# Multifractal Cross Correlation Analysis of Agro-Meteorological Datasets (Including Reference Evapotranspiration) of California, United States

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}and an albedo of 0.23. Different meteorological variables influence the estimation of ET0 and they were considered as potential inputs for modeling of ET0 [46]. Numerous modeling practices ranging from conventional statistical approaches to hybrid decomposition algorithms and data driven techniques were proposed to model ET0 [47,48,49,50,51]. The complex structure of ET0 and other agro-meteorological variables makes the accurate modelling a challenging task. Therefore, any efforts for exploring alternative modeling practices which can contribute towards improving the accuracy in predictions of such variables are appreciable. Fractal or multifractal modeling is another promising approach which can be adopted for modeling complex systems, for which the characterization of such time series is an essential prerequisite to proceed with such modeling practices. Understanding the inherent features of different time-series and their lagged influence have vital role in the accuracy of modeling ET0 and other time series following different alternative approaches. Fractal signatures are such features which can disseminate proper knowledge on the predictability and complexity of the time-series, which may eventually help in accurate modeling of agro-meteorological time series. Multifractal modeling is one potential alternative modeling approach and proper understanding of the persistence and multifractal properties is to be determined a priori before proceeding with such modeling. Moreover, the multifractal properties of time series may distinctly differ depending on terrain conditions. Understanding such differences in multifractal properties from stations of diverse terrain conditions may provide necessary scientific insights to find the reason behind such variability.

^{H}, or not. In that case, the data is said to be scaling in that range i.e., there exists a certain degree of statistical scale-invariance. Persistence can be defined as the existence of some statistical dependence among successive values of the same variable, or among successive occurrences of a given event [52]. The fractal theory quantifies the diminishing nature of correlations (dependence), the autocorrelation function declines as a power-law. Thus, the persistence properties of hydro-meteorological signals can give useful insights into the autocorrelation structure of the time-series [53], which in turn may help in predictor selection in time-series approach for modeling of the agro-meteorological variables.

## 2. Materials and Methods

#### 2.1. MFDFA Method

- (1)
- Determine the accumulated deviation of the time series signal X (x1, x2… xL) from its mean (so called ‘profile’) as:$$P(i)={\displaystyle \sum _{k=1}^{i}\left[{x}_{k}-\overline{x}\right]}$$
- (2)
- Identify different independent sub-series of different lengths (called scale s) both in forward and backward directions.
- (3)
- Fit polynomial to each sub-series and find the difference between original sub-series and its fitted polynomial as:$${f}^{2}(s,\upsilon )=\frac{1}{s}{{\displaystyle \sum _{i=1}^{s}\left\{P[(\upsilon -1)s+i]-{y}_{\upsilon}(i)\right\}}}^{2}\text{}\mathrm{for}\text{}\upsilon =1,2,\dots ,N$$$${f}^{2}(s,\upsilon )=\frac{1}{s}{{\displaystyle \sum _{i=1}^{s}\left\{P[L-(\upsilon -N)s+i]-{y}_{\upsilon}(i)\right\}}}^{2}\text{}\mathrm{for}\text{}\upsilon =N+1,\dots ,2N.$$
_{υ}(i) is the polynomial used for fitting in fragment υ. - (4)
- Determine the fluctuation function (FF) for different combination of scale s and statistical moment order q, based on step 3:$$F{F}_{q}(s)={\left\{\frac{1}{2N}{\displaystyle \sum _{\upsilon =1}^{2N}{\left[{f}^{2}(s,\text{}\upsilon )\right]}^{q/2}}\right\}}^{1/q}$$
- (5)
- Develop the log-log plot of fluctuation function FF
_{q}(s) vs. scale s. The slope of logarithmic plot of FF versus s is called generalized Hurst exponent (GHE) h(q) for different moment order q. The non-linear behavior of the plot between h(q) and q indicates the multifractality of the time series. For monofractal time series h(q) is independent of q. Multifractal behavior (a significant dependence of h(q) on q) will manifest itself only if small and large fluctuations scale differently in the time series. For positive values of q, h(q) describes the scaling behavior of the segments with large fluctuations, whereas for negative values of q, the scaling behavior of the segments with small fluctuations is described. While the precision of h(q) estimation depends mostly on the length of the time series, for large negative values of q (very small fluctuations) the precision of measurement is also starting to play a role. A detailed analysis on the precision of h(q) estimation for the MFDFA method was performed by López and Contreras [71]. The GHE value for q = 2 is considered to be equivalent to the classical Hurst exponent (H) [27,28]. From h(q) other useful exponents like mass exponent τ(q) or singularity exponent f(α) can be derived as follows:$$\tau (q)=qh(q)-1$$$$\alpha =\frac{d\tau (q)}{dq}$$$$f(\alpha )=q\alpha -\tau (q)$$

_{max}− α

_{min}) is an indicative of the strength of the multifractality in such a way that larger width indicates higher multifractal strength and vice versa. The symmetry of the spectrum (symmetric, left or right-skewed) indicates the dominancy for high (or low) fluctuations. It is quantified by a parameter called asymmetry index (AI), computed based on the width of right and left wings of the parabola as:

_{left}= α

_{0}− α

_{min}and ∆α

_{right}= α

_{max}− α

_{0}, respectively. If AI is zero, the spectrum is symmetric, positive values indicates a right skewed and negative values indicates a left-skewed spectrum. Negative values of AI suggest the existence of complex structures at the level of large fluctuation amplitudes in the respective series; they also suggest that the time series are characterized by a multifractal structure, which is insensitive to local fluctuations with small amplitudes. This means that a low weight can be attributed to low fractal exponents, which in turn suggests the frequent occurrence of extreme events [72]. The value of α at q = 0 (indicated as α

_{0}and called as Holder exponent) infers the complexity of the time series. The mathematical expressions for the computation of multifractal properties and the guidelines for selection of algorithm specific control parameters such as polynomial order, range of moment order, scaling range (s

_{min}− s

_{max}) etc. can be can be found in literature [73].

#### 2.2. MFCCA Method

- (1)
- For two time series x
_{i}and y_{i}(I = 1, 2,…, N), determine the profiles as:$${P}_{x}(j)={\displaystyle \sum _{i=1}^{j}\left[{x}_{i}-\overline{x}\right]}$$$${P}_{y}(j)={\displaystyle \sum _{i=1}^{j}\left[{y}_{i}-\overline{y}\right]}$$_{i}and y_{i}. - (2)
- Divide each profile P
_{x}(j) and P_{y}(j) into N_{s}non-overlapping segments both in progressive and retrograde directions, to avoid any omission of time series data from the head or tail end of the series. - (3)
- For each 2N
_{s}segments, compute local trend of both “profiles” P_{x}(j) and P_{y}(j) by fitting polynomials p_{X,υ}^{m}(j) and p_{Y,υ}^{m}(j) of appropriate order m. The subtraction of the fitted polynomial from the original segment gives the covariance as given below:$${{\phi}_{XY}}^{2}(\upsilon ,s)=\frac{1}{s}{\displaystyle \sum _{k=i}^{s}\left\{\left[{P}_{x}((\upsilon -1)s+k)-{{p}_{X,\upsilon}}^{m}(k)\right]\times \left[{P}_{y}((\upsilon -1)s+k)-{{p}_{Y,\upsilon}}^{m}(k)\right]\right\}}$$ - (4)
- Calculate detrended covariance by summation over all segments:$${{F}^{q}}_{XY}(s)=\frac{1}{2{N}_{s}}{\displaystyle \sum _{\upsilon =0}^{2{N}_{s}-1}sign\left[{{\phi}_{XY}}^{2}(\upsilon ,s)\right]}{\left|{{\phi}_{XY}}^{2}(\upsilon ,s)\right|}^{q/2}$$
- (5)
- Check if F
^{q}_{XY}(s) behaves as a power-law function of s (the scaling behavior), where s is the segmental sample size:$${\left\{{{F}^{q}}_{XY}\left(s\right)\right\}}^{1/q}={F}_{XY}\left(s\right)~\text{}{s}^{{\lambda}_{XY}\left(q\right)}.$$

_{XY}(q) is similar to the generalized Hurst exponent h(q) in MFDFA and it can be obtained by observing the slope of log-log plot of F(s) versus s by ordinary least squares. The ratio between the detrended covariance function F

^{q}

_{XY}(s) obtained from MFCCA and the detrended variance functions F

^{q}

_{X}(s) and F

^{q}

_{Y}(s) obtained from MFDFA defines the detrended cross correlation ρ

_{XY}(s) as:

_{XY}(s) ranges between −1 ≤ ρ

_{XY}(s) ≤ 1. The MFCCA analysis facilitate the estimation of scale dependent correlation between two candidate time series, which can provide better insight into the physical association between the variables.

#### 2.3. Study Area and Data

## 3. Results and Discussion

#### 3.1. MFDFA of Agro-Meteorological Time Series

_{min}greater than (m + 2), where ‘m’ is the polynomial order, is a widely accepted thumb rule. The maximum scale s

_{max}is suggested to be chosen as up to L/10 [27]. Here, a rigorous trial and error exercise is made in the selection of the scale range considering the physics of the problem. The fluctuation functions are developed by considering the statistical moment orders from −4 to 4 are fixed and the first order polynomial (m = 1) was chosen for invoking MFDFA [41]. The fitting is considered to be acceptable on getting an R

^{2}≥ 0.98 [81]. Thus a scaling range of [8:600] days was found to be appropriate to capture the scaling behavior, which corresponds from weekly to inter-annual period.

_{0}) were estimated and the values were provided in Table 2.

_{0}is indicating the complexity of the time series, and delivers valuable information about the structure of the studied process, with a high value indicating that it is less correlated and possesses fine structure. From Table 2, it is noted that there exists a strong association between α

_{0}and value of H. On considering the values of these exponents of different stations, the correlation values are found to be 0.966, 0.912, 0.986, 0.99, 0.966 and 0.992 respectively for ET0, T, U, SR, RH and P time series, indicating a strong association between the two exponents in all the time series. This is strongly in agreement with the observation reported by Burgeano et al. [34] for daily temperature series of Catalonia, Spain.

_{min}is fixed at weekly scale of 8 days considering the stability. The Hurst exponent estimate from weekly to inter seasonal range showed a larger persistence in temperature and relative humidity; medium persistence (0.7–0.8) in ET0 and air pressure, and low persistence in wind speed time series.

#### Origin of Multifractality

#### 3.2. MFCCA of Meteorological Variables with ET0

_{XY}(90) and annual ρ

_{XY}(365)) are calculated separately to investigate the change in magnitude and sign along with the overall Pearson correlation (ρ

_{o}). The graphical representations such as GHE plot, multifractal spectra and cross correlation plots for each link of different stations were prepared. We presented the plots for Dagget and San Diego (highest and lowest altitude stations) as representatives in the Figure 7 and Figure 8, whereas MFCCA plots for other stations are presented in the Supplementary Materials, Figures S9–S11. Detailed examination of different graphical representations of multifractal exponents is made to get an insight into the spatial diversity associations of different variables with ET0 in a multifractal perspective.

## 4. Conclusions

## Supplementary Materials

**a**) reference evapotranspiration; (

**b**) temperature; (

**c**) wind speed; (

**d**) solar radiation; (

**e**) relative air humidity; and (

**f**) air pressure, Figure S2: Spectral density plots of different agro-meteorological time series of Santa Maria station: (

**a**) reference evapotranspiration; (

**b**) temperature; (

**c**) wind speed; (

**d**) solar radiation; (

**e**) relative air humidity; and (

**f**) air pressure, Figure S3: Spectral density plots of different agro-meteorological time series of Los Angeles station: (

**a**) reference evapotranspiration; (

**b**) temperature; (

**c**) wind speed; (

**d**) solar radiation; (

**e**) relative air humidity; and (

**f**) air pressure, Figure S4: Spectral density plots of different agro-meteorological time series of San Diego station: (

**a**) reference evapotranspiration; (

**b**) temperature; (

**c**) wind speed; (

**d**) solar radiation; (

**e**) relative air humidity; and (

**f**) air pressure, Figure S5: GHE plots of original, shuffled and surrogate series of different agro-meteorological variables of Bakersfield station: (

**a**) reference evapotranspiration; (

**b**) temperature; (

**c**) wind speed; (

**d**) solar radiation; (

**e**) relative air humidity; and (

**f**) air pressure, Figure S6: GHE plots of original, shuffled and surrogate series of different agro-meteorological variables of Santa Maria station: (

**a**) reference evapotranspiration; (

**b**) temperature; (

**c**) wind speed; (

**d**) solar radiation; (

**e**) relative air humidity; and (

**f**) air pressure, Figure S7: GHE plots of original, shuffled and surrogate series of different agro-meteorological variables of Los Angeles station: (

**a**) reference evapotranspiration; (

**b**) temperature; (

**c**) wind speed; (

**d**) solar radiation; (

**e**) relative air humidity; and (

**f**) air pressure, Figure S8: GHE plots of original, shuffled and surrogate series of different agro-meteorological variables of San Diego station: (

**a**) reference evapotranspiration; (

**b**) temperature; (

**c**) wind speed; (

**d**) solar radiation; (

**e**) relative air humidity; and (

**f**) air pressure, Figure S9: GHE plots (upper panels), multifractal spectra (middle panels) and cross-correlation plots (lower panels) obtained by MFCCA of different links at Bakersfied station: (

**a**) T-ET0; (

**b**) U-ET0; (

**c**) SR-ET0; (

**d**) RH-ET0; (

**e**) P-ET0, Figure S10: GHE plots (upper panels), multifractal spectra (middle panels) and cross-correlation plots (lower panels) obtained by MFCCA of different links at Santa Maria station: (

**a**) T-ET0; (

**b**) U-ET0; (

**c**) SR-ET0; (

**d**) RH-ET0; (

**e**) P-ET0, Figure S11: GHE plots (upper panels), multifractal spectra (middle panels) and cross-correlation plots (lower panels) obtained by MFCCA of different links at Los Angeles station: (

**a**) T-ET0; (

**b**) U-ET0; (

**c**) SR-ET0; (

**d**) RH-ET0; (

**e**) P-ET0.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Time series plots of agro-meteorological datasets of different stations in California: (

**a**) Dagget; (

**b**) Bakersfield; (

**c**) Santa Maria; (

**d**) Los Angeles; (

**e**) San Diego.

**Figure 3.**Fluctuation function plots of different agro-meteorological time series of different stations for q = 2. (

**a**) reference evapotranspiration; (

**b**) temperature; (

**c**) wind speed; (

**d**) solar radiation; (

**e**) relative air humidity; and (

**f**) air pressure.

**Figure 4.**Multifractal plots of different agro-meteorological time series: (

**a**) reference evapotranspiration; (

**b**) temperature; (

**c**) wind speed; (

**d**) solar radiation; (

**e**) relative air humidity; and (

**f**) air pressure. The upper panels show the GHE plots, middle panels show the mass exponent plots and lower panels show the multifractal spectra.

**Figure 5.**Fourier spectrum of different agro-meteorological time series of Dagget station. (

**a**) reference evapotranspiration; (

**b**) temperature; (

**c**) wind speed; (

**d**) solar radiation; (

**e**) relative air humidity; and (

**f**) air pressure.

**Figure 6.**GHE plots of original, shuffled and surrogate series of different agro-meteorological variables of Dagget station: (

**a**) reference evapotranspiration; (

**b**) temperature; (

**c**) wind speed; (

**d**) solar radiation; (

**e**) relative humidity; and (

**f**) air pressure.

**Figure 7.**GHE plots (upper panels), multifractal spectra (middle panels) and cross-correlation plots (lower panels) obtained by MFCCA of different links at Dagget station: (

**a**) T-ET0; (

**b**) U-ET0; (

**c**) SR-ET0; (

**d**) RH-ET0; (

**e**) P-ET0.

**Figure 8.**GHE plots (upper panels), multifractal spectra (middle panels) and cross-correlation plots (lower panels) obtained by MFCCA of different links at San Diego station: (

**a**) T-ET0; (

**b**) U-ET0; (

**c**) SR-ET0; (

**d**) RH-ET0; (

**e**) P-ET0.

Station | Variable | Maximum | Minimum | Mean | Standard Deviation |
---|---|---|---|---|---|

Dagget | ET0 (mm) | 18 | 0.2 | 6.42 | 3.37 |

T (°C) | 37.2 | −5.3 | 19.72 | 8.47 | |

U (m/s) | 15.973 | 0.174 | 5.04 | 2.22 | |

SR (Wm^{−2}) | 1576.2 | 102 | 993.9 | 359.5 | |

RH (%) | 100 | 7 | 30.20 | 14.29 | |

P (kPa) | 96.7 | 92.7 | 94.66 | 0.48 | |

Bakersfield | ET0 (mm) | 13 | 0.2 | 4.68 | 2.78 |

T (°C) | 37 | −2.8 | 18.58 | 8.04 | |

U (m/s) | 14.564 | 0 | 3.49 | 1.32 | |

SR (Wm^{−2}) | 1521.3 | 129.6 | 895.3 | 400.7 | |

RH (%) | 100 | 12 | 48.10 | 18.56 | |

P (kPa) | 101.7 | 97.7 | 99.83 | 0.50 | |

Santa Maria | ET0 (mm) | 10.6 | 0.3 | 3.21 | 1.22 |

T (°C) | 33.3 | 5.7 | 16.72 | 3.36 | |

U (m/s) | 12.719 | 1.257 | 3.77 | 1.00 | |

SR (Wm^{−2}) | 1492.7 | 49.6 | 850.7 | 327.4 | |

RH (%) | 100 | 7 | 65.22 | 16.07 | |

P (kPa) | 103 | 99.1 | 101.19 | 0.37 | |

Los Angeles | ET0 (mm) | 9 | 0.5 | 3.03 | 1.21 |

T (°C) | 26.7 | 0.5 | 13.35 | 2.80 | |

U (m/s) | 12.53 | 0.453 | 4.07 | 1.31 | |

SR (Wm^{−2}) | 1527.7 | 157.2 | 889.6 | 346.7 | |

RH (%) | 98 | 16 | 65.34 | 13.78 | |

P (kPa) | 102.5 | 98.6 | 100.80 | 0.38 | |

San Diego | ET0 (mm) | 9.4 | 0.5 | 3.31 | 1.18 |

T (°C) | 32.2 | 6.7 | 17.55 | 3.43 | |

U (m/s) | 10.897 | 0.697 | 3.55 | 0.99 | |

SR (Wm^{−2}) | 1488.5 | 156.8 | 864.5 | 314.1 | |

RH (%) | 100 | 10 | 63.99 | 14.02 | |

P (kPa) | 103.1 | 99.5 | 101.46 | 0.35 |

Variable | Multifractal Properties | |||||
---|---|---|---|---|---|---|

H | W | AI | ∆f(α) | ∆h(q) | α_{0} | |

Dagget | ||||||

ET0 | 0.758 | 0.458 | 0.362 | 0.370 | 0.240 | 0.804 |

T | 0.851 | 0.270 | 0.596 | 0.344 | 0.133 | 0.867 |

U | 0.616 | 0.383 | 0.454 | 0.371 | 0.212 | 0.655 |

SR | 0.687 | 0.551 | 0.260 | 0.232 | 0.313 | 0.747 |

RH | 0.814 | 0.472 | 0.308 | 0.271 | 0.264 | 0.864 |

P | 0.578 | 0.348 | 0.320 | 0.223 | 0.199 | 0.619 |

Bakersfield | ||||||

ET0 | 0.810 | 0.279 | 0.529 | 0.307 | 0.147 | 0.831 |

T | 0.748 | 0.325 | 0.403 | 0.275 | 0.173 | 0.778 |

U | 0.710 | 0.246 | 0.300 | 0.109 | 0.120 | 0.728 |

SR | 0.701 | 0.523 | 0.377 | 0.385 | 0.293 | 0.755 |

RH | 0.820 | 0.269 | 0.527 | 0.305 | 0.138 | 0.840 |

P | 0.634 | 0.309 | 0.210 | 0.132 | 0.172 | 0.672 |

Santa Maria | ||||||

ET0 | 0.742 | 0.207 | 0.411 | 0.176 | 0.105 | 0.759 |

T | 0.843 | 0.381 | 0.390 | 0.292 | 0.218 | 0.885 |

U | 0.626 | 0.464 | 0.522 | 0.460 | 0.260 | 0.665 |

SR | 0.652 | 0.421 | 0.405 | 0.350 | 0.224 | 0.690 |

RH | 0.706 | 0.477 | 0.373 | 0.336 | 0.286 | 0.765 |

P | 0.648 | 0.413 | 0.286 | 0.268 | 0.220 | 0.693 |

Los Angeles | ||||||

ET0 | 0.685 | 0.380 | 0.245 | 0.196 | 0.198 | 0.724 |

T | 0.812 | 0.501 | 0.185 | 0.162 | 0.289 | 0.876 |

U | 0.735 | 0.239 | 0.381 | 0.203 | 0.148 | 0.772 |

SR | 0.644 | 0.417 | 0.528 | 0.471 | 0.229 | 0.682 |

RH | 0.674 | 0.611 | 0.378 | 0.419 | 0.389 | 0.760 |

P | 0.636 | 0.337 | 0.209 | 0.138 | 0.182 | 0.673 |

San Diego | ||||||

ET0 | 0.678 | 0.336 | 0.128 | 0.070 | 0.172 | 0.712 |

T | 0.826 | 0.426 | 0.121 | 0.078 | 0.247 | 0.883 |

U | 0.685 | 0.205 | 0.111 | 0.047 | 0.109 | 0.709 |

SR | 0.616 | 0.441 | 0.510 | 0.498 | 0.235 | 0.655 |

RH | 0.684 | 0.497 | 0.247 | 0.212 | 0.321 | 0.763 |

P | 0.642 | 0.338 | 0.213 | 0.145 | 0.179 | 0.678 |

**Table 3.**MFCCA parameters of different agro-meteorological datasets of California, supplemented with Pearson correlation coefficient (ρ

_{o}).

Link | MFCCA Parameters | Pearson Correlation Coefficient | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Scaling Exponent | Spectral Width | Asymmetry Index | Cross Correlation Coefficient | |||||||||

λ_{XX} | λ_{YY} | λ_{XY} | W_{XX} | W_{YY} | W_{XY} | AI_{XX} | AI_{YY} | AI_{XY} | ρ_{XY}(90) | ρ_{XY}(365) | ρ | |

Dagget | ||||||||||||

T-ET0 | 0.896 | 0.778 | 0.837 | 0.300 | 0.426 | 0.218 | −0.522 | −0.325 | −0.473 | 0.575 | 0.610 | 0.560 |

U-ET0 | 0.619 | 0.778 | 0.699 | 0.417 | 0.426 | 0.276 | −0.452 | −0.325 | −0.454 | 0.256 | 0.416 | 0.465 |

SR-ET0 | 0.708 | 0.778 | 0.743 | 0.548 | 0.426 | 0.365 | −0.348 | −0.325 | 0.071 | 0.615 | 0.610 | 0.447 |

RH-ET0 | 0.868 | 0.778 | 0.823 | 0.482 | 0.426 | 0.273 | −0.341 | −0.325 | −0.018 | −0.652 | −0.633 | −0.508 |

P-ET0 | 0.619 | 0.778 | 0.699 | 0.413 | 0.426 | 0.139 | −0.369 | −0.325 | −0.237 | 0.033 | 0.019 | −0.169 |

Bakersfield | ||||||||||||

T-ET0 | 0.824 | 0.804 | 0.814 | 0.374 | 0.440 | 0.276 | −0.379 | −0.468 | −0.138 | 0.707 | 0.585 | 0.617 |

U-ET0 | 0.690 | 0.805 | 0.748 | 0.369 | 0.515 | 0.351 | −0.786 | −0.468 | 0.068 | 0.136 | 0.310 | 0.402 |

SR-ET0 | 0.719 | 0.804 | 0.762 | 0.491 | 0.440 | 0.317 | −0.396 | −0.468 | −0.019 | 0.668 | 0.684 | 0.571 |

RH-ET0 | 0.855 | 0.804 | 0.830 | 0.309 | 0.440 | 0.210 | −0.445 | −0.468 | −0.287 | −0.784 | −0.706 | −0.700 |

P-ET0 | 0.675 | 0.804 | 0.739 | 0.345 | 0.440 | 0.075 | −0.220 | −0.468 | −0.745 | −0.145 | −0.113 | −0.118 |

Santa Maria | ||||||||||||

T-ET0 | 0.881 | 0.748 | 0.815 | 0.351 | 0.281 | 0.249 | −0.392 | −0.463 | −0.193 | 0.462 | 0.452 | 0.505 |

U-ET0 | 0.630 | 0.715 | 0.672 | 0.483 | 0.375 | 0.288 | −0.450 | −0.220 | −0.325 | 0.084 | 0.163 | 0.159 |

SR-ET0 | 0.672 | 0.748 | 0.710 | 0.469 | 0.281 | 0.272 | −0.432 | −0.463 | −0.354 | 0.742 | 0.627 | 0.679 |

RH-ET0 | 0.739 | 0.748 | 0.743 | 0.503 | 0.281 | 0.282 | −0.381 | −0.463 | −0.432 | −0.811 | −0.822 | −0.791 |

P-ET0 | 0.679 | 0.748 | 0.713 | 0.405 | 0.281 | 0.154 | −0.180 | −0.463 | −0.315 | 0.087 | 0.088 | 0.081 |

Los Angeles | ||||||||||||

T-ET0 | 0.853 | 0.708 | 0.781 | 0.514 | 0.425 | 0.400 | −0.116 | −0.177 | 0.102 | 0.428 | 0.454 | 0.522 |

U-ET0 | 0.708 | 0.708 | 0.708 | 0.326 | 0.425 | 0.193 | −0.158 | −0.177 | −0.219 | 0.057 | −0.011 | 0.172 |

SR-ET0 | 0.676 | 0.708 | 0.692 | 0.400 | 0.425 | 0.266 | −0.566 | −0.177 | −0.328 | 0.764 | 0.716 | 0.681 |

RH-ET0 | 0.712 | 0.708 | 0.710 | 0.604 | 0.425 | 0.456 | −0.286 | −0.177 | −0.358 | −0.863 | −0.853 | −0.848 |

P-ET0 | 0.659 | 0.708 | 0.684 | 0.407 | 0.425 | 0.255 | −0.176 | −0.177 | −0.233 | 0.031 | −0.003 | 0.060 |

San Diego | ||||||||||||

T-ET0 | 0.857 | 0.711 | 0.784 | 0.462 | 0.344 | 0.377 | −0.049 | 0.003 | 0.289 | 0.385 | 0.397 | 0.460 |

U-ET0 | 0.681 | 0.711 | 0.696 | 0.268 | 0.344 | 0.135 | −0.121 | 0.003 | −0.147 | −0.110 | −0.013 | −0.009 |

SR-ET0 | 0.662 | 0.711 | 0.686 | 0.437 | 0.344 | 0.263 | −0.564 | 0.003 | −0.326 | 0.807 | 0.745 | 0.738 |

RH-ET0 | 0.722 | 0.711 | 0.717 | 0.626 | 0.344 | 0.335 | −0.337 | 0.003 | −0.132 | −0.798 | −0.768 | −0.768 |

P-ET0 | 0.668 | 0.711 | 0.689 | 0.410 | 0.344 | 0.181 | −0.168 | 0.003 | −0.285 | 0.037 | −0.025 | 0.103 |

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Sankaran, A.; Krzyszczak, J.; Baranowski, P.; Devarajan Sindhu, A.; Kumar, N.P.; Lija Jayaprakash, N.; Thankamani, V.; Ali, M.
Multifractal Cross Correlation Analysis of Agro-Meteorological Datasets (Including Reference Evapotranspiration) of California, United States. *Atmosphere* **2020**, *11*, 1116.
https://doi.org/10.3390/atmos11101116

**AMA Style**

Sankaran A, Krzyszczak J, Baranowski P, Devarajan Sindhu A, Kumar NP, Lija Jayaprakash N, Thankamani V, Ali M.
Multifractal Cross Correlation Analysis of Agro-Meteorological Datasets (Including Reference Evapotranspiration) of California, United States. *Atmosphere*. 2020; 11(10):1116.
https://doi.org/10.3390/atmos11101116

**Chicago/Turabian Style**

Sankaran, Adarsh, Jaromir Krzyszczak, Piotr Baranowski, Archana Devarajan Sindhu, Nandhineekrishna Pradeep Kumar, Nityanjali Lija Jayaprakash, Vandana Thankamani, and Mumtaz Ali.
2020. "Multifractal Cross Correlation Analysis of Agro-Meteorological Datasets (Including Reference Evapotranspiration) of California, United States" *Atmosphere* 11, no. 10: 1116.
https://doi.org/10.3390/atmos11101116