# Statistical and Fractal Approaches on Long Time-Series to Surface-Water/Groundwater Relationship Assessment: A Central Italy Alluvial Plain Case Study

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Study Area

^{−9}to 10

^{−7}m/s), below the more permeable alluvial deposits (k values ranging from 10

^{−5}to 10

^{−3}m/s) suggests the presence of shallow, single and multi-layered aquifers [29,30].

^{2}(Figure 1), which can be subdivided in two main parts: the first one is the western mountainous sector, and the second one is the eastern sector, where the alluvial aquifer under study is located. The Pescara River mean discharge measured in the Pescara Santa Teresa gauge (PST) is about 50 m

^{3}/s (31 m

^{3}/s comes from the western part of its basin).

## 3. Datasets

## 4. Methods

#### 4.1. Analyses in Time Domain

#### 4.1.1. Seasonal-Trend Decomposition Procedure Census I

#### 4.1.2. The Autocorrelation Function

#### 4.1.3. The Cross-Correlation Function

#### 4.2. Analyses in Frequency Domain

#### 4.3. Fractal Analysis

_{F}in Figures 13 and 15), the following example can be useful. Two rainfall time series are analyzed. The data were obtained by daily measurements over 30 years (10,920 values), at two gauges, A and B, representative of two different sites (A and B). We are interested to study how the occurrences of daily rainfall above or equal to 50 mm (F_threshold) are distributed over the selected interval of time. If we consider a time scale of one year, the entire interval is subdivided into 30 subintervals, each lasting one year. Accordingly, N

_{g}= 30 and $\epsilon $ = 365, as resulting from the application of the Cantor Dust method. Suppose that for 40 times (n-events = 40) the intensity of the rainfall, measured at A and B stations exceeded the value of the selected F_threshold over, respectively, 2 and 22 different years, even non-continuously. Consequently, N(ε) = 2 and N(ε) = 22, respectively, for time series A and B.

_{g}= 60 and $\epsilon $ = 182). Moreover, assume that, as the time scale decreases, N(ε) = 2 remains the same for Station A, while N(ε) = 37, increases (thus almost 1 for each semester) for Station B. Repeating this operation by the means of the Cantor Dust approach, the resulting value of the D fractal dimension related to the time series A would be low (in this case close to 0), while for the other time series, D would be close to 1.

## 5. Results

#### 5.1. Descriptive Statistics

#### 5.2. The Residual Analysis

#### 5.3. Autocorrelation

#### 5.4. Raw Data Cross-Correlation

#### 5.5. Residual Cross-Correlation

#### 5.6. Spectral Analysis

#### 5.7. Cantor-Dust Fractal Analysis

## 6. Discussion

#### 6.1. Analyses in Time Domain

#### 6.2. Analyses in Frequency Domain

#### 6.3. Fractal Analysis

#### 6.4. Conceptual Model

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**(

**Top**) The study area; and (

**Bottom**) the hydrogeological framework and the monitoring network are shown (see Table 1 for initials and abbreviations).

**Figure 2.**Schematic geolithological section with a 10× vertical exaggeration (Cross section 1–1′ in Figure 1).

**Figure 3.**Morphologic and hydraulic characteristics of the river profile assessed in this study: (

**A**) distribution of the river width; (

**B**) plan view of the elevation of the river water profile (GPS survey, April 2015); and (

**C**) elevation profile of the river water head.

**Figure 5.**Rainfall (

**A**); river level (

**B**); and piezometric level (

**C**) for 1986–2009 three-day time series.

**Figure 6.**Monthly average: for rainfall and hydrometric level (

**A**); and for normalized piezometric level (

**B**), 1986–2009 period.

**Figure 8.**Autocorrelation functions: rainfall-river level (

**A**); and piezometric levels (

**B**). The dash-dotted line indicates the decorrelation threshold. The initials and the abbreviations refer to Table 1.

**Figure 9.**Cross-Correlations Function: rainfall–piezometric level (

**A**); and river level–piezometric level (

**B**). The dash-dotted lines indicate the significant thresholds.

**Figure 10.**Residual Cross-Correlation Function: rainfall–hydrometric level. The dash-dotted lines indicate the significant thresholds.

**Figure 11.**Residual Cross-Correlation Functions: rainfall–piezometric level (

**A**); and river level–piezometric level (

**B**). The dash-dotted lines indicate the significant thresholds.

**Figure 12.**Spectral density functions: rainfall and hydrometry levels (

**A**); and piezometric levels (

**B**).

**Figure 13.**Rainfall and river level outcomes; fractal dimensions related to different F_threshold events; rainfall F_threshold > 40%: D

_{F}$\approx 0.00$; river level F_threshold < 5%: D

_{F}$\approx 0.9999$.

**Figure 14.**Su, Sa, DN outcomes related to piezometric levels; Fractal dimensions related to different F_threshold events.

**Figure 16.**Comparison of borehole hydraulic head changes and the distribution of river level elevations. The borehole piezometric heads, which vary most frequently between the 25th and 75th percentile, correspond to the river bed elevations along the blue sections of the river profile.

**Figure 17.**Refined conceptual model of groundwater flow paths and groundwater/surface water relationships.

Name | ID | Type | Elevation (m a.s.l.) | Distance from River (m) | Distance from Sea (km) |
---|---|---|---|---|---|

Spoltore | Sp | Rain gauge | 165.00 | 2650 | 4.4 |

Surricchio | Su | Well | 18.10 | 250 | 8.3 |

Sanità | Sa | Well | 8.68 | 170 | 4.1 |

De Nicola | DN | Well | 11.68 | 1960 | 3.7 |

Pescara S. Teresa | PST | Hydrometer | 4.51 | 0 | 7.0 |

**Table 2.**Average and standard deviation of the piezometric level, rainfall and hydrometric level time series data.

Type | Name | Piezometric Level (m a.s.l.) | Water Depth (m) | |||||

Average | St. dev. | Min | Max | Average | ||||

Well | Su | 12.06 | 1.42 | 9.29 | 16.28 | 6.70 | ||

Sa | 5.49 | 0.43 | 4.41 | 7.28 | 3.89 | |||

DN | 10.46 | 0.19 | 9.85 | 11.18 | 2.02 | |||

Type | Name | Rainfall (mm) | ||||||

Average | St. dev. | Min | Max | |||||

Rain Gauge | Sp | 5.35 | 11.9 | 0 | 199.2 | |||

Type | Name | Hydrometric Level (m a.s.l.) | ||||||

Average | St. dev. | Min | Max | |||||

Hydrometer (m) | PST | 3.33 | 0.37 | 2.74 | 7.97 |

**Table 3.**Details of the Fractal analysis; d: days, w: week, m: months, y: years, F_threshold: the actual level (m) corresponding to the selected F_threshold ${f}_{i,j}(\%)$ level (percentage of the whole variability range).

Selected F_threshold Level | Sp Rainfall | PST Hydrometry | Su Well | Sa Well | DN Well |
---|---|---|---|---|---|

Range = | (0–199.2) (mm) | (2.74–7.97) (m) | (9.29–16.28) (m) | (4.41–7.28) (m) | (9.85–11.18) (m) |

L_thresh = min + 0.5 × range/100 (m) = | 0.001 | 2.77 | 9.33 | 4.42 | 9.86 |

Fluctuation (m) = | 0.001 | 0.03 | 0.04 | 0.01 | 0.01 |

n_events: f 0.5% =_{i,j} ≥ F_threshold = | 1170 | 2876 | 2877 | 2879 | 2879 |

percentage (%) = | 1170/2880 = 40.6 | 99.86 | 99.89 | 99.96 | 99.97 |

distribution return time (DRT) (d) = | 17 (2 w) | 3 | 3 | 3 | 3 |

L_thresh = min + 1.0 × range/100 (m) = | 0.002 | 2.79 | 9.36 | 4.44 | 9.86 |

Fluctuation (m) = | 0.002 | 0.05 | 0.07 | 0.03 | 0.01 |

n_events::f 1.0% =_{i,j} ≥ F_threshold = | 998 | 2860 | 2874 | 2879 | 2879 |

percentage (%) = | 34.7 | 99.31 | 99.79 | 99.96 | 99.97 |

DRT (d) = | 30 (1 m) | 3 | 3 | 3 | 3 |

L_thresh = min + 2.5 × range/100 (m) = | 0.005 | 2.87 | 9.47 | 4.48 | 9.88 |

Fluctuation (m) = | 0.005 | 0.13 | 0.18 | 0.07 | 0.03 |

n_events: f 2.5% =_{i,j} ≥ F_threshold = | 730 | 2792 | 2862 | 2872 | 2878 |

percentage (%) = | 25.3 | 96.94 | 99.38 | 99.72 | 99.93 |

DRT (d) = | 47 (1.7 m) | 3 | 3 | 3 | 3 |

L_thresh = min + 5.0 × range/100 (m) = | 0.01 | 3.00 | 9.64 | 4.55 | 9.92 |

Fluctuation (m) = | 0.01 | 0.26 | 0.31 | 0.14 | 0.07 |

n_events::f 5% =_{i,j} ≥ F_threshold = | 474 | 2491 | 2794 | 2866 | 2876 |

percentage (%) = | 6.5 | 86.49 | 97.01 | 99.51 | 99.86 |

DRT (d) = | 71 (2.4 m) | 3 | 3 | 3 | 3 |

L_thresh = min + 10 × range/100 (m) = | 0.02 | 3.26 | 9.99 | 4.70 | 9.98 |

Fluctuation (m) = | 0.02 | 0.52 | 0.7 | 0.29 | 0.13 |

n_events: f 10% =_{i,j} ≥ F_threshold = | 252 | 1472 | 2684 | 2798 | 2861 |

percentage (%) = | 8.8 | 51.11 | 93.19 | 97.15 | 99.34 |

DRT (d) = | 107 (3.6 m) | 95 (3.2 m) | 3 | 3 | 3 |

L_thresh = min + 20 × range/100 (m) = | 0.04 | 3.79 | 10.69 | 4.99 | 10.12 |

Fluctuation (m) = | 0.04 | 1.05 | 1.4 | 0.58 | 0.27 |

n_events: f 20% =_{i,j} ≥ F_threshold = | 74 | 218 | 2313 | 2671 | 2837 |

percentage (%) = | 2.6 | 7.569 | 80.31 | 92.74 | 98.51 |

DRT (d) = | 361 (12 m) | 428 (14.3 m) | 69 (2.3 m) | 3 | 3 |

L_thresh =min + 30 × range/100 (m) = | 0.06 | 4.32 | 11.40 | 5.27 | 10.25 |

Fluctuation (m) = | 0.06 | 1.58 | 2.11 | 0.86 | 0.40 |

n_events: f 30% =_{i,j} ≥ F_threshold = | 26 | 47 | 1824 | 1947 | 2618 |

percentage (%) = | 0.9 | 1.632 | 63.33 | 67.60 | 90.9 |

DRT (d) = | 727 (24 m) | 572 (19 m) | 176 (5.9 m) | 130 (4.3 m) | 30 |

L_thresh = min + 40 × range/100 (m) = | 0.08 | 4.84 | 12.10 | 5.56 | 10.38 |

Fluctuation (m) = | 0.08 | 2.1 | 2.81 | 1.15 | 0.53 |

n_events: f 40% =_{i,j} ≥ F_threshold = | 8 | 17 | 1382 | 1139 | 1859 |

percentage (%) = | 0.28 | 0.590 | 47.98 | 39.55 | 64.55 |

DRT (d) = | 1995 (67 m) | 861 (29 m) | 589 (19.6 m) | 360 (12 m) | 282 |

L_thresh = min + 60 × range/100 (m) = | 0.12 | 5.88 | 13.48 | 6.13 | 10.65 |

Fluctuation (m) = | 0.12 | 3.14 | 4.19 | 1.72 | 0.8 |

n_events: f 60% =_{i,j} ≥ F_threshold = | 1 | 7 | 518 | 216 | 457 |

percentage (%) = | 0.035 | 0.243 | 17.98 | 7.5 | 15.87 |

DRT (d) = | 8640 (24 y) | 1392 (46 m) | 764 (25.5 m) | 1272 (42 m) | 727 (24 m) |

L_thresh = min + 80 × range/100 (m) = | 0.160 | 6.92 | 14.88 | 6.71 | 10.91 |

Fluctuation (m) = | 0.160 | 4.18 | 5.59 | 2.3 | 1.06 |

n_events: f 80% =_{i,j} ≥ F_threshold = | 1 | 3 | 58 | 35 | 63 |

percentage (%) = | 0.035 | 0.104 | 2.014 | 1.215 | 2.19 |

DRT (d) = | 8640 (24 y) | 2694 (7.5 y) | 2694 (7.5 y) | 2860 (7.9 y) | 8640 (24 y) |

L_thresh = min + 90 × range/100 (m) = | 0.180 | 7.45 | 15.58 | 6.99 | 11.05 |

Fluctuation (m) = | 0.180 | 4.71 | 6.29 | 2.58 | 1.20 |

n_events: f 90% =_{i,j} ≥ F_threshold = | 1 | 2 | 9 | 13 | 21 |

percentage (%) = | 0.035 | 0.069 | 0.313 | 0.451 | 0.73 |

DRT (d) = | 8640 (24 y) | 4310 (12 y) | 3636 (10 y) | 4186 (11.5 y) | 8640 (24 y) |

© 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**

Chiaudani, A.; Di Curzio, D.; Palmucci, W.; Pasculli, A.; Polemio, M.; Rusi, S.
Statistical and Fractal Approaches on Long Time-Series to Surface-Water/Groundwater Relationship Assessment: A Central Italy Alluvial Plain Case Study. *Water* **2017**, *9*, 850.
https://doi.org/10.3390/w9110850

**AMA Style**

Chiaudani A, Di Curzio D, Palmucci W, Pasculli A, Polemio M, Rusi S.
Statistical and Fractal Approaches on Long Time-Series to Surface-Water/Groundwater Relationship Assessment: A Central Italy Alluvial Plain Case Study. *Water*. 2017; 9(11):850.
https://doi.org/10.3390/w9110850

**Chicago/Turabian Style**

Chiaudani, Alessandro, Diego Di Curzio, William Palmucci, Antonio Pasculli, Maurizio Polemio, and Sergio Rusi.
2017. "Statistical and Fractal Approaches on Long Time-Series to Surface-Water/Groundwater Relationship Assessment: A Central Italy Alluvial Plain Case Study" *Water* 9, no. 11: 850.
https://doi.org/10.3390/w9110850