# Evaluation of Karst Spring Discharge Response Using Time-Scale-Based Methods for a Mediterranean Basin of Northern Algeria

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## Abstract

**:**

## 1. Introduction

- Understand the internal structure of aquifers, storage capacity and obtain information about periodic characteristics.
- Classify the karst systems of the basin based on memory effect and regularization time.
- Study the random process of daily hydroclimatic time series using statistical fractals.
- Assess the causal links and linearity between rainfall and runoff for each sub-basin of the study area.
- Extract the significant coherence and covariance through isolated components developed between rainfall and runoff time series at a time-scale domain and identify dry and wet periods as well as anthropological impacts on the daily streamflow of the Sebaou River basin.

## 2. Study Area and Database

^{2}. Its elevations reach more than 2030 m above sea level (Figure 1b).

## 3. Materials and Methods

#### 3.1. Correlation and Spectral Analysis

#### 3.1.1. Simple Analysis

_{k}(k = 0, 1, 2, 3, …, m) are the autocorrelation coefficients obtained. According to Box and Jenkins (1976) [26], the choice of truncation (m) is not based on theoretical concepts, but it can be set as m = N/2, m = N/3 or m = 2N/3. The following expression gives r

_{k}:

#### 3.1.2. Cross-Analysis

- Cross correlograms

_{t}and Y

_{t}of N observations [23,25,57]. When ${r}_{+k}\ne {r}_{-k}$, this explains that the cross-correlation function is not symmetric:

- Cross spectrum

#### 3.2. Cross Wavelet Transform

_{n}) and runoff (Y

_{n}) is defined by the cross-wavelet power spectrum W

^{xy}= W

^{x}W

^{y*}, where (*) explains the conjugate complex W

^{xy}, and is given as follows [58,59]:

_{v}(P) is the significance level for the probability (P) density function. For XWT, the user must be aware that a coefficient of XWT can be high because the wavelet power spectrum of the two signals is high [60].

#### 3.3. Wavelet Coherence Transform

## 4. Results and Discussion

#### 4.1. Overview of the Rainfall Trends

#### 4.2. Univariate Correlation and Spectrum Analysis

#### 4.3. Cross Analyses

- -
- A transmissive function corresponding to the peak explains a well-developed drainage.
- -
- A slower decrease explains the capacitive effect.

^{−1}, the covariance becomes negligible, with the exception of some frequencies lower than 0.2, which have some characteristic peaks. According to El Hakim (2005) [77], the high covariance indicates that these periods correspond to a good cause-and-effect relationship. The amplitude function shows that the impulse response is valid for low and medium frequencies, especially for the A’N boubhir and Rabta sub-basins. This confirms the low inertia behavior of these systems (Figure 8a).

^{3}/year).

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Spatial analysis of (

**a**) the average intra-annual rainfall and (

**b**) coefficient of variation between 1972–2010 for the study area.

**Figure 3.**Representation of daily time series of rainfall (Ben Yenni station) and streamflow (RN30 station) in Aïssi River sub-basin (1985–2010).

**Figure 6.**Noise analysis of (

**a**) rainfall time series (Ait Aicha, Beni Yenni and Dra’a El Mizen) and (

**b**) different runoff time series of Sebaou River basin based on log-log representation spectra.

**Figure 7.**Cross correlogram of rainfall-runoff relationship of the main sub-basin of Sebaou River basin (windows of 125 days).

**Figure 8.**(

**a**) Cross amplitude function, (

**b**) phase function, (

**c**) gain function, and (

**d**) coherence function of the rainfall-runoff relationship of the main sub-basin of the Sebaou River basin (windows of 125 days).

**Figure 9.**Cross wavelet spectra (XWT) of daily rainfall and streamflow of (

**a**) Acsif N’Boubhir, (

**b**) Sebaou Rabta, and (

**c**) Aïssi sub-basins and wavelet coherence spectra (WCT) of daily rainfall and streamflow of (

**d**) Acsif N’Boubhir, (

**e**) Sebaou Rabta, and (

**f**) Aïssi sub-basins (the dark line denotes the “cone of influence”, where edge effects become important).

**Figure 10.**Representation analyses of (

**a**) cross wavelet spectra, (

**b**) square real part coherence spectra, and (

**c**) phase function spectra between daily rainfall and streamflow for three representative sub-basins of the Sebaou River for the different isolation components of short-, medium- and long-term processes.

**Figure 11.**Some pictures regarding the impacts and modifications of anthropological activities due to the extraction of large amounts of sand and rock from the riverbed of the Sebaou River.

Rainfall Station/ANRH ID | Runoff Station/ANRH ID | River Names | Sub-Basin/ANRH Code | Area of the Sub-Basin in ha | Study Periods |
---|---|---|---|---|---|

Ait Aicha 02-15-09 | Boubhir 02-15-13 | Boubhir | Acif N’boubhir 02-15 | 53,830 | 1987–2011 |

Fréha 02-16-03 | Fréha 02-16-05 | Dis | Sebaou Rabta 02-16 | 43,330 | 1987–2002 |

Benni Yenni 02-17-12 | RN 30 02-17-15 | Aïssi | Aïssi 02-17 | 47,020 | 1985–2010 |

Tizi Ouzou 02-18-10 | Belloua 02-18-03 | Sebaou | Sebaou Sebt 02-18 | 30,630 | 1987–1998 |

DEM 02-19-02 | RN 25 02-19-09 | Boughdoura | Boughdoura 02-19 | 53,450 | 1973–1993 |

Baghlia 02-20-02 | Baghlia 02-20-01 | Sebaou | Sebaou Maritime 02-20 | 22,890 | 1964–1998 |

**Table 2.**Classification of six karstic systems of Sebaou River basin and some representative Mediterranean karstic system based on CSA of daily flow.

Author | Region | Karstic System | Memory Effect (Day) (r _{k} = 0.1–0.2) | Spectral Band: Cutoff Frequency | Regularization Time (Day) |
---|---|---|---|---|---|

Mangin (1984) [27] | Pyrenees (France) | Aliou | Poor (5 days) | Very large (0.3) | 10–15 |

Baget | Small (10–15 days) | large (0.2) | 20–30 | ||

Fontestorbes | Large (50–60 days) | narrow (0,1) | 50 | ||

Torcal | Extensive (70 days) | Very narrow (0.05) | 70 | ||

Bouchaou (1995) [68] | The pleated Middle Atlas (Maroc) | Asserdoune | Extensive (70–80 days) | Very narrow (0.04–0.05) | 70–80 |

Larocque et al. (1997) [69] | Western France | Rochfoucauld | — | — | 76 |

Amraoui et al. (2004) [54] | The tabular Middle Atlas (Maroc) | Bittit | Large (37–45 days) | Very Wide | 35 |

Ribaa | Extensive (70 days) | large (0.14) | 57 | ||

Chettih and Mesbah (2010) [70] | Saharan Atlas (Algeria) | Seklafa | 2.5 | 0.4 | 1.5 |

Kerakda | 3.5 | ||||

Rhouiba | 4 | ||||

Bouanani (2004) [71] | basin Tafna (Western Algeria) | Sebdou | Small | — | 5 |

Mouilah | Large | 0.025 | 21 | ||

Isser | Extensive | 0.018 | 43 | ||

This work | Sebaou River (Algeria) | Boughdoura River | Reduced (18 days) | large (0.21) | 20–30 |

Aïssi River | Extensive (53–84 days) | Very narrow (0.032) | 50 | ||

Acif N’boubhir | Poor (9 days) | Large (0.22) | 15 | ||

Sebaou Sebt River | Small (16 days) | Large (0.19) | 20–30 | ||

Sebaou Rabta River | Poor (3 days) | Very large (0.44) | 5 | ||

Sebaou maritime River | Extensive (66 days) | Very narrow (0.067) | 60 |

Time Series | Stations | Period | Slope (β1) | Scale Invariance Ranges | Slope (β2) | Scale Invariance Ranges |
---|---|---|---|---|---|---|

Daily rainfall (mm/day) | Tizi Ouzou | 1990–2009 | −0.21 | 14 days–1 year | −0.66 | 1–13.5 days |

Ait Aicha | 1972–1991 | −0.15 | 9 days–1 year | −1.10 | 1–8.5 days | |

1991–2010 | −0.32 | 11 days–1 year | −1.03 | 10–13 days | ||

DEM | 1967–1988 | −0.26 | 16 days–1 year | −0.82 | 1–15 days | |

1988–2010 | −0.002 | 16 days–1 year | −0.88 | 1–15 days | ||

Freha | 1972–1991 | −0.27 | 10 days–1 year | −0.89 | 1–9 days | |

1991–2010 | 0.07 | 11 days–1 year | −0.88 | 1–10 days | ||

Beni Yenni | 1972–1991 | −0.09 | 10 days–1 year | −1.10 | 1–9 days | |

1991–2010 | −0.10 | 11 days–1 year | −0.73 | 1–10 days | ||

Daily runoff (m^{3}/s) | Belloua | 1949–1958 | −0.26 | 11days–1 year | −1.25 | 1–10 days |

1972–1983 | −0.22 | 12 days–1 year | −1.14 | 1–11 days | ||

1987–2000 | −0.37 | 12 days–1 year | −2.98 | 1–11 days | ||

Baghlia | 1963–1985 | −0.32 | 12 days–1 year | −2.85 | 1–13 days | |

1985–1997 | −0.01 | 13 days–1 year | −2.24 | 1–12 days | ||

Freha | 1986–2001 | −0.28 | 20 days–1 year | −1.60 | 1–19 days | |

Boubhir | 1987–2002 | −0.13 | 13 days–1 year | −1.45 | 1–12.5 days | |

RN25 | 1973–1994 | −0.75 | 14 days–1 year | −2.21 | 1–15 days | |

RN30 | 1985–1998 | −0.48 | 20 days–1 year | −2.43 | 1–19 days | |

1998–2010 | −0.39 | 30 days–1 year | −1.61 | 1–29 days |

**Table 4.**Extracted significant correlations between daily rainfall and streamflow time series using XWT and WCT.

Spectral Bands | Years with Significant Correlations Based on cross Wavelet Spectra (XWT) Analysis | |||||

Acif N’boubhir River | Aïssi River | Boughdoura River | Sebaou Rabta River | Sebaou Sebt River | Sebaou Maritime River(Outlet) | |

6–8 month | 1998, 2000 2006, 2007 | 1997, 2007 | 1974, 1976 | 1993, 1995, 1998, 2000 | 1994–1995 | 1974, 1978, 1983, 1987 1989 |

1 year | 1987–2010 | 1986–2002 2002–2010 | 1974–1994 | 1989–2001 | 1990–1997 | 1974–1998 |

1–3 year | 1996–2004 | 1996–2004 | 1983–1991 | 1990–1999 | 1993 1994–1997 | 1975–1981 1986–1993 |

3–6 year | 1996–2005 | 1997–2005 | — | 1995–1999 | 1994–1997 | 1979–1983 |

6–8 year | 1996–2004 | 1993–2004 | 1981 | — | — | 1984–1987 |

8–12 year | 1996–2004 | 1996–2001 | — | — | — | — |

Spectral Bands | Years with Significant Correlations Based on Wavelet Coherence Spectra | |||||

6–8 month | 1988–2000 | 1986 | 1977, 1979, 1981, 1984, 1987, 1990 | 1988–1990 1992–1994 1998–2001 | 1991, 1997 | 1973, 1976 1983, 1985 |

1 year | 1988–2010 | 1991–1999 | 1974–1976 1979–1994 | 1988–2001 | 1990–2000 | 1973–1975 1979–1993 |

1–3 year | 1988–2010 | 2003–2010 | — | 1988–2001 | — | 1987–1993 |

3–6 year | 1995–1999 | — | 1978–1990 | 1992–1999 | — | — |

6–8 year | 1996–2004 | 1999–2004 | — | — | — | 1984–1987 |

8–12 year | — | — | — | — | — | — |

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## Share and Cite

**MDPI and ACS Style**

Zerouali, B.; Chettih, M.; Alwetaishi, M.; Abda, Z.; Elbeltagi, A.; Augusto Guimarães Santos, C.; E. Hussein, E.
Evaluation of Karst Spring Discharge Response Using Time-Scale-Based Methods for a Mediterranean Basin of Northern Algeria. *Water* **2021**, *13*, 2946.
https://doi.org/10.3390/w13212946

**AMA Style**

Zerouali B, Chettih M, Alwetaishi M, Abda Z, Elbeltagi A, Augusto Guimarães Santos C, E. Hussein E.
Evaluation of Karst Spring Discharge Response Using Time-Scale-Based Methods for a Mediterranean Basin of Northern Algeria. *Water*. 2021; 13(21):2946.
https://doi.org/10.3390/w13212946

**Chicago/Turabian Style**

Zerouali, Bilel, Mohamed Chettih, Mamdooh Alwetaishi, Zaki Abda, Ahmed Elbeltagi, Celso Augusto Guimarães Santos, and Enas E. Hussein.
2021. "Evaluation of Karst Spring Discharge Response Using Time-Scale-Based Methods for a Mediterranean Basin of Northern Algeria" *Water* 13, no. 21: 2946.
https://doi.org/10.3390/w13212946