 Research
 Open access
 Published:
Classification of drought severity in contiguous USA during the past 21 years using fractal geometry
EURASIP Journal on Advances in Signal Processing volume 2024, Article number: 3 (2024)
Abstract
Drought is characterized by a moisture deficit that can adversely impact the environment, economy, and society. In North America, like many regions worldwide, predicting the timing of drought events is challenging. However, our novel study in climate research explores whether the Drought Monitor database exhibits fractal characteristics, represented by a single scaling exponent. This database categorizes drought areas by intensity, ranging from D0 (abnormally dry) to D4 (exceptional drought). Through vibration analysis using power spectral densities (PSD), we investigate the presence of powerlaw scaling in various statistical moments across different scales within the database. Our multifractal analysis estimates the multifractal spectrum for each category, and the Higuchi algorithm assesses the fractal complexity, revealing that D4 follows a multifractal pattern with a wide range of exponents, while D0 to D3 exhibit a monofractal nature with a narrower range of exponents.
1 Introduction
Drought, a natural segment of climate variability, is characterized by prolonged periods of reduced precipitation, affecting various aspects of life and natural resources, including water supplies, businesses, economic stability, social wellbeing, and the environment [1,2,3,4,5,6]. The escalating trend of global warming has led to a more frequent occurrence of drought, presenting significant challenges to both humanity and society. This has a profound impact on people’s lives, property security, food security, and the availability of water resources [7]. Not only municipal water suppliers could be affected by drought, but also, businesses [1, 2], economic [1], social life and welfare implications [3], and environmental interests which are dependent on wildlife with essential need of precipitation and water [4], may be influenced by drought in the same way [1]. While the USA boasts a robust agricultural system that has historically protected its citizens from severe droughtrelated impacts, recent droughts have posed significant challenges for farmers across different regions [5].
A typical drought inflicts significant economic losses on American farmers and businesses, ranging from 6 to 8 billion dollars annually. Surprisingly, these financial impacts surpass those incurred from floods and hurricanes. The consequences tend to be even more severe in regions where comprehensive planning for natural hazards is lacking and agriculture serves as the primary economic driver [8]. Various studies have been conducted to assess the relative effects of drought in different parts of the USA, revealing the increasingly apparent impact of drought on human life, wildlife, and agriculture [9,10,11,12,13,14]. According to these studies, during recent years, we can see clearly the impact of drought on human life, wildlife and agriculture. These studies on different part of the USA which experienced extreme drought discussed that it is likely unprecedented to know about when the next drought will happen. Some predictive models like linear regression have been used to determine the correlation between different factors and the effects of drought in different regions. Many studies have documented increasing the number of drought and severity during recent years in USA which have arisen many new concerns [5, 9, 15,16,17,18,19,20,21,22,23]. For example, agricultural economic in Nebraska is the first and the most common sector in economic which has been affected adversely by drought [16]. The problem of scarcity of water which happened in the agricultural and energy sectors in a multiyear drought in California (2007–2009) was a reaction to the massive gap between empirical research and the adaptive capacity of social and environmental systems to climate changes [17]. The unprecedented drought in Texas in 2011 led to very dried seasons and intensive wildfires and increasing hardship for ranchers [20, 21]. Climate change in the Pacific Northwest (PNW) of America has caused more frequent droughts, rising air temperature, reducing winter snowfall, increasing earlier snowmelt, reducing summer flows, and longer cropgrowing season [23]. In North Carolina, forest ecosystem sectors such as clean water, wildlife habitat, and timber supplies are progressively affected by drought during recent years [9]. Based on these recent records of severe drought impacts in different parts of USA and many more, it is crucial to better understanding the drought features and patterns to decrease the environmental and economical costs and damages.
To track drought conditions and their related environmental factors, resources such as the National Drought Mitigation Center (NDMC), National Oceanic and Atmospheric Administration (NOAA), and the United States Department of Agriculture (USDA) provide valuable information. Early detection of drought allows for proactive measures to reduce its impacts and economic losses. According to the US Drought Monitor (USDM) reports, there are different ways to recognize drought such as comparison between observed precipitation, soil moisture and crop conditions with their regular time and so on. The US Drought Monitor (USDM) is a collaborative effort started from 1999 and produced by the National Drought Mitigation Center (NDMC) at the University of NebraskaLincoln, the National Oceanic and Atmospheric Administration (NOAA), and the US Department of Agriculture (USDA).
Drought experts regularly estimate precipitation levels, comparing them to longterm averages, while considering various variables like temperature, soil moisture, water levels in streams and lakes, snow cover, and meltwater runoff. They also identify areas experiencing drought impacts, including water shortages and business disruptions. Experts use multiple indicators to assess regionalscale drought conditions and consult with other specialists before releasing weekly drought maps known as the US Drought Monitor (USDM) maps, see Fig. 1 (Source(s): National Drought Mitigation Center (NDMC), National Oceanic and Atmospheric Administration (NOAA), United States Department of Agriculture (USDA)).
Figure 1 displays the US Drought Monitor (USDM) categorization, which classifies drought into five levels: (1) D0, indicating areas abnormally dry but not yet in drought or recovering from it; (2) D1, representing moderate drought, the least severe level; (3) D2, marking severe drought; (4) D3, signifying extreme drought; and (5) D4, the most severe level, exceptional drought. This classification is vital for risk assessment and drought management, aiding in the quantification and evaluation of potential issues [Source(s): National Drought Mitigation Center (NDMC), National Oceanic and Atmospheric Administration (NOAA), United States Department of Agriculture (USDA)].
Various indices are employed to assess drought severity and its impacts across different timescales, including widely recognized measures like the Standardized Precipitation Index (SPI) and the Palmer Drought Severity Index (PDSI). These indicators have been utilized in operational drought management for many years, and their characteristics and performance characteristics are welldocumented and understood [24]. Another commonly used index for drought monitoring is the Normalized Difference Vegetation Index (NDVI) [25,26,27]. This index aids in assessing drought severity by tracking variations in vegetation cover within a specific region over a defined time frame. Satellite databases, including data from Advanced VeryHighResolution Radiometer (AVHRR), Terra Moderate Resolution Imaging Spectroradiometer (MODIS), and Landsat sensors, record and quantify changes in vegetation coverage due to evolving climate conditions. Positive NDVI values indicate vegetated areas, while zero and negative values correspond to bare soil and water bodies [25]. Significant progress has been achieved in understanding the impact of drought on vegetation dynamics by examining the correlations within their response characteristics. Several scientists have assessed vegetation productivity in response to droughts at various timescales, employing measures such as the 3, 6, 12, and 24month Standardized Precipitation Evapotranspiration Index (SPEI) and Normalized Difference Vegetation Index (NDVI) [28].
Figure 2 displays the time series of the average NDVI for Arizona, encompassing regions with moderate to exceptional drought (D1–D4), over ten years (2010–2021) for each month. The NDVI data, sourced from the opensource Google Earth Enterprise and derived from Terra Moderate Resolution Imaging Spectroradiometer (MODIS), serves as a valuable resource for predicting future vegetation changes in Arizona.
Each state experiences a different set of impacts during a drought. We have also demonstrated the table of reported impacts during past droughts in Arizona for each level of drought on the US Drought Monitor in Fig. 3 (Source(s): National Drought Mitigation Center (NDMC), National Oceanic and Atmospheric Administration (NOAA), United States Department of Agriculture (USDA)).
When studying realworld time series, one often encounters databases exhibiting nonlinear powerlaw properties, indicative of selfsimilar or fractallike patterns across various scales [6, 29,30,31,32,33,34,35]. ]. In mathematical terms, a fractal is a subset of Euclidean space characterized by a fractal dimension higher than its topological dimension, as introduced by Mandelbrot in 1983 [29, 30, 36, 37]. Using fractal geometry, the selfsimilarity and spacefilling properties of dynamical systems can be extracted [38]. Timeseries data can be classified as fractal or monofractal if it can be characterized by a single scaling exponent or as a linear function of moments. The presence of these scalefree databases has been observed in various fields, including biology, geophysics, stock markets, and finance [30, 39,40,41,42,43,44,45]. To analyze the nonlinear structure of scalefree timeseries data effectively, it is essential to employ analytical and computational tools that can characterize their complexity and selfsimilarity [41, 46]. Traditional timeseries analysis methods may fail when dealing with datasets exhibiting a wide range of scaling features. For such data, which may require multiple scaling exponents to describe their scaling structure, the scaling behavior follows a nonlinear function of moments (including vibration analysis using scaling power law and power spectral density (PSD) and continues and discrete wavelet analysis) [47]. However, there exist other type processes which require a large number of scaling exponents to characterize their scaling structure. For this class of phenomena, the scaling behavior follows a function which is nonlinear in the moments. For these processes which are called multifractal, the variability in data exhibits selfaffine multifractal properties and multifractal analysis needs to be applied to determine the complexity of consecutive time intervals in timeseries data. In multifractal analysis, we discover whether some type of powerlaw scaling exists for various statistical moments at different scales [48,49,50,51].
In this novel study, we leverage the concept of fractal geometry to classify drought severity in the USA from 2000 to the present, recognizing the significance of drought characterization for future forecasting. Multifractal analysis is employed to investigate whether powerlaw scaling is present for various statistical moments at different scales in the dataset. By plotting multifractal spectra and applying quantitative analysis through the fractal dimension (FD) using the Higuchi algorithm, we illustrate the fractal complexity of drought severity. Our findings suggest that fractal geometry can serve as a mathematical framework for the analysis and characterization of drought severity at different levels, offering a computational tool for comparing the complexity of each class of drought severity, ultimately aiding in predicting future drought occurrences.
2 Materials, methods, and results
2.1 Data
Here, data has been taken using ”Drought Monitor” from all US Drought Monitor categories for each week of the selected time period (January 2000 to Nov 2021) and location (contiguous USA), see Figs. 4 and 5. The US Drought Monitor which started from 1999, is a partnership between the National Drought Mitigation Center (NDMC) at the University of NebraskaLincoln, the United States Department of Agriculture (USDA), and the National Oceanic and Atmospheric Administration (NOAA). Each Thursday, the US Drought Monitor (USDM) will be updated to demonstrate the location and intensity of drought across the country. Using the experts’ assessments, drought categories display conditions related to dryness and drought such as observations of how much water is available in streams, lakes, and soils compared to usual time of year (Source(s): National Drought Mitigation Center (NDMC), National Oceanic and Atmospheric Administration (NOAA), United States Department of Agriculture (USDA)) [52].
2.2 Time–frequency analysis and continuous wavelet transform (CWT)
To represent nonstationary timeseries databases, continuous wavelet transform (CWT) provides a clear visualization platform by computing a linear time–frequency called scalogram which breaks the dataset into scales by preserving time shifts and time scales. Therefore, when we are working with timeseries data in different frequency ranges, the wavelet transform facilitates extracting useful information from the time intervals between consecutive waves of time series and makes analysis of data easier [53]. The continuous wavelet transform (CWT) of a dataset h(t) is given by (Mallat, 1998) [53]
where s is the scale, u is the displacement, \(\Phi\) is the mother wavelet used, and \(*\) means the complex conjugate. The CWT is therefore a convolution of the data with scaled version of the mother wavelet. Of course, the time coordinate t in Eq. (2.1) could equally well be the spatial coordinate x if profile data were being analyzed.
There is a propose a new classification method called wavelet transformbased smooth ordering (WTSO) which uses the WTSO wavelet transform to reduce the high dimensionality, the computational cost, and also perform classification [54]. In [55], the authors introduced a framework which performs better in downsampling balance and signal compression. Their wavelet decomposition method has application in application on synthetic and realworld graph. Another wavelet called Chebyshev wavelet has been used in fractional calculus and fractal geometry [56]. Guariglia et al. further developed mother wavelet in Taylor series using the differential properties of Chebyshev wavelets. Mallat et al. offered the wavelet representation as an orthogonal multiresolution representation which is defined between the spatial and Fourier domains [57]. In [58], the fractional derivative of the Gabor–Morlet wavelet has been used to obtain a characterization of the complex fractional derivative through the distribution theory. Sparse representation by frames has shown promising results in signal analysis via a concise approach through practical numerical experiments [59].
2.3 Vibration frequency analysis using power spectral densities (PSD)
Discrete Fourier transform (DFT) or fast Fourier transform (FFT) is one of the most frequently used vibration frequency analysis algorithms in frequency analysis and computing Fourier transform. This method works very well when we have a finite number of dominant frequency components; however, it fails when our data includes random vibrations. To solve this problem, one may apply power spectral densities (PSD) technique which is perfect to analyze the signal vibration. The spectral densities (PSD) acts by multiplying each frequency bin of FFT to its complex conjugate to derive the real spectrum, and next, it normalizes the results to frequency bin width. Because the drought monitor database displays nonlinearity and has nonstationary structure, the welch (PSD) method with overlapped segmentation, which is an averaging estimator technique, has been applied to study the complex fluctuations in drought timeseries structures.
2.4 Multifractal analysis and discrete wavelet transform (DWT)
There are different types of phenomena with scaling law behaviors which can be completely characterized using fractal theory. However, there exist some other processes which cannot be fully explained using fractal theory tools because they follow complex scaling behaviors of many irregular objects. For this group of phenomena, we may need to perform multifractal analysis which gives a spectrum of singularity exponents to describe the complex scaling behaviors. In general, fractal dimension determines the complexity of a fractal object by measuring the changes of coverings relative to the scaling factor. It also specifies the spacefilling capacity of a fractal object with respect to its scaling properties in the space. The relationship between scaling and covering is often hard to be characterized. The variation in the number of coverings, \(N(\epsilon )\), with respect to the scaling factor \(\epsilon\), can be written as
where D is the fractal dimension. Relation (2.2) is called scaling law that has been used to demonstrate the size distribution of many objects in nature. The boxcounting formula which has been widely applied to approximate the fractal dimension of an irregular object is defined as
However, this monofractal dimension is not able to fully characterize complex scaling behaviors of many irregular objects in the real world. That is why to study irregular objects like ECG signals one may need to apply the multifractal algorithm. The multifractal analysis used a spectrum of singularity exponents to provide a detailed and local description of complex scaling behaviors. In order to quantify local densities of the fractal set, we approximate the mass probability using the following formula
where \(N_i(a)\) is the number of mass in the ith subset of measure a and N is the total mass of the set. When we scale the mass probability \(P_i(a)\) with measure a of a multifractal set, it also demonstrates the powerlaw behavior:
where \(\alpha _i\) is the singularity exponent characterizing the local scaling in the ith subset. The multifractal spectrum \(f(\alpha )\) provides a statistical distribution of singularity exponents \(\alpha _i\). In general, \(f(\alpha )\) may be estimated using the Legendre transformation
where q is the moment and \(\tau (q)\) is the mass exponent of the qth order moment. In addition, the multifractal measures may be specified by scaling of qth moments of \(P_i(a)\) as
where \(D_q=\dfrac{\tau (q)}{(q1)}\) is the generalized fractal dimension. For \(q=0\) , Eq. (2.4) becomes
which is similar to formula (2.2).
To approximate the multifractal spectrum, wavelet analysis has been used extensively with promising results for noisy timeseries data [60,61,62,63,64,65]. This method utilizes discrete wavelet transform (DWT) technique which is robust enough to characterize the distribution of scaling exponents and provides a good approximation of the changes in regularity of data. The wavelet leader multifractal (WLM) analysis corresponds to the dimension of fractal sets to Holder exponent \(\mathscr {H}(\tau )\) to quantify the spectrum of singularity of the pointwise regular function F [64]. The Holder exponent of a fractal process \(F(\tau )\) is defined as follows:
Definition 2.1
[65] A fractal process \(F(\tau )\) satisfies a Holder condition, when there exist \(\mathscr {H}(\tau )>0\), such that
We can find \(\mathscr {H}(\tau )\) for constant F from the coarse Holder exponents as
The following sets have been introduced to discover the geometry of timeseries data
With varying d, these sets describe the local regularity of data. Next, we define the map
as the multifractal spectrum of F which is a compact form of the singularity structure of the fractal process [65]. To describe the complexity of timeseries data in a global setting, we may need to count the intervals over which the fractal process F evolves with Holder exponent \(\mathscr {H}(\tau )\) and it provides an estimation of \(\text {dim}(\mathscr {E}^{[d]})\). Then, we introduce the grain exponent which is a discrete approximation to \(h_{\xi }(\tau )\) [65]:
Thus, the grain multifractal spectrum has the following form [66,67,68,69]
where
2.5 Higuchi fractal dimension algorithm
Many different methods have been developed to measure the selfsimilarity of a fractal process. In fractal geometry, the Minkowski dimension or boxcounting dimension is one of the most common used techniques to approximate the fractal dimension of a fractal set in any metric space [70]. However, this method cannot catch the sudden changes happen in the irregular timeseries datasets [71]. To explore the complexity of scalefree timeseries data, a variety of different nonlinear techniques such as Higuchi algorithm, power spectrum analysis, and Katz algorithm have been highlighted in different areas [72,73,74,75,76,77]. To approximate the complexity index of Drought Monitor Categories database, we utilize the Higuchi Algorithm [72]. We start with a finite time series \(Y_{1}, \, Y_{2},\, Y_{3},\ldots,Y_{N}\). Then, we build k new time series \(Y_{m}^{k}\) of the form
where \(A=(Nm)/k\). For each time interval k and the initial time m such that \(m=1,\,2,\,\dots \,,k\), we calculate the length of \(Y_{m}^{k}\) using
where \(R=(N1)/[A]k\) is the curve length normalization factor. Then, we estimate the mean of \(L_{m}^{k}\) for \(m=1,\,2,\ldots ,k\) to find the average of curve length for each k. After finding the average values for \(k=1,\ldots ,k_{max}\), we plot \(\log (L_{m}^{k})\) versus \(\log (1/k)\) for different time interval k. At the end, we calculate the slope of each regressed line. To find the slope, we use the least squares approximation technique for an optimal value of time interval \(k=500\) when there is no change in fractal dimension after this value.
3 Discussion of results
Scalogram visualizes the dataset which is a function of time and frequency, by taking several steps: At first, it splits the data into overlapping segments, then computes the absolute value of the continuous wavelet transform coefficients for each segment, and finally plots it. We have demonstrated the continuous wavelet transform (CWT) plots of all drought categories database in Figs. 6, 7, and 8.
Here, we can see the nonlinear features of the timeseries data are encoded in the frequency domain of the vibrations.
To find whether for various statistical moments, powerlaw scaling behavior governs on the structure of the drought timeseries data at different scales, we compute the power spectral density of each drought category timeseries data using welch (PSD) technique, and then, we use least square method to fit linear regression to the logarithm of power spectral density results. In Fig. 9, we can see the fitted least squares approximation to the logarithm of power spectral density of different drought categories. In fractal processes, there exists a scaling relationship between power and frequency f in the spectral domain. These graphical representations in Fig. 9 reveal the fractal processes by a linear, negative slope of fitted least square lines, which means that the series cannot be generated by one or a finite set of subsystems, but for these processes different components act at different time scales. The results of power spectral density revealed the presence of longrange selfsimilar correlations extending over steps in a scalefree (fractal) powerlaw fashion. However, power spectral density fails to classify these five drought categories, and it may require to test this method with more databases.
Scaling exponent graphs are useful tools to demonstrate whether selfsimilar process is monofractal or multifractal. We also plot the scaling exponents of Drought Monitor Categories (2000–present) in Fig. 10. The nonlinear exponents for these signals may exhibit the multifractal structure of them; however, we need to apply multifractal analysis to check if this is a confirmed conclusion.
However, scaling exponent of Drought Monitor Categories does not give enough information to classify different drought categories for this limited database.
From multifractal analysis results (see Fig. 11), we can easily see that we have a wide range of exponents for extreme drought \(D_4\), which is a sign of multifractal structure of this drought level. This multifractal timeseries data needs to be indexed by different exponents as we decompose it into different subsets and also requires much more exponents to characterize its scaling properties. In addition, we can find a clear loss of multifractality for abnormally dry to extreme drought \(D_0  D_3\), which means they are homogeneous and monofractal since their spectrum displays a narrow width of scaling exponent. Using multifractal analysis of drought timeseries data, we can explore when moderate to extreme drought (\(D_1\) or \(D_3\)) and exceptional drought (\(D_4\)) are present. Although multifractal analysis could well separate moderate to extreme drought from extreme drought and we could monitor the complexity of each drought category dataset in terms of monofractal and multifractality, these results do not give us a clear framework to differentiate \(D_0\), \(D_1\), \(D_2\), and \(D_3\) from each other, and we need to find other tools and techniques to successfully classify different drought categories.
We have approximated the fractal dimension of Drought Monitor Categories database and plotted their regression models for each time series in Fig. 12.
From fractal dimension results in Fig. 12, we can compare the fractal dimension of different drought levels as
Although fractal dimension using the Higuchi algorithm is a good index for comparing the selfsimilarity and powerlaw structures of different drought categories, it fails to separate these five groups of datasets, and also, we need to try different drought databases to find a threshold for classification of different levels of drought.
4 Conclusion
Drought, often perceived as a gradual and inconspicuous climatic phenomenon, has historically been underestimated in terms of its environmental and economic ramifications. However, once its profound impacts on the environment and the economy are recognized, it becomes evident that drought is as significant as fastmoving natural disasters like tornadoes and hurricanes. Therefore, efforts to comprehend and predict drought are of great importance and should be supported by relevant organizations.
In the USA, the National Integrated Drought Information System (NIDIS), a multiagency partnership, is dedicated to enhancing drought monitoring, prediction, risk management, and planning at the national level. Given the substantial consequences of drought on agriculture, water supply, energy production, public health, and wildlife, our research aimed to identify analytical and computational techniques capable of classifying different drought severity levels using data from the US Drought Monitor database. Since this database displays irregular data structures, we opted for nonlinear techniques, such as vibration analysis and wavelet methods, specifically designed for this type of data. Our objective was to determine whether these techniques could effectively classify drought levels, ranging from moderate drought (D1) to exceptional drought (D4).
Our time–frequency analysis method, the continuous wavelet transform (CWT), successfully visualized the nonlinear structure of five distinct drought severity levels in the frequency domain of data oscillation. Utilizing the wellestablished vibration analysis techniques of powerlaw exponent and power spectral density, we uncovered powerlaw and selfsimilarity behaviors in the structure of the drought database. This discovery prompted us to proceed with fractal geometry techniques to further analyze the complexity of the various drought levels.
The nonlinear scaling exponents of the drought database suggested the presence of multifractality. Consequently, we conducted multifractal analysis using the discrete wavelet transform (DWT), a reputable method in timeseries data analysis. The results showed a broad range of scaling exponents for exceptional drought (D4) and a narrower range for abnormally dry (D0) to severe drought (D3). This effectively differentiated the monofractal dynamics of these levels from the multifractal nature of exceptional drought (D4).
As a result, the wavelet leader multifractal (WLM) analysis can serve as a classifier method for distinguishing exceptional drought (D4) from other levels. However, it fails to differentiate between moderate drought (D1), severe drought (D2), and extreme drought (D3). To achieve a comprehensive understanding of complexity in drought timeseries data, we conducted a fractal dimension analysis using the Higuchi algorithm, an appropriate technique for determining the fractal dimension of irregular, scalefree databases. The Higuchi fractal dimension revealed that abnormally dry (D0) had the highest fractal dimension, while exceptional drought (D4) had the lowest. Although it helped compare the selfsimilarity of different drought levels, this complexity index did not provide clear differentiation between the groups. Further analysis and effort are required to establish a specific threshold at which the fractal dimension can be considered a classification tool in such studies.
While the proposed algorithms have shown promising performance in various literature and evidence, a significant limitation lies in the limited amount of data available from the existing online database. Moreover, understanding the mechanisms behind the longrange correlations in the complex fluctuations of drought data remains a challenge. Developing an appropriate mathematical model, whether deterministic or stochastic, to describe the complex dynamics of drought timeseries data is essential. This endeavor calls for collaborative research between experimental and theoretical scientists to uncover effective strategies for forecasting drought and mitigating its impacts on the environment, wildlife, and the economy.
Availability of data and materials
The data that support the findings of this study (Fig. 4) are available in/from US Drought Monitor Categories database (2000–present), the drought status of areas represented by points; Source(s): National Drought Mitigation Center (NDMC), the US Department of Agriculture (USDA), and the National Oceanic and Atmospheric Administration (NOAA), https://droughtmonitor.unl.edu/DmData/TimeSeries.aspx.
Abbreviations
 D0:

Abnormally dry
 D1:

Moderate drought
 D2:

Severe drought
 D3:

Extreme drought
 D4:

Exceptional drought
 PSD:

Power spectral densities
 NDMC:

National Drought Mitigation Center
 NOAA:

National Oceanic and Atmospheric Administration
 USDA:

United States Department of Agriculture
 USDM:

US Drought Monitor
 NDVI:

Normalized difference vegetation index
 AVHRR:

Advanced veryhighresolution radiometer
 MODIS:

Terra moderate resolution imaging spectroradiometer
 LST:

Land surface temperature
 FD:

Fractal dimension
 CWT:

Continuous wavelet transform
 DFT:

Discrete Fourier transform
 FFT:

Fast Fourier transform
 WLM:

Wavelet leader multifractal
References
Y. Ding, M.J. Hayes, M. Widhalm, Measuring economic impacts of drought: a review and discussions. Disaster Prev. Manag. Int. J. 20(4), 434–446 (2011)
R.A. Lawes, R.S. Kingwell, A longitudinal examination of business performance indicators for droughtaffected farms. Agric. Syst. 106(1), 94–101 (2012)
M. Alston, J. Kent, Social impacts of drought, in Centre for Rural Social Research (Charles Sturt University, Wagga Wagga, NSW, 2004)
L.M. Mosley, Drought impacts on the water quality of freshwater systems: review and integration. EarthSci. Rev. 140, 203–214 (2015)
D.A. Wilhite, M.D. Svoboda, M.J. Hayes, Understanding the complex impacts of drought: a key to enhancing drought mitigation and preparedness. Water Resour. Manag. 21(5), 763–774 (2007)
S. Azizi, T. Azizi, The fractal nature of drought: power laws and fractal complexity of Arizona drought. Eur. J. Math. Anal. 2, 17 (2022)
M.M. Alimullah, Droughts in Asian least developed countries: vulnerability and sustainability. Weather Climate Extremes 7, 8–23 (2015)
V.K. Vidyarthi, A. Jain, Knowledge extraction from trained ANN drought classification model. J. Hydrol. 585, 124804 (2020)
Y. Yang, M. Anderson, F. Gao, C. Hain, A. Noormets, G. Sun, R. Wynne, V. Thomas, L. Sun, Investigating impacts of drought and disturbance on evapotranspiration over a forested landscape in North Carolina, USA using high spatiotemporal resolution remotely sensed data. Remote Sens. Environ. 238, 111018 (2020)
S.R. Evett et al., Precision agriculture and irrigation: current US perspectives. Trans. ASABE 63(1), 57–67 (2020)
M.S. Lisboa, R.L. Schneider, P.J. Sullivan, M.T. Walter, Drought and postdrought rain effect on stream phosphorus and other nutrient losses in the Northeastern USA. J. Hydrol. Reg. Stud. 28, 100672 (2020)
R. Yaddanapudi, A.K. Mishra, Compound impact of drought and COVID19 on agriculture yield in the USA. Sci. Total Environ. 807, 150801 (2022)
D.J.N. Young, M. Meyer, B. Estes, S. Gross, A. Wuenschel, C. Restaino, H.D. Safford, Forest recovery following extreme drought in California, USA: natural patterns and effects of predrought management. Ecol. Appl. 30(1), e02002 (2020)
N.T. Shephard, O. Joshi, C.R. Meek, R.E. Will, Longterm growth effects of simulateddrought, midrotation fertilization, and thinning on a loblolly pine plantation in southeastern Oklahoma, USA. For. Ecol. Manag. 494, 119323 (2021)
M. PeñaGallardo, S.M. VicenteSerrano, F. DomínguezCastro, S. Quiring, M. Svoboda, S. Beguería, J. Hannaford, Effectiveness of drought indices in identifying impacts on major crops across the USA. Climate Res. 75(3), 221–240 (2018)
H. Wu, D.A. Wilhite, An operational agricultural drought risk assessment model for Nebraska, USA. Nat. Hazards 33(1), 1–21 (2004)
J. ChristianSmith, M.C. Levy, P.H. Gleick, Maladaptation to drought: a case report from California, USA. Sustain. Sci. 10(3), 491–501 (2015)
H. Chang, M.R. Bonnette, Climate change and waterrelated ecosystem services: impacts of drought in California, USA. Ecosyst. Health Sustain. 2(12), e01254 (2016)
T. Tadesse, D.A. Wilhite, S.K. Harms, M.J. Hayes, S. Goddard, Drought monitoring using data mining techniques: a case study for Nebraska, USA. Nat. Hazards 33(1), 137–159 (2004)
J.W. NielsenGammon, The 2011 texas drought. Texas Water J. 3(1), 59–95 (2012)
B. Breyer, S.C. Zipper, J. Qiu, Sociohydrological impacts of water conservation under anthropogenic drought in Austin, TX (USA). Water Resour. Res. 54(4), 3062–3080 (2018)
J.S. Clark, E.C. Grimm, J.J. Donovan, S.C. Fritz, D.R. Engstrom, J.E. Almendinger, Drought cycles and landscape responses to past aridity on prairies of the northern Great Plains, USA. Ecology 83(3), 595–601 (2002)
I.W. Jung, H. Chang, Climate change impacts on spatial patterns in drought risk in the Willamette River Basin, Oregon, USA. Theor. Appl. Climatol. 108(3), 355–371 (2012)
Y. Zhang, Z. Hao, Y. Jiang, V.P. Singh, Impactbased evaluation of multivariate drought indicators for drought monitoring in China. Global Planetary Change 228, 104219 (2023)
R. Albarakat, V. Lakshmi, Comparison of normalized difference vegetation index derived from Landsat, MODIS, and AVHRR for the Mesopotamian marshes between 2002 and 2018. Remote Sens. 11(10), 1245 (2019)
K.I. Wheeler, M.C. Dietze, A statistical model for estimating midday NDVI from the geostationary operational environmental satellite (GOES) 16 and 17. Remote Sens. 11(21), 2507 (2019)
Y. Wang, C. Zhang, F.R. Meng, C.P.A. Bourque, C. Zhang, Evaluation of the suitability of six drought indices in naturally growing, transitional vegetation zones in Inner Mongolia (China). PLoS ONE 15(5), e0233525 (2020)
C. Wu et al., An evaluation framework for quantifying vegetation loss and recovery in response to meteorological drought based on SPEI and NDVI. Sci. Total Environ. 906, 167632 (2023)
B.B. Mandelbrot, The Fractal Geometry of Nature, vol. 1 (WH freeman, New York, 1982)
B.B. Mandelbrot, Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot (American Mathematical Society, Providence, 2004)
E. Guariglia, Harmonic Sierpinski gasket and applications. Entropy 20(9), 714 (2018)
T. Azizi, Measuring fractal dynamics of FECG signals to determine the complexity of fetal heart rate. Chaos Solitons Fractals X 9, 100083 (2022)
T. Azizi, On the fractal geometry of different Heart rhythms. Chaos Solitons Fractals X 9, 100085 (2022)
T. Azizi, Mathematical modeling of stress using fractal geometry: the power laws and fractal complexity of stress. Adv. Neurol. Neurosci. 5(3), 140–148 (2022)
S. Azizi, T. Azizi, Noninteger dimension of seasonal land surface temperature (LST). Axioms 12(6), 607 (2023)
K. Falconer, Fractal Geometry: Mathematical Foundations and Applications (Wiley, New York, 2004)
J. Briggs, Fractals: The Patterns of Chaos: A New Aesthetic of Art, Science, and Nature (Simon and Schuster, New York, 1992)
E. Guariglia, Primality, fractality, and image analysis. Entropy 21(3), 304 (2019)
M. Meyer, O. Stiedl, Selfaffine fractal variability of human heartbeat interval dynamics in health and disease. Eur. J. Appl. Physiol. 90(3), 305–316 (2003)
R. Acharya, P.S. Bhat, N. Kannathal, A. Rao, C.M. Lim, Analysis of cardiac health using fractal dimension and wavelet transformation. ItbmRbm 26(2), 133–139 (2005)
R. Martin, Noise power spectral density estimation based on optimal smoothing and minimum statistics. IEEE Trans. Speech Audio Process. 9(5), 504–512 (2001)
A.J. Barbour, R.L. Parker, psd: Adaptive, sine multitaper power spectral density estimation for R. Comput. Geosci. 63, 1–8 (2014)
S. Di Matteo, N.M. Viall, L. Kepko, Power spectral density background estimate and signal detection via the multitaper method. J. Geophys. Res. Space Phys. 126(2), e2020JA028748 (2021)
C. Heneghan, G. McDarby, Establishing the relation between detrended fluctuation analysis and power spectral density analysis for stochastic processes. Phys. Rev. E 62(5), 6103 (2000)
T. Di Matteo, Multiscaling in finance. Quant. Finance 7(1), 21–36 (2007)
P. Laguna, G.B. Moody, R.G. Mark, Power spectral density of unevenly sampled data by leastsquare analysis: performance and application to heart rate signals. IEEE Trans. Biomed. Eng. 45(6), 698–715 (1998)
C.K. Peng, S. Havlin, H.E. Stanley, A.L. Goldberger, Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series. Chaos Interdiscip. J Nonlinear Sci. 5(1), 82–87 (1995)
I.P. Ch, L.A.N. Amaral, A.L. Goldberger, S. Havlin, M.G. Rosenblum, Z.R. Struzik, H.E. Stanley, Multifractality in human heartbeat dynamics. Nature 399(6735), 461–465 (1999)
R. Sassi, M.G. Signorini, S. Cerutti, Multifractality and heart rate variability. Chaos Interdiscip. J. Nonlinear Sci. 19(2), 028507 (2009)
D.C. Lin, A. Sharif, Common multifractality in the heart rate variability and brain activity of healthy humans. Chaos Interdiscip. J. Nonlinear Sci. 20(2), 023121 (2010)
V. Tamas, Fractal growth phenomena. Asakura Shoten (1992)
M. Svoboda, D. LeComte, M. Hayes, R. Heim, K. Gleason, J. Angel, B. Rippey, R. Tinker, M. Palecki, D. Stooksbury, The drought monitor. Bull. Am. Meteorol. Soc. 83(8), 1181–1190 (2002)
S. Mallat, A Wavelet Tour of Signal Processing (Elsevier, Amsterdam, 1999)
L. Yang, H. Su, C. Zhong, Z. Meng, H. Luo, X. Li, Y.Y. Tang, Y. Lu, Hyperspectral image classification using wavelet transformbased smooth ordering. Int. J. Wavelets Multiresolut. Inf. Process. 17(06), 1950050 (2019)
X. Zheng, Y.Y. Tang, J. Zhou, A framework of adaptive multiscale wavelet decomposition for signals on undirected graphs. IEEE Trans. Signal Process. 67(7), 1696–1711 (2019)
E. Guariglia, R.C. Guido, Chebyshev wavelet analysis. J. Funct. Spaces (2022). https://doi.org/10.1155/2022/5542054
S.G. Mallat, A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Mach Intell. 11(7), 674–693 (1989)
E. Guariglia, S. Silvestrov, FractionalWavelet Analysis of Positive definite Distributions and Wavelets on D’(C) D’(C) (Springer, Berlin, 2016), pp.337–353
C. Baker, Sparse representation by frames with signal analysis. J. Signal Inf. Process. 7(1), 39–48 (2016)
L. RodríguezLiñares, A.J. Méndez, M.J. Lado, D.N. Olivieri, X.A. Vila, I.V. GómezConde, An open source tool for heart rate variability spectral analysis. Comput. Methods Programs Biomed. 103(1), 39–50 (2011)
H. Wendt, P. Abry, Multifractality tests using bootstrapped wavelet leaders. IEEE Trans. Signal Process. 55(10), 4811–4820 (2007)
H. Wendt, P. Abry, S. Jaffard, Bootstrap for empirical multifractal analysis. IEEE Signal Process. Mag. 24(4), 38–48 (2007)
S. Jaffard, B. Lashermes, P. Abry, Wavelet Analysis and Applications (Springer, Berlin, 2006), pp. 201–246
S. Jaffard, Wavelet techniques in multifractal analysis. PARIS UNIV (FRANCE), (2004)
R.H. Riedi, Multifractal processes. Rice Univ Houston Tx Dept Of Electrical And Computer Engineering, (1999)
B.B. Mandelbrot, Possible refinement of the lognormal hypothesis concerning the distribution of energy dissipation in intermittent turbulence, in Statistical Models and Turbulence. Springer, pp 333–351 (1972)
R. Riedi, An improved multifractal formalism and selfsimilar measures. J. Math. Anal. Appl. 189(2), 462–490 (1995)
H.G.E. Hentschel, I. Procaccia, The infinite number of generalized dimensions of fractals and strange attractors. Physica D Nonlinear Phenomena 8(3), 435–444 (1983)
T.C. Halsey, M.H. Jensen, L.P. Kadanoff, I. Procaccia, B.I. Shraiman, Fractal measures and their singularities: the characterization of strange sets. Phys. Rev. A 33(2), 1141 (1986)
C. Panigrahy, A. GarciaPedrero, A. Seal, D. RodríguezEsparragón, N.K. Mahato, C. GonzaloMartín, An approximated box height for differentialboxcounting method to estimate fractal dimensions of grayscale images. Entropy 19(10), 534 (2017)
Y. Öztürk, Fractal Dimension as a Diagnostic Tool for Cardiac Diseases (2019)
T. Higuchi, Approach to an irregular time series on the basis of the fractal theory. Physica D Nonlinear Phenomena 31(2), 277–283 (1988)
E. Shamsi, M.A. AhmadiPajouh, T.S. Ala, Higuchi fractal dimension: an efficient approach to detection of brain entrainment to theta binaural beats. Biomed. Signal Process. Control 68, 102580 (2021)
R.S. Gomolka, S. Kampusch, E. Kaniusas, F. Thürk, J.C. Széles, W. Klonowski, Higuchi fractal dimension of heart rate variability during percutaneous auricular vagus nerve stimulation in healthy and diabetic subjects. Front. Physiol. 9, 1162 (2018)
M.J. Katz, Fractals and the analysis of waveforms. Comput. Biol. Med. 18(3), 145–156 (1988)
A. Gil, V. Glavan, A. Wawrzaszek, R. Modzelewska, L. Tomasik, Katz fractal dimension of geoelectric field during severe geomagnetic storms. Entropy 23(11), 1531 (2021)
T. Azizi, On the fractal geometry of gait dynamics in different neurodegenerative diseases. Phys. Med. 14, 100050 (2022)
Acknowledgements
We are deeply grateful to all those who played a role in the success of this project.
Funding
The author(s) received no financial support for the research.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Azizi, S., Azizi, T. Classification of drought severity in contiguous USA during the past 21 years using fractal geometry. EURASIP J. Adv. Signal Process. 2024, 3 (2024). https://doi.org/10.1186/s1363402301094z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1363402301094z