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The Fractal Nature of Drought: Power Laws and Fractal Complexity of Arizona Drought

Sepideh Azizi1, Tahmineh Azizi2,∗
1Department of Urban Planning and Design, Shiraz University, Iran

sepid.azizi97@gmail.com
2Department of Mechanical Engineering, Florida State University, USA

tazizi@fsu.edu
∗Correspondence: tazizi@fsu.edu

Abstract. In this study, we explore the possibility that the Drought Monitor database belongs to classof fractal process which can be characterized using a single scaling exponent. The Drought Monitormap identifies areas of drought and labels them by intensity: D0 abnormally dry, D1 moderatedrought, D2 severe drought, D3 extreme drought, and D4 exceptional drought. The vibration analysisusing power spectral densities (PSD) method has been carried out to discover whether some typeof power-law scaling exists for various statistical moments at different scales of this database. Weperform multi-fractal analysis to estimate the multi-fractal spectrum of each group. We apply Higuchialgorithm to find the fractal complexity of each group and then compare them for different timeintervals. Our findings reveal that we have a wide range of exponents for D0-D4. Therefore, D0-D4belong to class of multi-fractal process for which a large number of scaling exponents are required tocharacterize the scaling structure.

1. Introduction
Drought is defined as a moisture deficit bad enough to have social, environmental or economiceffects. Drought is a recurring feature of nearly every climate on the planet [1–5]. In many partsof the world, including North America, we have little ability to predict exactly when drought willhappen next. But if we look at history and climate data, we can be sure that drought will happenagain at some point. In the United States, a well-developed economy and agricultural systemgenerally protect citizens from the most critical effects of drought such as shortages of food andwater. However, drought still causes extreme hardship for farm and ranch families, and individualwells may run dry. Besides affecting municipal water suppliers, drought affects businesses andenvironmental interests that are reliant on adequate and timely amounts of precipitation and wa-ter, such as habitat for fish and wildlife, outdoor recreation outfitters, and landscaping and car
Received: 2 Mar 2022.
Key words and phrases. fractional geometry; power spectral densities (PSD); multifractal analysis; fractal dimension;drought. 1

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Eur. J. Math. Anal. 10.28924/ada/ma.2.17 2wash services [5–14]. The Drought Monitor map identifies areas of drought and labels them byintensity. D1 is the least intense level and D4 the most intense. D0 areas are not in drought,but are experiencing abnormally dry conditions that could turn into drought or are recovering fromdrought but are not yet back to normal.There are different indices which have been used to asses drought severity and impacts in differenttime-scales. The normalized difference vegetation index (NDVI) is one of the most widely uti-lized drought indices to determine different drought levels [15–17]. Satellite databases have beenextensively used to record and quantify the changes may happen in vegetation coverage due tochanging climate conditions. The NDVI is estimated using visible and near-infrared (NIR) bandsfrom Advanced Very-High-Resolution Radiometer (AVHRR), Terra Moderate Resolution ImagingSpectroradiometer (MODIS), and Landsat sensors. In general, positive NDVI values demonstratevegetated areas, zero and negative values are associated with bare soil and water bodies [15]. Thetime series of the average NDVI for Arizona (Arizona includes regions with moderate to exceptionaldrought D1-D4) shows the highest and lowest NDVI values during 10 years (2010–2020) (for themonth of Jan-Dec each year) (see figure (1)). The NDVI data selected from the Google Earth En-terprise open source which is derived using Terra Moderate Resolution Imaging Spectroradiometer(MODIS) and would be useful to forecast the future changes in vegetation in Arizona. Each stateexperiences different set of impacts during a drought. We have also displayed the table of reportedimpacts during past droughts in Arizona for each level of drought on the U.S. Drought Monitor intable (2) (Source(s): NDMC, NOAA, USDA). When we study real world time series data, dependingon scale and higher order moments, we may confront with data that display nonlinear power-lawbehaviours. For these type data, we need to apply multifractal analysis. In multifractal analysis wediscover whether some type of power-law scaling exists for various statistical moments at differentscales. A process called mono-fractal, if it can be characterized using a single scaling exponent,or this process is a linear function of the moments. Likewise, a process called multi-fractal, if wesee the scaling behavior follows a function which is non-linear in the moments. When we studyscale invariant time series data, or data with different scaling behavior, we are not able to use theclassical time series analysis and we need to perform fractal analysis.In this study, we use fractal geometry to classify drought severity from 2000 to 2021 in Arizona.We perform multifractal analysis to discover whether some type of power-law scaling exists forvarious statistical moments at different scales of these data sets. We plot the multifractal spectrato compare the width of the scaling exponent for each spectrum. A quantitative analysis commonlyknown as the Fractal Dimension (FD) using Higuchi algorithm.
2. Materials, Methods and results

2.1. Data. Here, data has been collected using U.S. Drought Monitor for each week of the selectedtime period (January 2000 to Nov 2021) and location (Arizona, USA), see figures (3) and (4). The

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Figure 1. Normalized Difference Vegetation Index (NDVI) Arizona between 2010-2020; Google Earth Enterprise Open Source
U.S. Drought Monitor which started from 1999, is a partnership between the National DroughtMitigation Center (NDMC) at the University of Nebraska-Lincoln, the United States Departmentof Agriculture (USDA), and the National Oceanic and Atmospheric Administration (NOAA). EachThursday, the U.S. Drought Monitor (USDM) will be updated to demonstrate the location andintensity of drought across the country. Using the experts’ assessments, drought categories displayconditions related to dryness and drought such as observations of how much water is available instreams, lakes, and soils compared to usual time of year (Source(s): NDMC, NOAA, USDA) [18].
2.2. Time-Frequency Analysis and Continuous Wavelet Transform (CWT). Continuous WaveletTransform (CWT) provides a linear time-frequency representation of non-stationary signals calledscalogram by breaking the data into scales by preserving time shifts and time scales. Therefore, thewavelet transform makes the analysis of the data in different frequency ranges easier and we can

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Figure 2. Table of the reported impacts during past droughts in Arizona for eachlevel of drought on the U.S. Drought Monitor; (Source(s): NDMC, NOAA, USDA)
extract useful information from the time intervals between its consecutive waves of the data [19].To compute the scalogram of data which is function of time and frequency, at first we split thetime series data into overlapping segments, then we need to compute the absolute value of thecontinuous wavelet transform coefficients of each segment and finally, plot it. We have displayedthe scalogram plots of Drought Monitor Categories Arizona database (2000 - present) in figures(5)-(6).
2.3. Vibration frequency analysis using Power spectral densities (PSD). The fast Fourier trans-form (FFT) has been used widely to analysis of vibration frequency in computing discrete Fouriertransform (DFT). However, FFT only works accurately if we have a finite number of dominant fre-quency components. To overcome this problem, we use the power spectral densities (PSD) which

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Figure 3. Arizona Percent Area in U.S. Drought Monitor Categories database (2000- present); National Drought Mitigation Center (NDMC), the U.S. Department ofAgriculture (USDA), and the National Oceanic and Atmospheric Administration(NOAA)

Figure 4. Histogram of Arizona Percent Area in U.S. Drought Monitor Categoriesdatabase (2000 - present); National Drought Mitigation Center (NDMC), the U.S.Department of Agriculture (USDA), and the National Oceanic and AtmosphericAdministration (NOAA)

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Figure 5. Time-frequency representations of Drought Monitor Categories database(2000 - present) using Continuous Wavelet Transform (CWT) in two dimensionalTime-Frequency space

Figure 6. Time-frequency representations of Drought Monitor Categories database(2000 - present) using Continuous Wavelet Transform (CWT) in three dimensionalTime-Frequency-Magnitude space
is applied to characterize random vibration in time series data. To compute PSD, we multiply eachfrequency bin of FFT by its complex conjugate to get a real spectrum and then normalize the results

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Eur. J. Math. Anal. 10.28924/ada/ma.2.17 7to frequency bin width. Here, we have applied the (PSD) method for our database and then we fitthe logarithm power spectral densities to their frequencies in log format using least squares ap-proximation method. Finally, we calculate the slope for each regression line captures the linearityof data. In figure (7), we can see the fitted least squares approximation to the logarithm of powerspectral density of Arizona drought database.Moreover, we have plotted the scaling exponent graphs for Arizona drought database in figure (8).

Figure 7. Fitted least squares approximation to the logarithm of power spectraldensity of Arizona drought database (2000 - present) obtained by wavelet tech-niques

2.4. Multifractal Analysis and Discrete Wavelet Transform (DWT). Fractal dimension is one of themost often used algorithm to describe the complexity of a fractal object by measuring the changesof coverings relative to the scaling factor [20–25]. It also specifies the space filling capacity of afractal object with respect to its scaling properties in the space [26–29]. The relationship betweenscaling and covering is often hard to be characterized. The variation in the number of coverings,
N(ε), with respect to the scaling factor ε, can be written as

N(ε) ∝ ε−D (1)
where D is the fractal dimension. The relation (1) is called scaling law that is used to demonstratethe size distribution of many objects in nature. The box counting formula which has been widely

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Figure 8. Scaling exponent of power spectral density for Drought Monitor Cate-gories database (2000 - present).
applied to approximate the fractal dimension of an irregular object is defined as

DB = lim
a→0

ln(N(a))

ln(1/a)
(2)

However, this monofractal dimension is not able to fully characterize complex scaling behaviors ofmany irregular objects in the real world. That’s why to study irregular objects we need to applythe multifractal algorithm. The multifractal analysis utilizes a spectrum of singularity exponents toprovide a detailed and local description of complex scaling behaviors. In order to quantify localdensities of the fractal set, we approximate the mass probability using the following formula
Pi(a) =

Ni(a)

N
(3)

where Ni(a) is the number of mass in the i th subset of measure a, N is the total mass of the set.When we scale the mass probability Pi(a) with measure a of a multifractal set, it also demonstratesthe power law behavior:
Pi(a) ∝ aαi (4)

where αi is the singularity exponent characterizing the local scaling in the i th subset. The mul-tifractal spectrum f (α) provides a statistical distribution of singularity exponents αi . In general,

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Eur. J. Math. Anal. 10.28924/ada/ma.2.17 9

f (α) may be estimated using the Legendre transformation
f (α) = q α− τ(q)

α(q) =
d τ(q)

d qwhere q is the moment and τ(q) is the mass exponent of the qth order moment. In addition, themultifractal measures may be specified by scaling of qth moments of Pi(a) as
N(a)∑
i=1

P
q(a)
i ∝ aτ(q) = a(q−1)Dq (5)

where Dq = τ(q)

(q − 1) is the generalized fractal dimension. For q = 0 equation (2.4) becomes
N(a) ∝ a−D0

which is similar to formula (1).From multifractal analysis results of Arizona drought database (see figure (9)), we can easilysee that we have a wide range of exponents for D0-D4, which indicates they have multifractalstructure. The drought database needs to be indexed by different exponents as we decompose theminto different subsets. Therefore, D0-D4 require much more exponents to characterize their scalingproperties.

Figure 9. The multi-fractal spectrum analysis of Arizona Drought Monitor Cate-gories database (2000 - present) shows the occurrence of multi-fractality with abroad range of exponents in data structure of D0-D4.

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Eur. J. Math. Anal. 10.28924/ada/ma.2.17 102.5. Higuchi Fractal Dimension Algorithm. When we Use box counting method, we compute thefractal dimension and or the complexity of a fractal process in two dimensional space [30]. However,when we are working with many real world time series data, it fails to recognize the sudden changeshappen in data [31]. To solve this problem, there are different methods such as Higuchi algorithm,power spectrum analysis, and Katz algorithm that help to analyze the complexity of irregulardata [32–34]. Here we use Higuchi Algorithm. We start with a finite time series x1, x2, x3, . . . , xN .Then, we create k new time series xkm of the form
xm, xm+k , xm+2k , . . . , x[m+Ak]where A = (N −m)/k . For each time interval k and the initial time m such that m = 1, 2, . . . , k ,we calculate the length of xkm using

Lkm =

∑[A]
i=1 |xm+i k − xm+(i−1)k |

k
R

where R = (N − 1)/[A]k is the curve length normalization factor. To compute the average of curvelength for each k , we calculate the mean of Lkm for m = 1, 2, . . . , k and take the average for
k = 1, . . . , kmax . Next, we plot log(Lkm) versus log(1/k) for different time interval k . Finally, wecalculate the slope of regressed line which is obtained by the least-squares approximation as theHiguchi fractal dimension for time interval k = 500. We have estimated the fractal dimension ofthe Arizona drought database and plotted their regression models for each data in figure (10).

Figure 10. Plots of log(Lkm) versus log(k) for time interval k = 500, the logarithmicscale and the corresponding slope of fitted regression line (the Higuchi fractaldimension) for Arizona Drought Monitor Categories database (2000 - present).

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Eur. J. Math. Anal. 10.28924/ada/ma.2.17 113. Discussion
Drought as a slowly progressed and hidden disaster takes place in normal cycles of climate andit affects adversely environment and economic the same as other disasters. Therefore, character-izing the complexity of its nature will help to predict and recognize its different stages before itdamages our societies. In U.S., the National Integrated Drought Information System (NIDIS) whichis a multi-agency partnership, works on facilitating drought recording, predicting, risk managementand planning at different national levels. Because of obvious impacts of drought on agriculture,water supply, energy production, public health, and wildlife, we decided to find analytical andcomputational techniques to characterize the complexity of different levels of drought from mod-erate drought D1 to exceptional drought D4 using the Arizona Drought Monitor databases. Weapplied the time-frequency analysis using Continuous Wavelet Transform (CWT) to visualize thenon-linearity in the structure of five different groups of drought levels in the frequency domain ofdata vibration. Moreover, we carried out the vibration analysis using the power-law exponent and(PSD) to discover the power-law and self-similarity behaviors in the structure of drought database.we performed the multi-fractal analysis in studding the multi-fractal properties of our time seriesdata. This analysis revealed the presence of a wide range of scaling exponent for D0-D4 andmulti-fractal structure of the drought database. We continued our study by measuring the fractalcomplexity in drought time series data using Higuchi algorithm. This analysis helped to comparethe self-similarity of different drought levels. Although these methods helped to characterize thecomplexity in the nature of our database, however it requires further studies to find an appropriatemathematical model (deterministic or stochastic) governing the complex dynamics of drought timeseries data.

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	1. Introduction
	2. Materials, Methods and results
	2.1. Data
	2.2. Time-Frequency Analysis and Continuous Wavelet Transform (CWT)
	2.3. Vibration frequency analysis using Power spectral densities (PSD)
	2.4. Multifractal Analysis and Discrete Wavelet Transform (DWT)
	2.5. Higuchi Fractal Dimension Algorithm

	3. Discussion
	References

