The distribution of images shows geographical biases, such as the fact that there are many more images acquired near the Phoenix landing site. Despite these biases, following Korteniemi and Kreslavsky [2013], we believe that the chosen images are reasonably representative of all patterned ground terrains within 50°N–70°N.

Table 1. Measured Characteristic Scale
Image NumberLatitude (°N)Longitude (°E)Characteristic Scale λ (m)
PSP_001404_249069.0254.05.0
PSP_001413_249569.310.85.4
PSP_001413_249569.310.85.4
PSP_001415_247066.7317.35.8
PSP_001415_247066.7317.35.8
PSP_001418_249569.2234.24.8
PSP_001418_249569.2234.24.8
PSP_001421_247066.8153.65.1
PSP_001421_247066.8153.65.1
PSP_001422_246566.0125.05.4
PSP_001422_246566.0125.05.4
PSP_001430_247067.1266.06.0
PSP_001430_247067.1266.06.0
PSP_001448_247067.0136.05.8
PSP_001448_247067.0136.05.8
PSP_001464_246065.660.54.5
PSP_001464_246065.660.54.5
PSP_001467_249068.7337.85.5
PSP_001467_249068.7337.85.5
PSP_001473_248067.7174.55.7
PSP_001473_248067.7174.55.7
PSP_001475_246566.2122.45.0
PSP_001475_246566.2122.45.0
PSP_001477_247067.067.15.3
PSP_001477_247067.067.15.3
PSP_001481_241060.7318.55.1
PSP_001481_241060.7318.55.1
PSP_001482_249068.8288.44.8
PSP_001482_249068.8288.44.8
PSP_001483_246566.5262.36.2
PSP_001483_246566.5262.36.2
PSP_001484_245565.5235.15.0
PSP_001484_245565.5235.15.0
PSP_001491_246566.443.85.5
PSP_001497_248068.0241.45.5
PSP_001505_248568.521.26.0
PSP_001520_247066.6332.94.7
PSP_001523_246566.2251.65.3
PSP_001525_247567.3196.15.0
PSP_001531_247066.832.64.7
PSP_001535_249068.8282.53.9
PSP_001556_246065.672.55.3
PSP_001559_248568.2348.45.8
PSP_001565_249068.7184.55.1
PSP_001575_246566.4272.44.8
PSP_001576_246065.7245.73.9
PSP_001588_246566.4277.75.6
PSP_001591_247567.3195.74.9
PSP_001611_248568.39.85.2
PSP_001614_249068.9297.54.7
PSP_001615_247066.5255.26.2
PSP_001629_245565.5239.74.1
PSP_001635_246566.474.05.4
PSP_001655_246065.6251.03.9
PSP_001663_249068.631.15.2
PSP_001668_246065.8253.54.5
PSP_001669_246065.7231.45.1
PSP_001681_245565.3256.44.7
PSP_001682_246566.2233.64.9
PSP_001695_245565.1240.55.7
PSP_001708_245565.2246.64.9
PSP_001721_246065.5248.04.8
PSP_001734_247066.6253.64.1
PSP_001738_234554.0149.55.8
PSP_001741_239559.466.35.3
PSP_001742_237056.739.75.8
PSP_001744_235555.2345.79.5
PSP_001748_246065.6236.04.3
PSP_001753_246065.797.14.0
PSP_001761_249569.1237.55.0
PSP_001785_233052.5307.76.1
PSP_001801_245565.2234.56.6
PSP_001803_247567.1172.55.0
PSP_001813_245565.4254.94.9
PSP_001814_245565.1237.05.0
PSP_001880_248568.2232.04.8
PSP_001946_248568.3232.35.3
PSP_002183_249569.1236.34.9
PSP_002317_242061.8181.33.8
PSP_002396_234554.4186.67.3
PSP_002455_236055.414.06.5
PSP_002620_234554.0186.77.5
PSP_007803_237557.3162.15.4
PSP_008123_231050.865.417.0
PSP_008219_247066.6320.16.9
PSP_008266_244564.5117.36.3
PSP_008370_235555.4159.28.9
PSP_008372_245565.3102.35.3
PSP_008778_232552.0178.08.6
PSP_009086_236055.848.216.7
PSP_009112_234554.258.715.5
PSP_009604_231050.9306.312.3
PSP_009664_230550.1108.313.6
PSP_009713_241561.4208.06.0
PSP_010141_236055.643.14.3
PSP_010164_237056.8136.73.9
PSP_010394_240560.0336.76.0
PSP_010434_250069.6320.45.5
PSP_010625_236055.6150.65.8
ESP_011512_233052.654.65.7
ESP_016956_231050.6109.215.6
ESP_016957_242562.177.67.4
ESP_017112_244063.8166.96.5
ESP_017131_248568.06.77.0
ESP_017146_238558.2320.26.9
ESP_017166_246566.2131.87.0
ESP_017494_235054.7180.98.9
ESP_017543_233052.5282.39.4
ESP_017632_247567.19.26.6
ESP_018117_234554.090.65.2
ESP_018125_244564.3231.55.7
ESP_018126_244564.0202.47.4
ESP_018139_231550.9210.97.7
ESP_018157_236555.978.28.6
ESP_018158_243563.149.07.0
ESP_018173_238558.30.34.9
ESP_018211_237056.943.88.5
ESP_018245_242562.3194.17.4
ESP_018259_238558.1171.97.6
ESP_018264_237557.237.614.3
ESP_018294_236556.1298.214.3
ESP_018573_241561.0238.75.4
ESP_018646_244063.944.95.7
ESP_018746_235054.8196.95.7
ESP_019068_233553.246.39.3

[12] We take each chosen HiRISE image and identify a subimage with polygons representative of the entire image via visual inspection. We choose locations with as few boulders as possible to minimize their effects on further processing. We strive to avoid obvious albedo variations such as those which might be caused by other morphological features. Patterned ground terrain on Mars largely occurs on flat surfaces, in which case changes in illumination are due only to the polygons themselves.

[13] The illumination conditions (direction of the Sun) directly influence the patterns of bright and dark within an image: Surface facets tilted toward the Sun are systematically brighter than those tilted in the opposite direction. Because of this, the 2-D power spectra resulting from the 2-D Fourier transform are not isotropic: Illumination direction is seen as a particular direction in the spatial frequency domain. To analyze chosen subimages consistently, we rotate them so that the Sun direction is the same in all images (directly from the right) and then cut out a 512 × 512 pixel region (128 m × 128 m) and a 1024 × 1024 pixel region (256 m × 256 m) from the center of the original 1460 × 1460 subimage. This gives us a set of images of the proper size for an efficient 2-D Fourier transform and with uniform lighting conditions.

[14] With our images now ready for analysis, we run them through code presented by Taylor Perron (available at www.opentopography.org/index.php/resources/short_courses/lidar2_2010/) which we modified to work on brightness values in grayscale images. In our case, we look at patterns of brightness within an image rather than patterns of elevation in Lidar topographic data. Like Perron et al. [2008], we use a Hann window to remove edge effects; then we perform a 2-D Fourier transform on both the 512 × 512 pixel and 1024 × 1024 pixel subimages providing us with a spatial power spectrum (Figure 2), which we then examine.

Figure 2.

Example of a two-dimensional spatial power spectrum. Colors correspond to the power associated with a given spatial frequency. The illumination of the image from which we take the Fourier transform is from the right.

[15] In Figure 2, the color corresponds to the spectral power, P(kx, ky) at a given vector of spatial frequency (wave number) with Cartesian components kx, ky. It is clearly seen that the spectrum is not spatially isotropic: At some middle frequencies, the power of horizontally oriented components (small ky) is noticeably higher than the power of vertically oriented components (small kx) at the same distance from the center. This anisotropy is caused by directionally illuminated topography. The characteristic spatial frequency of the most pronounced anisotropy corresponds to the characteristic spatial scale of the dominant topographic features, the polygons. We extract this spatial scale from the spectrum with the following procedure.

[16] Due to symmetry in the spectra, we only look at the positive x frequencies. Then we artificially set P to zero for the pixels adjacent to the zero frequency because the spectral power at these frequencies is dominated by the Hann window and larger-scale (> ~ 100 m) albedo patterns. We also disregard frequencies with either a kx or ky component greater than 1 m−1, because the wavelengths corresponding to these frequencies (<1 m) are at, close to, or below the resolution of the original images; in result, P at these frequencies is strongly affected by details of interpolation algorithms used in map-projecting HiRISE images and subsequent rotation and bear little information about the real surface.

[17] We use reciprocal power-weighted-mean x component of the spatial frequency as a measure of the spatial scale (λ) most representative of the image:

display math(1)

[18] We chose this measure after a series of trials. The use of x component was dictated by the fact that in this direction, the topography-controlled variations of brightness are greatest (x direction is the direction to the Sun), and our goal is topographic rather than albedo pattern. We assessed the choice of the characteristic spatial scale measure performing four tests.

[19] For the first test, we magnify the image by a factor of two and expect to double the original characteristic spatial scale. We performed this analysis on every image in our survey. First, we magnified the image and then cropped it to the same size as the original image. Not all polygons in the original image are shown in the magnified image after cropping to the original size. Then we compare the characteristic spatial scale from the original image to the magnified image (Figure 3). In over 95% of comparisons of images north of 60°N, we find characteristic spatial scales within our expected range λmo = 1.6–2.4, where the subscript “o” is for the original image and the subscript “m” is for the magnified image. We consider 20% deviation from the expected λmo = 2 acceptable due to natural variability of the polygon size. When we include those images south of 60°N, our results fall into the appropriate range in 80% of cases. The outliers with λmo < 1.6 are the cases where the characteristic scale is large and not sharply defined: In these cases, after magnification, our 512 × 512 pixel window becomes comparable to the largest polygons and we lose power at long wavelengths. We find longer characteristic spatial scales at the lower latitudes often (see section 3 below), which explain the latitudinal differences in λmo.

Figure 3.

Comparison of characteristic spatial scales λ derived from original (λo) and twofold-magnified (λm) images. The shaded region shows the range of expected values.

[20] For the second test, we compare the characteristic spatial scale of similar polygons at different locations within the same HiRISE image and expect a characteristic scale within 20% of the original subimage. There are many boulders in our surveyed region; for the main survey, we chose portions of the HiRISE image with as few boulders as possible, but some boulders still appear in many of the subimages. In this test, we simultaneously determine how boulders influence the measured characteristic spatial scale. For this test, we use three additional subimages from four HiRISE images in our survey (Figure 4) and analyze them in the same fashion as our survey images. We specifically choose HiRISE images to look at the presence of boulders on small polygons, boulders on large polygons, rubble piles on small polygons, and additional boulder-free areas on small polygons. We show the results of this analysis in Figure 5. For small polygons, the presence or lack of boulders does not significantly alter the measured characteristic spatial scale. However, for larger-scale polygons, the presence of boulders greatly reduces the measured characteristic spatial scale. This means that more care is necessary for picking images displaying the larger-scale polygons to be sure boulders do not significantly influence the analysis. The characteristic spatial scale for each additional subimage fell within the 20% variation we predicted.

Figure 4.

Four subimages from each of the four HiRISE images. (a–d) Four subimages from PSP_001482_2490 used to show the influence of rubble piles on small polygons. Figure 4a is the image used in the survey. (e–h) Four subimages from PSP_001491_2465 used to show the influence of small boulders on small polygons. Figure 4e is the image used in the survey. (i–l) Four subimages from PSP_001556_2460 used to show other locations without boulders. Figure 4i is the image used in the survey. (m–p) Four subimages from ESP_018264_2375 used to show the influence of boulders on large polygons. Figure 4m is the image used in the survey.

Figure 5.

Results of multiple locations within a single image. We look at the cases where boulders appear on small polygons (circles), rubble piles form on small polygons (triangles), there are additional cases of no boulders (diamonds), and boulders appear on large polygons (cross symbols). For the small polygons, the additional images show good agreement with the survey image and the boulders do not seem to affect the characteristic spatial scale. For large polygons, the presence of boulders significantly lowers the characteristic spatial scale compared to a relatively boulder-free terrain.

[21] In the third test, we develop an artificial polygonal terrain (Figure 6) and analyze this image to see if the inferred characteristic spatial scale indeed represents the characteristic size of the polygons we developed. To develop this artificial terrain, we start with a hexagonal lattice of polygons with a 4 m diameter and give each polygon the topography of a bell-shaped (Gaussian) dome. Then we make the lattice irregular by randomly shifting the centers of each polygon in the top row up to one quarter the polygon diameter in both horizontal and vertical directions. For each subsequent row, we randomly shift the polygon center in the same way with respect to previous row; we introduced some additional “repulsion” between polygon centers in this random walk to prevent occasional reduction of polygon size to naught. On one hand, this procedure suppresses the long-range order of the lattice and prevents sharp lines in the power spectra; on the other hand, it preserves the characteristic scale of the original lattice. Then we arbitrarily rotate the obtained topography and artificially illuminate it directly from the right. Finally, we repeatedly add white noise and blur the image three times to mimic small-scale image texture. When we analyze this terrain using our method, we find a characteristic spatial scale of 4.3 m, which is less than 10% greater than the diameter of the polygons we generated.

Figure 6.

Artificially generated image of polygonal pattern with a characteristic polygon size of 4 m.

[22] Finally, we compared the characteristic spatial scale to previous studies which measured average polygon diameter [Levy et al., 2009; Korteniemi and Kreslavsky, 2013] and found a similar range of values. The measure λ defined in ((1)) was successful at these tests within less than 20% variation in all our test cases.

3 Results

[23] The characteristic spatial scales λ we report do not directly correspond to polygon size or spacing, but we interpret them as a proxy for these characteristics. The range of characteristic spatial scale we find is similar to the range in mean polygon diameter found in both Levy et al. [2009] and Korteniemi and Kreslavsky [2013], and we find them in similar locations. We also compare the characteristic spatial scale for different images with the polygons and find the diameter along the long axis of the majority of polygons are within 20% of the characteristic spatial scale. We do not expect every polygon diameter to match the characteristic spatial scale because there is variation between individual polygons. Figures 7 and 8 show the results of our analyses on 124 512 × 512 pixel images of patterned ground in the northern high latitudes of Mars.

Figure 7.

Map of the 50°N–70°N zone of Mars showing locations of images and characteristic spatial scale associated with each 512 × 512 pixel image. Colors correspond to characteristic spatial scale. Note primarily blue colors corresponding to low characteristic spatial scales above 60°N and a variety of colors below. Background shows a shaded relief map of the region derived from Mars Orbiter Laser Altimeter data.

Figure 8.

Characteristic spatial scale as a function of latitude measured with 512 × 512 pixel subimages. Each point corresponds to an image from our study. Note that no large-scale polygons are present above 60°N.

[24] We next look at how characteristic spatial scales λ can vary for polygons across the surface of Mars. Figure 7 shows the distribution of images across Mars' northern hemisphere with colored dots representing each of our chosen image locations, where the color of the dot corresponds to the characteristic scale λ we obtained for that image. Blue dots show the locations of short scales, red dots show the locations of long scales, and green or yellow dots show the locations of intermediate spatial scales. We find that all images northward of 60°N are at the lower range of our color scale indicating small polygons. Southward of 60°N, we find large, small, and intermediate spatial scales indicating a greater diversity in polygon appearances.

[25] Previous studies [Mangold, 2005; Levy et al., 2009] noted variations of patterned ground type with latitude, and so we looked for similar trends in our results. Figure 8 shows the characteristic scale of each image as a function of latitude. Like Figure 7, we clearly see a different trend in characteristic spatial scale between 50°N and 60°N and between 60°N and 70°N. The polygons at lower latitudes display a greater range in polygon sizes while those at higher latitudes cluster between ~4 and 8 m.

[26] To see if our results are robust, we compared 512 × 512 pixel images to the results for 1024 × 1024 pixel images of the same location. If we missed certain portions of the spatial power spectrum by using the 512 × 512 pixel image, we should find additional power at longer wavelengths in the larger image. Figure 9 shows that for most images, both sets of images provide nearly the same results. However, we find 11 images (denoted with triangles and grouped together with a box) in Figure 9 with significantly larger characteristic spatial scales in the 1024 × 1024 images. Upon inspection of the images (an example is shown in Figure 10), we see that the polygons present have multiple dominant scales: a small scale around 3–5 m and a larger scale comprising groups of these polygons. We can therefore identify, using our technique, that these polygons show multiple scales; however, we cannot reliably and objectively measure the larger scale.

Figure 9.

Characteristic spatial scale measured with 512 × 512 pixel subimages versus characteristic spatial scale measured with 1024 × 1024 pixel images. Squares represent large polygons, circles represent small polygons, and triangles represent multiscale polygonal patterns. Large-scale polygonal patterns are grouped within an ellipse and the multiscale polygonal patterns are grouped within a rectangle.

Figure 10.

Example of an image with multiple polygonal scales. In this image from PSP_001559_2485, there are at least two primary polygon sizes. The smallest diameter polygons have diameters of approximately 3–4 m, and sets of these polygons make larger polygons with diameters > 10 m. Illumination of the image is from the right and the image is ~125 m across. Image: NASA/JPL/University of Arizona.

4 Discussion

[27] Our results show substantial differences between patterned ground terrains poleward of 60°N and patterned ground terrains equatorward. Specifically, we find regions with large characteristic spatial scales in low latitudes and much shorter characteristic spatial scales in higher latitudes as shown in Figures 7 and 8. This same trend has previously been found in observational geomorphological surveys [Levy et al., 2009; Korteniemi and Kreslavsky, 2013]. Additionally, Levy et al. [2009] and Korteniemi and Kreslavsky [2013] found small-scale polygon morphologies extending across the entire range of latitudes we study. We also find short characteristic spatial scales across nearly all latitudes with our more objective method. This suggests to us that our objective method performs similarly to observational surveys in which terrains are classified by humans.

[28] Unlike Levy et al. [2009], however, we do not find trends within 60°N–70°N nor do we find trends within 50°N–60°N. This is not unexpected, though, because polygons of the same scale would give similar λ despite different morphological appearances, given that several of Levy et al.'s [2009] polygon classes have nearly the same diameter. Our inability to distinguish polygon morphology of similar scales may prove this method's greatest limitation because this means that visual inspection of polygonal surfaces is still necessary in mapping and characterizing the variety of patterned ground terrains. Differences in polygon morphology could indicate different formation mechanisms (thermal contraction, sublimation, or frost heave), different evolutionary stages in polygon development (polygons subdivide over time), or different environmental conditions [Levy et al., 2009; Ulrich et al., 2011].

[29] However, Korteniemi and Kreslavsky [2013] argue that the morphological variations of similar scale polygons do not indicate fundamentally different polygon types. Korteniemi and Kreslavsky [2013] find two primary types of polygons they call Type 1 (a larger, 10–20 m diameter, smoothly undulating or hummocky ground pattern) and Type 2 (clear, angularly connected sharp cracks) instead of the five or more classes of polygons in Levy et al. [2009]. Our findings support this type of classification scheme and we find a similar distribution of polygons amongst our survey images. Our method adds a more objective and quantitative approach and, when combined with observations, can prove more reliably predictive.

[30] Models [Mellon, 1997; Mellon et al., 2008] predict that the size of the polygons depend on the amplitude of seasonal temperature change, which depends in large part on climatic conditions. Larger temperature changes lead to greater thermal stresses, which should generate smaller polygons. Mellon [1997] estimated the tensile strength of ice-cemented Martian soil to be between 2 and 3 MPa at Martian temperatures and for length scales on the order of 10 m. More recently, Mellon et al. [2008] predict 3.5 MPa stresses in the center of a polygon with cracks 10 m apart for a seasonal temperature change less than 50 K. Under these conditions, Mellon et al. [2008] predict subdivision of such a polygon, so that polygon size decreases with the increase of the seasonal temperature amplitude, all other parameters being the same.

[31] Since, at the high latitudes, the summer day average temperature increases equatorward, while the winter day average temperatures are fixed at the CO2 frost point, the seasonal temperature amplitude increases from higher to lower latitudes. Thus, smaller characteristic polygon sizes are predicted at lower latitudes based on simple thermal contraction considerations. This is in contrast with our findings that show small polygons (< 6 m characteristic spatial scale) across almost the entire range of latitudes (~55°N–70°N) in the survey and large polygons only at low latitudes (< 60°N). A purely temperature-driven mechanism would predict size to vary with latitude in a linear fashion. Our results do not show such a trend, which suggests that other factors must play a role in polygon development.

[32] The extent of shallow (~1 m) ground ice down to ~60°N was deduced from the measurements obtained by the gamma ray and neutron spectrometer suite onboard the Mars Odyssey orbiter [Boynton et al., 2002; Feldman et al., 2002; Mitrofanov et al., 2002]. Southward from ~60°N, ground ice is also present, but the ice table is deeper than gamma rays and neutrons can detect [Byrne et al., 2009]. We suspect that this transition at ~60°N may cause polygons below this latitude to appear different from those at higher latitudes. The increased depth to the ice at lower latitudes leads to smaller seasonal temperature changes at the top of the ice table and hence smaller seasonal thermal contraction stresses, which in turn leads to larger polygons, if we assume that the strength of ice-rich material is the same.

5 Conclusions

[33] We show that our analysis of the spatial power spectrum produced from the 2-D FT of surface images is indeed capable of providing characteristic spatial scales comparable to the actual typical diameter of polygons forming patterned ground. This method improves on previous studies for two primary reasons. First, this technique gives an objective way to characterize the characteristic sizes of patterned ground terrains without any manual subjective input. Second, the automation of this technique allows for rapid and widespread analysis of terrain instead of decision making upon visual inspection of images. However, visual inspection would still be required for any subsequent morphological analyses.

[34] Our results mostly agree with those of Mangold [2005], Levy et al. [2009], and Korteniemi and Kreslavsky [2013] in that polygons vary between 50°N and 70°N. Like Korteniemi and Kreslavsky [2013], we find two primary types of polygons. We only find large-scale polygons at lower latitudes and small-scale polygons at higher latitudes (with some also at lower latitudes in Acidalia Planitia). Our method cannot discern differences in polygon morphology of the same scale, but we do find multiscale polygons in certain regions. Our results suggest that the depth to ground ice and seasonal temperature changes both influence polygon morphology. We conclude that spatial spectral characteristics can objectively determine features of landscapes in HiRISE images and will be a powerful tool of Mars patterned ground research.

Acknowledgments

[35] Research by T.O., M.K., and E.A. was supported by NASA Mars Fundamental Research Program award NNX08AT13G. M.K. was also supported by NASA Mars Data Analysis award NNX08AL07G. We appreciate fruitful discussions regarding the use of Fourier transforms on terrain with Noah Finnegan and Jon Perkins. We thank Tim Haltigin and an anonymous reviewer for their insightful advice, which greatly improved the manuscript. We also would like to acknowledge the HiRISE team for the spectacular images of the surface of Mars, which allowed us to perform this study.