# Spatial Modeling of Rainfall Patterns over the Ebro River Basin Using Multifractality and Non-Parametric Statistical Techniques

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{1}, α, and γ

_{s}UM parameters—was combined with non-parametric statistical techniques that allow spatial-temporal comparisons of distributions by gradients. The proposed combined approach was applied to a daily rainfall dataset of 132 time-series from 1931 to 2009, homogeneously spatially-distributed across a 25 km × 25 km grid covering the Ebro River Basin. A homogeneous increase in C

_{1}over the watershed and a decrease in α mainly in the western regions, were detected, suggesting an increase in the frequency of dry periods at different scales and an increase in the occurrence of rainfall process variability over the last decades.

## 1. Introduction

_{1}) and the Levy index (α), as well as a combination of them, the maximal probable singularity (γ

_{s}). With these parameters, any conservative data field can be statistically characterized and used to analyze its scaling nature.

_{1}over the Ebro River Basin and a decrease in the α index, mainly in the western areas of the watershed, were detected, which indicated that the frequency of dry periods at different scales increased over the last decades.

## 2. Materials and Methods

#### 2.1. Study Area and Data

^{2}, is located in Northeast Spain, and is characterized by the high spatial heterogeneity of its geology, topography, climatology, and land use. The Ebro River is one of the most important rivers for Spanish water policy and supplies water through water transfer projects to other river basins affected by desertification in some areas of Spain [34]. This area is characterized by the presence of mountains that border the north (Pyrenees) and south (Iberian System and Catalan Mountains coastal chain), allowing the river to pass through a large depression to the southeast. For additional details, see [34] and the references therein.

^{14}), to achieve higher multifractal analysis efficiency.

**Figure 1.**Geographical areas in the Ebro River Basin. Each area has an associated daily rainfall time-series from 1931 to 2009.

#### 2.2. Trend Test

_{0}: $\text{\tau}=0$ (no correlation between series and time) against H

_{a}: $\text{\tau}\ne 0$ (correlation between series and time). This test is very simple and calculates the difference, Dif, between the number of years i < j for which y(i) < y(j) and the number of years i < j for which y(i) > y(j), where y(k) represents the rainfall along the year k. Finally, because the data under study constitute a long time series, the Mann-Kendall statistic can be calculated as follows:

_{α/2}is the value that verifies Prob {Z > Z

_{α/2}} = α/2.

#### 2.3. Spectral Analysis

#### 2.4. UM Model

_{λ}

^{q}> ≈ λ

^{K}

^{(q)}, where λ is the scale ratio, which is inversely proportional to the size of the measurement interval. The scaling exponent function K(q) for the moments q of a cascade conserved process is obtained according to [46] as follows:

_{1}and α are aforementioned UM parameters. Then, these multifractal functions can be characterized using the following three parameters, where the last one is a combination of the two previous:

- 1)
- C
_{1}is the mean intermittency codimension, which estimates how concentrated the average of the measure is [45]. If the C_{1}value is low (near 0), the field (daily rainfall process in the present work) is similar to the average almost everywhere. However, when C_{1}is greater than 0.5, the field achieves very low values with respect to the average in most time series data, except in a few cases in which the measured value is much higher than the average value. This parameter can also be considered as an indicator of average fractality; - 2)
- α is the Levy index and indicates the distance from a monofractality case [47]. The range of this parameter is [0, 2]. When the value of this parameter is two, a lognormal distribution case occurs; if 1 < α < 2, log Levy processes with unbounded singularities occur; if α = 1, the log-Cauchy distribution occurs; if 0 < α < 1, the Levy process with bounded singularities occurs; and when α = 0, the monofractal process occurs, which indicates the grade of variance of the measure [48]. Higher Levy index values represent extreme measurements that are more frequent (extreme rainfall events in the present work);
- 3)
- γ
_{s}is the maximum probable singularity that can be observed from a unique sample of data and can be obtained from the other two parameters [24]. In addition, γ_{s}is directly related to the ratio of the range to the mean of the field. Here, the notion of singularity relates to an index used to characterize the variation of statistical behavior of data values as the measuring scale changes. Its distribution is consistent with the distribution of the anomalies of rainfall that result in anomalous amounts of energy releases at a fine (spatial and temporal) scale [46].

#### 2.5. Temporal Comparison Test of the UM Parameters

_{i,j,s}represents the estimated value of the j UM parameter in the i area and during the s period and V corresponds to the variance function, which was approximated by adding the variances of the two estimated values under comparison (different estimations of the UM parameters are generated by varying q in the final regression of the DTM method, allowing to obtain an estimation of the variance of each UM parameter in each period and geographical area).

#### 2.6. Spatial Comparison Test of the Distributions of UM Parameter Evolutions

_{1}, α, γ

_{s}). Therefore, nine variables were obtained, where the associated variables for each of the three UM parameters can be represented as follows:

_{1}, α or γ

_{s}), n

_{+,j}is the number of square areas with a positive trend for parameter j, n

_{−,j}is the number of areas with a negative trend for parameter j, and n

_{0,j}is the number of areas without a trend for parameter j.

_{1}, Ψ

_{2}, Ψ

_{3}, and Ψ

_{4}represent the generalized Cramer-Von Mises statistics evaluated using different methods depending on the approach in which the square areas were sorted (see Figure 2a,b). Obviously, the rest of the gradients used in this work, which were univariates, did not require this type of average evaluation of the Ψ statistic. Figure 2c illustrates the case in which the gradient of the distance to the main river basin axis is considered. This is a simplified gradient, considering as principal axis of variation the general course of the river. Using this gradient, the geographical areas were shorted from the smallest to the largest projection on the main river basin axis.

**Figure 2.**The cumulative criterion used in the geographical characterization of universal multifractal (UM) parameter evolutions: (

**a**) related to the spatial distribution of evolutions in an west–east gradient; (

**b**) related to a south–north gradient; and (

**c**) based on the geographical projection of the areas on the main river axis.

## 3. Results and Discussion

#### 3.1. Annual Rainfall Trend Tests

_{0.025}= −1.96 then the rainfall series is statistically significantly decreasing, and if the statistic τ is bigger than Z

_{0.025}= 1.96 then the rainfall series is statistically significantly increasing. It was found that most of the rainfall series did not show significant statistical trends, except in the case of the four series (areas), which showed a positive trend. Figure 3 shows a summary of these contrasts. Overall, it was concluded that no temporal trends exist in the Ebro River Basin, despite the four areas that exhibited significant trends in the northern watershed. Consequently, a stationary process was found and the spatial impact on trends in rainfall could not be determined in the watershed.

**Figure 3.**Annual rainfall trends results using the Mann-Kendall test for a significance level α = 0.05, for each time series from 1931 to 2009.

#### 3.2. Spectral Analysis

^{2}in the performed regressions were 0.967 and 0.713, respectively. The mean of the β spectral exponent in the first period was 0.700, with a standard deviation of 0.128, while the mean of β in the second period decreased until 0.611, with a standard deviation of 0.119. Based on these results, the proposed methodology was used to obtain a multifractal characterization of the watershed, and after this characterization, the rainfall patterns were modeled spatially. For the problematic area with β > 1 in the first period, the DTM model was applied for homogeneity with the rest of the series. Furthermore, focusing on the information provided by these spectral exponents, a higher number of exponents with low values in the second period can be observed in Figure 4.

**Figure 4.**Spectral slope (β) on the grid location and time period represented with different symbols, depending on their values. (

**a**) First period; (

**b**) Second period.

#### 3.3. UM Model

_{1}, α, γ

_{s}) in the two temporal periods under study, 1931–1975 and 1965–2009, that were obtained using the DTM technique. For the first regression in DTM method, the scaling break was reached for a scale lower than 150 days. The median and the minimum of R

^{2}were 0.967 and 0.901, respectively. On the other hand, the scaling break for the second regression used to estimate α parameter was 0.1 < μ < 2 and μq < 3.3 (where μ and q are the exponents of the field in the algorithm of DTM method). For these regressions the median and the minimum of R

^{2}were 0.997 and 0.9232, respectively. Table 1 summarizes the descriptive statistics of the UM parameters throughout the watershed.

**Figure 5.**UM parameters (C

_{1}, α and γ

_{s}) for each spatial localization during the first period (

**left**) and second period (

**right**).

_{1}is the lowest in the northwest and increases towards the southeast, which is opposite of the trend indicated by the Levy index α. With respect to the maximal probable singularity γ

_{s}, an increase from northwest to southeast is detected, which corresponds with the pattern observed in the intermittency. Therefore, it can be concluded that the effect of C

_{1}dominates that of α in the Ebro River Basin because γ

_{s}is graphically more strongly related to C

_{1}. The maps in Figure 5 show the spatial multifractal characterization of the watershed, which is symmetric with respect to the main river basin axis and increases in the direction of the river for the UM parameters C

_{1}and γ

_{s}.

**Table 1.**Descriptive statistics of universal multifractal (UM) parameters in the Ebro River Basin for 1931–2009.

Variable | Minimum | Median | Maximum | Mean | Std. Dev. | Coeff. of Variation |
---|---|---|---|---|---|---|

C_{1} 1st period | 0.192 | 0.314 | 0.434 | 0.314 | 0.058 | 18.590 |

C_{1} 2nd period | 0.213 | 0.324 | 0.457 | 0.330 | 0.059 | 17.847 |

α 1st period | 0.545 | 0.725 | 1.052 | 0.739 | 0.103 | 13.906 |

α 2nd period | 0.523 | 0.706 | 0.917 | 0.705 | 0.088 | 12.495 |

γ_{s} 1st period | 0.487 | 0.612 | 0.712 | 0.608 | 0.055 | 9.075 |

γ_{s} 2nd period | 0.487 | 0.622 | 0.738 | 0.618 | 0.056 | 9.118 |

_{1}values, which suggests a higher frequency of dry periods at several time scales. This increase seems especially important in the northwest region of the basin and, eventually, in the areas with higher annual average rainfall (in the north area of the basin) or with more frequent draught events (in the southeast area of the basin). In general, γ

_{s}shows a much more moderate increase. However, the α index exhibits opposite behavior and decreases over most of the watershed. These graphical findings were further contrasted by conducting rigorous statistical tests.

#### 3.4. Temporal Evolution of the UM Parameters

_{1}, and that 73 areas exhibited a significant positive evolution (+). The 55 remaining areas did not show any significant statistically evolution (0) of intermittency. The evolution of the α index was the opposite; that is, the α index generally decreased. However, this decrease was observed in fewer areas (34 in concrete) and was less intense (higher significant p-values). Only nine areas showed increasing α index values, and most of the basin areas (89 out of 132) did not show any statistically significant temporal evolution. Although Figure 5 suggests a slight increment in the γ

_{s}parameter in most areas of the Ebro River Basin, the temporal comparison statistical test did not confirm this assumption because no single area of the watershed resulted in statistically significant evolution (increasing or decreasing).

**Figure 6.**Statistically significant variations, for a significance level α = 0.05, of the three UM parameters (C

_{1}, α, and γ

_{s}) from the periods 1931–1975 to 1965–2009.

#### 3.5. Spatial Analysis of Rainfall Patterns

_{1}and α UM parameters could result from the spatial distribution along the Ebro River Basin, with potential information about the rainfall patterns in the watershed. To discover such information, the generalized Cramer-Von Mises non-parametric statistical test was performed in the basin over six different gradients. The target was to look for spatial patterns that could explain the significant differences found in the evolution of both UM parameters on the watershed. The test was applied using three possible temporal evolutions of UM parameters for rainfall processes (increase (+), decrease (–), and not significant (0)) by comparing the spatial distributions of the possible evolutions two by two; i.e., + versus –, + versus 0, and 0 versus –. The results of each comparison were considered statistically significant when their p-values were below 0.05.

_{1}and α) using a cumulative criterion west–east (Ψ

_{W–E}) and south–north (Ψ

_{S–N}) gradients (see Section 2.6). A west–east component evolution was detected for the C

_{1}and α UM parameters. Considering, i.e., the first row of the C

_{1}UM parameter and west-east gradient, Table 2 shows that areas with decreasing evolution (–) for the intermittency C

_{1}are distributed along the west–east axis in a different way compared with the areas with increasing evolution (+), at a p-value of 0.046 (<0.05); while considering i.e., the third row of such UM parameter and gradient, the areas with decreasing evolution (–) for C

_{1}are distributed along the west–east axis in a different way compared with the areas with no significant evolution (0), at a p-value of 0.018 (<0.05). These results are consistent with the fact that the four areas with negative evolution are located at a very specific longitude in the midwest area of the watershed (see Figure 6) and in the Aragón River valley before reaching the Ebro River. When studying the spatial distribution of the α UM parameter, significant differences can be observed (as shown in Table 2) along the west–east axis between the areas with a negative evolution and the areas that did not show significant differences, which included most of the areas (p-value 0.033). Furthermore, the spatial cumulative distributions shown in Figure 7a graphically show that these significant differences are due to a larger number of areas with decreasing evolution in the west area of the watershed. Thus, it can be observed that among the 50 most western areas of the basin, more than 60 percent of areas with decreasing evolution were accumulated and only near 30 percent of the areas with no significant evolution were accumulated. In contrast, the south–north axis showed no significant differences in the distributions of the evolutions of the UM parameters. Figure 7b graphically illustrates the absence of significant differences along the south–north axis among the three spatial distributions for α.

**Table 2.**Generalized Cramer-Von Mises statistic (Ψ) and the p-values used to study the UM parameters (C

_{1}and α) using a cumulative criterion west–east (Ψ

_{W–E}) and south–north (Ψ

_{S–N}) gradient, as shown in Figure 2a,b.

Pairwise Comparison | C_{1} | α | ||||||
---|---|---|---|---|---|---|---|---|

South–North | West–East | South–North | West–East | |||||

Ψ_{S–N} | p-value | Ψ_{W–E} | p-value | Ψ_{S–N} | p-value | Ψ_{W–E} | p-value | |

+ versus – | 3.261 | 0.667 | 14.997 | 0.046 * | 2.155 | 0.606 | 0.895 | 0.930 |

+ versus 0 | 1.410 | 0.170 | 1.097 | 0.182 | 1.541 | 0.677 | 2.666 | 0.399 |

0 versus – | 4.796 | 0.491 | 20.867 | 0.018 * | 1.418 | 0.269 | 3.783 | 0.033 * |

_{1}and α UM parameters. With respect to the intermittency C

_{1}, this table indicates that the distribution of the areas with significant decreasing evolution (–) differs from the distribution of the areas that do not exhibited significant evolution (0) when considering the distance to the main river basin axis (p-value 0.030) as its proximity to the river mouth (p-value 0.049). These results correspond with the fact that only four areas with significant decreasing evolution were detected in the watershed and that these areas were located far from the river mouth and next to the main river basin axis. However, significant differences in the distributions of the areas regarding the evolutions of the α Levy index were not detected for either of these gradients.

**Figure 7.**Spatial cumulative distributions of statistically significant evolutions for the α UM parameter considering different gradients: (

**a**) longitude gradient; (

**b**) latitude gradient; and (

**c**) altitude gradient.

**Table 3.**Generalized Cramer-Von Mises statistic (Ψ) and the p-values used to study the UM parameters (C

_{1}and α) using a cumulative criterion based on the progress of the main river basin (Ψ

_{P}) and the distance to the main river mouth (Ψ

_{M}) gradient, as shown in Figure 2c.

Pairwise Comparison | C_{1} | α | ||||||
---|---|---|---|---|---|---|---|---|

Projection | Mouth | Projection | Mouth | |||||

Ψ_{P} | p-value | Ψ_{M} | p-value | Ψ_{P} | p-value | Ψ_{M} | p-value | |

+ versus – | 11.349 | 0.111 | 10.428 | 0.139 | 9.299 | 0.062 | 1.328 | 0.828 |

+ versus 0 | 1.6248 | 0.077 | 1.566 | 0.089 | 5.533 | 0.124 | 3.389 | 0.292 |

0 versus – | 18.454 | 0.030 * | 15.893 | 0.049 * | 1.296 | 0.305 | 2.997 | 0.061 |

_{1}were not detected in either of the gradients.

**Table 4.**Generalized Cramer-Von Mises statistic (Ψ) and the p-values used to study the UM parameters (C

_{1}and α) using a cumulative criterion based on rainfall average (Ψ

_{R}) and the altitude (Ψ

_{A}) gradient of the river basin.

Pairwise Comparison | C_{1} | α | ||||||
---|---|---|---|---|---|---|---|---|

Mean Rainfall | Altitude | Mean Rainfall | Altitude | |||||

Ψ_{R} | p-value | Ψ_{A} | p-value | Ψ_{R} | p-value | Ψ_{A} | p-value | |

+ versus – | 10.633 | 0.130 | 13.440 | 0.075 | 3.996 | 0.302 | 7.700 | 0.102 |

+ versus 0 | 0.235 | 0.846 | 0.629 | 0.392 | 3.791 | 0.234 | 10.982 | 0.019 * |

0 versus – | 8.529 | 0.201 | 10.503 | 0.142 | 1.976 | 0.155 | 0.540 | 0.723 |

#### 3.6. Discussion

_{1}and α UM parameters obtained by these authors (C

_{1}= 0.25, α = 0.89) and the values of C

_{1}and α found in the Pyrenees region of the Ebro River Basin in this study (most C

_{1}values are less than 0.27 and most α values are between 0.79 and 1.05; Figure 5 and Table 1). Douglas et al. [57] found mean values for C

_{1}and α of 0.48 and 0.5, respectively, in a wide region in the Midwestern and Southeastern United States by using 327 daily time-series. These differences, with respect to the resultant values in the Ebro River Basin, occurred because the region studied by these authors includes a greater number of arid areas. This result is also supported by the study of De Lima et al. [58], who studied a semi-arid region of Portugal and obtained values of C

_{1}= 0.51 and α = 0.49, which were very similar to those of Douglas et al. [57]. Due to the results of the aforementioned references, it appears that the rainfall pattern in the Pyrenees region of the Ebro River Basin will be more similar to other semi-arid regions around the world if the value of C

_{1}gradually increases and the value of α decreases.

_{1}increased at more than half of the stations (reaching values above those obtained in the present study), the parameter γ

_{s}showed a general slightly increasing pattern at 12 of the 14 stations with values that were very similar to the values of the square geographical areas containing the analyzed stations. Thus, it seems that the γ

_{s}parameter best reflects the evolution of the multifractal properties of rainfall distribution. However, this issue should be tested in further studies that consider different length in the time series.

_{s}parameter reaches the higher values in the southeast of the Ebro River Basin (Figure 5) because this is the region of the watershed where major differences between the maximum values and the average values in rainfall events exist. Thus, the southeast region of the watershed has the greatest expected multifractality. This result indicates that the parameter γ

_{s}could be used as an indicator for detecting climate change from a multifractal perspective [24,49].

_{s}, a slight increase in γ

_{s}was found due to the temporal evolution of the C

_{1}and α parameters and by considering the higher increase in C

_{1}with respect to the decrease in α; however, this increase in γ

_{s}was not statistically significant (Figure 6). That is, the hypotheses suggested graphically in Figure 5 regarding γ

_{s}were not fully confirmed. Therefore, this result only provides a few small hints regarding the existence of climate change. On the other hand, the temporal evolution observed for the three UM parameters is in line with the results obtained by Royer et al. [24] on simulated data under a climate change scenario in France.

_{1}occurred in nearly 2/3 of the areas in which the Ebro River Basin was divided and a significant decrease in α was shown in 1/3 of the areas, we attempted to explain why these changes occurred in specific areas rather than throughout the watershed. That is, the research focused on determining which gradients (factors) were more sensitive to substantial changes in the UM parameters. Using the available data, spatial patterns, which can explain such differences, were searched for based on six gradients. From the results (Table 2, Table 3 and Table 4, and Figure 6), it was concluded that the evolution of C

_{1}is generally not influenced by any of the studied spatial factors because no spatial patterns were detected; in addition, the not significance in some areas may be explained by the typical deviations relatively large of the UM parameter estimations. The value of C

_{1}only decreased in a very small region located in the midwest region of the Ebro River Basin, with some specific characteristics that have not yet been discovered. To explain this unusual fact, the microclimatic factors affecting this particular region, which is located in a depression of the Aragon River valley, should be controlled. In summary, regarding the C

_{1}parameter, it is concluded that a generalized increase distributed throughout the watershed occurred that suggests an increase in the concentration of rainfall in the last decades over a smaller number of days in the Ebro River Basin [59]. On the other hand, regarding the α parameter, it was found evidence that the more easterly areas and the areas with more altitude appeared more resistant to significant changes (Table 2 and Table 4, Figure 6 and Figure 7a,c). Therefore, it is preferably in these areas in where it has remained so much the variability of rainfall at daily scale as the number of days with rainfall, because α is highly sensitive to this number and increases and decreases with such a number [59].

## 4. Conclusions

_{1}, α, and γ

_{s}UM parameters—and non-parametric statistical techniques, such as the generalized Cramer-Von Mises test. This methodological approach allows for the reproduction of the scale invariance and intermittency of rainfall processes and allows spatial distribution comparisons to determine the sensitivity of statistically significant factors that hydrologically characterize the watershed in response to the effects of climate change in the rainfall process. To apply this method experimentally, a daily rainfall dataset of 132 time series dating from 1931 to 2009 that was subdivided (1931–1975 and 1965–2009), spatially and homogeneously distributed across a grid of 25 km × 25 km over the entire watershed, was used. The following main conclusions can be extracted from the analytical results:

- 1)
- Most of the geographical areas in the Ebro River Basin exhibited a slight decrease in the mean annual rainfall over the period 1931–2009, although such a decrease is not statistically significant;
- 2)
- The watershed shows a spatial characterization of the C
_{1}and γ_{s}UM parameters that is symmetric with respect to the main river basin axis and increases in the river direction; - 3)
- The evolution of the intermittency C
_{1}over the last decades has preferably and homogeneously increased in the watershed, with a higher concentration of rainfall occurring over fewer days; - 4)
- In several areas, especially those located in the western area of the watershed, the α index has decreased in recent decades; accordingly, dry periods and rainfall process variability have increased in this zone;
- 5)
- The γ
_{s}UM parameter gradually, smoothly, and homogeneously increased in the watershed, although this temporal increase is not statistically significant. Since γ_{s}reacts positively to climate change effects in cases of increasing aridity, this result suggests that the effects of climate change have already begun to be perceived in the rain collected by the Ebro River Basin, although they are still very moderate.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Valencia, J.L.; Tarquis, A.M.; Saa, A.; Villeta, M.; Gascó, J.M.
Spatial Modeling of Rainfall Patterns over the Ebro River Basin Using Multifractality and Non-Parametric Statistical Techniques. *Water* **2015**, *7*, 6204-6227.
https://doi.org/10.3390/w7116204

**AMA Style**

Valencia JL, Tarquis AM, Saa A, Villeta M, Gascó JM.
Spatial Modeling of Rainfall Patterns over the Ebro River Basin Using Multifractality and Non-Parametric Statistical Techniques. *Water*. 2015; 7(11):6204-6227.
https://doi.org/10.3390/w7116204

**Chicago/Turabian Style**

Valencia, José L., Ana M. Tarquis, Antonio Saa, María Villeta, and José M. Gascó.
2015. "Spatial Modeling of Rainfall Patterns over the Ebro River Basin Using Multifractality and Non-Parametric Statistical Techniques" *Water* 7, no. 11: 6204-6227.
https://doi.org/10.3390/w7116204