# Testing the Beta-Lognormal Model in Amazonian Rainfall Fields Using the Generalized Space q-Entropy

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## Abstract

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## 1. Introduction

#### 1.1. Statistical Scaling and Multiplicative Random Cascades

#### 1.2. Multiplicative Random Cascades and the Beta-LogNormal Model

#### 1.3. q-Entropy

“An entropy of a system or of a subsystem is said extensive if, for a large number N of its elements (probabilistically independent or not), the entropy is (asymptotically) proportional to N. Otherwise, it is nonextensive. This means that extensivity depends on both the mathematical form of the entropic functional and the possible correlations existing between the elements of the system. Consequently, for a (sub)system whose elements are either independent or weakly correlated, the additive entropy S is extensive, whereas the nonadditive entropy ${S}_{q}$ ($q\ne 1$) is nonextensive. In contrast, however, for a (sub)system whose elements are generically strongly correlated, the additive entropy S can be nonextensive, whereas the nonadditive entropy ${S}_{q}$ ($q\ne 1$) can be extensive for a special value of q.”

#### 1.4. Generalized Space-Time q-Entropy in Rainfall Data

- ${S}_{q}\left(T\right)$ decreases monotonically with q for all values of T.
- For a given value of q, estimates are inversely related to T for $q<0$, but directly related for $q\ge 0$.
- Estimates of ${S}_{q}\left(T\right){\mid}_{q=1}$ recover the standard entropy for different values of T.
- Estimates of ${S}_{q}\left(T\right)$ increase with T for values of $q\ge 0$, up to a certain saturation value (maximum q-entropy).
- The function $S\left(T\right)$ vs. T in log–log space, for different values of q, can be considered an (time) entropy analogous of the (space) structure function in turbulence [54].
- The scaling exponents, $\mathsf{\Omega}\left(q\right)$, or the slope of the relation $S\left(T\right)$ vs. T in log–log space, for different values of q, exhibit a non-linear growth with q, such that ${\mathsf{\Omega}}_{sat}\approx 0.5$ for $q\ge 1$. This result allowed extending the conclusions from the standard Shannon entropy to the generalized q-entropy.
- The scaling exponents of saturation, ${\mathsf{\Omega}}_{sat}$, are different in time and space for hydrological data, such as time series of rainfall, for which ${\mathsf{\Omega}}_{sat}(q>1.0)=0.5$, for time series of streamflows, for which ${\mathsf{\Omega}}_{sat}(q>1.0)=0.0$, and for the spatial analysis of radar rainfall fields in Amazonia, for which ${\mathsf{\Omega}}_{sat}(q>1.0)=1.0$.

- Poveda [28] studied the time scaling properties of tropical rainfall in the Andes of Colombia upon temporal aggregation and introduced the Generalized Time q-Entropy Function (GTEF), as a time analogous for q-entropy of the structure function in turbulence [54]. He showed that the scaling exponents, $\mathsf{\Omega}\left(q\right)$ of the relation ${S}_{q}\left(T\right)$ vs. T in log–log space, for different values of q, exhibit a non-linear growth with q up to $\mathsf{\Omega}=0.5$ for $q\ge 1$, putting forward the conjecture that the time dependent q-entropy, ${S}_{q}\left(T\right)\sim {T}^{\mathsf{\Omega}\left(q\right)}$ with $\mathsf{\Omega}\left(q\right)\simeq 0.5$, for $q\ge 1$.
- Salas and Poveda [30] revisited results reported in [28], and analyzed the time scaling properties of Shannon’s entropy for the same data set in terms of the sensitivity to the record length, and the effect of zeros in rainfall data, and proposed the GTEF to study the scaling properties of river flows. They highlighted two important results: (i) The scaling characteristics of Shannon’s entropy differ between rainfall and streamflows owing to the presence of zeros in rainfall series; and (ii) the GTEF exhibits multi-scaling for rainfall and streamflows. For rainfall, the relation ${S}_{q}\left(T\right)$ vs. T in log–log space for different values of q, exhibits a non-linear growth with q, up to $\mathsf{\Omega}=0.5$ for $q\ge 1$, in contrast to the scaling properties of river flows which exhibit a non-linear growth with q, up to $\mathsf{\Omega}=0.0$ for $q\ge 1$.
- Poveda and Salas [31] studied diverse topics such as statistical scaling, Shannon entropy and Space-Time Generalized q-Entropy of Mesoscale Convective Systems (MCS) as seen by the Tropical Rainfall Measuring Mission (TRMM) over continental and oceanic regions of tropical South America, and in Amazonian radar rainfall fields. The main result of their study is that both the GTEF and GSEF exhibit linear growth in the range $-1.0<q<-0.5$, and saturation of the exponent ${\mathsf{\Omega}}_{sat}$ for $q\ge 1.0$, but for the spatial analysis (GSEF) the exponent tappers off at $\u2329{\mathsf{\Omega}}_{sat}\u232a\sim $ 1.0, whereas for the temporal analysis (GTEF) the exponent saturates at $\u2329{\mathsf{\Omega}}_{sat}\u232a=$ 0.5. In addition, results are similar for time series extracted from radar rainfall fields in Amazonia (radar S-POL) and in-situ rainfall series in the tropical Andes.

#### 1.5. Easterly and Westerly Regimes of Amazonian Rainfall

#### 1.6. Rationale and Objectives

- The presence of zeros in high resolution rainfall records constitute highly important information to understand, diagnose and forecast the dynamics of rainfall [28,60,61]. Salas and Poveda [30] argued that zeros (inter-storm periods) in time series of tropical convective rainfall are associated with the timescale required by nature to build up the dynamic and thermodynamic conditions of the next storm, as an atmospheric analogous of the time of energy build-up between earthquakes, avalanches and many other relaxational processes in nature [62]. Therefore, the role of zeros and their effect on scaling statistics must be investigated to further understand and model high resolution rainfall.
- The aforementioned previous works [28,30,31] are based on available rainfall data (S-POL radar, TRMM satellite and rain gauges), and it is difficult to understand differences of the q-statistics in temporal and spatial scales due to factors such as the intermittency of rainfall, record length, space-time resolution of data sets, and geographic setting.
- The number of bins in the probability mass function constitutes a central issue to quantify entropic measures. Previous studies have shown that the scaling exponent of Shannon entropy under aggregation in time it is not sensitive to either the number of bins [28] or record length [30]. Then, it is necessary to study the sensitivity of the q-entropic measures in order to check their robustness to characterizing 2-D tropical rainfall fields.

- To examine how the spatial structure of rainfall is reflected in the q-entropic scaling measures using the BL-Model and considering the influence of zeros in the GSEF through Montecarlo experiments, aimed at understanding the saturation of the exponent ${\mathsf{\Omega}}_{sat}$ reported by [28,30,31].
- To quantify the sensitivity of the q-entropic scaling statistics to the number of bins and to the variability of rainfall intensity, in an attempt to check the robustness of such statistical tools in the multi-scale characterization of rainfall.
- To link two important theoretical frameworks, namely stochastic processes (Multiplicative Cascades) and Information Theory (non-extensive statistical mechanics), to advance our understanding about the scaling properties of tropical rainfall.

## 2. Study Region and Data Sets

#### General Information

## 3. Methods

#### 3.1. Parameters of the BL-Model and Amazonian Precipitation Features

#### 3.2. Bin-Counting Methods and Entropic Estimators

#### 3.3. Sample-Size and Entropy Estimators

#### 3.4. Intermittency and q-Order

## 4. Results

#### 4.1. Linking Parameters of the BL-Model with Precipitation Features

- The cascade parameter, $\beta $, (Table 1), for the Easterly events is greater than the Westerly events, indicating more spatially concentrated rainfall fields (more zeros in the Easterly scans). This result is related to diverse precipitation features observed during the Easterly regime, given that the atmospheric conditions are relatively dry, with increased lightning activity and more intense and deeper convective systems [56,58,59].
- The cascade parameter, $\sigma $, (Table 1), exhibits smaller (larger) values during the Easterly (Westerly) regime, indicating that the variability of rainfall intensity for the Westerly events is higher than for Easterly events. This result is coherent with diverse features observed during the Westerly regime, which is characterized by less lightning activity, less deep convection and less intense precipitation rates [56,58,59].

#### 4.2. The Role of Zeros in the Generalized Space q-Entropy

#### 4.3. The Role of Rainfall Intensity Variability in Generalized Space q-Entropy

#### 4.4. Bin-Counting Methods and the Generalized Space q-Entropy

- The bin-counting method proposed by Dixon and Kronmal [70] is the nearest to the method presented by Gong et al. (2014) for Gaussian r.v. under aggregation, for a number of aggregation intervals greater than 70.
- The theoretical inequality given by Equation (17) is better captured by Scott’s method, although this method shows lower values than the theoretical expression, for aggregation intervals $T\ge 100$.
- The difference between the theoretical inequality (Equation (17)) and Gong et al.’s [67] method is explained because the “Discrete Entropy” and the “Continuous Entropy” (also referred to as “Differential Entropy”) are related as:$$\underset{\Delta \to 0}{lim}[{H}_{\Delta}\left({X}_{d}\right)+\mathrm{log}(\Delta )]=h\left(Xc\right),$$
- To check the sensitivity of the scaling exponents of the GSEF to the number of bins, we developed a numerical experiment using the BL-Model and 1000 independent simulations for each number of bins n = 10, 30, 50 and 100, with parameters $\beta $ = 0.10, 0.351, and 0.950 and $\sigma $ = 0.05, 0.40 and 0.80. Figure 5 and Figure 7 show that the GSEF is not statistically affected either by the number of bins or by the value of $\sigma $ when $nbins>30$ and the sample-size is bigger than 200 data. Consequently, the GSEF is not affect by the bin-counting method.

#### 4.5. Sample-Size and the Generalized Space q-Entropy

- The pmf to calculate ${S}_{q}$ was built considering all values in the synthetic rainfall field including zeros.
- The pmf to calculate ${S}_{q}$ was built considering all values separated in two subsets: (i) values greater than zero (rain), i.e., $P(x>0)$; and (ii) values equal to zero (dry), $P(x=0)$. For the subset (i), the pmf was built and then corrected by the probability of rainfall ($1-P(x=0)$) thus, the probability of occurrence of rainfall can be written as $P\left(x\right)=\left\{P(x=0),P(x>0)\right\}$.

#### 4.6. Rainfall Intermittency and q-Order

## 5. Discussion

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Schematic plot of the random cascade geometry taken from Gupta and Waymire [33].

**Figure 2.**Histograms for Beta-Lognormal Model parameters in rainfall scans of the S-POL radar: (

**a**,

**b**) Westerly events; and (

**c**,

**d**) Easterly events. (red) the Gaussian function for $\beta $ and the Generalized Extreme Value function for ${\sigma}^{2}$.

**Figure 3.**Empirical Cumulative Distribution Functions for cascade parameters: $\beta $ (

**a**); and ${\sigma}^{2}$ (

**b**) using 4227 scans of the S-POL radar considering climatic regimes of Amazonia (1867 scans for Westerly and 2360 for Easterly). The figure shows the $95\%$ confidence bounds using Greenwood’s formula.

**Figure 4.**Space Generalized q-Entropy for the S-POL radar scan 01/10/1999 18:23:15 LST: (

**a**) 3D plot of the Tsallis’ entropy, ${S}_{q}$, for different scale factors, $\lambda $, and q-values from −1.0 to 3.0; (

**b**) projection of S(q,$\lambda $) vs. q for different values of $\lambda $; (

**c**) projection of S(q,$\lambda $) vs. $\lambda $, for different values of q, or spatial structure function for entropy; and (

**d**) values of the regression slopes of the spatial structure function for entropy, $\mathsf{\Omega}$, as function of q, exhibiting a non-linear growth up to $\u2329\mathsf{\Omega}\u232a\sim 0.50$ for $q>2.5$.

**Figure 5.**Space Generalized q-Entropy Function for spatially distribute rainfall as a random cascade varying cascade’s parameters $\sigma $ and $\beta $ for 1000 simulated independently fields including zeros in the histogram. $LCI95\%$ and $UCI95\%$ are the lower and upper confidence intervals for Easterly and Westerly events (E-W) and the BL-Model model (BL): (

**a**–

**c**) varying the cascade’s parameter $\beta $; and (

**d**–

**f**) varying the cascade’s parameter $\sigma $. In all cases, varying the number of bins (nbins = 10, 30, 50 and 100).

**Figure 6.**Validation of the BL-Model using Generalized Space q-Entropy: (

**a**) comparison $\mathsf{\Omega}\left(q\right)$-Observed vs. $\mathsf{\Omega}\left(q\right)$-Simulated, (Solid line) relation 1:1; and (

**b**) box plots of the coefficients of determination, ${R}^{2}$, for the power fits ${S}_{q}\left(\lambda \right)\sim {\lambda}^{\mathsf{\Omega}}$ from 1000 synthetic fields of BL-Model with $\beta =0.351$ and $\sigma =0.245$. ${R}^{2}\ge 0.85$ in the intervals $-1.0\le q\le 0.0$ and $q\ge 2.5$. The histogram for estimate ${S}_{q}$ includes zeros.

**Figure 7.**Sensitivity analysis of the saturation ${\mathsf{\Omega}}_{sat}$, and minimum ${\mathsf{\Omega}}_{min}$ scaling exponents in the SGEF for 1000 independent rainfall fields generated by random cascade model [13]. Confidence intervals for $95\%$ in dash line and mean value in solid line. (

**a**) Varying cascade’s parameter $\beta $ and considering $\sigma $ = 0.25 constant; (

**b**) varying cascade’s parameter $\beta $ and considering $\sigma $ = 0.25 constant; (

**c**) varying cascade’s parameter $\sigma $ and considering $\beta =0.5$ constant; and (

**d**) varying cascade’s parameter $\sigma $ and considering $\beta =0.5$ constant.

**Figure 8.**Numerical estimation of Shannon’s Entropy using multiple bin-counting methods and the theoretical inequality (Equation (17), for a Gaussian r.v. for different levels of aggregation, T).

**Figure 9.**Boxplots for q-Entropy ${S}_{q=2.5}$ vs. sample sizes, $\xi $, in 1000 independent random cascade simulations of the Beta-Lognormal model with parameters $\beta =0.351$ and $\sigma =0.245$: (

**a**) q-Entropy including zeros in the histogram; and (

**b**) q-Entropy without including zeros in the histogram. In both cases, ($nbins=50$).

**Figure 10.**Comparison of q-Entropy ${S}_{q=2.5}$ in 1000 independent random cascade fields of the Beta-Lognormal model with parameters $\beta =0.351$ and $\sigma =0.245$: (

**a**) q-Entropy ${S}_{q}$ including zeros in the histogram vs. q-Entropy ${S}_{q}$ without zeros in the histogram, cascade’s level $=6$, i.e., $\xi =4096$; and (

**b**) q-Entropy ${S}_{q}$ including zeros in the histogram vs. q-Entropy ${S}_{q}$ without zeros in the histogram, cascade’s level $=8$, i.e., $\xi $ = 65,536. In both cases, ($nbins=50$).

**Figure 11.**Space Generalized q-Entropy Functions (SGEFs) for the climate regimes of Amazonian rainfall from the S-POL radar: (

**a**) Westerly events; and (

**b**) Easterly events. (circles) Average SGEF for scans with more than 200 values greater than zero, (dashed lines) $95\%$ confidence intervals (CI); (squares) average SGEF for all the scans available of each climate regime; (solid lines) $95\%$ confidence intervals (CI).

**Figure 12.**Coefficient of Determination, ${R}^{2}$, for the power fits ${S}_{q}\left(\lambda \right)\sim {\lambda}^{\mathsf{\Omega}}$ from scans radars of the climate regimes in Amazonian rainfall. ${R}^{2}\ge 0.85$ in the intervals $-1.0\le q\le 0.5$ and $q\ge 2.5$. The histogram for estimate ${S}_{q}$ includes zeros: (

**a**) Easterly events; and (

**b**) Westerly events.

**Figure 13.**Typical least-square regressions ${\widehat{\psi}}_{k}\left({q}_{1}\right)\sim {k}^{H\left({q}_{1}\right)}$ and ${\widehat{\psi}}_{k}\left({q}_{2}\right)\sim {k}^{H\left({q}_{2}\right)}$ with $H(q=1)=0.097$ and $H(q=2)=0.412$, for a synthetic 2-D rainfall field from the BL-Model with $\beta =0.351$, $\sigma =0.245$ and cascade level $n=8.0$.

**Figure 14.**Scale of fluctuation, $\theta \left(\tau \right)$, for a time serie of Amazonian rainfall from S-POL radar.

Description | Westerly | Easterly |
---|---|---|

Total number of scans | 2607 | 3884 |

Average $\beta $ for all scans | 0.421 | 0.491 |

Average $\sigma $ for all scans | 0.235 | 0.221 |

q-value where the SGEF saturates for all scans | 1.50 | 1.50 |

Average scaling exponent of saturation, ${\mathsf{\Omega}}_{sat}$, for all scans | 1.0 | 1.0 |

Scans with more than 200 values non-zero (Denoted as ∗) | 1867 | 2360 |

Scans with all values zeros | 86 | 21 |

Percentage of scans ∗ | 71.6% | 60.8% |

Percentage of scans with less than 200 values non-zero | 28.4% | 39.2% |

Average $\beta $ for scans ∗ | 0.336 | 0.365 |

Average $\sigma $ for scans ∗ | 0.248 | 0.242 |

q-value where the SGEF saturates for scans ∗ | 2.5 | 2.5 |

Average scaling exponent of saturation, ${\mathsf{\Omega}}_{sat}$, for scans ∗ | 0.38 ± 0.15 | 0.4 ± 0.15 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Salas, H.D.; Poveda, G.; Mesa, O.J. Testing the Beta-Lognormal Model in Amazonian Rainfall Fields Using the Generalized Space *q*-Entropy. *Entropy* **2017**, *19*, 685.
https://doi.org/10.3390/e19120685

**AMA Style**

Salas HD, Poveda G, Mesa OJ. Testing the Beta-Lognormal Model in Amazonian Rainfall Fields Using the Generalized Space *q*-Entropy. *Entropy*. 2017; 19(12):685.
https://doi.org/10.3390/e19120685

**Chicago/Turabian Style**

Salas, Hernán D., Germán Poveda, and Oscar J. Mesa. 2017. "Testing the Beta-Lognormal Model in Amazonian Rainfall Fields Using the Generalized Space *q*-Entropy" *Entropy* 19, no. 12: 685.
https://doi.org/10.3390/e19120685