The size distribution of sea-ice floes is an important descriptor of the sea-ice cover. Most studies report that floe sizes follow a power-law distribution over some size range, but the power-law exponents often differ substantially. Other studies report two power-law regimes over different size ranges, or more complicated behavior. We review the construction of power-law floe size distributions and compare the results of previous studies. Differences between studies may be due to spatial and temporal variability of the floe size distribution, sampling variability, inadequacy of the power-law model, or flaws in the mathematical analysis. For a power-law model, the most accurate method for determining the exponent from data is Maximum Likelihood Estimation; least-squares methods based on log-log plots of the data yield biased estimates. After calculating the power-law exponent from data, a goodness-of-fit test should be applied to determine whether or not the power-law model actually describes the distribution of the data. These analysis principles have been described in the literature but have not generally been applied to floe size distributions. Numerical ice-ocean models are beginning to simulate the floe size distribution, which should give further insight into the interpretation of observational studies.
The sea-ice cover in the Arctic and Antarctic is often observed to consist of discrete pieces or floes. The sea-ice floe size distribution (FSD) is the number of floes in different size categories in a region, divided by the area of the region (Rothrock and Thorndike, 1984). This is also known as the floe number density. An alternative description of the FSD is the area covered by floes in different size categories, or the floe area density. The FSD is important when modeling physical processes of the sea-ice cover, and ice-ocean models have recently begun to include FSDs in their simulations (Zhang et al., 2015, 2016; Horvat and Tziperman, 2015, 2017). These models use the floe area density while most observational studies use the floe number density, but the two are mathematically related and there is no complication in switching from one to the other. In this work we consider only the floe number density.
This paper is a companion to Stern et al. (2018), in which we analyzed over 250 satellite images from NASA’s Moderate Resolution Imaging Spectroradiometer (MODIS, http://modis.gsfc.nasa.gov/), finding that in 95% of the images the FSD in the Beaufort and Chukchi seas follows a power-law distribution over a range of floe sizes from 2 to 30 km, according to a statistical goodness-of-fit test. The mean monthly power-law exponent undergoes a seasonal cycle reflecting floe break-up in spring and melting in summer. We also found, using a smaller sample of higher resolution images, that the FSD follows a power law for floe sizes from 10 m to 3 km, with a power-law exponent close to that found at the larger scale.
Previous observational studies (Table 1) have all reported power-law FSDs, but with widely varying exponents. Modeling studies (Section 4.3) have examined simulated FSDs (Zhang et al., 2015, 2016; Horvat and Tziperman, 2017). The purpose of the present work is to review the construction of power-law FSDs (Sections 2 and 3) and to compare the results of previous studies (Section 4) in order to assess their level of agreement or disagreement, given their sampling of the sea-ice cover at different locations, times, and resolutions. We conclude with recommended steps for analyzing floe size data, and a summary of the possible reasons for the wide variability in the power-law exponents of the FSDs seen in previous studies (Section 5).
Studies of the sea-ice floe size distribution. DOI: https://doi.org/10.1525/elementa.304.t1
Study
Location
Time period
Cum. or Non-cum.^{a}
Exponent^{b}
Floe size range (m)
Number of images^{c}
Rothrock and Thorndike (1984)
Beaufort Sea
June–Aug 1973–1975
Cum.
–1.7 to –2.5
100 to 20,000
7
Matsushita (1985)
Sea of Okhotsk
not given
Cum.
–2.2
not given
1
Kergomard (1989)
Fram Strait
June 1987
Cum.
–1.0 to –1.8as ice edge is approached
5 to 2000
5
Lensu (1990)
Weddell Sea, Antarctica
Feb 1990
Cum.
–1.4
1 to 100
1
Holt and Martin (2001)
Beaufort and Chukchi seas
Aug 1992
Cum.
–1.9 to –2.2 (Bfort)–2.2 to –2.6 (Chuk)
900 to 10,000
94
Paget et al. (2001)
East Antarctica
Aug 1995
Non-Cum.
–1.9 to –3.5as ice edge is approached
1 to 150
6
Inoue et al. (2004)
Sea of Okhotsk
Feb 18, 2000
Cum.
–1.5 to –2.1as ice edge is approached
7 to 53
2
Toyota et al. (2006)
Sea of Okhotsk
Feb 2003
Cum.
–1.2x < 40 m–1.9x > 40 m
1 to 1000
3
Toyota et al. (2011)
Weddell Sea, Antarctica
Sept–Oct 2006, 2007
Cum.
–1.0 to –1.5x < 40 m–3.2 to –7.6x > 40 m
2 to 100
122
Toyota et al. (2016)
Wilkes Land, East Antarctica
Sept–Nov 2012
Cum.
–1.3 to –1.4x < 100 m–2.9 to –3.1x > 1500 m
5 to 100and1500 to 10,000
16
Steer et al. (2008)
Weddell Sea, Antarctica
Dec 2004
Non-cum.
–1.9x < 20 m–2.8 to –3.4x > 20 m
2 to 400
130
Lu et al. (2008)
Prydz Bay, East Antarctica
Dec 2004 to Feb 2005
Cum.
–0.6 to –1.4as ice edge is approached
2 to 100
19
Perovich and Jones (2014)
Beaufort Sea
June–Sept 1998
Cum.
–2.0 to –2.2following a seasonal cycle
10 to 10,000
10
Gherardi and Lagomarsino (2015)
Barents/Kara seas and Weddell Sea
Spring 2000, 2001, 2003, 2009
Non-cum.
–2.0
2 to 5000
4
Geise et al. (2016)^{d}
East Siberian Sea
June–Aug 2000–2002
Cum.
–0.6 to –1.2x < 200–850 m–1.2 to –2.0x > 200–850 m
12 to 3900^{d}
6
Wang et al. (2016)
Beaufort and Chukchi seas
Summer–fall 2014
Cum.
–1.0 to –1.5
5 to 10,000
18
Hwang et al. (2017)
Beaufort Sea
Aug–Sept 2014
Non-cum.
–2.7 to –3.0
450 to 3000
29
Stern et al. (2018)
Beaufort and Chukchi seas
Mar–Oct 2013, 2014
Non-cum.
–1.9 to –2.8following a seasonal cycle
10 to 30,000
273
^{a} Cumulative (Cum.) or non-cumulative (Non-cum.) representation of FSD.
^{b} Exponent (or slope in log-log space) of power law describing FSD.
^{c} Number of images, photographs, or frames used to derive FSD. When multiple frames were stitched together to form a larger composite image from which one FSD was derived, only one image was counted. When several regions from the same image were analyzed separately, each region was counted as a separate image.
^{d} Geise et al. (2016) reported cumulative FSD versus floe area (A), not floe diameter (x). As A = kx^{2}, where k is a constant, we have multiplied their reported exponents by 2 for this table. Their reported floe areas range from 10^{2} m^{2} to 10^{7} m^{2}. We have converted these to floe sizes 12 to 3900 m using k = 0.66 (from Rothrock and Thorndike, 1984). Their reported transition scale separating two power-law regimes is 28 × 10^{3} to 485 × 10^{3} m^{2}. We have converted these to 200–850 m for this table (again using k = 0.66).
2. Mathematical considerations2.1. Cumulative and non-cumulative floe size distributions
Let x represent the size of an ice floe, such as its mean caliper diameter (Rothrock and Thorndike, 1984). For a collection of many floes, the FSD can be expressed in cumulative or non-cumulative form. The cumulative form, F(x), is the fractional number of floes larger than x. The non-cumulative form, f(x)dx, is the fractional number of floes between size x and x + dx. The two forms are related by f(x) = –dF/dx. Technically, F(x) should be called the complementary cumulative distribution function, and f(x) should be called the probability density function, but in the sea-ice literature they are both loosely referred to as the FSD, so we maintain that informal nomenclature in the present work.
Let a > 0 be the smallest floe size. A power-law formulation of the non-cumulative FSD is f(x) = cx^{–α} for a ≤ x < ∞, where α > 1 is a parameter and c = (α–1)a^{α–1} is a normalization constant that ensures that f(x) integrates to 1. The corresponding cumulative FSD is F(x) = Cx^{–α+1}, where C = c/(α–1), or F(x) = (x/a)^{–α+1}. The cumulative FSD is a power law with exponent one greater than that of the non-cumulative FSD. Many studies have used this form of F(x) as a model to fit to the cumulative distribution of their floe size data (Rothrock and Thorndike, 1984; Matsushita, 1985; Kergomard, 1989; Lensu, 1990; Holt and Martin, 2001; Inoue et al. 2004). An implicit assumption in using this form of F(x) is that the range of floe sizes extends to infinity, because F(x) → 0 only in the limit as x → ∞.
In reality, the value of x is bounded by a finite upper limit, x ≤ b, because there is a physical limit to the size of ice floes, and because satellite images are often unable to capture the largest floes. Suppose the non-cumulative FSD obeys a power law, f(x) = cx^{–α}, over the finite range a ≤ x ≤ b, with normalization constant c = (α–1)a^{α–1}/[1 – (b/a)^{–α+1}]. The corresponding cumulative FSD is F(x) = Cx^{–α+1} – R, where C = c/(α–1) and R is a constant of integration that ensures that F(b) = 0; i.e., R = Cb^{–α+1}. This form of F(x) is called an upper-truncated power law (Burroughs and Tebbens, 2001a, 2001b). In the limit as b → ∞, R → 0 and F(x) becomes a pure power law, Cx^{–α+1}. Several studies have used the upper-truncated power law as a model to fit to the cumulative distribution of their floe size data (Lu et al. 2008; Toyota et al. 2016; Wang et al. 2016).
The non-cumulative form of the power-law FSD, f(x) = cx^{–α}, is a straight line with slope –α in a log-log plot: log(f(x)) = log(c) – α log(x). However, on a finite interval a ≤ x ≤ b, the cumulative form F(x) = Cx^{–α+1} – R is not a straight line in a log-log plot. Instead, it has concave-down curvature (Figure 1; see also Figure 6 of Lensu et al. 2008). If the range of floe sizes is small, the upper-truncated power law curves steadily downward (Figure 1a). If the range of floe sizes is large, there is a power-law regime below some cutoff size and a fall-off regime above the cutoff size (which is about x = 500 m in (Figure 1b). This is purely the mathematical behavior of the upper-truncated power law.
Cumulative and non-cumulative power-law distributions. (a) Power-law number density with exponent –2 (black line) on the interval x = [2, 20]. Note the log-log axes. The cumulative distribution (solid red) is concave down. If the number density extended to infinite x, the limiting cumulative distribution (dashed red) would be a power law with exponent –1. (b) Same as (a) but for the interval x = [2, 2000]. Note that the cumulative distribution (solid red) now appears to have a power-law regime for x < 500 and a fall-off regime for x > 500. DOI: https://doi.org/10.1525/elementa.304.f1
Concave-down curvature of the cumulative FSD in log-log plots can be seen in real data (Figure 2; see also Figure 5 of Lensu et al., 2008, and Figures 6 and 7 of Toyota et al., 2016). Rothrock and Thorndike (1984) noted that some cumulative distributions “have steeper slopes for larger floes and more gradual slopes for smaller floes.” Wang et al. (2016) similarly noted: “All cumulative floe size distribution curves show a gradual bending toward shallower slopes for smaller floes.” These observations suggest that the cumulative FSD may be best characterized by an upper-truncated power law. However, the concave-down curvature at the largest floe sizes could also be due to the finite size effect, in which the largest floes are under-sampled due to the finite size of the domain or other limitations of the observations. This possibility is discussed in Section 3.1. Another possibility is that the cumulative FSD consists of two power-law regimes separated by a break-point (e.g., Toyota et al., 2006, 2011), as discussed in Section 4.1.
Cumulative and non-cumulative floe size distributions from real data. Using data from 4488 floes identified in a MODIS image of 18 May 2013 in the Beaufort Sea (see Stern et al., 2018), the non-cumulative number of floes per unit bin width (solid black) is plotted along with the least-squares fit in log-log space (dashed black), which has slope –2.04 and squared correlation 0.99. Fifteen logarithmically spaced bins between 1 and 13 km were used. The cumulative distribution (solid red) is concave down, and falls off at larger floe sizes. The limiting cumulative distribution (dashed red) is constructed to be a power law with slope –2.04 + 1 = –1.04. This distribution would be realized if the non-cumulative distribution extended to infinite floe size. Adding a constant R (see main text) to the values of the cumulative distribution (red dots) produces the green dots, which fall very close to the limiting cumulative distribution (dashed red). On the horizontal axis, mean caliper diameter is a measure of floe size (see Rothrock and Thorndike, 1984). DOI: https://doi.org/10.1525/elementa.304.f2
2.2. Determining the exponent of a power law from data
Given a set of floe sizes, how is the exponent of the hypothesized power-law distribution calculated? Fifteen of the studies in Table 1 used a curve-fitting method based on a log-log plot of the floe size data. Some of these studies fit a power law or an upper-truncated power law to the cumulative distribution; others fit a power law to the non-cumulative distribution. Some of the studies used a least squares method; others used a simple graphical method. The essential point is that the methods are all based on curve-fitting of data in log-log plots.
The three studies in Table 1 that use different methods are Perovich and Jones (2014) (PJ14 hereafter), Hwang et al. (2017), and Stern et al. (2018). In PJ14, the FSD was assumed to be a power law a priori, and individual floes were not identified in the imagery, so the power-law exponent is not based on floe size data; see PJ14 for further details. In Hwang et al. (2017) and Stern et al. (2018), the power-law exponent of the non-cumulative distribution was calculated from floe size data using the Maximum Likelihood Estimate (MLE).
In Appendix A of this paper we show how to construct the MLE of the power-law exponent (–α) for data drawn from a power-law distribution on a finite interval a ≤ x ≤ b. As the resulting equation for α cannot be solved analytically, we show how to solve it numerically in Appendix B. White et al. (2008; see their Table 1) gave the same equation for the MLE as we derive in Appendix A, but in a more complicated form. They stated that the equation must be solved numerically, but they did not give an explicit procedure for doing so.
Several mathematical studies have investigated the best way to determine the exponent of a power law from data (e.g., White et al., 2008; Clauset et al., 2009). The studies all agree that the MLE provides the best estimate of the exponent. According to White et al. (2008), the MLE of the power-law exponent has been shown mathematically to be the minimum-variance unbiased estimator. The studies also agree that binning the data to construct a non-cumulative distribution, followed by application of a least squares method in log-log space, leads to a biased estimate of the power-law exponent. The bias is especially large if evenly spaced bins are used. Logarithmically spaced bins can provide a reasonable estimate if there are an adequate number of samples in each bin. We confirmed these conclusions by running our own simulations with synthetic data drawn from power-law distributions. Note that the MLE does not involve binning the data.
2.3. Goodness of fit
It is important to note that simply calculating the best-fitting power-law for a data set does not answer the question of whether the data are in fact power-law distributed or not. That determination requires a goodness-of-fit test such as the Kolmogorov-Smirnov test (e.g., Clauset et al., 2009), which we used in our companion paper (Stern et al., 2018), finding that 244 out of 256 floe size data sets derived from MODIS imagery in the Beaufort and Chukchi seas in 2013–2014 were properly characterized by power laws. Hwang et al. (2017) also applied a goodness-of-fit test, but none of the other studies in Table 1 did so.
2.4. Summary of mathematical considerations
The following points should be considered when fitting a power-law model to floe size data. (1) The range of floe sizes in the model may be chosen to be finite or infinite. (2) The power-law exponent should be calculated by Maximum Likelihood Estimation, not by curve-fitting in log-log space. (3) A goodness-of-fit test should be applied to determine whether the data are in fact power-law distributed or not.
3. Sampling and binning3.1. Finite size effect and upper truncation
In dealing with real floe size data, sampling issues must be considered. Suppose that the “true” (perfectly sampled) non-cumulative FSD follows a power law on a finite interval (or equivalently, the cumulative FSD follows an upper-truncated power law). Now suppose that in practice, the largest floes are under-sampled because of the finite size of the domain or other limitations of the observations, such as cloud cover. This under-sampling will lead to concave-down curvature in a log-log plot of both the cumulative and non-cumulative FSD. Thus, if we are presented with the cumulative FSD in a log-log plot and we observe that it is linear over part of its range but it curves downward at the largest floe sizes, we cannot immediately say whether the downward curvature is due to the mathematical form of the upper-truncated power law or whether it is due to under-sampling of the largest floes. In contrast, if we are presented with the non-cumulative FSD in a log-log plot and we observe that it is linear over part of its range but it curves downward at the largest floe sizes, we know right away that the downward curvature is due to under-sampling of the largest floes. (Another possibility, of course, is that the data are not power-law distributed at all, but in this hypothetical example we presume that they are).
If under-sampling of the largest floes is believed to be a problem, the data should be cut off to exclude the under-sampled range. Assuming this cut has been made, if the cumulative FSD still shows concave-down curvature in a log-log plot, an upper-truncated power-law model of the form F(x) = Cx^{–α+1} – R could be plotted to see (visually) if it fits the data. But because the objective is to discern whether an underlying power law exists, a more visually intuitive approach is to calculate the constant R from the data, add it back to the observed cumulative FSD, and see whether the result approximates the limiting power law Cx^{–α+1}, which is a straight line in a log-log plot. This approach is illustrated in Figure 2 for a set of floes from a MODIS image (see Stern et al., 2018). It shows the relationship between the non-cumulative power-law distribution, the cumulative upper-truncated power-law distribution, and the limiting cumulative power-law distribution.
The calculation of R is as follows. From R = Cb^{–α+1} and C = c/(α–1) (see Section 2.1), together with the expression for c from the beginning of Appendix A, one finds (after some algebra) that R = 1/(r^{α–1}–1), where r = b/a. In Figure 1, which is an idealized situation, R exactly accounts for the difference between the cumulative distribution (solid red curve) and the limiting cumulative distribution (dashed red line): in panel (a), a = 2, b = 20 and α = 2, so the difference is R = 1/9 ≈ 1.1 × 10^{–1}; in panel (b), a = 2, b = 2000 and α = 2, so the difference is R = 1/999 ≈ 10^{–3}. In Figure 2, which uses real data, a = 1 km, b = 13 km and α = 2.04 (derived from the data), so R = 0.0746. Because the vertical axis is scaled according to the number of floes, rather than the fractional number, it is necessary to multiply by N = 4488 floes to obtain NR = 335. Adding this constant to the cumulative distribution (red symbols) gives an estimate of the limiting cumulative distribution (green symbols), which indeed is very close to the theoretical line with slope –1.04 (dashed red). Note that the line (dashed red) is not a fit to the limiting cumulative distribution, but rather is calculated by adding 1 to the slope of the non-cumulative distribution (dashed black). This approach shows that when the non-cumulative distribution is well characterized by a power law (with exponent –α), the cumulative distribution (which is an upper-truncated power law) can be “corrected” by adding a constant to recover the limiting cumulative distribution with power-law exponent –α + 1.
3.2. Power-law zone and fall-off zone
As noted in Section 2.1, many studies have used a pure power-law form of the cumulative FSD as a model to fit to the cumulative distribution of their floe size data. This model implicitly assumes that floe sizes are unbounded, x < ∞. For bounded floe sizes, a power-law density function on the finite interval a ≤ x ≤ b with exponent –α gives rise to a cumulative FSD that is an upper-truncated power law of the form F(x) = Cx^{–α+1} – R. This cumulative FSD will appear to be nearly a pure power law for Cx^{–α+1} >> R. To make this definite, we define the power-law zone of F(x) to be the range of x where Cx^{–α+1} ≥ 10 R. The fall-off zone is the range of x where Cx^{–α+1} < 10 R. The boundary x* between the zones satisfies Cx*^{(–α+1)} = 10 R. If random samples are drawn from this distribution, the fractional number of samples in the fall-off zone will be F(x*), and the fractional number in the power-law zone will be 1 – F(x*). Table 2 shows the percent of samples in the power-law zone; i.e., [1 – F(x*)] × 100% for different values of the power-law exponent (–α) and the ratio (b/a) of the largest to smallest floe size.
Percent of samples in power-law zone, by power law exponent and ratio of largest to smallest floe size^{a}. DOI: https://doi.org/10.1525/elementa.304.t2
Power law exponent
Ratio of largest to smallest floe size (b/a)
(b/a) = 10
(b/a) = 100
(b/a) = 1000
–1.5
0%
0%
71%
–2.0
0%
91%
99%
–2.5
71%
99%
99.97%
–3.0
91%
99.9%
99.999%
^{a} For a power-law density function f(x) = cx^{–α} defined on the finite interval a ≤ x ≤ b, the complementary cumulative distribution function is an upper-truncated power law of the form F(x) = Cx^{–α+1} – R. The power-law zone of F(x) is defined to be the range of x where F(x) is nearly a pure power law; i.e., where Cx^{–α+1} >> R. For random samples drawn from this distribution, the table shows the percent of samples that would fall in the power-law zone, as a function of the power-law exponent (–α) and the ratio (b/a) of the largest to smallest floe size. See Section 3.2 for further details.
Clearly the power-law zone grows larger as the power law steepens (larger α) and as the range of floe sizes characterized by the power law widens (larger b/a). If b/a is small (only one order of magnitude) and α ≤ 2, then there is no power-law zone at all: the entire range of x is the fall-off zone, as in Figure 1a. As b/a becomes larger, the power-law zone becomes larger. For b/a = 1000 and α = 2, as in Figure 1b, 99% of randomly drawn samples will be in the power-law zone (x ≤ 100a in this case). If 1000 samples are drawn, about 990 will be in the power-law zone and only 10 will be in the fall-off zone. That makes it easy to overlook the fact that a fall-off zone even exists, and to see the random samples as obeying a pure power law. If the samples are grouped into bins, that makes it even easier to mask the existence of the fall-off zone. This may explain why some studies that plot the cumulative FSD of their data find pure power-law behavior, rather than upper-truncated power-law behavior.
The above analysis assumes proper sampling from the power-law distribution. In practice, if the underlying (non-cumulative) FSD is a pure power law but the largest floes are under-sampled, the resulting data set will show a fall-off of both the non-cumulative and the cumulative distributions at the largest floe sizes. Thus the under-sampling problem becomes tangled with the mathematical fall-off behavior of the upper-truncated power law, because they both show up in the same way: as a constant subtracted from a power law.
3.3. Visual representation of the floe size distribution and binning floe size data
To construct a cumulative distribution from a set of data, it is not necessary to bin the data. If the n data values in ascending order are x_{1}, x_{2} … x_{n}, one simply constructs a piecewise constant function F(x) = 1 – k/n for x_{k} < x < x_{k+1}, for k = 1, 2 … n–1, with F(x) = 1 for x < x_{1} and F(x) = 0 for x > x_{n}. The function F(x) has a step of size 1/n at every data value x_{1}, x_{2} … x_{n}.
To construct a non-cumulative distribution, the data must be binned. One defines a set of floe-size bins, counts the number of floes in each bin, and divides the count by the bin width. Two common methods of binning are linear (all bins of equal width) and logarithmic (successive bins increase in width by a constant factor). In connection with power-law-distributed data, White et al. (2008) stated that “binning is useful for visualizing the frequency distribution, and normalized logarithmic binning performs well at this task” (their use of “normalized” means that bin counts are divided by bin widths). Indeed, one should always look at the distribution of the data before deciding that a power law might be a good representation of it. For this visual check, logarithmic binning should be used. In Appendix C we derive guidelines for the number of bins to use, in order to ensure an adequate number of samples in each bin.
Burroughs and Tebbens (2001a, 2001b) discussed upper-truncated power laws, and their work is cited by several of the studies in Table 1. They noted the fall-off of the cumulative distribution when the non-cumulative distribution is a power law, as in Figure 1. Their “General Fitting Function” to describe the shape of the cumulative distribution is F(x) = Cx^{–α+1} – R (where R = Cb^{–α+1}), i.e., the same function derived in Section 2.1 by simply integrating a power law over a finite range. They also discussed linear and logarithmic binning of data. For logarithmic binning, they did not divide the bin counts by the bin widths, leading to the conclusion that “binning affects the relationship between non-cumulative and cumulative distributions” and “Different methods of binning the same data (e.g., linear or logarithmic) produce different non-cumulative distributions”. This misleading conclusion is easily avoided by dividing the bin counts by the bin widths, as is normal practice (e.g., Lindgren, 1975; Freedman et al., 1998; Peck et al., 2012).
4. Comparing floe size distributions
Table 1 summarizes many of the FSD studies since the foundational work of Rothrock and Thorndike (1984), who introduced mathematical descriptions of the FSD, the relationship of floe diameter to area and perimeter, and error estimation when sampling the FSD. They plotted FSDs from several different data sets, finding that the cumulative distribution followed a power law for one data set, but for others it had shallower slopes for smaller floes and steeper slopes for larger floes (in log-log space), i.e., concave down (see their Figure 7). They wrote, “…we see no reason to expect a power law or any other simple analytical form to be valid for all d [floe sizes]. We find changes in the distribution from year to year and from one region to another.”
Nevertheless, many subsequent studies have characterized the FSD as a power law. Weiss (2003) explained the mechanics of the fracture and faulting of ice, its scaling properties, and power-law distributions of fracture lengths and fragment sizes. His Table I lists four studies of the sea-ice FSD, which are the first four entries in our Table 1.
4.1. Break-point analysis
In three separate studies, Toyota et al. (2006, 2011, 2016) found a break-point (change in slope in a log-log plot) in the cumulative FSD at a floe size of 40 m (2006, 2011), or between 100 and 1500 m (2016), with the shallower slope for floes smaller than the break-point, and the steeper slope for floes larger than the break-point. Across the three studies, the slopes of the FSD for the smaller floes ranged from about –1.0 to –1.5, while the slopes for the larger floes varied more widely, from –1.9 to –7.6. Toyota et al. (2006, 2011, 2016) hypothesized that the two different regimes of the FSD are due to wave-ice interaction. However, the FSD in Toyota et al. (2006) appears to us to be a steadily curving concave-down distribution (see their Figure 3), rather than two linear regimes separated by a break-point. Similarly, the FSD in Toyota et al. (2011) is concave down, possibly due to the mathematical or physical truncation effects discussed in Sections 2.1 and 3.1. In Figure 3 we have reproduced and annotated a figure from Toyota et al. (2011) (their Figure 9A), showing that by adding a constant R (see Section 3.1) to the cumulative distribution, a single power law suffices to describe the FSD at all scales. The calculation of R proceeds as in Section 3.1: from Figure 3 we estimate a = 2 m and b = 100 m, so r = 50. Then from Toyota et al. (2011) Table 2, we find that the slope of the (cumulative) power law in Figure 3 for l < 40 m is –1.05. Thus our α is 2.05, and so R = 1/(50^{1.05} – 1) = 0.0167. In order to scale R to the units of the vertical axis (number of floes per km^{2}), we find in Toyota et al. (2011; see their Table 1) that the number of floes is 697 and the area is 0.131 km^{2}. Thus the constant correction term for the curve of October 18 in Figure 3 would be (0.0167) × (697/0.131) = 88.8, which we round to 90. With this correction added to the curve of October 18, the data of October 18 could plausibly be fit by a single power law across all scales. Note that we are not asserting that the analysis of Toyota et al. (2011) is wrong and that a single power law is in fact correct – we are pointing out that an alternative explanation to that put forward by Toyota et al. (2011) is also consistent with the data, and therefore further analysis is needed. We note, for example, that Steer et al. (2008) found two power-law regimes in the non-cumulative FSD, so there may be a physical basis for the same behavior in the cumulative FSD.
FSD from Toyota et al. (2011) with additional annotation. This is Figure 9A from Toyota et al. (2011) with annotation added by us, as shown in the legend in the upper right corner. The figure shows concave-down cumulative FSDs (colored curves) with a regime shift postulated by Toyota et al. (2011) at a floe size of 30 to 40 m (vertical dashed lines). We selected three points on the curve of October 18 (blue) where it crosses the Y axis at values of 10^{0}, 10^{1}, and 10^{2} (blue circles). We added a constant value R = 90 (see main text) to these points to obtain Y-axis values of 91, 100, and 190 (yellow circles). These values lie close to the black line (added by us), the slope of which matches that of the blue curve for floe sizes less than 20 m. This suggests that the curvature of the FSDs may be a truncation effect rather than a regime shift, and that a single power law may suffice to describe the FSDs. DOI: https://doi.org/10.1525/elementa.304.f3
Toyota et al. (2016) discussed the truncation effect that is seen in their FSDs, and applied the General Fitting Function method of Burroughs and Tebbens (2001a) (see Section 3.3) to fit their data and deduce the slope of the underlying power law in each of their two regimes (x < 100 m and x > 1500 m). It is unfortunate that the regime shift occurred in the gap between the two measurement systems (helicopter photos and MODIS images). Whether the two power laws would join at a distinct break-point or whether they would continuously curve into one another is not known.
Herman (2010) discussed the break-points in the slope of the FSD found by Toyota et al. (2006) and Steer et al. (2008), who both explained the slope changes in terms of physical processes. Herman wrote: “However, contrary to how the above authors interpret their results, in both cases the change in slope of the FSD seems rather gradual than abrupt. Instead of a combination of two power laws ‘glued together’ at a highly arbitrarily chosen floe diameter, another type of distribution would be desirable. It should reflect the observed gradually increasing deviation from a power-law distribution for decreasing floe diameter.” Herman goes on to propose a higher-order fitting function for the cumulative FSD that fits the data better, rejecting the break-point analysis. We agree with Herman’s reasoning, but we believe that her higher-order fitting function is not necessary, because the upper-truncated power law seems to account for the shapes of the cumulative FSDs.
Gherardi and Lagomarsino (2015) used fragmentation theory to derive a floe size probability density function of the form f(x) = c x^{–2} exp(–x^{2}/Δ^{2}), where x is floe size, Δ is a length scale, and c is a normalization constant. For x < Δ, f(x) is approximately a power law with exponent –2. For x > Δ, f(x) falls quickly to 0 as x increases. Thus the FSD contains a characteristic length scale Δ that separates the power-law regime from the fall-off regime. Their derivation of f(x) was motivated by their analysis of four visible-band satellite images of sea ice in which all four non-cumulative FSDs contained the same three features: a power-law regime with exponent –2; a fall-off from the power-law regime at small scales which they attributed to the resolution limit of the images; and a fall-off from the power-law regime at large scales which they did not attribute to the finite size of the images, but which instead they fit with f(x) by appropriately choosing the length scale Δ. However, we note that if the largest floes were undercounted, this could explain the fall-off at large scales without postulating an additional parameter Δ. This may be an instance of the finite size effect affecting the non-cumulative FSD.
4.2. Variability of the floe size distribution
What accounts for the wide range of power-law exponents of the FSD in Table 1 and Figure 4? In the following we consider whether results from different studies are consistent with respect to region, season, and distance to ice edge.
Power-law exponents of the floe size distribution from 18 studies. The range of power-law exponents found in the studies in Table 1 (identified by first author only) are shown by horizontal bars. Red denotes studies that used the cumulative distribution; blue denotes studies that used the non-cumulative distribution. Note that one branch of “Toyota 2011” with exponent ranging from –3.2 to –7.6 (see Table 1) has been cut off at –3.6 in this figure. DOI: https://doi.org/10.1525/elementa.304.f4
Six of the studies in Table 1 are for Antarctic waters, and all include floe sizes from 2 to 100 m (or larger). The studies of Lensu (1990), Toyota et al. (2011), and Toyota et al. (2016) are consistent with one another for floes up to x = 40 m in size, with slopes of the cumulative FSD in the range of –1.0 to –1.5. However, Toyota et al. (2011) found that the slope of the FSD steepens for x > 40 m, whereas Toyota et al. (2016) found that the slope remains shallow for x < 100 m. This could reflect a difference between the marginal ice zone studied in Toyota et al. (2011) and the interior ice pack studied in Toyota et al. (2016). Steer et al. (2008) found a break-point in the slope of the non-cumulative FSD at x = 20 m. If we add 1 to their slopes as a theoretical conversion to a cumulative power-law distribution, we obtain slopes of –0.9 for x < 20 m and –1.8 to –2.4 for x > 20 m, which are shallower than those of Toyota et al. (2011) and Toyota et al. (2016). The results of Steer et al. (2008) were obtained during the Antarctic summer (December), while those of Toyota et al. (2011, 2016) were obtained during spring (September–November). Whether the power-law exponent of the Antarctic FSD has a seasonal cycle as in the Arctic is not known, but the cycle found in the Arctic (Stern et al., 2018) has steeper slopes during summer and shallower slopes during spring, and so could not explain the discrepancy between Steer et al. (2008) and Toyota et al. (2011, 2016).
Considering distance to the ice edge, Lu et al. (2008) found that the slope of the cumulative FSD steepens from –0.6 to –1.4 as the ice edge is approached. If we convert the slopes of Paget et al. (2001) to cumulative FSD (by adding 1), they become –0.9 to –2.5 as the ice edge is approached, which are partially consistent with those of Lu et al. (2008). Note again that the seasons are different: summer (December–February) in Lu et al. (2008) versus winter (August) in Paget et al. (2001).
The slopes of the non-cumulative FSDs of Paget et al. (2001) and Steer et al. (2008) have very similar ranges (–1.9 to –3.5), but they tell different stories: the slopes of Paget et al. (2001) steepen as the ice edge is approached, where the floes become smaller, but the slopes of Steer et al. (2008) are shallower for smaller floes. This apparent contradiction holds across all studies in Table 1: four studies found steepening slopes as the ice edge is approached, yet five studies found shallower slopes for smaller floes. If the Arctic is any guide to the Antarctic, our finding that the slope of the FSD steepens in spring as large floes break up (Stern et al., 2018) supports the idea that the slope of the FSD should steepen as the ice edge is approached. However, lateral melting would make the slope of the FSD shallower at smaller floes sizes (Perovich and Jones, 2014). In summary, it is not clear why some studies find that the slope of the FSD steepens as the ice edge is approached, where smaller floes are found, while other studies find that the slope of the FSD is shallower for smaller floe sizes.
Moving to the Arctic, six of the studies in Table 1 include the Beaufort Sea, with a common range of floe sizes from 900 to 10,000 m for five of the six. The studies of Rothrock and Thorndike (1984), Holt and Martin (2001), Perovich and Jones (2014), and Hwang et al. (2017) are consistent with one another, with slopes of the cumulative power-law FSD in the broad range of –1.7 to –2.5. The range is narrower for the latter three studies: –1.9 to –2.2 for Holt and Martin (2001), –2.0 to –2.2 for Perovich and Jones (2014), and –1.7 to –2.0 for Hwang et al. (2017; after converting their non-cumulative power-law slopes to cumulative by adding 1). On the other hand, Wang et al. (2016) found mean slopes in the range –1.0 to –1.5, considerably shallower than the first four studies. Finally, if we convert the non-cumulative power-law slopes of Stern et al. (2018) to cumulative FSDs (by adding 1), the range of slopes becomes –0.9 to –1.8, in agreement with Wang et al. (2016). Thus we have a group of four studies that are consistent with one another, and another group of two studies that are consistent with one another, but the two groups are not consistent with each other.
Perovich and Jones (2014) found that the power-law FSD in the Beaufort and Chukchi seas in summer 1998 followed a seasonal cycle in which the slope steepened in mid-July, peaked in mid-August, then became shallower in September. This pattern agrees well with Stern et al. (2018). However, the values of the slopes do not agree, as noted in the previous paragraph.
Toyota et al. (2016) is the only Antarctic study that lends itself to comparison with the five studies from the Arctic’s Beaufort Sea, with floe sizes up to 10,000 m. The Antarctic FSDs, with slopes from –2.9 to –3.1, are steeper than all the Arctic FSDs, implying relatively more small floes and relatively fewer large floes in the Antarctic than in the Arctic.
To investigate whether the exponent of a power-law FSD depends on the range of floe sizes from which it is determined, we plotted the exponent versus the floe size range (Figure 5) for the studies in Table 1 that use the cumulative FSD. The figure shows that the studies that include larger floes tend to have more highly negative power-law exponents (steeper slopes in log-log plots). This tendency is consistent with both the mathematical upper-truncation property and the finite size effect from under-sampling the largest floes (Section 3.1). There are several possible explanations: (1) the cumulative FSD does not follow a single power law across all scales; (2) an upper-truncated power law may provide a better model for the cumulative FSD; or (3) the data processing procedures of some previous studies may need to be re-examined. We note that there are too few studies in Table 1 that use the non-cumulative FSD to determine whether a similar pattern exists for them.
Range of exponents of power-law FSD versus range of floe sizes. Range of exponents of power-law FSD versus range of floe sizes over which the FSD was calculated. Only those studies that used the cumulative FSD are plotted (see Table 1). Horizontal segments indicate the range of floe sizes; intersecting vertical segments indicate the range of the power-law exponent. Each black dot is placed at the arithmetic mean of the exponent range (Y axis) and the geometric mean of the floe size range (X axis). Note log scale on X axis. The studies that included larger floes tend to have more highly negative power-law exponents of the FSD. The branch of Toyota et al. (2011) with exponent ranging from –3.2 to –7.6 (see Table 1) is not plotted in this figure. DOI: https://doi.org/10.1525/elementa.304.f5
4.3. Modeling studies
Two ice-ocean models that simulate the FSD have been developed (Horvat and Tziperman, 2015, 2017; Zhang et al., 2015, 2016), with different results. The Zhang model has 11 categories of floe sizes with diameters from 5 to 2500 m. The model is configured to include the Arctic Ocean and peripheral seas. Model components include an ice thickness distribution, an ice enthalpy distribution, and a mechanical redistribution function that determines ice ridging. The model was run with realistic forcing to simulate the seasonal evolution of the Arctic FSD in 2014, when FSD observations were available for model calibration and validation. The model found that in mid-spring and summer, the cumulative FSD resembled an upper-truncated power law. A comparison of model-derived and data-derived power-law exponents showed a correlation of 0.57 with a mean difference of only 5%, but with model-derived FSDs tending to be shallower (in log-log plots) than data-derived FSDs.
The model of Horvat and Tziperman (2015, 2017) has 90 categories of floe sizes with diameters from 1 to 3300 m. The model includes an ice thickness distribution as well as thermodynamic and mechanical components. The domain and forcing are idealized in order to examine the influence of four forcing factors: advection of sea ice, thermodynamic forcing from the ocean and atmosphere, mechanical interactions between colliding floes, and floe fracture due to ocean surface waves. The model found the emergence of different behavior at different length scales: for floe sizes 5–50 m, a shallow power-law regime; for floe sizes 50–150 m, a steep power-law regime; and for floe sizes 150–1500 m, an intermediate power-law regime. We note that the characterization of the FSD as a power law over a narrow range of floe sizes (such as 50–150 m) is subject to considerable uncertainty.
The realistic simulations of Zhang et al. (2016) are consistent with co-located data and with other studies that report cumulative FSDs that resemble upper-truncated power laws (Lu et al. 2008; Toyota et al. 2016; Wang et al. 2016). The idealized simulations of Horvat and Tziperman (2017) are consistent with the two power-law regimes reported by Toyota et al. (2011) with break-point around 50 m, which Horvat and Tziperman found is set by the ocean surface wave spectrum. They also found that the FSD may deviate from a power law due to mechanical interactions when the sea ice is subjected to transient rather than steady forcing, and they concluded that “it is difficult to justify using a single power law for representing the FSD, because of the different processes active at different scales”. Thus we have different conclusions from different models, although both may be correct over different regimes of ice conditions and external forcing. Proper and consistent analysis of floe size data, as outlined in Sections 2 and 5, should lead to better model validation, which can then give further insight into the interpretation of observational studies.
4.4. Mean floe size
The exponent of a power-law FSD controls the relative number of small and large floes, but it is not intuitively easy to interpret changes in the value of the exponent. The mean floe size, on the other hand, is easy to understand and compute, and so might seem like a natural statistic to use for comparing FSDs. However, the mean floe size is not a meaningful statistic for power-law FSDs. As Clauset et al. (2009) wrote, power laws “are not well characterized by their typical or average values.” This assertion is explained as follows.
Consider the non-cumulative power-law distribution f(x) = cx^{–α} where α > 2, c is a normalization constant, and x ≥ a for some minimum floe size a below which it is not possible to detect floes. The mean value of this distribution is proportional to a. Therefore, if we were to improve the resolution of the measurement system so that we could detect floes of size a/2, and the power-law relationship continued to hold, the mean floe size would be reduced by a factor of 2. Every reduction in a would be accompanied by a proportional reduction in the mean floe size; i.e., the mean floe size is directly tied to the resolution of the measurement system. In contrast, suppose the FSD followed an exponential distribution: f(x) = (1/λ) exp[–(x–a)/λ] for x ≥ a where λ is a positive parameter. The mean value of this distribution is a + λ, which is insensitive to a as long as a << λ (which always holds in the limit as a → 0). Therefore, the mean value is a useful statistic for an exponential FSD. For power-law distributions, it is meaningful to compare mean floe sizes across data sets with the same spatial resolution, but not across data sets with different spatial resolutions. In general, a power-law FSD is best characterized by its exponent, together with the range of floe sizes over which it is valid, as determined by the data from which it is derived.
4.5. Normalization of the floe size distribution
Distributions (whether cumulative or non-cumulative) may be normalized in several different ways. For example, the non-cumulative distribution may be normalized so that it integrates to 1, or integrates to the total number of floes. Different normalizations lead to different vertical scales in plots of the FSD, but often the important property is the shape of the distribution, not its absolute values. On a log scale, a normalization factor (multiplicative constant) translates into a constant shift of the distribution up or down without changing its shape. Therefore, the normalization of power-law distributions is not so important, because it does not affect the power-law exponent.
4.6. Advantages and disadvantages of using the cumulative or non-cumulative distribution
The purpose of constructing a cumulative or non-cumulative distribution from floe size data is to see patterns and anomalies in the data, i.e., to answer questions such as: do the data appear to be power-law distributed? Over what range of floe sizes? Do the largest floes appear to be under-sampled? The idea is to provide guidance for the subsequent analysis. The actual determination of the power-law exponent (if a power-law model is chosen) should use the MLE (Section 2.2 and Appendices A and B), not curve-fitting of the cumulative or non-cumulative distribution. The MLE does not depend on plots of the distribution or binning of the data.
4.6.1. Cumulative distribution
Advantages. Data need not be binned in order to construct a cumulative distribution (Section 3.3). The cumulative distribution is smoother than the non-cumulative distribution because it integrates (accumulates) successive values. The first study of the FSD (Rothrock and Thorndike, 1984) used the cumulative distribution, as have most subsequent studies, facilitating inter-comparisons.
Disadvantages. If we suspect power-law behavior and we observe concave-down curvature of the cumulative distribution in a log-log plot, we cannot say whether the curvature is due to the mathematical upper-truncation property or the finite size effect from under-sampling the largest floes (Section 3.1). (If the non-cumulative distribution also curves downward this would implicate under-sampling). If we cut off the downward-curving regime of the cumulative distribution in order to highlight the power-law regime at smaller floe sizes, we may be throwing away potentially useful information.
4.6.2. Non-cumulative distribution
Advantages. If the data come from a power law, the non-cumulative distribution will be a straight line in a log-log plot.
Disadvantages. Data need to be binned in order to construct the distribution. This must be done carefully to ensure an adequate number of samples in each bin, but making the bins too wide degrades the resolution of the distribution.
5. Conclusions5.1. Analysis of floe size data
The analysis of floe size data should proceed in several steps:
(1) (Optional step) Plot the data to see whether a power law or other mathematical form appears to describe the distribution. Either the cumulative or non-cumulative form of the distribution may be constructed from the data and plotted. The cumulative form does not require binning the data. If the non-cumulative form is to be constructed, use logarithmically spaced bins (Section 3.3 and Appendix C). In the following steps we assume that a power-law model will be applied to the data.
(2) Select the range of floe sizes over which the power-law model will apply. The objective is to include the size range that is adequately sampled by the data. The choice may be based on the visual appearance of the plotted data, or physical reasoning (such as the spatial resolution of the data and their geographical extent), or a statistical procedure for choosing the minimum floe size x = a, as outlined in Clauset et al. (2009) (and denoted x_{min} therein). Another criterion might be consistency across multiple data sets. For example, if floe size data from several different images acquired by the same sensor in the same region are being analyzed separately, it would make sense to impose the same range of floe sizes on the analysis of all the data sets.
(3) Calculate the exponent of the power law using the MLE (Section 2.2 and Appendices A and B). This calculation does not require binning the data or making any plots.
(4) Apply a goodness-of-fit test (Section 2.3) to determine whether or not the modeled form of the distribution (i.e., power law with best-fitting exponent) is in fact a good description of the actual data distribution.
(5) (Optional step) Go back to step (2), select a new range of floe sizes, and repeat.
(6) A final step may also be taken: test alternative models of the FSD, such as exponential or Weibull, to determine whether they fit the data (in case a power law did not) or whether they fit even better than a power law (in case a power law did pass the goodness-of-fit test). See Clauset et al. (2009) for further details. Given that a possible physical basis for power-law FSDs exists in the theory of self-organized criticality (Bak et al., 1988; Korsnes et al., 2004), whereas alternative distributions may lack such a physical basis, the utility or necessity of this final step is debatable.
5.2. Assessment of studies of the floe size distribution
There are substantial differences in the power-law FSDs from one study to another (Table 1 and Figure 4). Reasons may include:
(1) Spatial and temporal variability of the FSD arising from natural causes. For example, some studies found that the power-law exponent depends on the distance to the ice edge. Also, the power-law exponent of the Arctic FSD goes through seasonal transitions reflecting floe break-up and melting (Perovich and Jones, 2014; Stern et al., 2018), so the time of year (month) must be taken into account when comparing studies. The range of the power-law exponent found by Stern et al. (2018), depicted at the bottom of Figure 4, shows that seasonal variability could account for a sizeable portion of the overall variability across the 18 studies, but certainly not all of it or even most of it.
(2) Sampling variability. Many of the studies in Table 1 analyzed only a handful of images.
(3) Inadequacy of the power law as a model of the FSD. Although visual inspection shows that in many cases the FSD does appear to be linear in log-log space over several orders of magnitude in floe size, only two of the studies in Table 1 included goodness-of-fit tests of the power law. Furthermore, measurements of the FSD by Rothrock and Thorndike (1984), parameterizations of the FSD by Perovich and Jones (2014), and modeling of the FSD by Horvat and Tziperman (2017) have all found instances of non-power-law behavior.
(4) Use of a pure power law to represent the cumulative FSD on a finite interval when an upper-truncated power law should have been used (Section 2.1).
(5) Biased results due to the use of a curve-fitting procedure in log-log space to estimate the power-law exponent. White et al. (2008) wrote in the journal Ecology: “Because of the use of biased statistical methods for estimating the exponent, the conclusions of several recently published papers should be revisited.” Perhaps the same problem applies to some sea-ice FSD investigations.
5.3. Is it possible to reconcile disparate studies of the sea-ice floe size distribution?
The answer is a partial ‘yes’: some of the studies in Table 1 are consistent with one another. A first step toward further reconciliation might be a reanalysis of the data along the lines suggested in Section 5.1. But perhaps the question should be: is it necessary to reconcile disparate studies of the FSD? Sea-ice models do not need to represent the FSD as a power law or any other distribution – they only need to simulate the FSD according to the relevant physics, and compare the results to observations. But if the observations do in fact exhibit simple characteristics like a power-law size distribution, then this provides insight into the physical processes governing sea ice, and therefore it is worth trying to determine when and where power-law floe size distributions are valid.
Appendix AMaximum likelihood estimate of power-law exponent on finite interval
Consider a power-law probability density function f (x) = c x^{–α} on the finite interval a ≤ x ≤ b, where a > 0. (This is called a truncated Pareto distribution by White et al., 2008). The normalization constant c is determined from the requirement that
Suppose the values x_{1}, x_{2} … x_{n} are drawn from this power-law distribution, but α is unknown. The problem is to determine the most likely value of α that would give rise to these observations.
The standard maximum likelihood estimate (MLE) of α is constructed as follows (e.g., Wilks, 1995). The likelihood of observing the given values is the product L = f (x_{1})f (x_{2})…f (x_{n}). To maximize L, one simply sets dL/dα = 0 and solves for α. In practice, it is more convenient to calculate ln(L) first and then set its derivative to 0, which yields the same value of α because ln(L) is a monotone increasing function of L. We have:
First we consider the special case in which b → ∞ (i.e., r → ∞), and then (in Appendix B) we show how to solve equation (A2) numerically for α using fixed-point iteration.
Assuming α > 1 and allowing r → ∞, the last term in square brackets in equation (A2) goes to 0. Thus:
in agreement with White et al. (2008; see their Table 1). Equation (A3) gives the MLE of α for the case in which f (x) is defined on the semi-infinite interval a ≤ x < ∞. Note that in this case, the normalization constant c simplifies to c = (α–1) aα^{–1}, and f (x) can be written as:
The cumulative distribution function is then F(x) = (x/a)^{–(α–1)}.
For finite r, equation (A1) is the same as that given by White et al. (2008; see their Table 1) for the truncated Pareto distribution, in which their λ is our –α, and their ln(x)¯
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is our ln(G). The correspondence becomes clearer if ln(a) is subtracted from both sides of their equation.
Appendix BNumerical solution for maximum likelihood estimate of power-law exponent
We wish to solve for α in equation (A2), which we repeat here:
where we have defined the right-hand side of the equation to be the function H(α). The method of solution is simple: start with an initial guess α = α_{0} and calculate α_{1} = H(α_{0}). Continue with α_{2} = H(α_{1}), and so on. This procedure is called fixed-point iteration (e.g., Conte and de Boor, 1972) because it seeks to find a fixed point of the function H(α). To make the iteration more explicit, we re-write equation (B1) as:
where α_{j+1} appears on the left-hand side and α_{j} appears on the right-hand side. The initial guess α_{0} generates α_{1}, which generates α_{2}, etc.
Let the solution of equation (B1) be denoted α = α*. Then if the initial guess α_{0} is sufficiently close to α*, the iteration in equation (B2) converges if |H′(α*)| < 1 where H′ is the derivative of H with respect to α (Conte and de Boor, 1972). With each iteration, the error in α_{j} is multiplied by |H′(α*)|.
To examine the convergence of the iteration procedure, we calculate H′(α) from equation (B1), making use of equation (A1) to simplify the result, obtaining (after some algebra):
We observe that if α > 1 then in the limit as b → ∞ (i.e., r → ∞), H′(α) → 0. Thus, fixed-point iteration always converges for sufficiently large b, as long as α* > 1. But for fixed r, H′(α) → 1 as α → 1, so the rate of convergence is very slow if α* is close to 1.
Consider a representative case in which α* = 2 (note that most of the non-cumulative exponents in Table 1 are at least this large in magnitude) and r = 100 (i.e., the range of x spans two orders of magnitude). Then from equation (B3), H′(α*) ≈ 0.2, so the fixed-point iteration in equation (B2) converges when α_{0} is sufficiently close to α*, and the error at each iteration is reduced by a factor of 5.
Consider a worst-case example in which α* = 1.5 (which is smaller than any non-cumulative exponent in Table 1) and r = 10 (i.e., the range of x only spans one order of magnitude). Then from equation (B3), H′(α*) ≈ 0.9, so the fixed-point iteration still converges. Approximately 65 iterations would be required to reduce the initial error by a factor of 1000 (since 0.9^{65} ≈ 0.001).
The fixed-point iteration in equation (B2) has three favorable properties: (1) it is very easy to program, (2) it executes quickly (hundreds of iterations can be done in a fraction of a second on most machines), and (3) it converges under practically all parameter values that are likely to be encountered in practice.
Appendix CChoosing the number of logarithmically spaced bins for a power-law distribution
There is no optimal number of bins to represent a density function, and most “rules of thumb” apply to evenly spaced bins with Gaussian-distributed data (e.g., Wilks, 1995). Here we consider logarithmically spaced bins with power-law-distributed data. The number of bins should be large enough to show the features of the distribution but small enough to ensure that each bin contains an adequate number of samples. For logarithmically spaced bins with power-law-distributed data, the last bin (largest scale) contains the fewest number of samples, so we focus on that bin.
We assume the data follow a power-law probability density function f (x) = c x^{–α} on the finite interval a ≤ x ≤ b where a > 0, α > 1, and the normalization constant is c = (α–1)a^{α–1}/[1 –(a/b)^{α–1}] as in Appendix A. To construct m logarithmically spaced bins, we first define h = (b/a)^{1/m}. The bin boundaries are then a × h^{k} for k = 0, 1 …, m. In particular, the last bin (largest scale) extends from x = b/h to x = b. Integrating f (x) over these limits, and defining r = b/a, we find that the fraction of the distribution in the last bin is:
For the case m = 1, h = r so F_{last} = 1; i.e., a single bin contains the whole distribution. In the limit as m → ∞, h → 1 so F_{last} → 0; i.e., as the number of bins increases, the fraction in the last bin goes to 0.
Let N = the total number of data values. Then the expected number of data values in the last bin is N·F_{last}. We will require this to be at least as large as some minimum value n: N·F_{last} ≥ n. This inequality will allow us to set an upper bound on m, the number of bins.
A common assumption for a histogram bin count is that the uncertainty (standard deviation) of the count is the square root of the count (e.g., Aggarwal and Caldwell, 2012). While this estimate is imperfect, it is nevertheless used in practice for its simplicity (e.g., by the Statistics Committee at Fermilab; https://www-cdf.fnal.gov/physics/statistics/notes/pois_eb.txt). If the minimum bin count n is required to be at least two standard deviations above 0, then n > 2n^{1/2}, so n > 4. We set n = 5 as an absolute minimum value, with perhaps n = 10 as a more conservative value.
In this equation, the values of r (=b/a) and N are known, the value of n is taken to be 5 or 10, and the value of α can be calculated from the data by the method in Appendices A and B. For example, suppose that r = 10^{3}, N = 10^{4}, α = 2, and we set n = 10. Then from equation (C2) we find m ≤ 10 bins.
Of course it is also possible to determine m experimentally by simply choosing a candidate value m′, binning the data, observing the number of data values in the largest bin, adjusting m′ (smaller if there are too few values, larger if there are too many values), and repeating.
Acknowledgements
We thank Jérôme Weiss for providing some of the papers referenced in Table 1, and we thank Christopher Horvat for discussions about the floe size distribution. We thank Takenobu Toyota, one anonymous reviewer, and the guest editor for helpful comments.
Funding information
This work was funded by the Office of Naval Research under the programs Emerging Dynamics of the Marginal Ice Zone, grant N00014-12-1-0112, and Seasonal Ice Zone Reconnaissance Surveys, grants N00014-12-1-0232 and N00014-17-1-2545. Stern, Schweiger, and Zhang also acknowledge funding from the NASA Cryospheric Sciences Program, grant NNX17AD27G.
Competing interests
The authors have no competing interests to declare.
Author contributions
Contributed to conception and design: All
Contributed to acquisition of data: No data acquired
Contributed to analysis and interpretation: All
Drafted and/or revised the article: All
Approved the submitted version for publication: All
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