1. Introduction

2. Materials, Methods and Results

2.2. Data: The Land Surface Temperature (LST) of Riley County for February, April, July, and October (2021)

The present study investigates the non-linear patterns of daytime (LST) databases which have been collected using Google Earth Engine and over Riley County in the U.S. state of Kansas. Google Earth Engine is a combination of multi-petabyte catalog of satellite imagery and Geo-spatial databases with planetary-scale analysis capabilities [20]. Google Earth Engine helps to track changes and or measuring the differences on the Earth’s surface and has been used widely to find map trends. (LST) databases are remotely sensed data and taken from MODerate Resolution Imaging Spectroradiometer (MODIS) sensor in the form of 16-day composites of daytime values at a resolution of one km during February, April, July and October (2021) . See Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17.

2.2.1. Time Series Regression Analysis

To explore and quantify the trends and patterns in (LST) database, we perform a time series regression analysis. Time series regression is a statistical technique which uses experimental data to predict the future (LST) based on the current (LST) responses by transferring the dynamics of our predictors which is the height. This forecasting method provides a great understanding of the structure of the dynamic system which is (LST) database [21].

We start with defining the design matrix $H_t$ of current and past height data and ordered bt time and we call this matrix as regressor matrix. Next, we use ordinary least squares (OLS) method for the linear regression (LR) equation:

$$\begin{array}{c}\hfill L_t=H_t\beta +{ϵ}_t\end{array}$$

(1)

where $\beta$ is the linear estimator parameter, $L_t$ represent the (LST) database. The model (1) helps to find the linearity between the (LST) variable $L_t$ and regressor matrix $H_t$.

If we replace the linear part of equation (1) with a non-linear function $f(H_t,{ϵ}_t)$, we can better understand the relationship between the (LST) variable $L_t$ and regressor matrix $H_t$ as the following form:

$$\begin{array}{c}\hfill L_t=f(H_t,{ϵ}_t)\end{array}$$

(2)

See Figure 2.

Figure 5. Time series non-linear regression model (2) using the Land Surface Temperature (LST) of Riley County during February, April, July and October (2021) obtained via Google Earth Engine.

Figure 5. Time series non-linear regression model (2) using the Land Surface Temperature (LST) of Riley County during February, April, July and October (2021) obtained via Google Earth Engine.

2.2.2. Time-Frequency Analysis using Continuous Wavelet Transform (CWT) and Wavelet-Based Semblance Analysis

When we are working with stationary databases with constant frequencies throughout the time intervals of data, Fourier transform helps to visualize the data over time, however, this technique fails when we have datasets which are changing in time in response to a particular stimulus, i.e. they have time varying features [15,22]. The reason is the main limitation of Fourier transform which gives a broad power spectrum by integrating the frequency of time series data along the whole data. Because our Days (LST) databases are non-stationary, we will apply time-frequency methods such as Continuous Wavelet Transform (CWT). Wavelet-based approaches provide the ability to account for temporal (or spatial) variability in spectral character. Although wavelet analysis is relatively new compared with Fourier analysis, its use has become widespread in recent years, and so the theory will be described only briefly here [16]. Mallat (1998) or Strang and Nguyen (1996) contain excellent and detailed summaries of wavelet analysis. The continuous wavelet transform (CWT) of a dataset h(t) is given by (Mallat, 1998)[16,32]

$$\begin{array}{c}\hfill \mathrm{CWT}(u,s)={\int }_{-\infty }^{\infty }h\left(t\right){\displaystyle \frac{1}{{\left|s\right|}^{0.5}}}{\mathsf{\Phi }}^{*}\left({\displaystyle \frac{t-u}{s}}\right)dt\end{array}$$

(3)

where s is scale, u is displacement, $\mathsf{\Phi }$ is the mother wavelet used, and * means 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 equation (3) could equally well be the spatial coordinate x if profile data were being analyzed.

Figure 6. Time-frequency representations of the Land Surface Temperature (LST) of Riley County during February, April, July and October (2021) using Continuous Wavelet Transform (CWT) in three dimensional Time-Frequency-Magnitude space.

Figure 6. Time-frequency representations of the Land Surface Temperature (LST) of Riley County during February, April, July and October (2021) using Continuous Wavelet Transform (CWT) in three dimensional Time-Frequency-Magnitude space.

Similarity measures are becoming increasingly commonly used in comparison of multiple datasets from various sources. Semblance filtering compares two datasets on the basis of their phase, as a function of frequency. Semblance analysis based on the Fourier transform [11]

$$\begin{array}{c}\hfill \mathrm{FT}\left(f\right)={\int }_{-\infty }^{+\infty }h\left(t\right)e^{-2\pi ift}dt\end{array}$$

(4)

where $h\left(t\right)$ is dataset, f represents the frequency and t is the time, suffers from problems associated with that transform, in particular its assumption that the frequency content of the data must not change with time (for time-series data) or location (for data measured as a function of position). To overcome these problems, semblance is calculated here using the continuous wavelet transform. When calculated in this way, semblance analysis allows the local phase relationships between the two datasets to be studied as a function of both scale (or wavelength) and time [16,45].

When we define the Fourier transform (4) of time series datasets $h_1\left(t\right)$ and $h_2\left(t\right)$, the semblance S or the difference in the phase angle at each frequency can be obtained using the following formula [14,47]:

$$\begin{array}{c}\hfill S=cos\theta \left(f\right)={\displaystyle \frac{R_1\left(f\right)R_2\left(f\right)+I_1\left(f\right)I_2\left(f\right)}{\sqrt{R_1^2\left(f\right)+I_1^2\left(f\right)}\times \sqrt{R_2^2\left(f\right)+I_2^2\left(f\right)}}}\end{array}$$

(5)

where $R_1\left(f\right)$ and $R_2\left(f\right)$ represent the real parts of the Fourier transform of dataset 1 and dataset 2 respectively, $I_1\left(f\right)$ and $I_2\left(f\right)$ demonstrate the imaginary parts of the Fourier transform of dataset 1 and dataset 2 respectively, and the semblance S varies between $-1$ to $+1$. If $S=+1$, there exist a perfect phase correlation, for $S=0$ there is no correlation, and for $S=-1$ we have a perfect anti-correlation. However, the the Fourier-based semblance method (5) does not work well in real world application fails to present the correlation between time series data. Therefore, for our study to find the correlation between two time series data, we use a wavelet based method called the cross-wavelet transform which is introduced by (Torrence et al. (1998)) has the form [45]:

$$\begin{array}{c}\hfill {\mathrm{CWT}}_{a,b}={\mathrm{CWT}}_a\times {\mathrm{CWT}}_b^{*}\end{array}$$

(6)

This is a complex quantity and its amplitude which is called the cross-wavelet power defined as

$$\begin{array}{c}\hfill \mathrm{AMP}=|{\mathrm{CWT}}_{a,b}|\end{array}$$

(7)

with local phase $\theta$

$$\begin{array}{c}\hfill \theta =arctan\left({\displaystyle \frac{\mathrm{Im}\left({\mathrm{CWT}}_{a,b}\right)}{\mathrm{Re}\left({\mathrm{CWT}}_{a,b}\right)}}\right)\end{array}$$

The local phase $\theta \in (-\pi ,\pi )$ measures the phase correlation between two time series data. Now, we define the wavelet-based semblance as the form

$$\begin{array}{c}\hfill S={cos}^n\left(\theta \right)\end{array}$$

(8)

Here $n>0$ is odd and integer. For $S=-1$, our time series datasets are inversely correlated, for $S=0$ there are un-correlated, and for $S=+1$, they are correlated. However, the wavelet-based semblance (8) fails to demonstrate the correlation between two noisy time series data because of of the lack of amplitude information [16]. Therefore, we define the vector dot product of the two complex wavelet vectors at each point in scale and position as [16]

$$\begin{array}{c}\hfill \mathrm{D}={cos}^n\left(\theta \right)|{\mathrm{CWT}}_a\times {\mathrm{CWT}}_b^{*}|\end{array}$$

(9)

As we can see from equation (9), D is a combination of the wavelet-based semblance (8) and the cross-wavelet power (7), which helps to find the phase correlations of the larger amplitude components of the noisy dataset.

In Figure 7, we have plotted the Dot product D of Land Surface Temperature (LST) of Riley County (Figure 4) during February, April, July and October (2021) using the formula (9) where dark red corresponds to large-amplitude signals with semblance equals +1, and yellow corresponds to large-amplitude signals with positive semblance between 0 and +1. The semblance results show that (LST) dataset and height datasets are highly correlated.

Figure 7. Semblance of (LST) datasets shown in Figure 4 to find the correlation between (LST) data and height of each grid point in Figure 2 computed using the vector dot product of the two complex wavelet vectors (9), the correlation varies as a function of time and wavelength. Strongly correlated on larger wavelengths ($S=+1$) in dark red, Positively correlated ($0<S<+1$) in yellow, zero correlation ($S=0$) in green. The results show mostly dark red, implying strong positive correlation between (LST) data and height of each grid point in Figure 2.

Figure 7. Semblance of (LST) datasets shown in Figure 4 to find the correlation between (LST) data and height of each grid point in Figure 2 computed using the vector dot product of the two complex wavelet vectors (9), the correlation varies as a function of time and wavelength. Strongly correlated on larger wavelengths ($S=+1$) in dark red, Positively correlated ($0<S<+1$) in yellow, zero correlation ($S=0$) in green. The results show mostly dark red, implying strong positive correlation between (LST) data and height of each grid point in Figure 2.

2.2.3. Vibration Frequency Analysis Via Power Spectral Densities (PSD)

Power spectral density (PSD) analysis is one of the methods in frequency-domain analysis which has been used frequently to analyze the signal vibration. Power defines the magnitude of the PSD which is the mean square value of the given time series data. PSD as a function of frequency demonstrates the distribution of a time series data over a spectrum of frequencies and its magnitude is normalized to a single Hertz bandwidth. PSD curves display the continuous probability density function of each measure, and it is obtained via multiplying each frequency bin of fast Fourier transform (FFT) by its complex conjugate to get a real spectrum. Because, the nature of (LST) database is non-linear and non-stationary, the welch (PSD) method with overlapped segmentation which is an averaging estimator technique has been applied to study the complex fluctuations in (LST) structures. Then, we use least square approximation technique to fit a linear regression model to the logarithm of power spectral density of (LST) database. Finally, we compute the slope for each regression model of (LST) data. In Figure 8, we have shown the fitted least squares approximation to the logarithm of power spectral density of the Land Surface Temperature (LST) of Riley County during February, April, July and October (2021).

Figure 8. Power-Law or self-similarity patterns; Fitted least squares approximation to the logarithm of power spectral density of the Land Surface Temperature (LST) of Riley County during February, April, July and October (2021) obtained by wavelet techniques.

Figure 8. Power-Law or self-similarity patterns; Fitted least squares approximation to the logarithm of power spectral density of the Land Surface Temperature (LST) of Riley County during February, April, July and October (2021) obtained by wavelet techniques.

In fractal processes, we can find a scaling relationship between power and frequency f in the spectral domain. The PSD results in Figure 8 represent the fractal structure in (LST) database via a linear, negative slope of fitted least square lines. As result, the (LST) time series data cannot be obtained by one or a finite set of subsystems, and different components in each fractal process act at different time scales. Even though, PSD revealed the power law structure in (LST) databases, however, it failed to classify these four groups of data.

2.3. Fractal Geometry and World’s Weather and Climate Patterns

2.3.1. Multi-Fractal Analysis and Discrete Wavelet Transform (DWT)

We have shown that complex structures in (LST) databases in Figure 4 exhibit long-range correlation, i.e. they are self-similar: statistically, vibrations at one time scale are similar to those at multiple other time scales. To discover whether these four groups of (LST) data belong to the class of multi-fractal processes or they are mono-fractal, we have plotted the scaling exponents of the Land Surface Temperature (LST) of Riley County during February, April, July and October (2021) in Figure 9.

Figure 9. Scaling exponent of power spectral density of the Land Surface Temperature (LST) of Riley County during February, April, July and October (2021).

Figure 9. Scaling exponent of power spectral density of the Land Surface Temperature (LST) of Riley County during February, April, July and October (2021).

The non linear relationships between scaling exponents and the statistical moments could be a sign of multi-fractality in the structure of data, however, we need to check this claim via multi-fractal analysis. Multi-fractal analysis helps us to understand weather the (LST) databases are multi-fractal, i.e. means that a larger number of scaling exponents are required to characterize the scaling structures of time series data. To investigate scaling law of a multi-fractal process, the standard partition multi-fractal function was first introduced which was based on generalized fluctuation analysis. However, it failed to fully characterize the scaling behavior of the non-stationary time series data [10]. Arneodo et al., 1995b [4], Bacry et al., 1993 [9], Muzy et al., 1993, 1994 [36], developed a statistical techniques based on the Continuous Wavelet Transform (CWT) to characterize the multi-fractal description of singular measures. Arneodo et al., 2002 developed a method called Wavelet Transform Modulus Maxima (WTMM) which solved the deficiency of partition multi-fractal function and provided a wide insight into a variety of problems. Abry et al., 2000, 2002a,b [1,2,3], Veitch and Abry, 1999 [46], developed multi-fractal analysis based on Discrete Wavelet Transform (DWT) including the recent use of Wavelet Leaders (WL) (Jaffard et al., 2006 [28], Wendt and Abry, 2007 [48], Wendt et al., 2007 [49]). In this study, we focus on multi-fractal analysis using (DWT) and (WL) [23,24,28,40,41,48,49]. The wavelet analysis associates the dimension of the fractal sets to Hölder exponent $H\left(\tau \right)$ to quantify the spectrum of singularity of the pointwise regular function F[28]. The Hölder exponent of a fractal process $F\left(\tau \right)$ can be defined as follows:

Definition 2.1.

[41] A fractal process $F\left(\tau \right)$ satisfies a Hölder condition, when there exist $H\left(\tau \right)>0$, such that

$$\begin{array}{c}\hfill |F\left({\tau }_1\right)-F\left({\tau }_2\right)|\simeq |{\tau }_1-{\tau }_2{|}^{H\left(\tau \right)}\end{array}$$

(10)

We can find $H\left(\tau \right)$ for constant F from the coarse Hölder exponents as

$$\begin{array}{c}\hfill h_{\xi }\left(\tau \right)={\displaystyle \frac{1}{log\xi }}log\underset{|{\tau }_1-{\tau }_2|<\xi }{sup}|F\left({\tau }_1\right)-F\left({\tau }_2\right)|\end{array}$$

(11)

The following sets may be defined to extract the geometry of a signal

$$\begin{array}{c}\hfill E_h^{\left[d\right]}=\{\tau :H\left(\tau \right)=d\}\end{array}$$

(12)

with varying d, these sets describe the local regularity of signal. We call the map

$$\begin{array}{c}\hfill d\mapsto \dim \left(E^{\left[d\right]}\right)\end{array}$$

(13)

which is a compact form of the singularity structure of the fractal process F, the multi-fractal spectrum of F[41]. In a global setting, to describe the complexity of a signal, we may need to count the intervals over which the fractal process F evolves with Hölder exponent $H\left(\tau \right)$ and it gives an estimation of $\dim \left(E^{\left[d\right]}\right)$. The grain exponent which is a discrete approximation to $h_{\xi }\left(\tau \right)$ can be written as the following form [41]

$$\begin{array}{c}\hfill h_k^{\left(n\right)}:=-{\displaystyle \frac{1}{n}}{log}_{2}sup\{|F\left(\eta \right)-F\left(\tau \right)|:(k-1)2^{-n}\le \eta \le \tau \le (k+2)2^{-n}\}\end{array}$$

(14)

Therefore, the grain multi-fractal spectrum has the form [23,24,40]

$$\begin{array}{c}\hfill F\left(d\right)=\underset{\xi \to 0}{lim}\underset{n\to \infty }{lim}{\displaystyle \frac{log{N}^n(d,\xi )}{nlog2}}\end{array}$$

(15)

where

$$\begin{array}{c}\hfill N^n(d,\xi )=\#\{k:|h_k^{\left(n\right)}-d|<\xi \}\end{array}$$

(16)

From multi-fractal analysis in Figure 10, we can easily see a clear loss of multi-fractality in all four groups of (LST) time series data, i.e. they are homogeneous and mono-fractal since their spectrum displays a narrow width of scaling exponent.

Figure 10. The multi-fractal spectrum analysis of the Land Surface Temperature (LST) of Riley County during February, April, July and October (2021) shows occurrence of mono-fractal process with a narrow range of exponents for all four groups of (LST) time series data.

Figure 10. The multi-fractal spectrum analysis of the Land Surface Temperature (LST) of Riley County during February, April, July and October (2021) shows occurrence of mono-fractal process with a narrow range of exponents for all four groups of (LST) time series data.

Higuchi Fractal Dimension Algorithm

One of the key concepts in theory of fractal geometry is the concept of fractal dimension. There are a variety of techniques to approximate the dimension of an irregular object. For example, the dimension of an object can be obtained via covering it with small boxes. The Hausdorff-Besicovitch dimension or box-counting dimension is one of the most often used methods to estimate the fractal dimension which works in this way [39]. To find box-counting dimension, we consider a square $L\times L$ and partition it into grids of linear length $ϵ$. If $N={(L/ϵ)}^2$ defines the total number of boxes, then, the total measure would be [19]:

$$\begin{array}{c}\hfill M=M_{\mu }=L^2{ϵ}^{\alpha -2}\end{array}$$

(17)

where we denote the measuring unit by $\mu ={ϵ}^{\alpha }$ and try out a number of different $\alpha$. If we choose $\alpha =1$ then $M\to \infty$ as $ϵ\to 0$. This means that the length of a square is infinite, which makes sense. If we try $\alpha =3$, then $M\to 0$ as $ϵ\to 0$. This means that the volume of a square is 0, which is also correct. For $\alpha =2$, we can find the true dimension of square. Using this argument, we define the Hausdorff-Besicovitch dimension as follows: first, we need $\alpha$-covering measure, which is the summation of the total measure in all the "boxes" denoted by $V_i$, $i=1,2,\cdots$, subject to the condition that the union of the boxes covers the object E, while the size of the largest $V_i$ is not greater than $ϵ$:

$$\begin{array}{c}\hfill m^{\alpha }\left(E\right)=\underset{ϵ\to 0}{lim}inf\{\sum {\left(diam{V}_i\right)}^{\alpha }:E\subset \cup V_i,diam{V}_i\le ϵ\}\end{array}$$

(18)

Using (18), $diamE$ can be defined by

$$\begin{array}{c}\hfill diamE=inf\{\alpha :m^{\alpha }\left(E\right)=0\}=sup\{\alpha :m^{\alpha }\left(E\right)=\infty \}\end{array}$$

(19)

Although, this method can successfully measure the self-similarity of a fractal process, however, it fails when the sudden changes happen in the irregular time series data sets [37]. To solve this problem, a variety of different non-linear methods such as Higuchi algorithm, power spectrum analysis, and Katz algorithm have been developed by different researchers [25,29]. To find the fractal dimension of each (LST) timeseries data, we apply the Higuchi Algorithm [25]. At first, we define a finite time series

$$\begin{array}{c}\hfill X_1,X_2,X_3,\cdots ,X_N\end{array}$$

and we create k new time series $X_m^k$ of the form

$$\begin{array}{c}\hfill X_m,X_{m+k},X_{m+2k},\cdots ,X_{[m+Ak]}\end{array}$$

such that $A=(N-m)/k$. For each time interval k, and the initial time $m=1,2,\cdots ,k$, we compute the length of $X_m^k$ via

$$\begin{array}{c}\hfill L_m^k={\displaystyle \frac{{\sum }_{i=1}^{\left[A\right]}|X_{m+ik}-X_{m+(i-1)k}|}{k}}R\end{array}$$

here, $R=(N-1)/\left[A\right]k$ called the curve length normalization factor. Next, we calculate the mean of $L_m^k$ for $m=1,2,\cdots ,k$ to find the average of curve length for each $k=1,\cdots ,k_{max}$. Then, we plot $log\left(L_m^k\right)$ versus $log(1/k)$ for different time interval k. Finally, we use the least-squares approximation to estimate the slope of each regressed line as Higuchi fractal dimension for an optimal value of time interval $k=500$ (optimal means there is no change in Higuchi fractal dimension after this value).

We have estimated the fractal dimension of the Land Surface Temperature (LST) of Riley County during February, April, July and October (2021) and plotted their regression models for each time series in Figure 11.

From the results in Figure 11, we can compare the fractal dimension of different (LST) data as follows

$$\begin{array}{c}\hfill F{D}_{\mathrm{February}}<F{D}_{\mathrm{July}}<F{D}_{\mathrm{October}}<F{D}_{\mathrm{April}}\end{array}$$

Figure 11. Plots of $log\left(L_m^k\right)$ versus $log\left(k\right)$ for time interval $k=500$, the logarithmic scale and the corresponding slope of fitted regression line (the Higuchi fractal dimension) of the Land Surface Temperature (LST) of Riley County during February, April, July and October (2021).

Figure 11. Plots of $log\left(L_m^k\right)$ versus $log\left(k\right)$ for time interval $k=500$, the logarithmic scale and the corresponding slope of fitted regression line (the Higuchi fractal dimension) of the Land Surface Temperature (LST) of Riley County during February, April, July and October (2021).

Although, fractal dimension using the Higuchi algorithm is a good index for comparing the self-similarity and complexity in fractal structure of different (LST) data sets, however, it fails to classify four groups of (LST) data sets.

Discussion

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