Power Scaling of Ice Floe Areas in the Weddell Sea, Southern Ocean With a Summary of Previous Ice Floe Scaling Studies

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Coffey, Christopher C Barton, Sarah F Tebbens This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3784463/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The distribution of cumulative number as a function of floe area of seasonal ice floes from four satellite images covering the summer season (November - February) in the Weddell Sea, off Antarctica, during the summer ice breakup were well fit by two scale-invariant power functions. For a power function of the form N = C x -β , the scaling exponents -β for the larger floe areas range from − 1.5 to -1.8. Scaling exponents -β for the smaller floe areas range from − 0.8 to -1.0. The inflection point between the two scaling regimes ranges from 58 x 10 6 to 155 x 10 6 m 2 and generally moves from larger to smaller floe areas through the summer season. We propose that the two power scaling regimes and the inflection between them are established during the initial breakup of sea ice solely by the process of fracturing. Floe areas range from 3 x 10 6 to 550 x 10 6 m 2 . The distributions of floe size regimes retain approximately the same scaling exponents as the floe pack evolves from larger to smaller floe areas from the initial breakup through the summer season, due to scale-independent processes of fracturing, grinding and melting. The scaling exponents for floe area distribution are in the same range as those reported in previous studies of Antarctic and Arctic floes. A probabilistic model of fragmentation is presented that generates a single power scaling distribution of fragment size. Ice floes power distribution scaling Antarctic Southern Ocean Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 1. Introduction In winter season (March-October), sea ice forms and floats on the surface of the Southern Ocean (Fig. 1 ) and ranges up to 3 m thick (Shi et al., 2021 ). In November, the ice begins to break up into free floating angular fragments of different shapes and sizes. A floe is defined as a fragment of sea ice whose upper surface area is larger than 1 m 2 (Gherardi and Lagomarsino, 2015 ). Newly formed floes exhibit rough angular edges, wavy edges, and sharp corners. The surface area of the floes observed in the Weddell Sea in November 2016 range in area from 4 x 10 6 to 550 x 10 6 m 2 . Once formed, the floes sizes are reduced by additional fracturing caused by impact, crushing, and grinding due to wave motion, wind, and ocean currents and by melting and smoothing along their edges and faces. By the end of the summer season, as observed in the Weddell Sea in February 2015, the surface area of the floes range from 3 x 10 6 to 390 x 10 6 m 2 . Larger floes are surrounded by smaller floes which cushion the larger floes protecting them from further fracturing (e.g., Fig. 4 a). The cushioning extends over a wide range of size scales. In the process of fracturing, only particles of approximately the same size or larger can fracture any given particle (Sammis & King, 2007 ). This observation is the basis for the probabilistic model put forward in Geise et al. ( 2017 ) to explain the power distribution of floe sizes in the Arctic Ocean. The size distribution of floes and its evolution over the Antarctic summer is the subject of this paper. This topic is of relevance to marine vessels that encounter floes, to the calculation of sea ice albedo, to the determination of Antarctic heat exchange which is strongly influenced by ice concentrations and the amount of open water between floes (Worby & Allison, 1991 ), and to photosynthetic marine organisms which are dependent upon sunlight penetrating the spaces between floes (Grose & McMinn, 2003 ). 2. Previous Studies Previous studies of the distribution of floe surface areas plot the cumulative or the non-cumulative number as a function of either the floe diameter or the upper surface area of the floes, both of which are fit by a power function. The power function fit to a cumulative distribution is: $$N=C {x}^{-\beta }$$ 1 where N is the number of floes greater than size x, -β is the scaling exponent, and C is a constant of proportionality. C is also referred to as the activity level and is equal to the value of N when x = 1. In many studies, the upper floe surface area was approximated by the caliper-diameter method where one approximates the diameter d of a circle of equivalent area A using the relation A = 0.7854 d 2 . Note, the diameter is a one-dimensional measure of a two-dimensional object (upper floe surface area). Since floe upper surfaces are rarely circular, the equivalent diameter of a floe may be a poor approximation of floe area and may lead to a different scaling exponent for the distribution of floe areas. The scaling exponents of the cumulative distribution of floe area can be converted to obtain the equivalent scaling exponent of the caliper-diameter method by multiplying the scaling exponent of the area method by two (Stern et al., 2018 ). A more precise method, the group pixel method, groups image pixels of ice and water and uses these to determine areas. A few previous studies have used the grouped pixel method to approximate the area (a two-dimensional measure) of each upper floe surface. 2.1 Southern Ocean Lensu ( 1990 ) analyzed one image taken of the Weddell Sea during the summer ice pack break up from February 1990. Using ice floe surface areas, he found the distribution of cumulative number as a function of floe area was well fit by a single power function with a scaling exponent of -0.68. The floe areas ranged from 0.1 to 20 m 2 . Lu et al. ( 2008 ) analyzed nineteen aerial photos taken of the Prydz Bay, East Antarctic [Southern] Ocean in December 2004 through February 2005. They used the caliper-diameter method. Distributions of cumulative number as a function of floe diameter are found were well fit by a single power function with scaling exponents ranging between − 0.6 to -1.4. The floes ranged in diameter from 2 to 100 m. Steer et al. ( 2008 ) analyzed 130 aerial images taken of the Weddell Sea in December 2004. They used the caliper-diameter method. They found the distributions of non-cumulative, log-binned histograms of caliper-diameter were well fit by two scaling exponents. A scaling exponent of -1.9 was found for floes smaller than 20 m and − 2.8 to -3.4 for floes larger than 20 m. The floes sizes ranged from 2 to 120 m. Toyota et al. ( 2011 ) analyzed 122 helicopter images taken of the Weddell Sea on three dates from September through October 2006 and 52 images of the Southern Ocean off of Wilkes Land on three dates from September through October 2007. For both regions, they used the caliper-diameter method. In the Weddell Sea, the distribution of cumulative number as a function of floe diameter were well fit by two scaling exponents: -1.05 to -1.39 were found for floes smaller than 30–40 m and − 5.18 to -7.59 for floes larger than 30–40 m. The floe diameters ranged from 2 to 120 m. Off of Wilkes Land, the distribution of cumulative number as a function of floe diameter were well fit by two scaling exponents. Scaling exponents from − 1.03 to -1.52 were found for floes smaller than 15 to 25 m and − 3.15 to -5.51 for floes larger than 15 to 25 m. The floes sizes ranged from 2 to 100 m. Gherardi and Lagomarsino ( 2015 ) analyzed one satellite image taken of the Weddell Sea in October 2003. They grouped white pixels for ice floes and black pixels for water in the image. They used the caliper-diameter d , of a circle of equivalent area A , to represent the upper surface area of each floe. They found the distribution of a non-cumulative log-binned histogram of caliper-diameter was well fit by a single power function with a scaling exponent of -2.0 (Gherardi and Lagomarsino, 2015 , Fig. 2 ). The floe diameters ranged from 2 to 100 m. Toyota et al. ( 2015 ) analyzed four helicopter and twelve satellite images taken off of Wilkes Land, East Antarctica from September-November 2012. They used the caliper-diameter method. The distribution of cumulative number as a function of floe diameter was well fit by two scaling exponents. Scaling exponents from − 2.9 to -3.1 were found for floes smaller than 100 m and from − 1.3 to -1.4 for floes larger than 100 m. The floe diameters ranged from 5 to 10,000 m. Paget et al. ( 2017 ) analyzed six aerial images taken of East Antarctic sea ice in August 1995. They used the caliper-diameter method. They found the distributions of non-cumulative linearly-binned histograms of caliper-diameter were well fit by power functions with scaling exponents ranging from − 1.9 to -3.5 (Paget et al., 2017 , Fig. 5 ). The floes sizes ranged from 1 to 150 m. 2.2 Arctic Ocean Rothrock and Thorndike ( 1984 ) analyzed seven aerial and satellite images taken of the Beaufort Sea from March to October 1973–1975. They used the caliper-diameter method. They found the distributions of cumulative number as a function of floe diameter were well fit by a single power function with scaling exponents ranging from − 1.7 to -2.5. The floe diameters ranged from 1,000 to 20,000 m. Matsushita ( 1985 ) analyzed one satellite image taken of the Sea of Okhotsk in December 1984. He used the caliper-diameter method. The distribution of cumulative number as a function of floe diameter was well fit by a single power function with a scaling exponent of -2.2. The floe diameters ranged from 5 to 30 m. Holt and Martin ( 2001 ) analyzed fifteen satellite images taken of the Beaufort, Chukchi, and East Siberian Seas from August 1992. They used the caliper-diameter method. The distributions of cumulative number as a function of floe diameter were well fit by a single power function with scaling exponents ranging from − 1.9 to -2.6. The floe diameters ranged from 1,000 to 20,000 m. Inoue et al. ( 2004 ) analyzed two aerial images taken of the Sea of Okhotsk in February 18, 2000. They used the caliper-diameter method. They found the distributions of cumulative number as a function of floe diameter were well fit by a single power function with scaling exponents ranging from − 1.5 to -2.1. The floe diameters ranged from 10 to 100 m. Toyota et al. ( 2006 ) analyzed four helicopter, icebreaker, and satellite images taken of the Sea of Okhotsk in February 2003. They used the caliper-diameter method. They found the distributions of cumulative number as a function of floe diameter were well fit by two scaling exponents. A scaling exponent of -1.15 was found for floes smaller than 40 m and − 1.87 for floes larger than 40 m. The floe diameters ranged from 1 to 1500 m. Perovich and Jones ( 2014 ) analyzed nine aerial and satellite images taken of the Beaufort Sea between June and September 1998. They used the caliper-diameter method. They found the distributions of cumulative number as a function of floe diameter were well fit by a single power function with scaling exponents ranging from − 2.0 to -2.2. The floe diameters ranged from 10 to 10,000 m. Gherardi and Lagomarsino ( 2015 ) analyzed three satellite images taken of the Kara Sea, Svalbard area, and Barents Sea from the Spring seasons of 2000, 2001, and 2009, respectively. They used the caliper-diameter method. They found the distributions of non-cumulative log-binned histograms of caliper-diameter were well fit by a single power function with a scaling exponent of -2.0 (Gherardi and Lagomarsino, 2015 ). The floe areas ranged from 2 to 5,000 m. Wang et al. ( 2016 ) analyzed eighteen satellite images taken of the Beaufort and Chukchi Seas from summer through fall 2014. They used the caliper-diameter method. They found the distributions of cumulative number as a function of floe diameter were well fit by a single power function with scaling exponents ranging from − 2.77 to -4.12. The floe diameters ranged from 1 to 40,000 m. Geise et al. ( 2017 ) analyzed six satellite images taken of the East Siberian Sea from June-August 2000–2002. They used pixel grouping to measure the floe surface area. The distribution of cumulative number as a function of floe area were well fit by two power functions with scaling exponents for the smaller floes ranged from − 0.3 to -0.6 and scaling exponent for the larger floes ranged from − 0.6 to -1.0. The size at the inflection point ranged from 280 x 10 3 to 485 x 10 3 m 2 . The floe areas ranged from 30 m 2 to 28.4 x 10 6 m 2 . Hwang et al. ( 2017 ) analyzed multiple satellite images taken of the Beaufort Sea from July 19, August 15, and August 23, 2014. They used the caliper-diameter method. They found the distributions of non-cumulative log-binned histograms of caliper-diameter were well fit by a single power function with scaling exponents ranging from − 2.7 to -3.0 (Hwang et al, 2017 , Fig. 1 ). The floe diameters ranged from 245 to ~ 4000 m. Stern et al. ( 2018 ) analyzed 273 satellite images taken of the Beaufort and Chukchi Seas from March to October 2013 and 2014. They used the caliper-diameter method. They found the distributions of non-cumulative log-binned histograms of caliper-diameter were well fit by a single power function with scaling exponents ranging from − 2 to -2.9 (Stern et al., 2018 , Fig. 5 ). The floe diameters ranged from 10 to 30,000 m. 3. Calculation of Equivalent Scaling Dimensions Table 1 provides a list of the results from our study and previous studies of ice floe scaling in the Antarctic and Arctic regions. To directly compare scaling exponents, we convert scaling exponents reported in the literature to equivalent scaling exponents of cumulative number as a function of floe area distributions. For each study, the uppermost value reported in Table 1 is the scaling exponent provided in the original research paper. To obtain equivalent scaling exponents, the order of operations is important. The conversion from caliper-diameter to area was determined for a cumulative distribution by Stern et al. ( 2018 ). For studies that reported the scaling exponent for a non-cumulative histogram distribution, we first obtain the equivalent dimension for the cumulative distribution. Second, for studies that reported a dimension for caliper-diameter, we convert to the equivalent dimension for floe area. Conversion of scaling exponents from non-cumulative histogram distributions to cumulative number versus size distributions Non-cumulative linearly-binned histogram distributions. The scaling exponent for a non-cumulative linearly binned histogram distribution that follows a power function has a slope that is steeper by an integer value of one (1) as compared to the same data plotted as a cumulative number versus size distribution (e.g., Fig. 1 of Burroughs and Tebbens, 2001 ). To obtain the equivalent scaling exponent for a cumulative number versus size distribution, when starting with the scaling exponent for a non-cumulative linearly binned histogram distribution, we add an integer value of one (1). As a specific example, Paget et al. ( 2017 ) reported scaling exponents of -1.9 to -3.5 for a non-cumulative linearly binned distribution, and we provide the equivalent scaling exponent for the cumulative number versus size distribution of -0.9 to -2.5 in square brackets [ ] (Table 1). Non-cumulative logarithmically-binned histogram distribution. The scaling exponent for a cumulative number versus size distribution that follows a power function has a slope that is equal to the slope of the same data plotted as a non-cumulative linearly binned histogram distribution (e.g., Fig. 2 of Burroughs and Tebbens, 2001 ). In Table 1, for studies that reported the non-cumulative linearly binned histogram distributions (e.g., Gherardi & Lagomarsino, 2015 and Stern et al, 2018 ), the scaling exponents are the same as for the cumulative number vs. size distributions. Therefore, there is no conversion of the value of the scaling exponent. Conversion from cumulative number versus caliper-diameter to cumulative number versus area The scaling exponents derived from a 1-dimensional representation of the surface area (caliper-diameter) of a floe are not equivalent to those calculated from 2-dimensional measures of floe area. The scaling exponents calculated by the caliper-diameter method can be converted to the equivalent surface area scaling exponents by dividing the caliper-diameter scaling exponent by two (Stern et al., 2018 ). In Table 1, we show the equivalent surface area scaling exponent in parentheses ( ). 4. Data This study is based on four mostly or completely cloud free satellite images of four floe packs at different stages of fragmentation during the Antarctic summer months (November 2016, December 2015, January 2015, and February 2015) in the Weddell Sea (Fig. 2 ) obtained from the USGS Earth Explorer website (United States Geological Survey, 2016 ). The Weddell Sea was selected because of the image clarity, quality, and cloud coverage less than twenty percent. The Weddell Sea is a preferred research location for many ice floe studies because it is protected by the Antarctic peninsula. The images used in this study were collected by the Landsat 8 satellite. To measure the upper surface areas of individual floes, ideally the portions of the images analyzed must be cloud free because the bands used to view the ice floes do not penetrate through clouds. Melting ponds on the surface of the floes were not removed from the images. Satellite images were greyscale at 30 m per pixel resolution. We use floe area as the measure of floe size in contrast to the caliper-diameter method used in many previous studies. A visual inspection of the satellite images reveals no characteristic floes size (e.g., Figs. 3 , 4 , 5 , 6 , and 7 ), suggesting qualitatively that the distribution of floe sizes is scale invariant and might, therefore, be described by a power function. The satellite images obtained from Earth Explorer were imported into ESRI’s geographic information system (GIS) program, ArcPro (2.2). Tiff pixels from these images were classified into two classes, ice and water, using ArcPro’s Supervised Classification tool. The high brightness pixels represent the ice, while the low brightness pixels represent water. Individual floes larger than 3 x 10 6 m 2 were identified by grouping of ice class pixels and are surrounded by water class pixels. The Raster to Polygon tool was used to automatically identify the edges of each floe. The resulting floe polygons form the floes seen in the raster image (Fig. 7 ). The area and perimeter length of each floe in the raster image were automatically tabulated in the attribute table by ArcPro. Floes cut off by the edge of the image, regions obscured by cloud cover, and slush/finely crushed ice were not included in the analysis. 5. Analysis A plot of the cumulative number versus floe area is shown in Fig. 8 . The data for each image are well fit by two power functions (Eq. 1 ), with an inflection point between them. The choice of an algorithm for fitting a distribution that appears to be a power function has been debated in recent years (Deluca & Corral, 2013 ; Holme, 2019 ). One method is that of Clauset et al. ( 2009 ). Corral & Gonzalez (2019) enumerated three issues with the Clauset et al. (2019) method: the method cannot be extended to truncated power distributions; there is no justification why the minimization of the Kolmogorov-Smirnov distance should work to find a meaningful scaling exponent; and the method was found to fail when applied to simulated data with power function tails, despite the data being a known power function. For this paper, the specific scaling exponent is not as important as the result that the data are well represented by a power function. Further, we are not extrapolating the results beyond the range of the data, which would amplify any errors in the reported values. Given these issues, we have used the same method used in our previous paper studying ice floe distributions in the Artic (Geise et al., 2017 ), to facilitate direct comparison of results. For each of the data sets we analyze, we find the parameters C and β using the Levenberg-Marquardt algorithm to minimize Chi squared (Press et al., 2001 ) using the Wavemetrics software IgorPro version 6.37. This method is appropriate for fitting the power functions to data and calculating the associated error; the method assumes that the errors in fitting the functions to the data are normally distributed (Press et al., 2001 ). In all analyses for this study, the data were fit using IgorPro by the power function y 0 + Cx -β , where y 0 was set to zero and the fit was found to converge. Errors are reported to plus/minus one standard deviation in Table 1. Each of the four data sets analyzed and plotted on Fig. 8 exhibit two power scaling regimes where larger floes have a distribution whose scaling exponent is greater than that of the small floes. The range of floe areas on either side of the inflection point is greater than one order of magnitude for the smaller floes and nearly one order of magnitude for the larger floes, where one order of magnitude is considered to be the minimum range for uniquely fitting a power function to a dataset. The scaling exponents -β exhibit a range of values (Table 1). The scaling region for smaller floes has scaling exponents -β ranging from − 0.8 to -1.0. The larger floes have scaling exponents -β ranging from − 1.5 to -1.8. The inflection point between the two regions ranges from 86 x 10 6 to 174 x 10 6 m 2 . The number of floes analyzed in each image varies from 303 (February 2015) to 1,078 (December 2015). Inflection point sizes decrease roughly sequentially from November to February as sea ice evolves from larger to smaller floes throughout the season (Fig. 8 ). Note that cushioning of larger floes by smaller floes is present for almost all floe sizes. This suggests that once the floe size distribution is established at the time of initial breakup, further size reduction is primarily by grinding and melting of the sides of floes, not by fracturing across the floes (Fig. 3 b). Summing the areas of the larger floes, those above the inflection point, and dividing by the total floe area for each image indicates that the area of the large floes is 21–44% of the total floe area (Table 1). Note that this differs from the percentage of total floe area above and below the inflection points found in the Arctic by Geise et al. ( 2017 ) where the larger floes, those greater than the inflection points, accounted for 72–95% of the total floe area. 6. Fragmentation Model Geise et al. ( 2017 ) proposed a model that produces a power distribution of fragment sizes, which may explain the power distribution of floe areas but not the existence of an inflection point between the larger and smaller floe size distributions. Figure 9 illustrates their probabilistic model for fragmentation of a cubic volume for any material where fracture is the method of fragmentation. At iteration n = 1 the volume is broken into eight smaller cubes of equal size. At iteration n = 2 some of the eight cubes are randomly selected and broken into eight smaller cubes. The probability that a cube will be fragmented is an adjustable variable. In Fig. 9 , the probability of fragmentation is p = 0.25. At iteration n = 3 the model bifurcates based on whether (A) only the smallest of the cubes or (B) a cube of any size, can be randomly selected and broken. With a large number of iterations, the rule for A or B determines the cumulative frequency versus size distribution of the fragments. Rule A leads to a power fragment size distribution and Rule B leads a lognormal fragment size distribution. As we observe power distributions in the size (area) of floes, we propose that a process following Rule A is appropriate for ice floes. The choice of probability of fragmentation controls the scaling exponent of the power law distribution (e.g., p = 0.25 yields a b of -0.5). We note that this fragmentation model generates a single power distribution without an inflection point. This model could be appropriate for ice floe distributions reporting a single power distribution of floe areas (Table 1, studies that report “b with no inflection”). Sammis and King ( 2007 ) propose a physical process that results in power scaling of fragmentation sizes of geologic materials in a fault gauge. Adjacent fragments must be approximately the same size to break each other by fracture. They observe that in the process of fragmentation by grinding and crushing, larger fragments are surrounded by smaller fragments that cushion them and protected them thus greatly reducing the probability of further fragmentation. They propose that such a process leads to a power distribution in fragmentation size. Visual inspection of Figs. 3 a, 4 a, 5 , and 6 reveal that for ice floes the large fragments are surrounded by smaller fragments which may similarly have cushioned them and reduced the probability of further fragmentation. 7. Discussion and Conclusions The distribution of floe area as a function of cumulative number versus floe area in each image were well fit by two power functions that meet at an inflection point between the larger floes and the smaller floes (Fig. 8 , Table 1). The floe areas range from 3 x 10 6 to 550 x 10 6 m 2 . Floe areas larger than the inflection points are well fit by a power function with scaling exponents of -1.5 to -1.8 while the floes areas smaller than the inflection points are well fit by a power function with a scaling exponents of -0.8 to -1.0. The inflection points range from 58 x 10 6 to 155 x 10 6 m 2 . This study supports some of the previous studies that found that the cumulative number versus floe area distribution of seasonal floes are distributed in two power scaling regimes in the Antarctic region (Steer et al., 2008 ; Toyota et al., 2011 ; and Toyota et al., 2015 ) and in the Arctic (Toyota et al., 2006 and Geise et al., 2017 ). We suggest that the two power scaling regimes and the inflection between the large and small floes are established during the initial breakup of the sea ice solely by the process of fracture. The distributions of floe size regimes retain the same scaling exponent (-b are within 2 standard deviations for three out of four of the images; see Fig. 8 and Table 1) as the sea ice evolves from larger to smaller floes throughout the season due to grinding and melting as seen in Fig. 8 . When the initial ice breaks up by fracture the floe size distribution and the scaling exponents are established. The scaling exponents established during the initial ice break-up are maintained through the summer season and change minimally (by not more than 0.1) during the reduction of floe sizes at all scales through the processes of grinding and melting. While the scaling exponents b do not appreciably change during the season in our study, the activity level C tends to decrease (Fig. 8 , Eq. 1 ). As C decreases, the number of floes of any and all sizes present is smaller. If there is not change is slope, the decrease in size is equal in log space at all sizes. indicating the floe sizes have systematically decreased. This can be seen in Fig. 8 where the data points for Feb 2015 are below those of Jan 2015, and the largest sized floe is also smaller. Declarations The authors declare that no funds, grants, or other support were received during the preparation of this manuscript. The authors have no relevant financial or non-financial interests to disclose. Author Contribution Coffey obtained the satellite data, analyzed the images for floe area, and created many of the figures.Barton conceived and coordinated the project.Tebbens lead the statistical analysis of the data sets.We all worked on writing the manuscript. Acknowledgements The authors thank Greg Geise and Dan Khoel for their help in processing the GIS data. We also thank two anonymous reviewers for constructive criticism that greatly improved an early version of this manuscript. References Burroughs, S.M. & Tebbens, S.F. (2001). Upper-truncated Power Law Distributions, Fractals, 9, 209-222. https://doi.org/10.1142/S0218348X01000658 Clauset, A., Shalizi, C. R. & Newman., M. E. J. (2009). Power-Law Distributions in Empirical Data. SIAM Review , 51 (4), 661–703. JSTOR . https://doi.org/10.1137/070710111 Corral, Á., & Gonzálex, Á., (2019). Power Law Size Distributions in Geoscience Revisited. Earth and Space Science, 6 (5). https://doi.org/10.1029/2018EA000479 Deluca, A., & Corral, Á. (2013). Fitting and goodness-of-fit test of non-truncated and truncated power law distributions. Acta Geophysica , 61 , 1351–1394. https://doi.org/10.2478/s11600-013-0154-9 Geise, G. R., Barton, C. C. & Tebbens, S. F. (2017). Power Scaling and Seasonal Changes of floe areas in the Arctic East Siberian Sea, Pure and Applied Geophysics , 174 (387). https://doi.org/10.1007/s00024-016-1364-2 Gherardi, M., & Lagomarsino, M. C. (2015). Characterizing the size and shape of sea ice floes. Scientific Reports, 5 , 10226. https://doi.org/10.1038/srep10226 Grose, M., & McMinn, A. (2003). Algal biomass in East Antarctic Pack ice: How much is in the East? In A. H. L. Huiskes, W. W. C. Gieskes, J. Rozema, R. M. L. Schorno, S. M. van der Vies & W. J. Wolff (Eds.) Antarctic biology in a global context (pp. 182–186). Backhuys Publishers. https://doi.org/ 10.1017/S0954102003251725 Holme, P. (2019). Rare and everywhere: Perspectives on scale-free networks. Nature Communications , 10 , 1016. https://doi.org/10.1038/s41467-019-09038-8 Holt, B. and Martin, S. (2001). The effect of a storm on the 1992 summer sea ice cover of the Beaufort, Chukchi, and East Siberian Seas, J. Geophys. Res. Ocean. , 106 , 1017–1032. https://doi.org/10.1029/1999JC000110 Hwang, B., Wilkinson, J., Maksym, T. Graber, H.C., Schweiger, A., Horvat, C., Perovich, D.K., Arntsen, A.E., Stanton, T.P., Ren, J., & Wadhams, P. (2017). Winter-to-summer transition of Arctic sea ice breakup and floe size distribution in the Beaufort Sea. Elementa: Science of the Anthropocene, 5 (40), https://doi.org/10.1525/elementa.232 Inoue, J., Wakatsuchi, M., & Fujiyoshi, Y. (2004). Ice floe distribution in the Sea of Okhotsk in the period when sea-ice extent is advancing. Geophysical Research Letters, 31 (20), L20303. https://doi.org/10.1029/2004GL020809 Lensu, M. (1990, August 22-23). The fractality of sea ice cover . IAHR, the 10 th International Symposium on Ice, Helsinki University of Technology, Espoo, Finland. Proceedings pp. 300–313. Lu, P., Li, Z. J., Zhang, Z. H. & Dong, X. L. (2008). Aerial observations of floe size distribution in the marginal ice zone of summer Prydz Bay. Journal of Geophysical Research: Oceans, 113 (C2). https://doi.org/10.1029/2006JC003965 Matsushita, M. (1985). Fractal viewpoint of fracture and accretion, Journal of the Physical Society of Japan , 54 (3), 857–860. https://doi.org/10.1143/JPSJ.54.857 National Aeronautics and Space Administration (2016). https://earthobservatory.nasa.gov/features/SeaIce/page4.php Paget, M. J., Worby, A. P. & Michael, K. J. (2017). Determining the floe-size distribution of East Antarctica sea ice from digital aerial photographs. Annals of Glaciology, 33, 94–100. https://doi.org/10.3189/172756401781818473 Perovich, D. K., & Jones, K. F. (2014). The seasonal evolution of sea ice floe size distribution, Journal of Geophysical. Research - Oceans , 119 , 8767–8777, https:/doi.org/10.1002/2014JC010136 Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (2001). Numerical recipes in FORTRAN: The art of scientific computing , Cambridge University Press. Rothrock, D. A., & Thorndike, S. (1984). Measuring the sea ice floe size distribution. Journal of Geophysical Research: Oceans 89, 6477-6486. http://doi.org/10.1029/JC089iC04p06477 Sammis, C. G., & King, G. C. P. (2007). Mechanical origin of power law scaling in fault zone rock. Geophysical Research Letters, 34 , L04312. https://doi.org/10.1029/2006GL028548 Shi, Q., Yang, Q., Mu, L., Wang, J., Massonnet, F., & Mazloff, M. R. (2021). Evaluation of sea-ice thickness from four reanalyses in the Antarctic Weddell Sea, The Cryosphere, 15 , 31–47. https://doi.org/10.5194/tc-15-31-2021 Steer, A., Worby, A., & Heil, P. (2008). Observed changes in sea-ice floe size distribution during early summer in the western Weddell Sea, Deep-Sea Research II, 55 , 933–942. https://doi.org/10.1016/j.dsr2.2007.12.016 Stern, H.L., Schweiger, A. J., Stark, M., Zanj, J., Steele, M., & Hwang, B. (2018). Seasonal evolution of the sea-ice floe size distribution in the Beaufort and Chukchi seas. Elementa Science of the Anthropocene, 6 (48). https://doi.org/10.1525/elementa.305 Toyota, T., Haas, C. & Tamura, T. (2011). Size distribution and shape properties of relatively small sea-ice floes in the Antarctic marginal ice zone in late winter. Deep-Sea Res II, 58 , p. 1182–1193. https://doi.org/10.1016/j.dsr2.2010.10.034 Toyota, T., Kohout, A. & Fraser, A. (2015). Formation processes of sea ice floe size distribution in the interior pack and its relationship to the marginal ice zone off East Antarctica. Deep Sea Research Part II Topical Studies in Oceanograp hy, 131 , 28-40. https://doi.org/10.1016/j.dsr2.2015.10.003 Toyota, T., Takatsuji, S., & Nakayama, M. (2006). Characteristics of sea ice floe size distribution in the seasonal ice zone, Geophysical Research Letters , 33 , L02616, http://doi.org/10.1029/2005GL024556 United States Geological Survey, 2016, Earth Explorer, https://earthexplorer.usgs.gov/ Wang, Y., Holt, B., Erick Rogers, W., Thomson, J., & Shen, H. H. (2016). Wind and wave influences on sea ice floe size and leads in the Beaufort and Chukchi Seas during the summer-fall transition 2014, Journal of Geophysical Research - Oceans , 121 , 1502–1525, https://doi.org/10.1002/2015JC011349 Worby, A. P. & Allison, I. (1991). Ocean-atmosphere energy exchange over thin, variable concentration Antarctic pack ice. Annals of Glaciology ,15 , 184–190. https://doi.org/10.3189/1991AoG15-1-184-190 Zhang, Q., & Skjetne, R. (2018). Sea Ice Image Processing with MATLAB, CRC Press, Boca Raton, FL, 272p. Table Table 1 is available in the Supplementary Files section. Additional Declarations No competing interests reported. Supplementary Files AntarcticIceFloe.Table1.Dec202023.pdf Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3784463","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":263310786,"identity":"544515e0-75bc-43a0-8192-b4deb5147793","order_by":0,"name":"Tristan J. Coffey","email":"","orcid":"","institution":"Wright State University","correspondingAuthor":false,"prefix":"","firstName":"Tristan","middleName":"J.","lastName":"Coffey","suffix":""},{"id":263310787,"identity":"b81acf3a-b4af-4154-921c-9b7f77531742","order_by":1,"name":"Christopher C Barton","email":"","orcid":"","institution":"Wright State University","correspondingAuthor":false,"prefix":"","firstName":"Christopher","middleName":"C","lastName":"Barton","suffix":""},{"id":263310788,"identity":"f815423b-7c5b-4ae7-8eaa-9cbf42177a7f","order_by":2,"name":"Sarah F Tebbens","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABCElEQVRIiWNgGAWjYJACZhBhIAEiKxgY2Bh44IL4tBhAtZwBa2FsIF4LYxuIT0CLfPvZh58LGP7ImUs3H3v4c97hfD7+tccfMFRYJzbg0GJwJt1YegaDgbHlnGPpxrzbDlu2SbwDqj6TjlsLQxqDNA+DQeKGGzlm0ozbbhuwSZwxbGBsO4xTi3z/M+bfQC31G27kf5P8OQem5R9uLQw30thAtiQY3Mhhk+BtAGrh7wFqacCtxeDGMzZrHgNjw51zjplJ8xz7D7SFx3BGAtBjuB2Wxnybp0JOHhhizyR/1KQZyPefMfjwocZaFqfDoIGABCQSGBgS8CrHAPwHSFM/CkbBKBgFwx4AAMWnVCFPSfnGAAAAAElFTkSuQmCC","orcid":"","institution":"Wright State University","correspondingAuthor":true,"prefix":"","firstName":"Sarah","middleName":"F","lastName":"Tebbens","suffix":""}],"badges":[],"createdAt":"2023-12-21 02:59:26","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3784463/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3784463/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":49079361,"identity":"b4c92bf3-efad-4dd9-8253-172106c94f6c","added_by":"auto","created_at":"2024-01-02 19:47:52","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":446014,"visible":true,"origin":"","legend":"\u003cp\u003eSatellite image showing the extent of Antarctic sea ice maximum on October 6, 2016 (left) and minimum on February 19, 2016 (right). Water is black, sea ice is white, land is dark grey, and ice shelfs are light grey. Images are from https://earthobservatory.nasa.gov/features/SeaIce/page4.php. The Weddell Sea is in the upper left quadrant of each image.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-3784463/v1/ba0a065abf65f07a76c057a7.png"},{"id":49081543,"identity":"3f3a5908-eed0-48f8-b293-b0d7c501300c","added_by":"auto","created_at":"2024-01-02 20:03:52","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":224003,"visible":true,"origin":"","legend":"\u003cp\u003eLocation the ice floe images in the Weddell Sea analyzed in this study. Figure was created in Arc PRO (ESRI). The ice floe images were obtained from USGS Earth Explorer for November 2016 (Image Number LC08_L1GT_202104_20161105), December 2015 (Image Number LC08_L1GT_200106_20151207), January 2015 (Image Number LC08_L1GT_198105_20150107), and February 2015 (Image Number LC08_L1GT_202108_20150220).\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-3784463/v1/52121859ad76a04cc6210bcc.png"},{"id":49079365,"identity":"588d4f60-cdbb-4b04-bf41-6bef92081941","added_by":"auto","created_at":"2024-01-02 19:47:52","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":2044093,"visible":true,"origin":"","legend":"\u003cp\u003ea. Satellite image of ice floes in Weddell Sea on November 5, 2016. The white rectangular inset marks location of Figure 3b. Image is from USGS Earth Explorer (Image Number LC08_L1GT_202104_20161105).\u003c/p\u003e\n\u003cp\u003eb: Satellite image of inset region in Figure 3a showing larger floes (grey) being cushioned by smaller floes (white). Cushioning of the larger floes reduces the probability of size reduction by fracturing. Note that the cushioning is present for almost all sizes which suggests that once the floe size distribution is established at the time of initial breakup further size reduction is primarily by grinding and melting of the floe sides and not by fracturing across the floes. Image is from USGS Earth Explorer (Image Number LC08_L1GT_202104_20161105).\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-3784463/v1/3c3b5492d08a43a8e2fb1b0f.png"},{"id":49080377,"identity":"c70e172f-cf58-42be-8773-73622601e49a","added_by":"auto","created_at":"2024-01-02 19:55:52","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":1836737,"visible":true,"origin":"","legend":"\u003cp\u003ea: Satellite image of ice floes in Weddell Sea from December 7. 2015. The black rectangular inset marks location of Figure 4b. Image is from USGS Earth Explorer (Image number LC08_L1GT_200106_20151207).\u003c/p\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003eb: Satellite image of inset region in Figure 3b. Image was taken soon after the initial breakup of sea ice by the process of fracture. The irregular lines visible within the floes are fused boundaries between old floes that comprise the ice pack. Image is from USGS Earth Explorer (Image Number LC08_L1GT_200106_20151207).\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-3784463/v1/6c4379498b0ede110e1ca0f2.png"},{"id":49079370,"identity":"40a60bec-50af-4c07-a5eb-09e8070aba0e","added_by":"auto","created_at":"2024-01-02 19:47:52","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":834501,"visible":true,"origin":"","legend":"\u003cp\u003eSatellite image of ice floes in Weddell Sea on January 7, 2015. Top center and left regions of image appear out of focus as a result of cloud cover.Image is from USGS Earth Explorer (Image Number LC08_L1GT_198105_20150107).\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-3784463/v1/aca6da7b4cd845820776d44d.png"},{"id":49080376,"identity":"c8e20ad7-4616-4fe1-93e9-8adadf965d00","added_by":"auto","created_at":"2024-01-02 19:55:52","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":886345,"visible":true,"origin":"","legend":"\u003cp\u003eSatellite Image of ice floes in the Weddell Sea on February 20, 2015. Top and left regions of image appear out of focus as a result of cloud cover. Image is from USGS Earth Explorer (Image Number LC08_L1GT_202108_20150220).\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-3784463/v1/14975be349af1068b630dd7a.png"},{"id":49079364,"identity":"5c904dc5-a8fe-4fe4-aef5-e649c4c1c1df","added_by":"auto","created_at":"2024-01-02 19:47:52","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":770999,"visible":true,"origin":"","legend":"\u003cp\u003eRaster images (shape files) of satellite images of four floe packs at different stages of fragmentation and floe pack density from November 2016 (Figure 3a), December 2015 (Figure 4a), January 2015 (Figure 5), and February 2015 (Figure 6). Floes with areas greater than the inflection point are gray, floes with areas smaller than the inflection point are white, and water is black. Floes partially cut off by image boundaries, regions obscured by cloud cover, and slush/finely crushed ice were not included in the analysis.\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-3784463/v1/e89ff1a78b3cff991ab5dc31.png"},{"id":49079362,"identity":"f0f228c9-d39e-44c6-9047-c6b1514be583","added_by":"auto","created_at":"2024-01-02 19:47:52","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":76594,"visible":true,"origin":"","legend":"\u003cp\u003ePlot of ice floe areas versus cumulative number for the Weddell Sea ice floes from four images ranging over the Antarctic summer from early November to mid-February for 2015 and 2016 (Figure 7). The data from each image is fit by two power functions. The scaling exponents for the larger floes (1.5-1.8) are consistently greater than those for the smaller floes (0.8-1.0). The power scaling exponents are listed in Table 1.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-3784463/v1/95ebf12a49055a808dc0f6e8.png"},{"id":49079367,"identity":"dfd84b2d-62a1-479c-a9f0-7148c303f28a","added_by":"auto","created_at":"2024-01-02 19:47:52","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":291193,"visible":true,"origin":"","legend":"\u003cp\u003eFour iterations (n) in the fragmentation of a unit cube. On left, the constant probability of fragmentation p = 0.25 is only for the smallest cubes to be broken at each iteration, resulting in a power distribution of fragment sizes. On right, the constant probability of fragmentation p = 0.25 is for any cube to be broken at each iteration, resulting in a lognormal distribution of fragment sizes.\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-3784463/v1/cf98f9b1d70b7eaee7e44839.png"},{"id":53799113,"identity":"605929aa-149a-4dcb-948a-dc09154dba5a","added_by":"auto","created_at":"2024-03-31 05:37:37","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":4745070,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3784463/v1/c784e0ca-aea5-4d93-a258-43d56399a753.pdf"},{"id":49080375,"identity":"271e9f22-2c49-41e4-9c84-5adb41f6abf3","added_by":"auto","created_at":"2024-01-02 19:55:52","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":108710,"visible":true,"origin":"","legend":"","description":"","filename":"AntarcticIceFloe.Table1.Dec202023.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3784463/v1/e4f7c9f9bd6339a7c9d5372d.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Power Scaling of Ice Floe Areas in the Weddell Sea, Southern Ocean With a Summary of Previous Ice Floe Scaling Studies","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eIn winter season (March-October), sea ice forms and floats on the surface of the Southern Ocean (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) and ranges up to 3 m thick (Shi et al., \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). In November, the ice begins to break up into free floating angular fragments of different shapes and sizes. A floe is defined as a fragment of sea ice whose upper surface area is larger than 1 m\u003csup\u003e2\u003c/sup\u003e (Gherardi and Lagomarsino, \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2015\u003c/span\u003e). Newly formed floes exhibit rough angular edges, wavy edges, and sharp corners. The surface area of the floes observed in the Weddell Sea in November 2016 range in area from 4 x 10\u003csup\u003e6\u003c/sup\u003e to 550 x 10\u003csup\u003e6\u003c/sup\u003e m\u003csup\u003e2\u003c/sup\u003e. Once formed, the floes sizes are reduced by additional fracturing caused by impact, crushing, and grinding due to wave motion, wind, and ocean currents and by melting and smoothing along their edges and faces. By the end of the summer season, as observed in the Weddell Sea in February 2015, the surface area of the floes range from 3 x 10\u003csup\u003e6\u003c/sup\u003e to 390 x 10\u003csup\u003e6\u003c/sup\u003e m\u003csup\u003e2\u003c/sup\u003e. Larger floes are surrounded by smaller floes which cushion the larger floes protecting them from further fracturing (e.g., Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e4\u003c/span\u003ea). The cushioning extends over a wide range of size scales. In the process of fracturing, only particles of approximately the same size or larger can fracture any given particle (Sammis \u0026amp; King, \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2007\u003c/span\u003e). This observation is the basis for the probabilistic model put forward in Geise et al. (\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2017\u003c/span\u003e) to explain the power distribution of floe sizes in the Arctic Ocean. The size distribution of floes and its evolution over the Antarctic summer is the subject of this paper. This topic is of relevance to marine vessels that encounter floes, to the calculation of sea ice albedo, to the determination of Antarctic heat exchange which is strongly influenced by ice concentrations and the amount of open water between floes (Worby \u0026amp; Allison, \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e1991\u003c/span\u003e), and to photosynthetic marine organisms which are dependent upon sunlight penetrating the spaces between floes (Grose \u0026amp; McMinn, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2003\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"2. Previous Studies","content":"\u003cp\u003ePrevious studies of the distribution of floe surface areas plot the cumulative or the non-cumulative number as a function of either the floe diameter or the upper surface area of the floes, both of which are fit by a power function. The power function fit to a cumulative distribution is:\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$N=C {x}^{-\\beta }$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere N is the number of floes greater than size x, -β is the scaling exponent, and C is a constant of proportionality. C is also referred to as the activity level and is equal to the value of N when x\u0026thinsp;=\u0026thinsp;1.\u003c/p\u003e \u003cp\u003eIn many studies, the upper floe surface area was approximated by the caliper-diameter method where one approximates the diameter \u003cem\u003ed\u003c/em\u003e of a circle of equivalent area \u003cem\u003eA\u003c/em\u003e using the relation \u003cem\u003eA\u003c/em\u003e\u0026thinsp;=\u0026thinsp;0.7854 \u003cem\u003ed\u003c/em\u003e \u003csup\u003e2\u003c/sup\u003e. Note, the diameter is a one-dimensional measure of a two-dimensional object (upper floe surface area). Since floe upper surfaces are rarely circular, the equivalent diameter of a floe may be a poor approximation of floe area and may lead to a different scaling exponent for the distribution of floe areas. The scaling exponents of the cumulative distribution of floe area can be converted to obtain the equivalent scaling exponent of the caliper-diameter method by multiplying the scaling exponent of the area method by two (Stern et al., \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). A more precise method, the group pixel method, groups image pixels of ice and water and uses these to determine areas. A few previous studies have used the grouped pixel method to approximate the area (a two-dimensional measure) of each upper floe surface.\u003c/p\u003e \u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Southern Ocean\u003c/h2\u003e \u003cp\u003eLensu (\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e1990\u003c/span\u003e) analyzed one image taken of the Weddell Sea during the summer ice pack break up from February 1990. Using ice floe surface areas, he found the distribution of cumulative number as a function of floe area was well fit by a single power function with a scaling exponent of -0.68. The floe areas ranged from 0.1 to 20 m\u003csup\u003e2\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eLu et al. (\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2008\u003c/span\u003e) analyzed nineteen aerial photos taken of the Prydz Bay, East Antarctic [Southern] Ocean in December 2004 through February 2005. They used the caliper-diameter method. Distributions of cumulative number as a function of floe diameter are found were well fit by a single power function with scaling exponents ranging between \u0026minus;\u0026thinsp;0.6 to -1.4. The floes ranged in diameter from 2 to 100 m.\u003c/p\u003e \u003cp\u003eSteer et al. (\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2008\u003c/span\u003e) analyzed 130 aerial images taken of the Weddell Sea in December 2004. They used the caliper-diameter method. They found the distributions of non-cumulative, log-binned histograms of caliper-diameter were well fit by two scaling exponents. A scaling exponent of -1.9 was found for floes smaller than 20 m and \u0026minus;\u0026thinsp;2.8 to -3.4 for floes larger than 20 m. The floes sizes ranged from 2 to 120 m.\u003c/p\u003e \u003cp\u003eToyota et al. (\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2011\u003c/span\u003e) analyzed 122 helicopter images taken of the Weddell Sea on three dates from September through October 2006 and 52 images of the Southern Ocean off of Wilkes Land on three dates from September through October 2007. For both regions, they used the caliper-diameter method. In the Weddell Sea, the distribution of cumulative number as a function of floe diameter were well fit by two scaling exponents: -1.05 to -1.39 were found for floes smaller than 30\u0026ndash;40 m and \u0026minus;\u0026thinsp;5.18 to -7.59 for floes larger than 30\u0026ndash;40 m. The floe diameters ranged from 2 to 120 m. Off of Wilkes Land, the distribution of cumulative number as a function of floe diameter were well fit by two scaling exponents. Scaling exponents from \u0026minus;\u0026thinsp;1.03 to -1.52 were found for floes smaller than 15 to 25 m and \u0026minus;\u0026thinsp;3.15 to -5.51 for floes larger than 15 to 25 m. The floes sizes ranged from 2 to 100 m.\u003c/p\u003e \u003cp\u003eGherardi and Lagomarsino (\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2015\u003c/span\u003e) analyzed one satellite image taken of the Weddell Sea in October 2003. They grouped white pixels for ice floes and black pixels for water in the image. They used the caliper-diameter \u003cem\u003ed\u003c/em\u003e, of a circle of equivalent area \u003cem\u003eA\u003c/em\u003e, to represent the upper surface area of each floe. They found the distribution of a non-cumulative log-binned histogram of caliper-diameter was well fit by a single power function with a scaling exponent of -2.0 (Gherardi and Lagomarsino, \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2015\u003c/span\u003e, Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e2\u003c/span\u003e). The floe diameters ranged from 2 to 100 m.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eToyota et al. (\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e2015\u003c/span\u003e) analyzed four helicopter and twelve satellite images taken off of Wilkes Land, East Antarctica from September-November 2012. They used the caliper-diameter method. The distribution of cumulative number as a function of floe diameter was well fit by two scaling exponents. Scaling exponents from \u0026minus;\u0026thinsp;2.9 to -3.1 were found for floes smaller than 100 m and from \u0026minus;\u0026thinsp;1.3 to -1.4 for floes larger than 100 m. The floe diameters ranged from 5 to 10,000 m.\u003c/p\u003e \u003cp\u003ePaget et al. (\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2017\u003c/span\u003e) analyzed six aerial images taken of East Antarctic sea ice in August 1995. They used the caliper-diameter method. They found the distributions of non-cumulative linearly-binned histograms of caliper-diameter were well fit by power functions with scaling exponents ranging from \u0026minus;\u0026thinsp;1.9 to -3.5 (Paget et al., \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2017\u003c/span\u003e, Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e). The floes sizes ranged from 1 to 150 m.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Arctic Ocean\u003c/h2\u003e \u003cp\u003eRothrock and Thorndike (\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e1984\u003c/span\u003e) analyzed seven aerial and satellite images taken of the Beaufort Sea from March to October 1973\u0026ndash;1975. They used the caliper-diameter method. They found the distributions of cumulative number as a function of floe diameter were well fit by a single power function with scaling exponents ranging from \u0026minus;\u0026thinsp;1.7 to -2.5. The floe diameters ranged from 1,000 to 20,000 m.\u003c/p\u003e \u003cp\u003eMatsushita (\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e1985\u003c/span\u003e) analyzed one satellite image taken of the Sea of Okhotsk in December 1984. He used the caliper-diameter method. The distribution of cumulative number as a function of floe diameter was well fit by a single power function with a scaling exponent of -2.2. The floe diameters ranged from 5 to 30 m.\u003c/p\u003e \u003cp\u003eHolt and Martin (\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2001\u003c/span\u003e) analyzed fifteen satellite images taken of the Beaufort, Chukchi, and East Siberian Seas from August 1992. They used the caliper-diameter method. The distributions of cumulative number as a function of floe diameter were well fit by a single power function with scaling exponents ranging from \u0026minus;\u0026thinsp;1.9 to -2.6. The floe diameters ranged from 1,000 to 20,000 m.\u003c/p\u003e \u003cp\u003eInoue et al. (\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2004\u003c/span\u003e) analyzed two aerial images taken of the Sea of Okhotsk in February 18, 2000. They used the caliper-diameter method. They found the distributions of cumulative number as a function of floe diameter were well fit by a single power function with scaling exponents ranging from \u0026minus;\u0026thinsp;1.5 to -2.1. The floe diameters ranged from 10 to 100 m.\u003c/p\u003e \u003cp\u003eToyota et al. (\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2006\u003c/span\u003e) analyzed four helicopter, icebreaker, and satellite images taken of the Sea of Okhotsk in February 2003. They used the caliper-diameter method. They found the distributions of cumulative number as a function of floe diameter were well fit by two scaling exponents. A scaling exponent of -1.15 was found for floes smaller than 40 m and \u0026minus;\u0026thinsp;1.87 for floes larger than 40 m. The floe diameters ranged from 1 to 1500 m.\u003c/p\u003e \u003cp\u003ePerovich and Jones (\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2014\u003c/span\u003e) analyzed nine aerial and satellite images taken of the Beaufort Sea between June and September 1998. They used the caliper-diameter method. They found the distributions of cumulative number as a function of floe diameter were well fit by a single power function with scaling exponents ranging from \u0026minus;\u0026thinsp;2.0 to -2.2. The floe diameters ranged from 10 to 10,000 m.\u003c/p\u003e \u003cp\u003eGherardi and Lagomarsino (\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2015\u003c/span\u003e) analyzed three satellite images taken of the Kara Sea, Svalbard area, and Barents Sea from the Spring seasons of 2000, 2001, and 2009, respectively. They used the caliper-diameter method. They found the distributions of non-cumulative log-binned histograms of caliper-diameter were well fit by a single power function with a scaling exponent of -2.0 (Gherardi and Lagomarsino, \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2015\u003c/span\u003e). The floe areas ranged from 2 to 5,000 m.\u003c/p\u003e \u003cp\u003eWang et al. (\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2016\u003c/span\u003e) analyzed eighteen satellite images taken of the Beaufort and Chukchi Seas from summer through fall 2014. They used the caliper-diameter method. They found the distributions of cumulative number as a function of floe diameter were well fit by a single power function with scaling exponents ranging from \u0026minus;\u0026thinsp;2.77 to -4.12. The floe diameters ranged from 1 to 40,000 m.\u003c/p\u003e \u003cp\u003eGeise et al. (\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2017\u003c/span\u003e) analyzed six satellite images taken of the East Siberian Sea from June-August 2000\u0026ndash;2002. They used pixel grouping to measure the floe surface area. The distribution of cumulative number as a function of floe area were well fit by two power functions with scaling exponents for the smaller floes ranged from \u0026minus;\u0026thinsp;0.3 to -0.6 and scaling exponent for the larger floes ranged from \u0026minus;\u0026thinsp;0.6 to -1.0. The size at the inflection point ranged from 280 x 10\u003csup\u003e3\u003c/sup\u003e to 485 x 10\u003csup\u003e3\u003c/sup\u003e m\u003csup\u003e2\u003c/sup\u003e. The floe areas ranged from 30 m\u003csup\u003e2\u003c/sup\u003e to 28.4 x 10\u003csup\u003e6\u003c/sup\u003e m\u003csup\u003e2\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eHwang et al. (\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2017\u003c/span\u003e) analyzed multiple satellite images taken of the Beaufort Sea from July 19, August 15, and August 23, 2014. They used the caliper-diameter method. They found the distributions of non-cumulative log-binned histograms of caliper-diameter were well fit by a single power function with scaling exponents ranging from \u0026minus;\u0026thinsp;2.7 to -3.0 (Hwang et al, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2017\u003c/span\u003e, Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). The floe diameters ranged from 245 to ~\u0026thinsp;4000 m.\u003c/p\u003e \u003cp\u003eStern et al. (\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2018\u003c/span\u003e) analyzed 273 satellite images taken of the Beaufort and Chukchi Seas from March to October 2013 and 2014. They used the caliper-diameter method. They found the distributions of non-cumulative log-binned histograms of caliper-diameter were well fit by a single power function with scaling exponents ranging from \u0026minus;\u0026thinsp;2 to -2.9 (Stern et al., \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2018\u003c/span\u003e, Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e). The floe diameters ranged from 10 to 30,000 m.\u003c/p\u003e \u003c/div\u003e"},{"header":"3. Calculation of Equivalent Scaling Dimensions","content":"\u003cp\u003eTable\u0026nbsp;1 provides a list of the results from our study and previous studies of ice floe scaling in the Antarctic and Arctic regions. To directly compare scaling exponents, we convert scaling exponents reported in the literature to equivalent scaling exponents of cumulative number as a function of floe area distributions. For each study, the uppermost value reported in Table\u0026nbsp;1 is the scaling exponent provided in the original research paper.\u003c/p\u003e \u003cp\u003eTo obtain equivalent scaling exponents, the order of operations is important. The conversion from caliper-diameter to area was determined for a cumulative distribution by Stern et al. (\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). For studies that reported the scaling exponent for a non-cumulative histogram distribution, we first obtain the equivalent dimension for the cumulative distribution. Second, for studies that reported a dimension for caliper-diameter, we convert to the equivalent dimension for floe area.\u003c/p\u003e \u003cp\u003e \u003cb\u003eConversion of scaling exponents from non-cumulative histogram distributions to cumulative number versus size distributions\u003c/b\u003e \u003c/p\u003e \u003cp\u003e \u003cem\u003eNon-cumulative linearly-binned histogram distributions.\u003c/em\u003e The scaling exponent for a non-cumulative linearly binned histogram distribution that follows a power function has a slope that is steeper by an integer value of one (1) as compared to the same data plotted as a cumulative number versus size distribution (e.g., Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e of Burroughs and Tebbens, \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2001\u003c/span\u003e). To obtain the equivalent scaling exponent for a cumulative number versus size distribution, when starting with the scaling exponent for a non-cumulative linearly binned histogram distribution, we add an integer value of one (1). As a specific example, Paget et al. (\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2017\u003c/span\u003e) reported scaling exponents of -1.9 to -3.5 for a non-cumulative linearly binned distribution, and we provide the equivalent scaling exponent for the cumulative number versus size distribution of -0.9 to -2.5 in square brackets [ ] (Table\u0026nbsp;1).\u003c/p\u003e \u003cp\u003e \u003cem\u003eNon-cumulative logarithmically-binned histogram distribution.\u003c/em\u003e The scaling exponent for a cumulative number versus size distribution that follows a power function has a slope that is equal to the slope of the same data plotted as a non-cumulative linearly binned histogram distribution (e.g., Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e2\u003c/span\u003e of Burroughs and Tebbens, \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2001\u003c/span\u003e). In Table\u0026nbsp;1, for studies that reported the non-cumulative linearly binned histogram distributions (e.g., Gherardi \u0026amp; Lagomarsino, \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2015\u003c/span\u003e and Stern et al, \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2018\u003c/span\u003e), the scaling exponents are the same as for the cumulative number vs. size distributions. Therefore, there is no conversion of the value of the scaling exponent.\u003c/p\u003e \u003cp\u003e \u003cb\u003eConversion from cumulative number versus caliper-diameter to cumulative number versus area\u003c/b\u003e \u003c/p\u003e \u003cp\u003eThe scaling exponents derived from a 1-dimensional representation of the surface area (caliper-diameter) of a floe are not equivalent to those calculated from 2-dimensional measures of floe area. The scaling exponents calculated by the caliper-diameter method can be converted to the equivalent surface area scaling exponents by dividing the caliper-diameter scaling exponent by two (Stern et al., \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). In Table\u0026nbsp;1, we show the equivalent surface area scaling exponent in parentheses ( ).\u003c/p\u003e"},{"header":"4. Data","content":"\u003cp\u003eThis study is based on four mostly or completely cloud free satellite images of four floe packs at different stages of fragmentation during the Antarctic summer months (November 2016, December 2015, January 2015, and February 2015) in the Weddell Sea (Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e) obtained from the USGS Earth Explorer website (United States Geological Survey, \u003cspan class=\"CitationRef\"\u003e2016\u003c/span\u003e).\u003c/p\u003e\n\u003cp\u003eThe Weddell Sea was selected because of the image clarity, quality, and cloud coverage less than twenty percent. The Weddell Sea is a preferred research location for many ice floe studies because it is protected by the Antarctic peninsula.\u003c/p\u003e\n\u003cp\u003eThe images used in this study were collected by the Landsat 8 satellite. To measure the upper surface areas of individual floes, ideally the portions of the images analyzed must be cloud free because the bands used to view the ice floes do not penetrate through clouds. Melting ponds on the surface of the floes were not removed from the images. Satellite images were greyscale at 30 m per pixel resolution. We use floe area as the measure of floe size in contrast to the caliper-diameter method used in many previous studies. A visual inspection of the satellite images reveals no characteristic floes size (e.g., Figs. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e, \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e, \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e, \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e, and \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e), suggesting qualitatively that the distribution of floe sizes is scale invariant and might, therefore, be described by a power function.\u003c/p\u003e\n\u003cp\u003eThe satellite images obtained from Earth Explorer were imported into ESRI\u0026rsquo;s geographic information system (GIS) program, ArcPro (2.2). Tiff pixels from these images were classified into two classes, ice and water, using ArcPro\u0026rsquo;s Supervised Classification tool. The high brightness pixels represent the ice, while the low brightness pixels represent water. Individual floes larger than 3 x 10\u003csup\u003e6\u003c/sup\u003e m\u003csup\u003e2\u003c/sup\u003e were identified by grouping of ice class pixels and are surrounded by water class pixels. The Raster to Polygon tool was used to automatically identify the edges of each floe. The resulting floe polygons form the floes seen in the raster image (Fig. \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e). The area and perimeter length of each floe in the raster image were automatically tabulated in the attribute table by ArcPro. Floes cut off by the edge of the image, regions obscured by cloud cover, and slush/finely crushed ice were not included in the analysis.\u003c/p\u003e"},{"header":"5. Analysis","content":"\u003cp\u003eA plot of the cumulative number versus floe area is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e8\u003c/span\u003e. The data for each image are well fit by two power functions (Eq.\u0026nbsp;\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e), with an inflection point between them.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe choice of an algorithm for fitting a distribution that appears to be a power function has been debated in recent years (Deluca \u0026amp; Corral, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Holme, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2019\u003c/span\u003e). One method is that of Clauset et al. (\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2009\u003c/span\u003e). Corral \u0026amp; Gonzalez (2019) enumerated three issues with the Clauset et al. (2019) method: the method cannot be extended to truncated power distributions; there is no justification why the minimization of the Kolmogorov-Smirnov distance should work to find a meaningful scaling exponent; and the method was found to fail when applied to simulated data with power function tails, despite the data being a known power function. For this paper, the specific scaling exponent is not as important as the result that the data are well represented by a power function. Further, we are not extrapolating the results beyond the range of the data, which would amplify any errors in the reported values. Given these issues, we have used the same method used in our previous paper studying ice floe distributions in the Artic (Geise et al., \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2017\u003c/span\u003e), to facilitate direct comparison of results.\u003c/p\u003e \u003cp\u003eFor each of the data sets we analyze, we find the parameters C and β using the Levenberg-Marquardt algorithm to minimize Chi squared (Press et al., \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2001\u003c/span\u003e) using the Wavemetrics software IgorPro version 6.37. This method is appropriate for fitting the power functions to data and calculating the associated error; the method assumes that the errors in fitting the functions to the data are normally distributed (Press et al., \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2001\u003c/span\u003e). In all analyses for this study, the data were fit using IgorPro by the power function \u003cem\u003ey\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u003cem\u003e+\u0026thinsp;Cx\u003c/em\u003e\u003csup\u003e\u003cem\u003e-β\u003c/em\u003e\u003c/sup\u003e,\u003cem\u003e \u003c/em\u003ewhere \u003cem\u003ey\u003c/em\u003e\u003csub\u003e\u003cem\u003e0\u003c/em\u003e\u003c/sub\u003e was set to zero and the fit was found to converge. Errors are reported to plus/minus one standard deviation in Table\u0026nbsp;1.\u003c/p\u003e \u003cp\u003eEach of the four data sets analyzed and plotted on Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e8\u003c/span\u003e exhibit two power scaling regimes where larger floes have a distribution whose scaling exponent is greater than that of the small floes. The range of floe areas on either side of the inflection point is greater than one order of magnitude for the smaller floes and nearly one order of magnitude for the larger floes, where one order of magnitude is considered to be the minimum range for uniquely fitting a power function to a dataset.\u003c/p\u003e \u003cp\u003eThe scaling exponents -β exhibit a range of values (Table\u0026nbsp;1). The scaling region for smaller floes has scaling exponents -β ranging from \u0026minus;\u0026thinsp;0.8 to -1.0. The larger floes have scaling exponents -β ranging from \u0026minus;\u0026thinsp;1.5 to -1.8. The inflection point between the two regions ranges from 86 x 10\u003csup\u003e6\u003c/sup\u003e to 174 x 10\u003csup\u003e6\u003c/sup\u003e m\u003csup\u003e2\u003c/sup\u003e. The number of floes analyzed in each image varies from 303 (February 2015) to 1,078 (December 2015).\u003c/p\u003e \u003cp\u003eInflection point sizes decrease roughly sequentially from November to February as sea ice evolves from larger to smaller floes throughout the season (Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e8\u003c/span\u003e). Note that cushioning of larger floes by smaller floes is present for almost all floe sizes. This suggests that once the floe size distribution is established at the time of initial breakup, further size reduction is primarily by grinding and melting of the sides of floes, not by fracturing across the floes (Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e3\u003c/span\u003eb).\u003c/p\u003e \u003cp\u003eSumming the areas of the larger floes, those above the inflection point, and dividing by the total floe area for each image indicates that the area of the large floes is 21\u0026ndash;44% of the total floe area (Table\u0026nbsp;1). Note that this differs from the percentage of total floe area above and below the inflection points found in the Arctic by Geise et al. (\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2017\u003c/span\u003e) where the larger floes, those greater than the inflection points, accounted for 72\u0026ndash;95% of the total floe area.\u003c/p\u003e"},{"header":"6. Fragmentation Model","content":"\u003cp\u003eGeise et al. (\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2017\u003c/span\u003e) proposed a model that produces a power distribution of fragment sizes, which may explain the power distribution of floe areas but not the existence of an inflection point between the larger and smaller floe size distributions. Figure\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e9\u003c/span\u003e illustrates their probabilistic model for fragmentation of a cubic volume for any material where fracture is the method of fragmentation. At iteration n\u0026thinsp;=\u0026thinsp;1 the volume is broken into eight smaller cubes of equal size. At iteration n\u0026thinsp;=\u0026thinsp;2 some of the eight cubes are randomly selected and broken into eight smaller cubes. The probability that a cube will be fragmented is an adjustable variable. In Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e9\u003c/span\u003e, the probability of fragmentation is p\u0026thinsp;=\u0026thinsp;0.25. At iteration n\u0026thinsp;=\u0026thinsp;3 the model bifurcates based on whether (A) only the smallest of the cubes or (B) a cube of any size, can be randomly selected and broken. With a large number of iterations, the rule for A or B determines the cumulative frequency versus size distribution of the fragments. Rule A leads to a power fragment size distribution and Rule B leads a lognormal fragment size distribution. As we observe power distributions in the size (area) of floes, we propose that a process following Rule A is appropriate for ice floes. The choice of probability of fragmentation controls the scaling exponent of the power law distribution (e.g., p\u0026thinsp;=\u0026thinsp;0.25 yields a b of -0.5). We note that this fragmentation model generates a single power distribution without an inflection point. This model could be appropriate for ice floe distributions reporting a single power distribution of floe areas (Table\u0026nbsp;1, studies that report \u0026ldquo;b with no inflection\u0026rdquo;).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eSammis and King (\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2007\u003c/span\u003e) propose a physical process that results in power scaling of fragmentation sizes of geologic materials in a fault gauge. Adjacent fragments must be approximately the same size to break each other by fracture. They observe that in the process of fragmentation by grinding and crushing, larger fragments are surrounded by smaller fragments that cushion them and protected them thus greatly reducing the probability of further fragmentation. They propose that such a process leads to a power distribution in fragmentation size. Visual inspection of Figs.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e3\u003c/span\u003ea, \u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e4\u003c/span\u003ea, \u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e, and \u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e6\u003c/span\u003e reveal that for ice floes the large fragments are surrounded by smaller fragments which may similarly have cushioned them and reduced the probability of further fragmentation.\u003c/p\u003e"},{"header":"7. Discussion and Conclusions","content":"\u003cp\u003eThe distribution of floe area as a function of cumulative number versus floe area in each image were well fit by two power functions that meet at an inflection point between the larger floes and the smaller floes (Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e8\u003c/span\u003e, Table\u0026nbsp;1). The floe areas range from 3 x 10\u003csup\u003e6\u003c/sup\u003e to 550 x 10\u003csup\u003e6\u003c/sup\u003e m\u003csup\u003e2\u003c/sup\u003e. Floe areas larger than the inflection points are well fit by a power function with scaling exponents of -1.5 to -1.8 while the floes areas smaller than the inflection points are well fit by a power function with a scaling exponents of -0.8 to -1.0. The inflection points range from 58 x 10\u003csup\u003e6\u003c/sup\u003e to 155 x 10\u003csup\u003e6\u003c/sup\u003e m\u003csup\u003e2\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eThis study supports some of the previous studies that found that the cumulative number versus floe area distribution of seasonal floes are distributed in two power scaling regimes in the Antarctic region (Steer et al., \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2008\u003c/span\u003e; Toyota et al., \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2011\u003c/span\u003e; and Toyota et al., \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e2015\u003c/span\u003e) and in the Arctic (Toyota et al., \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2006\u003c/span\u003e and Geise et al., \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). We suggest that the two power scaling regimes and the inflection between the large and small floes are established during the initial breakup of the sea ice solely by the process of fracture. The distributions of floe size regimes retain the same scaling exponent (-b are within 2 standard deviations for three out of four of the images; see Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e8\u003c/span\u003e and Table\u0026nbsp;1) as the sea ice evolves from larger to smaller floes throughout the season due to grinding and melting as seen in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e8\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eWhen the initial ice breaks up by fracture the floe size distribution and the scaling exponents are established. The scaling exponents established during the initial ice break-up are maintained through the summer season and change minimally (by not more than 0.1) during the reduction of floe sizes at all scales through the processes of grinding and melting. While the scaling exponents \u003cem\u003eb\u003c/em\u003e do not appreciably change during the season in our study, the activity level C tends to decrease (Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e8\u003c/span\u003e, Eq.\u0026nbsp;\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). As C decreases, the number of floes of any and all sizes present is smaller. If there is not change is slope, the decrease in size is equal in log space at all sizes. indicating the floe sizes have systematically decreased. This can be seen in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e8\u003c/span\u003e where the data points for Feb 2015 are below those of Jan 2015, and the largest sized floe is also smaller.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003eThe authors declare that no funds, grants, or other support were received during the preparation of this manuscript.\u003c/p\u003e\n\n\u003cp\u003eThe authors have no relevant financial or non-financial interests to disclose.\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eCoffey obtained the satellite data, analyzed the images for floe area, and created many of the figures.Barton conceived and coordinated the project.Tebbens lead the statistical analysis of the data sets.We all worked on writing the manuscript.\u003c/p\u003e\u003ch2\u003eAcknowledgements\u003c/h2\u003e \u003cp\u003eThe authors thank Greg Geise and Dan Khoel for their help in processing the GIS data. We also thank two anonymous reviewers for constructive criticism that greatly improved an early version of this manuscript.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eBurroughs, S.M. \u0026amp; Tebbens, S.F. (2001). Upper-truncated Power Law Distributions, \u003cem\u003eFractals,\u0026nbsp;\u003c/em\u003e9, 209-222. https://doi.org/10.1142/S0218348X01000658\u003c/li\u003e\n \u003cli\u003eClauset, A., Shalizi, C. R. \u0026amp; Newman., M. E. J. (2009). Power-Law Distributions in Empirical Data.\u003cem\u003e\u0026nbsp;SIAM Review\u003c/em\u003e, \u003cem\u003e51\u003c/em\u003e(4), 661\u0026ndash;703. \u003cem\u003eJSTOR\u003c/em\u003e.\u0026nbsp;https://doi.org/10.1137/070710111\u003c/li\u003e\n \u003cli\u003eCorral, \u0026Aacute;., \u0026amp; Gonz\u0026aacute;lex, \u0026Aacute;., (2019). Power Law Size Distributions in Geoscience Revisited. \u003cem\u003eEarth and Space Science, 6\u003c/em\u003e(5). https://doi.org/10.1029/2018EA000479\u003c/li\u003e\n \u003cli\u003eDeluca, A., \u0026amp; Corral, \u0026Aacute;. (2013). Fitting and goodness-of-fit test of non-truncated and truncated power law distributions. \u003cem\u003eActa Geophysica\u003c/em\u003e, \u003cem\u003e61\u003c/em\u003e, 1351\u0026ndash;1394. https://doi.org/10.2478/s11600-013-0154-9\u003c/li\u003e\n \u003cli\u003eGeise, G. R., Barton, C. C. \u0026amp; Tebbens, S. F. (2017). Power Scaling and Seasonal Changes of floe areas in the Arctic East Siberian Sea, \u003cem\u003ePure and Applied Geophysics\u003c/em\u003e, \u003cem\u003e174\u003c/em\u003e(387). https://doi.org/10.1007/s00024-016-1364-2\u003c/li\u003e\n \u003cli\u003eGherardi, M., \u0026amp; Lagomarsino, M. C. (2015). Characterizing the size and shape of sea ice floes. \u003cem\u003eScientific Reports, 5\u003c/em\u003e, 10226. https://doi.org/10.1038/srep10226\u003c/li\u003e\n \u003cli\u003eGrose, M., \u0026amp; McMinn, A. (2003). Algal biomass in East Antarctic Pack ice: How much is in the East? In A. H. L. Huiskes, W. W. C. Gieskes, J. Rozema, R. M. L. Schorno, S. M. van der Vies \u0026amp; W. J. Wolff (Eds.) \u003cem\u003eAntarctic biology in a global context\u003c/em\u003e (pp. 182\u0026ndash;186). Backhuys Publishers. https://doi.org/ 10.1017/S0954102003251725\u003c/li\u003e\n \u003cli\u003eHolme, P.\u0026nbsp;(2019). Rare and everywhere: Perspectives on scale-free networks. \u003cem\u003eNature Communications\u003c/em\u003e,\u003cem\u003e10\u003c/em\u003e, 1016. https://doi.org/10.1038/s41467-019-09038-8\u003c/li\u003e\n \u003cli\u003eHolt, B. and Martin, S. (2001). The effect of a storm on the 1992 summer sea ice cover of the Beaufort, Chukchi, and East Siberian Seas, \u003cem\u003eJ. Geophys. Res. Ocean.\u003c/em\u003e, \u003cem\u003e106\u003c/em\u003e, 1017\u0026ndash;1032. https://doi.org/10.1029/1999JC000110\u003c/li\u003e\n \u003cli\u003eHwang, B., Wilkinson, J., Maksym, T. Graber, H.C., Schweiger, A., Horvat, C., Perovich, D.K., Arntsen, A.E., Stanton, T.P., Ren, J., \u0026amp; Wadhams, P. (2017). Winter-to-summer transition of Arctic sea ice breakup and floe size distribution in the Beaufort Sea. \u003cem\u003eElementa: Science of the Anthropocene, 5 (40),\u0026nbsp;\u003c/em\u003ehttps://doi.org/10.1525/elementa.232\u003c/li\u003e\n \u003cli\u003eInoue, J., Wakatsuchi, M., \u0026amp; Fujiyoshi, Y. (2004). Ice floe distribution in the Sea of Okhotsk in the period when sea-ice extent is advancing. \u003cem\u003eGeophysical Research Letters, 31\u003c/em\u003e(20), L20303. https://doi.org/10.1029/2004GL020809\u003c/li\u003e\n \u003cli\u003eLensu, M. (1990, August 22-23). \u003cem\u003eThe fractality of sea ice cover\u003c/em\u003e. IAHR, the 10\u003csup\u003eth\u003c/sup\u003e International Symposium on Ice, Helsinki University of Technology, Espoo, Finland. Proceedings pp. 300\u0026ndash;313.\u003c/li\u003e\n \u003cli\u003eLu, P., Li, Z. J., Zhang, Z. H. \u0026amp; Dong, X. L. (2008). Aerial observations of floe size distribution in the marginal ice zone of summer Prydz Bay. \u003cem\u003eJournal of Geophysical Research: Oceans, 113\u003c/em\u003e(C2). https://doi.org/10.1029/2006JC003965\u003c/li\u003e\n \u003cli\u003eMatsushita, M.\u0026nbsp;(1985). Fractal viewpoint of fracture and accretion, \u003cem\u003eJournal of the Physical Society of Japan\u003c/em\u003e, \u003cem\u003e54\u003c/em\u003e\u003cem\u003e(3),\u003c/em\u003e 857\u0026ndash;860. https://doi.org/10.1143/JPSJ.54.857\u003c/li\u003e\n \u003cli\u003e\u003cem\u003eNational Aeronautics and Space Administration\u003c/em\u003e (2016). https://earthobservatory.nasa.gov/features/SeaIce/page4.php\u003c/li\u003e\n \u003cli\u003ePaget, M. J., Worby, A. P. \u0026amp; Michael, K. J. (2017). Determining the floe-size distribution of East Antarctica sea ice from digital aerial photographs. \u003cem\u003eAnnals of Glaciology, 33,\u003c/em\u003e 94\u0026ndash;100. https://doi.org/10.3189/172756401781818473\u003c/li\u003e\n \u003cli\u003ePerovich, D. K., \u0026amp; Jones, K. F. (2014). The seasonal evolution of sea ice floe size distribution, \u003cem\u003eJournal of Geophysical. Research - Oceans\u003c/em\u003e, \u003cem\u003e119\u003c/em\u003e, 8767\u0026ndash;8777, https:/doi.org/10.1002/2014JC010136\u003c/li\u003e\n \u003cli\u003ePress, W. H., Teukolsky, S. A., Vetterling, W. T., \u0026amp; Flannery, B. P. (2001). \u003cem\u003eNumerical recipes in FORTRAN: The art of scientific computing\u003c/em\u003e, Cambridge University Press.\u003c/li\u003e\n \u003cli\u003eRothrock, D. A., \u0026amp; Thorndike, S. (1984). Measuring the sea ice floe size distribution. \u003cem\u003eJournal of Geophysical Research: Oceans 89,\u0026nbsp;\u003c/em\u003e6477-6486. http://doi.org/10.1029/JC089iC04p06477\u003c/li\u003e\n \u003cli\u003eSammis, C. G., \u0026amp; King, G. C. P. (2007). Mechanical origin of power law scaling in fault zone rock. \u003cem\u003eGeophysical Research Letters, 34\u003c/em\u003e, L04312. https://doi.org/10.1029/2006GL028548\u003c/li\u003e\n \u003cli\u003eShi, Q., Yang, Q., Mu, L., Wang, J., Massonnet, F., \u0026amp; Mazloff, M. R. (2021). Evaluation of sea-ice thickness from four reanalyses in the Antarctic Weddell Sea, \u003cem\u003eThe Cryosphere, 15\u003c/em\u003e, 31\u0026ndash;47.\u0026nbsp;https://doi.org/10.5194/tc-15-31-2021\u003c/li\u003e\n \u003cli\u003eSteer, A., Worby, A., \u0026amp; Heil, P. (2008). Observed changes in sea-ice floe size distribution during early summer in the western Weddell Sea, \u003cem\u003eDeep-Sea Research II, 55\u003c/em\u003e, 933\u0026ndash;942. https://doi.org/10.1016/j.dsr2.2007.12.016\u003c/li\u003e\n \u003cli\u003eStern, H.L., Schweiger, A. J., Stark, M., Zanj, J., Steele, M., \u0026amp; Hwang, B. (2018). Seasonal evolution of the sea-ice floe size distribution in the Beaufort and Chukchi seas. \u003cem\u003eElementa Science of the Anthropocene, 6\u003c/em\u003e(48). https://doi.org/10.1525/elementa.305\u003c/li\u003e\n \u003cli\u003eToyota, T., Haas, C. \u0026amp; Tamura, T. (2011). Size distribution and shape properties of relatively small sea-ice floes in the Antarctic marginal ice zone in late winter. \u003cem\u003eDeep-Sea Res II, 58\u003c/em\u003e, p. 1182\u0026ndash;1193. https://doi.org/10.1016/j.dsr2.2010.10.034\u003c/li\u003e\n \u003cli\u003eToyota, T., Kohout, A. \u0026amp; Fraser, A. (2015). Formation processes of sea ice floe size distribution in the interior pack and its relationship to the marginal ice zone off East Antarctica. \u003cem\u003eDeep Sea Research Part II Topical Studies in Oceanograp\u003c/em\u003ehy, \u003cem\u003e131\u003c/em\u003e, 28-40. \u0026nbsp;https://doi.org/10.1016/j.dsr2.2015.10.003\u003c/li\u003e\n \u003cli\u003eToyota, T., Takatsuji, S., \u0026amp; Nakayama, M. (2006). Characteristics of sea ice floe size distribution in the seasonal ice zone, \u003cem\u003eGeophysical Research Letters\u003c/em\u003e, \u003cem\u003e33\u003c/em\u003e, L02616, http://doi.org/10.1029/2005GL024556\u003c/li\u003e\n \u003cli\u003eUnited States Geological Survey, 2016, Earth Explorer, https://earthexplorer.usgs.gov/\u003c/li\u003e\n \u003cli\u003eWang, Y., Holt, B., Erick Rogers, W., Thomson, J., \u0026amp; Shen, H. H. (2016). Wind and wave influences on sea ice floe size and leads in the Beaufort and Chukchi Seas during the summer-fall transition 2014, \u003cem\u003eJournal of Geophysical Research - Oceans\u003c/em\u003e, \u003cem\u003e121\u003c/em\u003e, 1502\u0026ndash;1525, https://doi.org/10.1002/2015JC011349\u003c/li\u003e\n \u003cli\u003eWorby,\u0026nbsp;A. P.\u0026nbsp;\u0026amp; Allison,\u0026nbsp;I. (1991). Ocean-atmosphere energy exchange over thin, variable concentration Antarctic pack ice. \u003cem\u003eAnnals of Glaciology\u003c/em\u003e\u003cem\u003e,15\u003c/em\u003e, 184\u0026ndash;190. https://doi.org/10.3189/1991AoG15-1-184-190\u003c/li\u003e\n \u003cli\u003eZhang, Q., \u0026amp; Skjetne, R. (2018). Sea Ice Image Processing with MATLAB, CRC Press, Boca Raton, FL, 272p.\u003c/li\u003e\n\u003c/ol\u003e"},{"header":"Table","content":"\u003cp\u003eTable 1 is available in the Supplementary Files section.\u003c/p\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Ice floes, power distribution, scaling, Antarctic, Southern Ocean","lastPublishedDoi":"10.21203/rs.3.rs-3784463/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3784463/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe distribution of cumulative number as a function of floe area of seasonal ice floes from four satellite images covering the summer season (November - February) in the Weddell Sea, off Antarctica, during the summer ice breakup were well fit by two scale-invariant power functions. For a power function of the form N\u0026thinsp;=\u0026thinsp;C x\u003csup\u003e-β\u003c/sup\u003e, the scaling exponents -β for the larger floe areas range from \u0026minus;\u0026thinsp;1.5 to -1.8. Scaling exponents -β for the smaller floe areas range from \u0026minus;\u0026thinsp;0.8 to -1.0. The inflection point between the two scaling regimes ranges from 58 x 10\u003csup\u003e6\u003c/sup\u003e to 155 x 10\u003csup\u003e6\u003c/sup\u003e m\u003csup\u003e2\u003c/sup\u003e and generally moves from larger to smaller floe areas through the summer season. We propose that the two power scaling regimes and the inflection between them are established during the initial breakup of sea ice solely by the process of fracturing. Floe areas range from 3 x 10\u003csup\u003e6\u003c/sup\u003e to 550 x 10\u003csup\u003e6\u003c/sup\u003e m\u003csup\u003e2\u003c/sup\u003e. The distributions of floe size regimes retain approximately the same scaling exponents as the floe pack evolves from larger to smaller floe areas from the initial breakup through the summer season, due to scale-independent processes of fracturing, grinding and melting. The scaling exponents for floe area distribution are in the same range as those reported in previous studies of Antarctic and Arctic floes. A probabilistic model of fragmentation is presented that generates a single power scaling distribution of fragment size.\u003c/p\u003e","manuscriptTitle":"Power Scaling of Ice Floe Areas in the Weddell Sea, Southern Ocean With a Summary of Previous Ice Floe Scaling Studies","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-01-02 19:47:47","doi":"10.21203/rs.3.rs-3784463/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"f04339a2-88e7-4951-9dbb-16d93b6647fc","owner":[],"postedDate":"January 2nd, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-03-31T05:29:17+00:00","versionOfRecord":[],"versionCreatedAt":"2024-01-02 19:47:47","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3784463","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3784463","identity":"rs-3784463","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}