The original definition of the fractal dimension Di as proposed by is that an individual cloud with perimeter pi can be related to a variable “ruler length” or resolution ξ through 1pi∝ξ1-Di. Provided Di is not a function of ξ, the object can be termed to be “self-similar” because the measured quantity pi is a power law function of the chosen spatial scale ξ respecting a symmetry of spatial scale. Visually, self-similarity implies that the shape of an object such as that shown in Fig.  repeats at many different spatial scales. Such exact repetition is not present in clouds, but there is still a sense in which small-scale cloud edge contours appear similar to their larger-scale counterparts. To precisely capture this appearance, we may modify the concept of self-similarity to only require that the statistics of the patterns – rather than the exact pattern – be similar across scales. This broader notion was termed “statistical self-similarity” by . More specifically, statistical self-similarity exists when 2pi∝ξ1-Di where the average is taken over some large collection of clouds with uniform size but not necessarily uniform perimeter (Fig. ). The requirement of uniform size is intended to isolate changes in perimeter that are due solely to changes in the “roughness” of cloud edge, or the fractal dimension (Fig. ; ). Size here is defined as a property that, unlike perimeter, is not a function of resolution. It could be defined in a number of ways, but for simplicity of argument's sake (as discussed in more detail below) we start by defining size as the cloud bounding box width l (Fig. ).

It is difficult to apply Eq. () directly to satellite images of cloud fields because clouds encompass a wide range of sizes and types, not to mention that satellite images are obtained at a fixed image resolution. It is for this reason that a different expression is usually used to determine the fractal dimension, one that relates satellite measures of individual cloud perimeters pi to their areas ai , namely 3pi∝aiDi/2. In this case, Di is estimated from a simple linear fit to a scatterplot of the easily calculated quantities of log⁡pi and log⁡ai.

Equation () for the fractal dimension is very different in form than the strict definition given by Eq. (). As justification, referred to the work by (p. 110), where it was proposed that the two expressions are interchangeable. It is worth reviewing Mandelbrot's subtle argument in more detail because, as we will show, its assumptions do not strictly apply to clouds.

On an intuitive level, any given cloud in a cloud field can be considered to have two independent geometric properties: size and shape. By shape, we mean the roughness or complexity of the cloud boundary, which is perhaps the most visually notable aspect that differentiates clouds, such as those shown in Fig. . To quantify cloud shape, suppose a subset of clouds within the field, chosen such that each cloud has the same size li but with varying perimeter pi. This collection, represented by panel (a) in Fig. , could be used to calculate the fractal dimension. One would “resample” each cloud by increasing ξ, creating a collection of coarsened images as in panel (b). Such a resampling process would change each cloud's shape as finer contours are no longer resolved in the coarsened images. Using a range of values for ξ and the resulting range of perimeters 〈pi〉(ξ) at each resolution, Di could then be obtained from a fit to Eq. ().

Alternatively, the size could be varied as the shape stays the same. Di could be calculated from Eq. () by taking a collection of clouds having a range of sizes li, as in panel (c), and then resizing each cloud by normalizing the image width and height dimensions by the cloud size, effectively zooming the image. Although all clouds were originally imaged with the same resolution, the “larger” clouds, once resized, would appear to have a finer resolution than the smaller clouds. In fact, all length dimensions would change during resizing by the same factor 1/li, including image width, cloud size, resolution, and perimeter. The resized clouds therefore have size li/li=1, resolution ξi/li, and perimeter pi/li. The fractal dimension Di could then be calculated by substituting the normalized perimeters and resolutions into Eq. (), leading to the expression pi/li∝ξi/li1-Di or 4pi∝ξi1-DiliDi. The size metric li must have dimensions of length given that it is used to normalize the length quantities pi and ξi.

Thus, there are two methods that can be used to calculate Di for a collection of clouds: resampling by varying shape and resizing by varying size. If the two methods yield the same value, then the clouds exhibit statistical self-similarity as defined above. Note that, provided the field is self-similar, calculating Di by resizing has the advantage of enabling a more statistically robust fit. This is because any cloud in the field can be resized by normalizing by its size li, whatever its original size, and so the number of clouds that can be considered is much larger than if only resampled clouds of some fixed size are considered.

For the length dimension li, proposed that the square root of the cloud area could be used, making the assumption that the cloud area is proportional to the area of the smallest bounding box aBB. If so, Eq. () becomes Eq. (). However, Mandelbrot's assumption that ai∝aBB requires that cloud area is not a function of resolution because aBB is not a function of resolution. If cloud area does in fact change with resolution, it would certainly be possible to calculate a statistical fit of pi to ai that yields a value of D using Eq. (). But, this value of D cannot be interpreted to be a fractal dimension defining the geometric properties of the cloud field , at least not one consistent with the original definition of the fractal dimension given by Eq. (). It is not even clear what the correct interpretation should be.

Are Mandelbrot's assumptions valid for clouds such that Eq. () is justified as a substitute for Eq. ()? One potential issue is that clouds viewed from space have interior holes where the underlying surface can be viewed beneath. Resampling the cloud at a coarser resolution would cause the cloud holes to disappear so that a becomes an increasing function of ξ. Cloud areas would not be resolution independent, and so the perimeter-area relationship given by Eq. () would not strictly hold. If a fit were calculated using Eq. () regardless, the calculated fractal dimension would be larger for large clouds that have more holes. Such an increase of fractal dimension with cloud size has in fact been observed in several prior studies .

To test whether the observed dependence of fractal dimension on cloud size is a real property of the clouds, or it is instead a consequence of a false assumption that cloud area is resolution independent, any cloud holes could be filled to define a “filled area” af . If instead Mandelbrot's assumption that cloud area is independent of ξ holds for clouds, then filling the holes should not be expected to affect the calculated fractal dimension. To illustrate, consider the case where filling cloud holes increased every cloud's area by a constant factor c such that af=ca. This would not affect Di because the coefficient Di/2 relating log⁡p to log⁡a on a scatter plot of cloud sizes would remain unchanged.

Figure  shows such a comparison between cloud perimeter and the square root of area, for filled and unfilled clouds observed using MODIS as described in Sect. . In the imagery, filled areas af and filled perimeters pf are calculated by first identifying all contiguous clear regions that are not connected to the largest contiguous clear region in the image (the clear-sky “background”), and then assigning those interior regions, or holes, as being cloudy. Then, af and pf are calculated from the hole-filled cloud mask in an equivalent manner as a and p as described in Sect. . To determine any size dependence of fractal dimension calculated from Eq. (), regressions between log⁡p and log⁡a are divided into four different decades in cloud area: 10 to 10², 10² to 10³, 10³ to 10⁴, and 10⁴ to 105km2.

Figure shows that the “unfilled” fractal dimension has a higher value than the “filled” dimension at all scales (a similar observation has been made of noctilucent clouds by ). More importantly, the unfilled dimension increases from 1.41±0.01 for the smallest cloud size class to 1.87±0.18 for the largest cloud size class. By contrast, the filled fractal dimension displays substantially less scale dependence, ranging only from 1.36±0.01 to 1.55±0.13.

For filled clouds, calculated values of Di change minimally with the reflectance threshold R as shown in Fig. . For six different thresholds ranging from R=0.1 to R=0.35, Di only ranges from 1.38±0.02 to 1.40±0.01. Values of Di are consistent with values obtained in the minority of prior studies of cloud fractal properties that explicitly mentioned that cloud holes were filled .

To summarize, a fractal dimension Di defining a collection of clouds may be calculated from a perimeter-area relationship through Eq. (), in which case it offers a succinct metric for understanding cloud dynamics or for evaluating or comparing measurements and models. However, for the perimeter-area exponent to represent a true “dimension” mathematically, there is an a priori requirement that cloud areas not be a function of sampling resolution. This mathematical requirement can be satisfied, but only if cloud holes are filled, a procedure that might be argued to neglect an important physical property of the cloud field.

An alternative approach for defining the fractal properties of a cloud field according to Eq. () is to employ what we term the “ensemble fractal dimension” De. The mathematical definition is analogous to Eq. (), but with the mean cloud perimeter pi being replaced by the total cloud perimeter P: 5P=∑ipi∝ξ1-De. This definition of the ensemble fractal dimension was employed by to demonstrate, using satellite observations, that the turbulent properties of the atmosphere are not 2D at large scales and 3D at smaller scales as is commonly supposed . In Appendix A, we show that De captures two orthogonal effects: individual cloud edge complexity and the size distribution of clouds in the field.

Curiously, where most past studies only measured Di , some have in fact determined De without discussion of how the two measures differ e.g.. By considering the lengths of idealized island coastlines, made the distinction clear: the ensemble fractal dimension De applies when “lumping all the islands' coastlines together”.

For a better understanding of the numerical distinction between Di and De, consider again that cloud fields have the combined properties of size and shape as shown in the idealized cloud field shown in Fig. . Given that increasing the fractal dimension of any single cloud increases its perimeter, and therefore the total P, we can see that De increases if Di increases. However, there is the added consideration of the relative numbers of small and large clouds – a subject now of considerable interest for studies of convective cloud organization. To see how, consider re-drawing some cloud field such that the number of small clouds increases while maintaining a fixed total cloud area, for example by dividing a few large clouds into a larger number of smaller ones. Necessarily, this too would increase the total perimeter, since one would need to draw additional cloud edges to divide up the cloud. Precisely how much P changes depends on the change to the cloud size distribution. From observations spanning a wide range of cloud sizes and climate states , a reasonable assumption is that the distribution follows a power-law distribution with some exponent β such that 6n(p)∝p-(1+β) where the total number of clouds N is given by N∝∫pminpmaxn(p)dp. In this case, a field with a large number of small clouds relative to large clouds would have a larger value of P and a higher value of β, assuming the total cloud area remains unchanged. An example showing this dependence is shown along the ordinate of Fig. ().

Thus, both shape and size affect P, numerically through Di and β. The ensemble fractal dimension must be a function of both parameters. As a first guess: 7De=βDi. In Appendix , we propose based on analytical reasoning that Eq. () is correct. Here, we take it as a conjecture to be tested using satellite observations. If true, the ensemble fractal dimension given by Eq. () provides a simple observational metric for quantifying cloud field geometries, one that has previously been related to the invisible turbulent processes that roughen each individual cloud's edge Di; as well as the physics controlling the competition for available energy among clouds of varying sizes β;.

To evaluate Eq. (), we employ two different methods to calculate the ensemble fractal dimension of cloud fields from satellite imagery that are consistent with the definition of De given by Eq. (). The most familiar approach is to use the “Minkowski-Bouligand” or “box” dimension , which is calculated by first overlaying an evenly spaced grid on the cloud mask and then counting the number N of square boxes required to cover all cloud edges (red squares in Fig. ). The procedure is performed using a range of box (grid) sizes, where each box side length ε covers some integer number of image pixels in both directions. The fractal dimension calculated in this manner Dbox is defined by the relation 8N(ε)∝ε-Dbox. From this equation, Dbox may be estimated using a least-squares linear regression to a plot of N(ε) vs. ε. Note that the box dimension is equal to the usual dimension for Euclidean objects; for example, a line has Dbox=1 and a disk has Dbox=2 .

Although the actual resolution of the cloud mask ξ is fixed, the box size ε serves as an effective resolution for the purpose of calculating the fractal dimension as defined by Mandelbrot, equivalent in the limit ε→0 to that definition given by Eqs. () or () since P∼εN∼ε1-Dbox. There are, however, two other limiting cases. For a cloud field resolved at some finite resolution ξ, the box dimension Dbox will tend to unity if the box length is smaller than the satellite resolution, that is ε≲ξ. This is because Dbox will correspond to the Euclidean dimension of a line, in this case a pixel edge. Similarly, for boxes larger than the domain Dbox=0 because a single box is always sufficient to cover all cloud edges, implying N(ε) is independent of ε for such large boxes.

A second approach commonly used to measure the fractal dimension is the “correlation integral” method . For clouds, this consists of first identifying all cloud edge pixels (i.e. cloudy pixels that are adjacent to a clear pixel). For each pixel with index j, a circle of varying radius r is drawn centered on that pixel (Fig. ). The total number of cloud edge pixels within the circle Cj(r) is computed as a function of r. The fractal dimension is then calculated using the “correlation integral” C(r)=∑jCj(r), which tends to be a power law of form 9C(r)∝rDc where Dc is the “correlation dimension”. Although not strictly equivalent to Mandelbrot's ensemble fractal dimension (Eq. ), Dc is empirically very close to De , perhaps unsurprisingly given that both represent a relationship between the scale under consideration (represented by ε or r) and the density of cloud edge points (represented by N(ε) or C(r)). In practice, C(r) follows a power law only for a finite range in r. As with the box dimension, the correlation dimension tends toward the Euclidean dimensions of a single pixel for small r and the domain shape for large r. Accordingly, below we will only consider circles with r≥3ξ.

An important subtlety for calculation of Dc is that, without proper care, some circles might extend beyond the domain boundary. If Cj(r) is computed including such circles, Dc can become biased towards the dimension of the artificially straight domain boundary of Dc=1 at large scales. To remedy this issue, here we ensure that no circles extend beyond the domain boundary by enforcing r≤rmax=min(L,W)/3 where W and L are the domain width and length, respectively. We then only draw circles when the distance between the circle center and the nearest domain edge point is at least rmax. Note that including contributions from small circles that are closer to the domain edge than rmax, even when they do not extend beyond the domain edge, can still introduce boundary artifacts. This is because a set of values for Cj(r) that contained data only from small circles would bias the sum C(r) to be too large for small r.

There is a statistical advantage of using the correlation dimension for analysis of cloud fields, which is that C(r) is created using a much greater number of measured values compared with N(ε). This is because C(r) is a sum over many circle center locations j located along cloud edge – even for a single cloud. Accordingly, the correlation dimension is better suited for satellite datasets with fewer clouds or those obtained at coarse resolution.

For the box dimension, Fig.  shows the number of cloud edge boxes N as a function of box length ε for all 72 MODIS images and six reflectance thresholds R as described in Sect. . As expected, Dbox≈1 for boxes that are small compared to the satellite resolution, and Dbox=0 for very large boxes comparable to the domain size. For intermediate sized boxes, Dbox varies from Dbox=1.68±0.05 for threshold R=0.1 to Dbox=1.50±0.07 for R=0.35 suggesting (plausibly) that the more reflective regions of clouds are some combination of either being more often large, with smaller β, or more Euclidean, with smaller Di.

For the correlation dimension, we find that observed values of C(r) for clouds scale with r, implying a measured correlation dimension for the cloud field ranging from 1.77±0.01 for a reflectance threshold of R=0.1 to 1.69±0.01 for R=0.35 (Fig. ). These values are somewhat higher than those obtained using the box dimension but display a similar trend of decreasing values with increasing R. As shown in Fig. , the correlation integral C(r) is better represented by a power-law when compared to the box-counting function N(ε) (Fig. ), implying estimates of Dc are more accurate when compared to Dbox.

To evaluate the hypothesis that De=βDi (Eq. ), β is determined directly from the MODIS cloud masks. Accurately determining the power-law distribution exponent in satellite data can be surprisingly challenging, primarily because the artificial truncation of clouds along the domain edge can influence the measured cloud size distribution. A comprehensive analysis of the possible biases arising from domain truncation was done by . Here, we implement their recommended methodology, which was to calculate β using a linear regression to a logarithmically binned and transformed histogram of cloud perimeter (Fig. ). As recommended by , fits are performed only for those bins for which at least 50 % of clouds in that bin are entirely contained within the measurement domain. This method ensures that β is calculated from only the unbiased portion of the distribution, i.e. the portion that is not dominated by large clouds that extend beyond the measurement domain. This filtering removes approximately 0.06 % to 0.4 % of all clouds depending on threshold.

Another consideration for the calculation of β is the presence of “nested perimeters” (Appendix ), or the perimeters of cloud holes and even nested clouds within cloud holes. Here, we take the approach that each nested perimeter is in effect a distinct cloud edge, for which there is a distinct value to be considered in the power law fit used to determine a value for β . In effect, a single cloud may possess multiple perimeter values if it contains holes. The alternative approach would be to sum the exterior perimeter of a cloud with the perimeter of all the holes within the cloud prior to fitting. Appendix  considers this distinction further and shows that the method employed here of including nested perimeters results in values of β that are larger by approximately 0.1.

Table compares the calculated values of the box dimension and correlation dimension to the product βDi. Generally, the fractal dimension decreases with threshold R, although the dependence is significant only for De with minimal sensitivity for Di (see Appendix for values of De calculated using a wider range of thresholds). Such dependence is an indication that cloud fields are “multifractal”, a term meaning that the ensemble fractal dimension is a function of the thresholding scheme used to binarize the cloud field. Such multifractal behavior has been used to shed light on turbulent intermittency in the atmosphere and indicates that the intensity of the turbulence varies spatially .

As shown in Table 1, all three methods for calculating the ensemble fractal dimension – the box dimension, correlation dimension, and βDi – point to a value roughly between 1.6 and 1.8, with a possible exception of the box dimension at higher reflectivity thresholds. Although Dc and Dbox decrease with increasing threshold, the product βDi displays less of a trend due to the relatively weak dependence of both β and Di on R. More importantly, whichever method is used to calculate De, the values are substantially higher than those for the individual fractal dimension Di, which has a value of approximately 1.4 (Fig. ).

For cloud fields, this distinction between the two fractal dimensions Di and De has been almost entirely overlooked by previous studies, most of which only considered Di . The few studies that measured De did so by box-counting but without noting that Di and De are different. Conflating the two different dimensions could lend a false impression of a discrepancy between studies, when in reality the two values simply reflect the difference between the geometries of individual clouds and those of a cloud field.