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:

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 m2.

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 103 to 485 x 103 m2. The floe areas ranged from 30 m2 to 28.4 x 106 m2.

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.