# Uncovering the Fractal Nature of Water Vapor Distribution above the Surface
of the Earth

Anirudha Menon  Physics and Applied Mathematics Unit, Kolkata 700108, India
<EMAIL_ADDRESS>Banasri Basu  Physics and Applied Mathematics Unit, Kolkata
700108, India<EMAIL_ADDRESS>
###### Abstract

Fractals have been at the heart of geophysical and geospatial studies in the
recent past. We examine the emergent fractal character of water vapor
distributions above the surface of the Earth as a function of both image
resolution (number of pixels) and moisture content percentile. We calculate
physically relevant quantities such as fractal dimension, number of clusters,
and size of the largest cluster with varying vapor percentile using
computational methods and algorithms. We test the applicability of Korcak’s
law on our system and determine the quality of the fit using the Kolmogorov-
Smirnoff satistic. We show that the fractal character of the distribution is
exact as a function of image resolution and approximate in some regimes as a
function of the vapor percentiles.

##  I Introduction

Initiated by a series of papers of Mandelbrot and Wallis 1 ; 2 ; 3 , the
fractal dimensional analysis of geophysical variables, over a time period, has
become an established tool for the investigation of the dynamics of the
systems. This has been followed by applications of the technique to various
geophysical phenomena 4 ; 5 ; 6 . The geophysical phenomena, particularly
climate variability and trends, have significant environmental and socio-
economic impacts. Understanding the emergent fractal structures embedded in
certain geographic distributions of physical quantities will lead to further
discoveries and characterizations of these quantities.

A fractal 6a  is a beautiful and infinite pattern that is self-similar from
the macroscopic to the microscopic. Such objects are not integer dimensional
in the conventional sense. Fractal dimensional analysis can identify and
quantify how the geometry of patterns repeat from one size to another. Scaling
laws are at the heart of fractal analyses. Many natural and social-economic
phenomena display a scale distribution 1 ; 2 ; 3 ; 6a , including coastline
and mountain formations. These distributions are approximate fractals, i.e.,
that they appear scale-invariant in a certain range of scales. It is well
understood that while ideal fractals exist as mathematical constructions, the
fractal structures found in nature are approximate.

Numerous attempts 7 ; 8 ; 9 ; 10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16 ; 17 ; 18  that
elucidate the spatio-temporal patterns of land surface temperature (LST) and
reveal the scaling properties in these distributions motivated us to study the
fractal analysis of the distribution of water vapour data. Based on theory,
fractal analysis can be used to decompose irregular data sets to many
microscopic pictures on multiple scales and reveal their scaling properties in
various geospatial distributions.

In this paper, we concentrate on the analysis of Earth’s primary greenhouse
gas, water vapour, which is a critical component of Earth’s climate system.
Accurate quantification of water vapour data and the interpretation of its
physical characteristics are crucial in understanding how global climate
systems, such as global warming, are changing over time. Urban land use 18b ;
18c , urban infrastructure 18d , and network distributions 18e ; 18f ; 18g
have been studied using fractal based analyses and techniques. Similar
techniques have also been developed for atmospheric sciences 18h . We,
however, seek to characterize water vapor distribution above the entire
planet, and not restrict to cities or urban areas. We also analyze the
applicability of Korcak’s law to our system and provide substantial data
driven evidence in the affirmative.

The structure of this manuscript is as follows: In section II we discuss the
methods of data acquisition, data extraction, and data processing. In section
III, we examine the fractal nature of the distribution as a function of the
resolution of the images obtained, namely at $180\times 360$ pixels and
$360\times 720$ pixels. In section IV we examine the fractal character of the
vapor distribution as a function of the vapor percentile. We discuss our
observations in both sections III and IV. In section V we summarize our
findings and conclude.

##  II Data Extraction and Processing

Data was obtained in the Geotiff format for the monthly global distribution of
water vapor from the TERRA MODIS (Moderate Resolution Imaging
Spectroradiometer) dataset 18a  for 2021 at $360\times 720$ pixels and for the
years 2012-2021 at $180\times 360$ pixels. The MODIS atmospheric water vapor
data estimates the total column water vapor made from integrated MODIS
infrared retrievals of atmospheric moisture profiles in clear scenes over land
and ocean. Note that the total column water vapor can only be obtained in
clear-sky conditions. Representative datafiles are plotted in Fig. 1, where
the data in the top row is for two different months at the resolutions of
$180\times 360$ and $360\times 720$ pixels, respectively, for the left and
right plots. This data was then extracted using the Rasterio module in python
v3.9 and converted to the numpy array format. Different values of vapor
percentiles were chosen from $0$ to $100$ and the numpy arrays were binned
binarily based on whether the value in a given pixel was greater or lesser
than the given vapor percentile. The bottom row in Fig. 1 represents the
binned data at different vapor percentiles. Once this was completed for all
data files, a clustering algorithm was used to to determine the clusters in
the binarily binned arrays. This algorithm was recursive and identified all
the pixels in the Moore neighborhood of a given pixel. Two pixels are said to
be in the same cluster if they both have value 1 and are in each others Moore
neigborhood. The pixels were added to a list if they satisfy the above
criteria, and the neighboring cells of these added cells were identified. The
whole process was repeated until the entire grid was covered. A separate
algorithm was used to calculate the perimeters and areas of the clusters.
While the areas of clusters were trivial to calculate, since they are simply
the number of cells in a cluster, the perimeters were harder to determine and
required a caching of the neighbors of each cell. The data produced from these
computations was then stored on disk for further analysis.

Figure 1: Visualization of the MODIS water vapor data: The top panel shows the
distributions at two different resolutions, namely $180\times 360$ pixels and
$360\times 720$ pixels. The bottom panel shows the same data after the binning
procedure described above. The left plot is at the 80th percentile while the
right plot is at the 20th percentile.

##  III Resolution based Numerical Analysis

The self-similar property underlying the original fractal model assumes that
the form or pattern of the spatial phenomenon remains unchanged throughout all
scales. Closely related to scale is the concept of resolution. Resolution
refers to the smallest distinguishable parts in an object or a sequence and is
often determined by the least count of the instrument or the sampling interval
used in a study. Moreover, fractals may be useful in identifying the
breakpoints in a pattern or distribution at a particular scale where the
processes contributing to these patterns become unstable. In this regard, it
is important to have a clear understanding of the effect of resolution in
fractal dimensional analysis.

We considered the binarily binned data with the clusters being formed by the
aggregation of the cells with value $1$. We examined data at two different
resolutions of $180\times 360$ pixels and $360\times 720$ pixels in order to
see how the nature of the distribution varied as a function of resolution.
Alternatively, we could also have binned the cells with value $0$ and this
would give us a complementary distribution to the one we are currently
studying. An analysis of this complimentary distribution, while potentially
interesting, is relegated to a future work.

###  III.1 Perimeter vs Area Plots

In order to quantify the fractal nature of the water vapor distribution, we
consider the aggregated area-perimeter fractal dimension 6a ; 18  of the
clusters at each vapor percentile $V$ using the following equation

$\displaystyle\sum_{c\subset C}P_{c}=k\big{[}\sum_{c\subset
C}A_{c}\big{]}^{D/2},$ (1)

where $c$ represtens each cluster in the total set of clusters $C$, $A_{c}$
and $P_{c}$ are the area and the perimeter of the cluster $c$,
$k=2\sqrt{\pi}$, and $D$ is the fractal dimension. The value of $k$ is
determined based on the limiting case of a circle. When the shapes of the
clusters cumuluatively are more elongated or convoluted, we approach the
linear limit of $P=A$ and $D=2$.

The distributions of clusters are obtained at different values of the water
vapor percentile ranging from $0$ to $100$. We present below a log-log plot of
the distribution of the cumulative perimeter vs cumulative area at different
percentiles for the year 2021. All 12 months of the year are considered in the
analysis at two different resolutions of $180\times 360$ pixels and $360\times
720$ pixels. The slope of the scatter plots represents the fractal dimension
up to an additive constant.

Figure 2: Aggregate area-perimeter scatter plot at 180x360 pixels and 360x720
pixels on the left and right, respectively. The fact that the distributions
look almost identical at these two resolutions leads us to explore further.

As a very interesting observation, we note that the distributions (and in
particular, the slopes) look visually identical at both resolutions,
indicating that there maybe some universality in the data as a function of
resolution. Thus, we find motivation to explore this aspect further.

###  III.2 Fractal Dimension

We plot the fractal dimensions as a function of water vapor percentiles for
the $12$ months in 2021 at both resolutions. The results are shown in Fig. 3.
We observe that the fractal dimensions are in general identical for the
cluster distribtuions at both resolutions for almost all the months. We find
that the January data presents a disagreement to the previous observation and
we allude this discrepancy to an anomaly in the data, since it is not a
prevalent phenomenon. The fact that the fractal dimension is independent of
resolution is a key result of this work, as the fundamental construction of
fractals states that fractal geometries are scale invariant. While we would
like to have explored this further at other resolutions, current computational
capacity limits this.

Figure 3: Fractal Dimension vs Vapor Percentile for 12 months in 2021: The
blue markers represent the $180\times 360$ pixel data and the orange markers
represent the $360\times 720$ pixel data.

###  III.3 Percolation Phenomena

The existence of percolation phenomena associated with the vapor distribution
clusters may be studied by examining the number of clusters at different vapor
percentiles. We examine the scaled number of clusters as a function of vapor
percentile. The scaled number of clusters is defined as follows

$\textrm{scaled \\# of clusters}=\frac{\textrm{\\# of
clusters}}{\textrm{maximum \\# of clusters}}$ (2)

The results are plotted in Fig. 4 for all 12 months in 2021 at both
resolutions. Strikingly, we find once again, an almost identical distribution
at both resolutions, further strengthening our claim of scale-invariance. We
observe a general trend that above $V=30\%$ the number of clusters remain
approximately constant, while below this percentile the number of clusters
drop linearly to $1$ at $V=0$. This indicates that a percolation phenomenon
exists, such that below the $30$th percentile the clusters aggregate to form
larger and larger clusters. The percolation threshold is thus estimated to be
$V_{thres}=30\%$ and is represented in Fig. 4 as the dashed vertical line. The
anomaly presented in the previous subsection is also present in this case for
the month of January.

Figure 4: Scaled Number of Clusters vs Vapor Percentile for all 12 months in
2021. The blue markers represent the $180\times 360$ pixel data and the orange
markers represent the $360\times 720$ pixel data.

###  III.4 Maximum Cluster Size

We also examine the variation of the scaled maximum cluster size as a function
of the vapor percentile and the results for both resolutions is presented in
Fig. 5. The scaled maximum cluster size is defined as

$\textrm{scaled maximum size of clusters}=\frac{\textrm{local maximum cluster
size}}{\textrm{global maximum cluster size}}$ (3)

The data in the plots below corroborate well with the story presented in the
previous subsection, with the scaled maximum cluster size decreasing from $1$
to $0$ as the vapor percentile increases from $0$ to $100$. As such, this
shows that the clusters are largest at $0\%$ and get smaller as the vapor
percentile increases. The curves presented in Fig. 5 appear to be decaying
exponentials though we have not attempted to fit the data to an exponentially
decaying fit function. We note that there appears to be a dip in the scaled
maximum cluster size for all months at the $60$th percentile, but we don’t
have a good physical reason for its occurence.

Figure 5: Scaled Maximum Cluster Size vs Vapor Percentile for all 12 months in
2021. The blue markers represent the $180\times 360$ pixel data and the orange
markers represent the $360\times 720$ pixel data.

###  III.5 Scaling Exponents

In fractal landscape structures, clusters are statistically self-similar at
the percolation threshold over certain size-ranges. The cluster areas follow a
probability distribution with a power-law tail 19  at the percolation
threshold. The physcist Korcak made this prediction, and the law is now known
as Korcak’s law 20 . We wish to examine the applicability of Korcak’s law to
our distribution, wherein the statement of the law reads

$P(a)\propto a^{\beta}$ (4)

where $a$ represents the area of a cluster, $P(a)$ represents the probability
of randomly selecting a cluster of size $a$, and $\beta$ is the scaling
exponent. Tempering may occur as a result of deviations from the powerlaw
above the percolation threshold 21 .

We plot the probabilities for March 2021 as a function of the cluster areas in
Fig. 6 at the percolation threshold for both resolutions, $180\times 360$
pixels on the left and $360\times 720$ pixels on the right. The data is
plotted as the orange scatter points, and the power-law fit is represented by
the blue line. We test for the quality of the fit by using the Kolmogorov-
Smirnoff (KS) statistic. These values are stated in the plots below. As we can
see, the fit to the data is close to perfect, with the KS statistic being
approximately $1$ (up to machine precision) and the p-value being $0$. This
verifies that our data is in agreement with Korcak’s law.

We also plot the scaling exponents for the data at the percolation threshold
for all months of the year 2021 in Fig. 7. The blue markers show the scaling
exponents for the $180\times 360$ pixel data and the orange dots show the same
for the $360\times 720$ pixel data. The scaling exponents are not identical
for each month at the two resolutions and there is no reason to believe that
there should be a correspondence. The range exhibited by the scaling exponent
is $\beta\in(1.6,2.3)$. We have verified that in each case the KS statistic is
very close to $1$ and the p-value is roughly $0$.

Figure 6: Power law fit: The plots show the power-law fit at two resolutions
of the data for March 2021, $180\times 360$ pixels (left) and $360\times 720$
pixels (right).  Figure 7: Scaling Exponents: The plot shows the variation of
the scaling exponents as a function of months in the year 2021 at both
resolutions. The blue markers show the scaling exponents for the $180\times
360$ pixel data and the orange dots show the same for the $360\times 720$
data.

##  IV Vapor Percentile based Numerical Analysis

In this section, we examine the fractal nature of the water vapor distribution
as a function of vapor percentile. We conduct the analysis at a data
resolution of $180\times 360$, and $10$ years worth of monthly data is
considered in the period $2012-2021$. We proceed with the analysis as in the
previous section and the results are presented below.

###  IV.1 Perimeter vs Area

We start by plotting the cumulative perimeter versus the cumulative area on a
log-log plot. The plots are presented in Fig. 8 where we average over each
month on the left plot and over each year for the plot on the right. The
distributions are very similar with the fractal dimension lying between $1$
and $2$ which are the limiting cases of a circle and a straight line. This is
the first indication that there is some fractal character to the distribution,
since the aggregated area-perimeter fractal dimension is not an integer.

Figure 8: Aggergated Area-Perimeter Plot for 10 years of data from 2012 to
2021 at a resolution of $180\times 360$ pixels.

###  IV.2 Fractal Dimension

Figure 9: Fractal Dimension vs Vapor Percentile for each month and all 10
years, $2012-2021$, at a resolution of $180\times 360$ pixels.

We next examine the fractal dimensions for all $12$ months of the years
$2012-2021$ as a function of the vapor percentile. Each subplot in Fig. 9
contains the data for all $10$ years for a given month. We notice an overall
trend in the behavior of the fratal dimension while the exact numerical values
are case dependent, but continue to lie in the interval $1<D<2$. For each of
the subplots in Fig. 9 we observe that the fractal dimension increases rapidly
for vapor percentiles $V\leq 30\%$, and then stabilizes to an approximately
fixed value in the range $30<V<50$. Then the fractal dimensions increase again
in the interval $50\leq V<80$, but more gradually than before. Finally, the
fractal dimensions remain steady for percentiles $80\leq V\leq 100$.

Figure 10: Fractal Dimension vs Vapor Threshold averaged monthly and yearly
for all 10 years, $2012-2021$, at a resolution of $180\times 360$ pixels.

We can thus conclude that the fractal dimension remains constant in two
regimes of vapor percentiles, and so the clusters exhibit an approximate
fractal behavior as a function of threshold in these two regimes. Outside
these regimes, since the fractal dimensions vary with vapor percentile, we
cannot conclude an approximate fractal behavior.

We further plot the monthly and yearly averaged distributions of fractal
dimensions as a function of vapor percentile in Fig. 10. Some interesting
trends arise as a result of this data representation. Firstly, we see that on
the left plot of Fig. 10 that fractal dimensions show a broad distribution at
$80\%\leq V\leq 100\%$ with the hottest and driest months of the year in the
Northern hemisphere, May and June showing the lowest values. A comparatively
drier month, December, shows the next highest value of fractal dimension in
this regime, while the rest of the months are clumped together. The yearly or
annual averages also show unobvious trends, with the fractal dimension
distributions being almost identical in the ranges $0\%\leq V\leq 40\%$ and
$80\%\leq V\leq 100\%$. In the intermediate range $40\%<V<80\%$, the
distributions grow in magnitude at different rates for different years, with
the peak growth rate being observed for the year 2018. The plot on the right
shows that the peak fractal dimension occurs for the year 2017 and this value
is reached at the $60$th percentile.

###  IV.3 Percolation Phenomena

We plot the number of clusters as a function of vapor percentile for each
month for all 10 years, $2012-2021$ in the plots in Fig. 11. While the
distributions of cluster sizes are narrow for each month for percentiles
$V<30\%$, they form a broad distribution above this percentile. The phenomenon
of percolation is once again observed here and we percolation the peroclation
threshold percentile as $V_{thres}=30\%$. Above the $30$th percentile, the
number of clusters is approximately constant for each year, while below it the
number of clusters falls in a linear manner to $1$ at $V=0\%$. Other features
exist in the plots in Fig. 11, however, we can make no ubiquitous statements
about these features and so we decide not to discuss them in any detail.

In Fig. 12, we plot the same distributions as in Fig. 11 averaged over years
and months on the left and right plots, respectively. For the plot on the
left, the cluster counts show a broad distribution at percentiles higher than
$V=50\%$. The general trend is for the number of clusters to increase with the
increase in vapor percentile, with the highest value being held by April at
the $100$th percentile. The plot on the right shows an overlap for the data at
percentiles lower than $V=30\%$. At percentiles higher than $V_{thres}$, the
distribution assumes a broader character, with the highest cluster count being
exhibited by 2015 at the $100$th percentile.

Figure 11: Number of Clusters vs Vapor Percentile for each month for all 10
years, $2012-2021$ at $180\times 360$ pixels.

Figure 12: Number of Clusters vs Vapor Percentile averaged over months and
years for 10 years, $2012-2021$, at a resolution of $180\times 360$ pixels.

###  IV.4 Maximum Cluster Size

As before, we also plot the size of the largest cluster as a function of vapor
percentile for each month for all $10$ years at a resolution of $180\times
360$ pixels. These plots are shown in Fig. 13. The distributions show an
exponential decay for all months and years, with significant overlap between
the data for different years. Some exceptions to this observations occur on
and around the $60$th percentile. The distributions agree well with the
percolation hypothesis, since at low vapor percentiles there exits one single
massive cluster while at very large percentiles, the average size of clusters
is small and approximately constant. While we observe that the decays are
likely exponential, we make no attempt at fitting them to some fit function.

Figure 13: Maximum Size of Cluster vs Vapor Percentile for each month for 10
years, $2012-2021$, at a resolution of $180\times 360$ pixels.

We also plot the above data averaged over years and months in Fig. 14 on the
left and right plots, respectivelty. The plot on the left shows a relatively
broad distribution for percentiles less than $30\%$, and a narrow or
convergent distribution for larger percentiles. The overall tendency of the
distribution is one of an exponential decay, although there exist some
inflection points in the interim. The plot on the right, which represents data
averaged anually, shows a large overlap for all years except around the $60$th
percentile. While this is an interesting observation, we have no hypotheses as
to appearance of the broadening and the relavance of the $60$th percentile to
it.

Figure 14: Maximum Size of Cluster vs Vapor Percentile averaged over months
and years for 10 years, $2012-2021$, at a resolution of $180\times 360$
pixels.

###  IV.5 Scaling Exponents

As discussed previously, we are interested in testing the validity of Korcak’s
law for our distributions. We proceed as before and calculate the
probabilities $P(a)$ for all areas of cluster size $a$. We plot the
probabilities for the March 2021 data as a function of the cluster areas in
Fig. 15 at the percolation threshold (left plot) and at the $75$th percentile
(right plot). The data is plotted as the orange scatter points, and the power-
law fit is represented by the blue line. We test for the quality of the fit by
using the Kolmogorov-Smirnoff (KS) statistic. These values are stated in the
plots below. As we can see from the plots below, the fit to the data is close
to perfect, with the KS statistic being approximately $1$ and the p-value
being $0$. This verifies that our data is in agreement with Korcak’s law, with
little deviation from the powerlaw behaviour even above the percolation
threshold.

We plot the scaling exponents for the data at the percolation threshold
averaged over years and months in Fig. 16 on the left and right plots
respectively. The scatter plots contain the mean scaling exponents and the
respective standard deviations. The range of the scaling exponents in both
cases lie between $1.90$ and $2.20$ with relatively small deviations from the
mean. We have verified that in each case the KS statistic is very close to $1$
and the p-value is roughly $0$.

Figure 15: Power law fit

Figure 16: Scaling Exponent

##  V Conclusions

In this work we have sought to unravel the underlying fractal nature of the
distribution of water vapor above the Earth. We have obtained data from the
TERRA MODIS dataset and processed it using python v3.9. Clustering and
numerical calculations were performed using author-designed algorithms.

We first investigated the variation of fractal dimension of the dataset as a
function of the resolution of the data at resolutions of $180\times 360$
pixels and $360\times 720$ pixels for the year 2021. We found that the fractal
dimension of the vapor distribution as a function of vapor percentile remained
universal with values between $1$ and $2$ for both resolutions for all months
except January. The anomalous result for January 2021 was attributed to some
inconsitency in constructing the dataset. Further, the scaled number of
clusters as a function of vapor percentile was plotted, and we found
percolation to occur at $V_{thres}=30\%$. This observation was further
supported by the results of plotting the scaled maximum cluster size versus
vapor percentile. Both sets of representations showed an identical overlap
between the data for both resolutions, further confirming the universality of
the fractal structure as a function of resolution. We then determined the
scaling exponent for this data distribution based on Korcak’s law and found
our numerical analysis to be in agreement with it. The quality of our fit was
tested using the Kolmogorov-Smirnov statistic which approached unity, with a
p-value of zero. This confirmed the excellent quality of our fit function and
parameters.

We then examined the variation of fractal dimension of the dataset as a
function of vapor percentile at a resolution of $180\times 360$ pixels for the
years 2012-2021. The fractal dimension of the vapor distribution as a function
of vapor percentile remained constant (as a function of vapor percentile) in
certain ranges with values between $1$ and $2$ for all months and years. We
therefore concluded that the data exhibited approximate fractal behavior in
these ranges of vapor percentiles. The number of clusters as a function of
vapor percentile was plotted, and we found percolation to occur at
$V_{thres}=30\%$. This observation was further supported by the results of
plotting the maximum cluster size versus vapor percentile. We plotted the
monthly averaged and yearly averaged version of all three quantities discussed
above and found some interesting trends in the distributions. The data was
then fitted to Korcak’s law and the scaling exponents were determined. We
tested the quality of our fit using the Kolmogorov-Smirnov statistic which
approached unity, with a p-value of zero. This affirmed the effectiveness of
our fit function and parameters.

In conclusion, we find that the water vapor distribution above the surface of
the Earth exhibits a fractal character as a function of data resolution and
vapor percentile, the former of which is exact while the latter is
approximate. The water vapour clusters display properties analogous to that of
a percolating system as we vary the vapour percentile. At the percolation
threshold, the size distribution of these water vapour clusters follows a
power law, namely Korkac’s law, with a scaling exponent of
$\beta\in(1.6,2.3)$. To the best of our knowledge, the examination of the
global distribution of water vapor from a fractal perspetctive is novel, which
underlies the importance of this work.

Acknowledgements. AM would like to thank the Science and Engineering Research
Board (SERB), Government of India, for financial support through the
fellowship Grant No. CRG/2019/001461. The research of BB is partially
supported by research grant No. CRG/2019/001461 from the Science and
Engineering Research Board (SERB), Government of India. The authors would also
like to thank Dr. Anamika Shreevastava (NASA Jet Propulsion Lab, Caltech) for
extensive and highly engaging discussions on the research associated with this
manuscript.

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