the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Seasonal Evolution of the Sea Ice Floe Size Distribution from Two Decades of MODIS Data
Abstract. The Arctic sea ice cover seasonally evolves from large plates separated by long, linear leads in the winter to a mosaic of smaller sea ice floes in the summer. The interplay between physical and thermodynamic mechanisms during this process ultimately sets the observed sea ice floe size distribution (FSD), an important metric for characterizing the sea ice cover and assessing model performance. Historically, FSDs have been studied at fixed locations over short periods, leaving a gap in our understanding of the spatial and temporal evolution of the FSD at large scales. Here, we present a new framework for image segmentation, allowing the identification and labeling of individual ice floes in Moderate Resolution Imaging Spectroradiometer (MODIS) data. Using this algorithm, we automatically process and segment 4,861 images, identifying more than 9.4 million floes over 23 years. The extracted characteristics of the floes–including area, perimeter, and orientation–evolve throughout the spring and summer in the Beaufort Sea. We find seasonal patterns of decreasing mean floe area, increasing FSD power law slope, and more variability in the floe orientation as the summer progresses.

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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.
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RC1: 'Comment on egusphere202489', Anonymous Referee #1, 13 Mar 2024
The authors processed 4861 MODIS images of sea ice in the Beaufort Sea spanning the months of March through September in the years 2000 through 2022, identifying more than 9 million ice floes. They then constructed and analyzed the seaice floe size distribution (FSD). They found that the FSD follows a power law with an exponent that varies seasonally. They also calculated the mean floe size and the orientation of floes.
Previous studies of the FSD have typically been based on the analysis of a handful of images, or at most a few hundred. The amount of imagery analyzed in this work exceeds all previous studies by a factor of 15 or more, and for that reason alone it is worthy of publication. But regarding the seasonal evolution of the FSD, the authors did not make a careful comparison of their results with previous studies, and there are some mathematical issues to resolve.
Specific Comments
Lines 8990. "The dynamic threshold value is the weighted mean for the 400pixel (100 km) neighborhood of a pixel subtracted by a constant." A few questions:
1. A neighborhood is an area. Is the neighborhood 400 x 400 pixels (100 x 100 km), or is it 20 x 20 pixels (400 total, 10 x 10 km)?
2. Is the neighborhood a square or some other shape?
3. What does "subtracted by a constant" mean? What constant?Lines 161167. I do not understand the bootstrapping procedure. Samples of (what?) are drawn from (what?).Ā
Lines 178182. These sentences discuss the mean and median floe areas as shown in Figure 5(a). It should be noted that for a powerlaw distribution, the mean and the median are directly proportional to the minimum floe area, Amin, which is 5 km^2 in the present work. The powerlaw pdf with exponent alpha is:
f(A) = ((alpha1)/Amin) * (A/Amin)^(alpha)
The integral of f(A) from Amin to infinity is 1.
The mean floe area is:
Amean = Amin * (alpha1)/(alpha2)
The median floe area is:
Amed = Amin * (2^(1/(alpha1)))
The point is this: The mean and median floe areas are completely dependent upon the resolution of the measuring device. If the resolution of the sensor were to allow the detection of floes 10 times smaller, the mean and median floe areas would be 10 times smaller (provided the data continued to follow the same power law). The mean and median floe areas reported here do not reflect any kind of objective reality but rather the limitations of the sensor. They cannot be meaningfully compared to the mean and median floe areas detected by sensors with different resolutions. It's fine to calculate and plot the mean and median floe areas, and to note their variation over time, but their direct proportionality to the resolution (Amin) of the detection mechanism should be noted, if in fact the floe areas are powerlaw distributed.The previous comment raises a further difficulty. For a powerlaw distribution with exponent alpha, the mean does not exist when alpha LE 2. In other words, the integral of A*f(A)dA from Amin to infinity is infinite when alpha LE 2. Now look at Figure 5(b). It shows that alpha is LE 2 quite often, and yet Figure 5(a) shows a finite mean floe area. It is not mathematically possible for a powerlaw distribution with the reported values of alpha to have a finite mean floe area. Fortunately, this is easily fixed by postulating that a maximum floe area (Amax) exists such that the distribution of floe areas follows a power law only for the finite range Amin LE A LE Amax. If this approach is taken, the authors should estimate Amax. If this approach is not taken, the authors should explain how to reconcile the mathematical contradiction.
Line 188. "The power law alpha is inversely correlated to the mean floe area". This finding is based on the data. It could also be checked analytically for a powerlaw distribution. As noted above, if alpha GT 2 then Amean = Amin * (alpha1)/(alpha2). As alpha becomes larger, Amean does indeed become smaller, although not linearly. But most of the values of alpha in this paper are LE 2. If the power law is defined on the finite interval [Amin,Amax] then Amean is a more complicated function of Amin, Amax, and alpha, but Amean still decreases as alpha increases, although not linearly. Whether the authors wish to include such an analytical comparison to the data is up to them, I am only pointing out the possibility to do so.
Lines 210218. This paragraph compares the powerlaw exponents of the present study to those found by Stern et al (2018a), but the comparison is not done correctly. The present study uses floe area as the measure of floe size, while Stern et al (2018a) used floe diameter. Assuming that floe area (A) is proportional to the square of floe diameter (x^2), a powerlaw distribution in A with exponent alpha is equivalent to a powerlaw distribution in x with exponent 2*alpha+1. With this conversion, the authors should make a quantitative monthbymonth comparison of their results with those of Perovich and Jones (2014), Hwang et al (2017), and Stern et al (2018a), all of which looked at the seasonal evolution of the FSD in the Beaufort Sea.
Regarding floe orientation (section 4.3), a couple of comments:
(1) It's possible that nearly circular floes could have a more variable orientation than elongated floes, because small changes in their perimeter could trigger large changes in their orientation. Therefore, as floe orientation becomes more variable (as in Figure 5(c) in June), how do we know whether it's due to floes becoming more randomly oriented or floes becoming more circular?
(2) Lines 227231 discusses the orientation of large rectilinear floes, concluding that their orientation is less variable in the early season compared to the full data set, but increasingly variable into the summer. It's not clear what the point of this comparison is. The reader is left hanging, with no figures or numbers or explanation of the significance of the result.Minor Comments
First line of the Introduction. Consider changing "from the atmosphere to the ocean" to "between the atmosphere and the ocean" because the fluxes go both ways  atm to ocean and ocean to atm.
Lines 5253. "The older and thicker multiyear ice is now melting out"  but the loss of older and thicker multiyear ice has been going on for a long time. It was first noted in 2007:
Maslanik, J. A., C. Fowler, J. Stroeve, S. Drobot, J. Zwally, D. Yi, and W. Emery (2007), A younger, thinner Arctic ice cover: Increased potential for rapid, extensive seaice loss, Geophys. Res. Lett., 34, L24501, doi:10.1029/2007GL032043.ĀLines 95102. This same segmentation procedure was also used by Stern et al (2018a)  see their section 3.2.
Lines 9596. "where where"
Lines 110111. "from from" should be "range from"
Line 119. "This quality assurance steps"  singular or plural?
Line 130. "The standard error of the fitted distribution"  is this supposed to say the standard error of alpha?
Line 150. "a correlation of 0.99"  is this supposed to be a squared correlation? Figure 4 says "R^2 = 0.99"
Line 166. "The the"
Line 167. "values increases"  singular or plural?
Figure 3 caption:
Line 3. "imaeg"
Line 4. Either write "captured" or delete the word "capture"Figure 5(b). The caption refers to "the slope anomaly" but the figure shows the actual slope, not the anomaly.
Figure 5(c). The caption refers to "low circularity floes" in green but I don't see a green curve.
Figure 5(d). The caption refers to "land (magenta)" but I don't see any magenta color.
Figure 5(d). Is it really necessary to show all that gray area (clouds)? That makes it harder to see the green and blue areas, which are really the quantities of interest.
Lines 203204. "the standard deviation calculated as in Equation 3." Is this supposed to say "standard error" or "standard deviation"? Does Equation 3 give the standard error or the standard deviation? See line 130.
Line 204. "Fig. 5b, shaded"  I don't see any shaded part of Figure 5b.
Line 211. 59400 should be 594
Line 225. "that are can be"
Line 239. "increasing power law slopes"  does this mean steeper or shallower? When alpha increases, the slope decreases (becomes more negative). When alpha decreases, the slope increases (becomes less negative). That's why it might be clearer to say steeper or shallower.
Lines 241242. This is the first mention of "discrete element models" in the paper. I would suggest that on line 37 the words "discrete element" be added just before the word "models".
Line 245. "new segmentation algorithm presented here"  the segmentation algorithm is not new. See the comment above for lines 95102.
There are other minor typos not noted here.
Ā
Citation: https://doi.org/10.5194/egusphere202489RC1 
AC1: 'Reply on RC1', Ellen Buckley, 08 May 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere202489/egusphere202489AC1supplement.pdf

AC1: 'Reply on RC1', Ellen Buckley, 08 May 2024

RC2: 'Comment on egusphere202489', Anonymous Referee #2, 26 Mar 2024
The comment was upload in the form of the supplement.

AC2: 'Reply on RC2', Ellen Buckley, 13 May 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere202489/egusphere202489AC2supplement.pdf

AC2: 'Reply on RC2', Ellen Buckley, 13 May 2024
Interactive discussion
Status: closed

RC1: 'Comment on egusphere202489', Anonymous Referee #1, 13 Mar 2024
The authors processed 4861 MODIS images of sea ice in the Beaufort Sea spanning the months of March through September in the years 2000 through 2022, identifying more than 9 million ice floes. They then constructed and analyzed the seaice floe size distribution (FSD). They found that the FSD follows a power law with an exponent that varies seasonally. They also calculated the mean floe size and the orientation of floes.
Previous studies of the FSD have typically been based on the analysis of a handful of images, or at most a few hundred. The amount of imagery analyzed in this work exceeds all previous studies by a factor of 15 or more, and for that reason alone it is worthy of publication. But regarding the seasonal evolution of the FSD, the authors did not make a careful comparison of their results with previous studies, and there are some mathematical issues to resolve.
Specific Comments
Lines 8990. "The dynamic threshold value is the weighted mean for the 400pixel (100 km) neighborhood of a pixel subtracted by a constant." A few questions:
1. A neighborhood is an area. Is the neighborhood 400 x 400 pixels (100 x 100 km), or is it 20 x 20 pixels (400 total, 10 x 10 km)?
2. Is the neighborhood a square or some other shape?
3. What does "subtracted by a constant" mean? What constant?Lines 161167. I do not understand the bootstrapping procedure. Samples of (what?) are drawn from (what?).Ā
Lines 178182. These sentences discuss the mean and median floe areas as shown in Figure 5(a). It should be noted that for a powerlaw distribution, the mean and the median are directly proportional to the minimum floe area, Amin, which is 5 km^2 in the present work. The powerlaw pdf with exponent alpha is:
f(A) = ((alpha1)/Amin) * (A/Amin)^(alpha)
The integral of f(A) from Amin to infinity is 1.
The mean floe area is:
Amean = Amin * (alpha1)/(alpha2)
The median floe area is:
Amed = Amin * (2^(1/(alpha1)))
The point is this: The mean and median floe areas are completely dependent upon the resolution of the measuring device. If the resolution of the sensor were to allow the detection of floes 10 times smaller, the mean and median floe areas would be 10 times smaller (provided the data continued to follow the same power law). The mean and median floe areas reported here do not reflect any kind of objective reality but rather the limitations of the sensor. They cannot be meaningfully compared to the mean and median floe areas detected by sensors with different resolutions. It's fine to calculate and plot the mean and median floe areas, and to note their variation over time, but their direct proportionality to the resolution (Amin) of the detection mechanism should be noted, if in fact the floe areas are powerlaw distributed.The previous comment raises a further difficulty. For a powerlaw distribution with exponent alpha, the mean does not exist when alpha LE 2. In other words, the integral of A*f(A)dA from Amin to infinity is infinite when alpha LE 2. Now look at Figure 5(b). It shows that alpha is LE 2 quite often, and yet Figure 5(a) shows a finite mean floe area. It is not mathematically possible for a powerlaw distribution with the reported values of alpha to have a finite mean floe area. Fortunately, this is easily fixed by postulating that a maximum floe area (Amax) exists such that the distribution of floe areas follows a power law only for the finite range Amin LE A LE Amax. If this approach is taken, the authors should estimate Amax. If this approach is not taken, the authors should explain how to reconcile the mathematical contradiction.
Line 188. "The power law alpha is inversely correlated to the mean floe area". This finding is based on the data. It could also be checked analytically for a powerlaw distribution. As noted above, if alpha GT 2 then Amean = Amin * (alpha1)/(alpha2). As alpha becomes larger, Amean does indeed become smaller, although not linearly. But most of the values of alpha in this paper are LE 2. If the power law is defined on the finite interval [Amin,Amax] then Amean is a more complicated function of Amin, Amax, and alpha, but Amean still decreases as alpha increases, although not linearly. Whether the authors wish to include such an analytical comparison to the data is up to them, I am only pointing out the possibility to do so.
Lines 210218. This paragraph compares the powerlaw exponents of the present study to those found by Stern et al (2018a), but the comparison is not done correctly. The present study uses floe area as the measure of floe size, while Stern et al (2018a) used floe diameter. Assuming that floe area (A) is proportional to the square of floe diameter (x^2), a powerlaw distribution in A with exponent alpha is equivalent to a powerlaw distribution in x with exponent 2*alpha+1. With this conversion, the authors should make a quantitative monthbymonth comparison of their results with those of Perovich and Jones (2014), Hwang et al (2017), and Stern et al (2018a), all of which looked at the seasonal evolution of the FSD in the Beaufort Sea.
Regarding floe orientation (section 4.3), a couple of comments:
(1) It's possible that nearly circular floes could have a more variable orientation than elongated floes, because small changes in their perimeter could trigger large changes in their orientation. Therefore, as floe orientation becomes more variable (as in Figure 5(c) in June), how do we know whether it's due to floes becoming more randomly oriented or floes becoming more circular?
(2) Lines 227231 discusses the orientation of large rectilinear floes, concluding that their orientation is less variable in the early season compared to the full data set, but increasingly variable into the summer. It's not clear what the point of this comparison is. The reader is left hanging, with no figures or numbers or explanation of the significance of the result.Minor Comments
First line of the Introduction. Consider changing "from the atmosphere to the ocean" to "between the atmosphere and the ocean" because the fluxes go both ways  atm to ocean and ocean to atm.
Lines 5253. "The older and thicker multiyear ice is now melting out"  but the loss of older and thicker multiyear ice has been going on for a long time. It was first noted in 2007:
Maslanik, J. A., C. Fowler, J. Stroeve, S. Drobot, J. Zwally, D. Yi, and W. Emery (2007), A younger, thinner Arctic ice cover: Increased potential for rapid, extensive seaice loss, Geophys. Res. Lett., 34, L24501, doi:10.1029/2007GL032043.ĀLines 95102. This same segmentation procedure was also used by Stern et al (2018a)  see their section 3.2.
Lines 9596. "where where"
Lines 110111. "from from" should be "range from"
Line 119. "This quality assurance steps"  singular or plural?
Line 130. "The standard error of the fitted distribution"  is this supposed to say the standard error of alpha?
Line 150. "a correlation of 0.99"  is this supposed to be a squared correlation? Figure 4 says "R^2 = 0.99"
Line 166. "The the"
Line 167. "values increases"  singular or plural?
Figure 3 caption:
Line 3. "imaeg"
Line 4. Either write "captured" or delete the word "capture"Figure 5(b). The caption refers to "the slope anomaly" but the figure shows the actual slope, not the anomaly.
Figure 5(c). The caption refers to "low circularity floes" in green but I don't see a green curve.
Figure 5(d). The caption refers to "land (magenta)" but I don't see any magenta color.
Figure 5(d). Is it really necessary to show all that gray area (clouds)? That makes it harder to see the green and blue areas, which are really the quantities of interest.
Lines 203204. "the standard deviation calculated as in Equation 3." Is this supposed to say "standard error" or "standard deviation"? Does Equation 3 give the standard error or the standard deviation? See line 130.
Line 204. "Fig. 5b, shaded"  I don't see any shaded part of Figure 5b.
Line 211. 59400 should be 594
Line 225. "that are can be"
Line 239. "increasing power law slopes"  does this mean steeper or shallower? When alpha increases, the slope decreases (becomes more negative). When alpha decreases, the slope increases (becomes less negative). That's why it might be clearer to say steeper or shallower.
Lines 241242. This is the first mention of "discrete element models" in the paper. I would suggest that on line 37 the words "discrete element" be added just before the word "models".
Line 245. "new segmentation algorithm presented here"  the segmentation algorithm is not new. See the comment above for lines 95102.
There are other minor typos not noted here.
Ā
Citation: https://doi.org/10.5194/egusphere202489RC1 
AC1: 'Reply on RC1', Ellen Buckley, 08 May 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere202489/egusphere202489AC1supplement.pdf

AC1: 'Reply on RC1', Ellen Buckley, 08 May 2024

RC2: 'Comment on egusphere202489', Anonymous Referee #2, 26 Mar 2024
The comment was upload in the form of the supplement.

AC2: 'Reply on RC2', Ellen Buckley, 13 May 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere202489/egusphere202489AC2supplement.pdf

AC2: 'Reply on RC2', Ellen Buckley, 13 May 2024
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Ellen Margaret Buckley
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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.
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