Disagreement in Observed and Simulated Cloud Perimeter Distributions
Perimeter distributions: expectation
Following [1], perimeter distributions can be derived from considering a "cascade" of clouds moving between perimeter classes as they grow and dissipate. At equilibrium, the total flux in each size class must be the same, otherwise one size class would tend to grow. Since the flux is proportional to the total perimeter in each bin, \begin{align} \mathrm{total \ flux} \propto np \tag{1} \label{eq:total-flux} \\ \end{align} where \(n\) is the number of clouds in the bin and \(p\) is their average perimeter. If the total flux in each bin is the same, \begin{align} \frac{d}{dp}np = 0 \end{align} and therefore \begin{align} \frac{dn}{dp}\propto \frac{p}{n} \end{align} which has the solution \begin{align} \frac{dn}{d\log p}\propto p^{-\beta}, \quad \quad \beta = 1 \tag{2} \label{eq:p-dist} \end{align} if bins are logarithmically spaced. Thus, in a log-log plot, the slope of the perimeter distribution is expected to be -1.
Area distributions
Cloud area \(a\) is similarly power-law distributed, and can be converted to perimeter using an established fractal dimension of 4/3. The perimeter distribution above implies \begin{align} \frac{dn}{d\log a}\propto a^{-\alpha}, \quad \quad \alpha = 2/3 \end{align} which also disagrees with previous measurements. For example, [3] finds empirically \(\frac{dn}{d\log a}\propto a^{-0.87\pm 0.03}\).
Why the disagreement?
The satellite-measured perimeter distribution slopes \(\beta\) notably converge on a value close to the fractal dimension of clouds, \(D=4/3\). Indeed, if we replace \(p\) with a distinct measured perimeter \(p_m\), where \(p\propto p_m^D\) in equation \ref{eq:total-flux}, equation \ref{eq:p-dist} becomes \begin{align} \frac{dn}{d\log p_m}\propto p_m^{-4/3}, \tag{3} \label{eq:p-dist-modified} \end{align} in excellent agreement with the satellite data.

Why is there a difference between \(p\) and \(p_m\)? Perhaps \(p\) represents a physical perimeter that does not scale fractally like \(p_m\), one that is relevant to fluxes between clouds and clear air. It is maybe not too surprising that model turbulence parameterizations would produce cloud perimeters along this "flux-relevant boundary" instead of the actual cloud boundary. However, how this discrepancy interacts with observed agreements in model cloud fractal dimension and observed cloud fractal dimension [6] is unclear.
Image edge bias
Any finite image of a collection of fractal objects imposes a characteristic scale, the scale of the image, on a fractal image that has no characteristic scale. In practice, this results in object size distributions, which are normally power law distributed, having a cut-off where the object size approaches the size of the image. It is not, however, a sharp cutoff, because objects of all size classes are affected by the edges of the image. Any object that touches the edge of the image cannot be correctly measured, as part of it is not seen.

Previous work has found significant disagreement in cloud area distributions, from the power law exponent (from \(\alpha = 1.3\) in [2] to \(\alpha = 1.87\) in [3]) to the upper limit the power law is valid (from 0.28 km\(^2\) [4] to above \(10^6\)km\(^2\) [3]). We hypothesize that at least some disagreement in previous literature are due to underappreciated effects caused by clouds touching the edge.

In the figure below, the dark green line is a histogram of cloud perimeters (and cloud areas show qualitatively similar effects) from 1 month of GOES WEST data. The blue, cyan, and red lines simulate a coarser resolution satellite, generated by taking subdomains of the original image (sizes listed in units of pixels). Every subdomain is treated as an independent image, and here any clouds touching the edge are removed. While each histogram comes from the same original data, the smaller images have a significant deviation from the power-law distribution due to the removal of edge-touching clouds. This may be misinterpreted as a scale break, as there is not simply a truncation of the power law but a gradual deviation. If edge cloud perimeters are included, the bias is comparably significant.

To account for this effect, here we omit any bins where the number of cloud perimeters touching the edge is more than 10% of the number not touching the edge. With this constraint, we note there is no evidence of a scale break in any satellite dataset, some of which extend to very large perimeters, corresponding to cloud areas larger than 10\(^6\)-10\(^7\) km\(^2\).
Cloud Perimeter Distribution Disagreements
Accounting for edge biases, perimeter distributions for each satellite dataset are shown below. Perimeters smaller than 10 times the nadir resolution are also omitted as these clouds cannot be correctly measured. The slopes for these histograms are shown below, along with theory and data from the SAM Giga-LES model in [1].
Data Sources
All data sources consist of level-2 cloud mask products produced by NASA, NOAA, or ICARE. The datasets fall into two broad categories: polar orbiting (MODIS, VIIRS, and POLDER) and geostationary (GOES WEST, GOES EAST, HIMAWARI, METEOSAT 9, METEOSAT 11, and GEORING). Polar-orbiting satellites are useful for latitude comparisons since the image geometry is not latitude dependent, while the geostationaries generally have a greater total pixel density, which is beneficial for mitigating the edge effect.
The GEORING dataset is a new product produced by ICARE and University of Lille. It is a composite dataset of geostationary satellites GOES WEST, GOES EAST, HIMAWARI, METEOSAT 9, METEOSAT 11, combined so as to produce a continuous seamless product. Images show impressive consistency without visible transitions between individual satellite images. The product is gridded onto a lat-lon grid at 0.1°\(\times\)0.1°.
Because the image does not have edges on the sides, it is advantageous to investigate size distributions because the edge effect is minimized (though there are still edges at the poles). Note the GEORING has bins at the largest perimeter values despite having relatively coarser resolution (which reduces the perimeter relative to finer resolution measurements).
Dates used
GOES WEST, GOES EAST, HIMAWARI, METEOSAT 9, METEOSAT 11: One image per day (local noon), Jan 1 2021 - Dec 31 2021
GEORING: Two images per day (0000Z & 1200Z), Jan 1 2021 - Dec 31 2021
MODIS: All data for every 10th day, Jan 1 2012 - Dec 31 2012
VIIRS: All data for day 351 2021 and day 361 2021
POLDER: All data, Jan 1 2012 - Dec 31 2012
Each dataset is imported as a binary array (where pixels are cloudy or not cloudy), and an algorithm is applied to identify connected regions using the convention that adjacent cloudy pixels are considered connected while diagonals are not [3]. The resolution of the satellite changes as the sensor zenith angle varies across the cloud field because 1) the distance to the satellite changes and 2) the pixel is projected onto the Earth at an angle. Accounting for these two effects, we independently determine the vertical and horizontal dimensions of each pixel. Once pixel sizes are known,the perimeter of each cloud is computed by adding up all the pixel side lengths between a cloudy and not cloudy pixel, and area by summing the areas of each individual pixel. Cloud holes add to the perimeter but reduce the area. Areas and perimeters of the smallest clouds cannot be reliably determined, so they are omitted, and as measurement of cloud fields at oblique angles becomes inaccurate, we truncate the images where the sensor zenith is greater than 60°.
[1] Timothy J Garrett, Ian B Glenn, and Steven K Krueger. “Thermodynamic constraints on the size distributions of tropical clouds”. In: Journal of Geophysical Research: Atmospheres 123.16 (2018), pp. 8832–8849.
[2] Ilan Koren et al. “How small is a small cloud?” In: Atmospheric Chemistry and Physics 8.14 (2008), pp. 3855–3864.
[3] Robert Wood and Paul R Field. “The distribution of cloud horizontal sizes”. In: Journal of Climate 24.18 (2011), pp. 4800–4816.
[4] Robert F Cahalan and Joachim H Joseph. “Fractal statistics of cloud fields”. In: Monthly weather review 117.2 (1989), pp. 261–272.
[5] S Lovejoy. “Area-perimeter relation for rain and cloud areas”. In: Science 216.4542 (1982), pp. 185–187.
[6] Siebesma, AP and HJJ Jonker. “Anomalous scaling of cumulus cloud boundaries”. In: Physical review letters 85.1 (2000), p. 214.