Contribution of cross time-frequency analysis in assessment of possible relationships between large-scale climatic fluctuations and rainfall of northern central Algeria

Abstract

This paper is proposed for the investigation of possible relationships between the large-scale atmospheric circulation phenomena such as the North Atlantic Oscillation (NAO), Southern Oscillation (SOI), Mediterranean Oscillation (MO), Western Mediterranean Oscillation (WeMO) and rainfall of Sebaou river watershed (Northern central Algeria), covering a period of 39 years at monthly scale. Several time and scale-based methods were used: correlation and spectral analysis (CSA), continuous wavelet transform (CWT), multiresolution wavelet analysis (MRWA), cross wavelet analysis (XWT), wavelet coherence transform (WCT) and cross multiresolution wavelet analysis (CMRWA). The rainfall analysis by CSA and CWT has been clearly demonstrating the dominance of 1 year and 1–3-year modes, which they explain 30 to 51% and 25 to 28% of the variance respectively. However, the indices have shown that inter-annual fluctuations up to long-term explain between 60 and 90%. CWT and MRWA indicated significant fluctuations materialising a dry period more marked between the 1980s and 1990s with strong trend towards drier conditions starting from the 1980s, explained by the decadal components D7 and the approximation A7. In addition to the annual component, the XWT spectrums reveal strong coefficients for the SOI between 1992–2005 and 1986–2000 for the modes of 5–10 years and higher than 10 years respectively and less intense for NAO. The WCT between NAO and rainfall indicated the most significant relationship for 1 year, 1–3 years and 3–5 years approximately from the early 1980s corresponding to the dry period. However, the SOI affects rainfall only locally and with significant values more or less localised in the time-frequency space between MO, WeMO and rainfall, but this influence could be significant for low-frequency events. CWMRA shows that the components of 5–10 years and higher than 10 years are the most effective to represent climate index-rainfall significant relationships, where change in Daubechies wavelet properties can improve the correlation across the scales. Furthermore, has indicated that the short-term processes dominate the relationship index-rainfall, which masks the long-term phenomena whose influence can sometimes be very distant. As such, the rainfall variability of the study area has shown fairly significant links, at least locally with large-scale atmospheric circulation phenomena.

Introduction

The knowledge of spatiotemporal changes of meteorological variables is a key and essential element for efficient water resources planning and management. These changes have become complex, difficult to manage and to control, particularly under the direct effects of human and his socio-economic activities, climate changes and large-scale climatic fluctuations that are sometimes responsible for the appearance of extreme events and natural disasters as floods, droughts and desertification, which are increasingly rampant around the world (Mühlbauer et al. 2016; Pozo-Vázquez et al. 2001).

In its fifth assessment report about regional changes in precipitation (Intergovernmental Panel on Climate Change (IPCC), 2007), the Intergovernmental Panel on Climate Change (IPCC) indicated a significant decrease in rainfall in several parts of the world over the last five decades, when the droughts will be more intense and more frequently for longer periods in the upcoming years in some regions, particularly around the Mediterranean basin, the centre of North America and Southern Europe. Furthermore, Sun et al. (2014) indicated that the southern part of Europe has become drier since the 1980s. From then, Ongoma and Chen (2017) analysed the spatial-temporal changes of temperature and precipitation over East Africa from 1951 to 2010 using Sequential Mann–Kendall (MK) test. The results indicated a decrease in annual rainfall and an increase in temperature was recorded in the 1960s. Hertig and Tramblay (2017) have developed a procedure of statistical downscaling for analysing the future climate change based on RCP 8 scenario through the Mediterranean area. The results showed a downward trend in monthly rainfall between 1970 and 2000 and substantial increase in drought severity and frequency to the horizon 2100. In addition, in the Mediterranean basin, a number of studies have been carried out showing a downward trend in rainfall, we will cite among others the works of (Elouissi et al. 2017; Valdes-Abellan et al. 2017; Soldini and Darvini 2017; Achite and Ouillon 2016). On the other hand, Da Silva et al. (2015) have applied nonparametric tests of MK and Sen Slope estimator to study spatio-temporal trends in hydrometeorological variables at annual scale at Cobres River basin (southern Portugal). The results showed a very high variation in the time and the space with decreasing trend during the period (1960–2000). The statistical analysis of precipitation of Marche region (Italy) using the MK test shows a negative trend in annual mean rainfall anomalies since the mid-1980s compared to the climatological reference period 1961–1990, with an increase over the period 2009–2013 (Soldini and Darvini 2017).

However, local changes in meteorological variables are mainly influenced by the large atmospheric circulation as El Niño-Southern Oscillation (ENSO), Western Mediterranean Oscillation (WEMO) and North Atlantique Oscillation (NAO), Atlantic Multidecadal Oscillation (AMO) as pointed out by Massei et al. (2017); Pomposi et al. (2016); Martín Vide et al. (2008); Brunetti et al. (2002). These phenomena can offer real support for understanding the physical variability ocean-atmosphere coupled system (Mateescu and Haidu 2006). The ENSO phenomenon characterises the interaction between the large-scale processes in the atmosphere and the Equatorial Pacific Ocean; its quantification is based on two main indexes, the first one is the Southern Oscillation Index (SOI) and the other one is the Sea Surface Temperature (SST). The main regional mechanisms contributing to the production of rainfall in the world are closely associated, statistically and physically with the changes themselves of the ENSO (Xu et al. 2004). Pomposi et al. (2016) clearly showed the covariance of the ENSO with African Sahel rainfall and it is the main cause of inter-annual variability. Pozo-Vazquez et al. (2001) indicated that the appearance of extreme hydrological events in the North Atlantic-European region can be attributed to the ENSO phenomenon, especially during winter months. The study of Mariotti et al. (2002) has shown that ENSO significantly influenced on inter-annual rainfall variability of the Euro-Mediterranean sector, particularly in Central and Eastern Europe during winter and spring seasons. According to Hurrell and van Loon (1997) the NAO phenomena, it should be considered among the main sources of the inter-annual variability of weather and climate. Several researches have documented that the rainfall of western Mediterranean region is strongly conditioned by NAO phases (Vergni et al. 2016; Turki et al. 2016). NAO is responsible for significant climate anomalies at the North Atlantic, mainly in winter, greatly affects the temperature (Zamrane et al. 2016), the precipitation at the regional level (Marchane et al. 2016; Ferrari et al. 2013), and is responsible for major ecological impacts on the different resources of the Mediterranean basin like birds migration (Gordo et al. 2011). In mainland Portugal during 1941–2007, De Lima et al. (2015) observed that the positive phase of the NAO was significantly related to the downward trend of daily rainfall index. On the other hand, the WeMO and MO are among the newly large-scale climatic fluctuations characterising the atmospheric circulation in the Mediterranean basin (Zeroual et al. 2017) and they also have a very significant environmental impact. For example, Angulo-Martínez and Beguería (2012) have observed in Ebro basin (NE of Spain), during 1955–2006, that the erosive force of rainfall is stronger during the negative phases of the NAO, WeMO and MO and it is lower during the positive phases. In Morocco, Filahi et al. (2016) have observed a stronger relationship between NAO, MO, WEMO and climate indices of precipitation compared to temperature indices, particularly in the Atlantic region. The approach of previous work on the Mediterranean basin showed that the spatio-temporal rainfall variability is closely associated with coupled modes of the atmospheric circulation configurations as NAO, MO and WeMO (Beranová and Kyselý 2016). Brunetti et al. (2002) reported a good correlation between MO index and total precipitation, especially with the number of rainy days in Italy. V icente-Serrano et al. (2009) observed that the negative WeMO events are responsible for producing the extreme daily precipitation during the winter months.

In Algeria, some studies performed during the recent years in the framework of climate change and rainfall analysis (Elouissi et al. 2017; Zeroual et al. 2017; Elmeddahi et al. 2014; Hamlaoui-Moulai et al. 2013) reveal the existence of two distinct periods. The first wet started from the mid-1940s to the mid-1970s and probably until the early 1980s. Then, a second dry period has been established in the country since the late 1970s until 1990s (Belarbi et al. 2017; Achite and Ouillon 2016; Megnounif and Ghenim 2016; Meddi et al. 2014). This period has been marked by significant rainfall decrease in the North of Algeria. (Taibi et al. 2017; Zeroual et al. 2017), particularly in the northern part close to the Mediterranean Sea (Elouissi et al. 2016; Elmeddahi et al. 2014). However, the nature and specificity of the hydrological signals and more specifically in arid and semi-arid regions suggest specific analytical tools in order to highlight the possible relationships between climate indices and hydroclimatological signals. According to Rodríguez-Iturbe (1991), Torrence and Compo (1998) and Labat et al. (2000), taking into account the non-linear, non-stationary and multi-scale nature of hydrological signals, the joint analysis “time-scale” appears as the preferred tool to fulfil the extraction functions, quantifying and linking information obtained at different scales. This type of analysis has already been carried out in meridional region of Algeria, where, it has allowed a more detailed and finer analysis (Chettih and Mesbah 2010). So far, no study has been carried the subject of cross time-scale analysis between climate indices and precipitation in Algeria in order to assess their possible links. As such, the paper proposes in this note to analyse the possible relationships between the climate indexes NAO, SOI, WeMO, MO and some representative rainfall series of Sebaou river watershed located in the North central of Algeria and this by using:

• 1. CSA based on simple and cross functions are used to find the cyclical components and the existence of causal links between variables at frequency space.

• 2. CWT applied for characterising stationary, non-stationary and periodic phenomena, as well as the identification of wet and dry periods.

• 3. WMRA used to confirm CWT results and to highlight short to long-term phenomena presented in the analysed signals and study more finely the multi-scale response.

• 4. XWT and WCT are used to extract the significant coherence and covariance between indices-rainfall time series at frequency-time space through common power spectrum.

• 5. CWMRA to determine the correlations and the degree of causality between each detail index and detail rainfall, approximation index and approximation rainfall, scale by scale for different Daubechies wavelets in order to highlight the main periods relating the dry and wet periods that have affected the study area.

Area of study

The watershed of Sebaou river, its position between latitudes North 36° 30 and 37° 00 and longitudes East 03° 30 and 04° 30, is part of coastal watersheds Algiers code (02) according to the nomenclature of the Algerian National Agency of Water Resources (Fig. 1). It extends over an area of 2500 km² limited in the north by the Mediterranean coastal, in the south by the Djurdjura mountain chain (wilaya of Bouira), in the East by the massif Akfadou and Beni -Ghobri (wilaya of Bejaia) and in the west by the mountains of Sidi Ali and Jebel Bounab Bouberak (wilaya of Boumerdes).

Fig. 1

Geographical location and Digital Elevation Model of Sebaou river watershed

In general, the climate is affected by various factors such as geography, topography, distance from the sea and atmospheric circulation. The climate of Mediterranean and study area are characterised by warm to hot, dry summers, and mild to cool, wet winters, and are classified in the sub-humid bioclimatological stage with alternation of two seasons during the year, a wet season that starts in October and ends at the end of May. Moreover, a dry season that begins at the end of May and ends at September. Northwest winds, meeting Screen Iberian chains (Spain) and one of the Moroccan Atlas Mountains arriving dry on the North-West of Algeria, then recharge on Mediterranean to provide heavy rainfall on the Northern Tell. For this rain decline from North to South superimposes a growth from Western to the East with longitude, this law being particular in Algeria, which is why the Kabylie (Tizi Ouzou, Bejaia and Bouira) is one of most watered areas of Northern Algeria (Meddour 2010).

Materials and methods

Materials

Data

The rainfall series database collected as part of the study includes 23 stations (Fig. 2), the operated stations are those with longest and do not present any gaps for the period 1972 to 2010. Rainfall data used in this study are taken from the Algerian National Agency of Water Resources.

Fig. 2

Selected (a) rainfall and (b) indices time series used for the analysis

North Atlantic Oscillation

Sir Gilbert Walker has discovered the North Atlantic Oscillation (NAO) for the first time in the years since 1920,;it characterises the atmospheric circulation of the Northern hemisphere. It is highlighted by the difference of normalised sea level pressure between the Azores high (35.1° N, 5.3° W) and the Icelandic low (65° N, 20° W) (Hurrell and van Loon 1997). The NAO is the principal mode of climate variability in winter season especially in northern Europe and has two phases namely a positive phase and a negative one; each phase is responsible for the distinct atmospheric conditions around the North Atlantic. In the case of positive NAO, there is a strong western current generating multiple depressions towards Northern Europe. Nevertheless, in the case of negative NAO, the pressure current is less intense and moves more to the South (Rossi et al. 2011).

Southern Oscillations

Walker and Bliss (1932) have proposed the term Southern Oscillation (SO) to describe a large-scale phenomenon. The SO characterises the difference in atmospheric pressure between the Eastern (Tahiti) and Western (Drawin) of the tropical Pacific Ocean. In the month of November, the pressure is higher in the East of Tahiti than in the West; this pressure difference along the equator generates a circulation of air masses towards the West and provides positive phase called “La Niña”, which is the opposite of “El Niño” (negative phase). In addition, the combination between the Southern Oscillation and El Niño gives the ENSO phenomenon.

Western Mediterranean Oscillation

The WeMO index proposed by Martin-Vide and Lopez-Bustins (2006), it is defined as the difference of the normalised surface pressures between San Fernando (Spain) (36.3° N, 6.1° W) and Padua (Italy) (45° N, 10° E) stations. It is characterised in its positive phase by the strengthening of Azores high and its extension on the southwestern quarter of the Iberian Peninsula, associated with low pressure in the Gulf of Ligure inducing a North-west and southeast general circulation of surface air masses. However, the negative index plays a reverse role.

Mediterranean Oscillation

Conte et al. (1989) and Palutikof et al. (1996) are the first researchers who have spoken about “Mediterranean Oscillation (MO)”. The MO is one of the phenomena describing characterising atmospheric circulation at the scale of the Mediterranean basin and is defined as the difference of normalised pressures between Algiers (Algeria) (36.4 ° N, 3.1 ° E) and Cairo (Egypt) (30.1 ° N, 31.4 ° E) at 500 hPa geopotential height. The monthly values of NAO, MO, SOI are obtained from Climatic Research Unit, University of East Anglia (https://crudata.uea.ac.uk). However, the WeMO is obtained from the Group of Climatology-University of Barcelona (http://www.ub.edu/gc).

Methods

Correlation and spectral analysis

The correlation and spectral analysis been developed by Jenkins and Watts (1968) and Box and Jenkins (1976). This theory makes the objective of identifying, simulating random processes and the nature of sequential relationships of variables based on the concepts of cross-correlation in time scale and cross-spectrum in frequency scale. The simple correlation analysis highlights the dependence of successive events for increasing time intervals, it based on the autocorrelation function for a time series of N observations.

Cross Analysis is a statistical tool used to examine the relationship between two time series; it provides two types of information: the degree of correlation depending on the time lag and the causal relationship between two series. Whereas, the cross spectrum is the Fourier transform of the cross correlogram. It corresponds to the decomposition of the covariance between inputs and outputs in the frequency domain. This technique is widely used in the analysis of behaviours and characteristics of hydrogeological and hydrological systems (Mangin 1984; Larocque et al. 1998). The reader will find sufficient detail and reference about the correlation and spectral analysis in the works of Jenkins and Watts (1968); Box and Jenkins (1976); Mangin (1984) and Larocque et al. (1998).

Continuous wavelet transform

Continuous wavelet transform is an analysis method; its applications are becoming more diverse, first introduced by the French geophysicist Jean Morlet for seismic signals analysis in the early 1980s (Morlet et al. 1982).

According to Anctil and Pelletier (2011), generally, the practical goal of the experimenter is a key element in wavelet type selection, so it is preferable to choose a wavelet whose shape of its convolution is similar to the fluctuations of the time series used. Which is the case of the Morlet wavelet that traces peaks and troughs in the manner of a wave, this feature is compatible with the hydroclimatic series fluctuations (Anctil and Pelletier 2011).

The continuous wavelet transform W_n(s) for a discrete series of X_n observations with a time interval between two observations equal δ t is defined by the convolution of X_n with a wavelet Ψ₀(η) consists of a complex wave modulated by a Gaussian function. W_n(s) and Ψ₀(η) are represented as follows respectively:

$$ {W}_n(s)=\sqrt{\frac{\delta t}{s}\sum \limits_{n=1}^N{x}_{n^{\prime }}}{\Psi}^{\ast}\left[\left({n}^{\prime }-n\right)\frac{\delta t}{s}\right] $$

(1)

$$ {\Psi}_0\left(\eta \right)={\pi}^{-0.25}{e}^{i{\omega}_0\eta }{e}^{-0.5{\eta}^2} $$

(2)

Knowing that, (*) is the complex conjugate, s is the scale parameter of the wavelet linked to frequency, n is a translation factor or time location. ω₀ is a dimensionless frequency, the value of ω₀ equal to 6 is a good choice for the Morlet wavelet to satisfy the admissibility condition because it provides a good stability between time and frequency localization and η is a dimensionless parameter (Torrence and Compo 1998, Grinsted et al. 2004). |W_n(s)|² defined the wavelet power spectrum, This function describes the fluctuations in the time series variance for selected spectral band (Farge 1992, Torrence and Compo 1998, Grinsted et al. 2004).

Discrete wavelet transform

According to Daubechies 1992, the discrete wavelet transform (DWT) is based on the decomposition of the original signal of data in a series of discrete functions of varying levels, the assembly which reconstructs the original signal. DWT of Daubechies (db wavelets) is one of the most usable transformations in the analysis of the variance of high-frequency phenomena of time series (Daubechies 1992). It is also seen as a process of decomposition of the signal in approximation and details (Mallat 1989), where, the approximation (A) corresponds to the high scale this means low-frequency components of the signal and the details (D) are corresponding to low scales, it is the high-frequency components. DWT has the form of:

$$ {\psi}_{m,n}\left(\frac{t-\tau }{s}\right)={s_0}^{-m/2}\kern0.5em \psi \kern0.5em \left(\frac{t-n{\tau}_0{s}_0^m}{s_0^m}\right) $$

(3)

where; ψ is the mother wavelet s is the scaling factor (dilation) while the parameter τ called translation factor (in time). m and n are responsible for the operations of dilation and translation respectively (m and n are integers), t is an independent value, s₀ is a specified fixed dilation step greater than 1 and t₀ is the position parameter (t₀> 0). In the case of m = 2 and n = 1, we express the dyadic wavelet transform, this is the most efficient case for practical reasons (Daubechies 1990).

For a discrete time series (x₁, x₂, x₃, .…, x_n), where x_i is observed at discrete time i, the DWT becomes:

$$ {W}_{m,n}={2}^{-m/2}\sum \limits_{i=0}^{N-1}{x}_i\kern0.5em \psi \kern0.5em \left({2}^{-m}\kern0.5em i-n\right) $$

(4)

where W_{m, n} is the wavelet coefficient for the discrete wavelet of scale s = 2^m and position. τ = 2^mn. So, DWT informs change in an input chronological series at various scales and position. For more details about DWT and its opposite version, lecture is invited to consult the papers of Mallat (1989), Daubechies (1990, 1992) and Labat et al. (2000, 2002).

Cross wavelet transform

Cross wavelet transform (XWT) is developed for the purpose of comparing two time series in order to evaluate the similarities between them far and to identify the degree of correlation through a cross-spectrum. The XWT between two time series X_n and Y_n is defined asW^xy = W^x W^y∗, where (*) is the conjugate complex and |W^xy| is the cross wavelet power spectrum given by:

$$ D\kern0.5em \left(\frac{\left|{W}_n^X(s){W}_n^{X\ast }(s)\kern0.5em \right|}{\delta_X{\delta}_Y}<p\right)\kern0.5em =\kern0.5em \frac{Z_{\nu }(p)}{\nu}\sqrt{P_k^X{P}_k^Y} $$

(5)

Where:

$$ {P}_k=\frac{1-{\alpha}^2}{{\left|1-\alpha \kern0.1em {e}^{-2 i\pi k}\right|}^2} $$

(6)

P_k is the Fourier power spectrum of a first-order autoregressive process (AR1) with autocorrelation αof lag-1 (${P}_k^X$ calculated for X_n of variance δ_X and ${P}_k^Y$ calculated for Y_n of variance δ_Y. Z_ν(p) is the significance level related to the probability p for the probability density function (PDF). In the case: ν = 2 and the significance level of 5%; Z_ν(p) will be Z₂(95%) = 3.999.

Wavelet coherence transform

In this approach, the coherence is measured between two wavelet cross transforms in space (time-frequency). The wavelet coherence function is defined as follows (Torrence and Webster 1998; Grinsted et al. 2004):

$$ {R}_n^2(s)=\kern0.5em \frac{{\left|S\left({s}^{-1}{W}_n^{XY}(s)\right)\kern0.5em \right|}^2}{S\left({\left|{s}^{-1}\Big({W}_n^X(s)\right|}^2\right)\kern0.5em .\kern0.5em S\left({\left|{s}^{-1}\left({W}_n^Y(s)\right)\right|}^2\right)} $$

(7)

Where S represents the smoothing operator in time and scale. Theoretical description for CWT, XWT and WTC is discussed in detail by Torrence and Compo (1998), Torrence and Webster (1998), and Grinsted et al. (2004).

Cross multiresolution wavelet analysis

Cross multiresolution analyses in the form of a multiresolution wavelet co-spectrum T_xy (m) can be defined between two discrete signals x (i)and y (i) of length 2^m at the scale mby the following expression:

$$ {T}_{xy}\kern0.1em (m)=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${2}^{M-n}$}\right.\kern0.1em \sum \limits_{n=1}^{2^{M-m}}\left|{D}_{m,n}^x\kern0.1em {D}_{m,n}^y\right| $$

(8)

It should be noticed that the signal variance of the spectra (T_x (m), T_y (m))and cospectra T_xy (m) can be normalised (Labat et al. 2000).

Results and discussion

In this study, two types of analysis have performed, the first univariate analysis was applied to rainfall and climatic indexes in order to characterise the main modes of variability, wet periods, dry periods, as well as the stationary and non-stationary events. The second cross analysis identified possible causal relationships between rainfall and index variables at space time-scale over the study period of 39 years.

Univariate analysis

Correlation and spectral analysis

A first analysis focused on rainfall series (stations: Aghrib, Dra’a El Mizen, Nezlioua and Beni Yenni) (Fig. 2a) and monthly atmospheric circulation indices NAO, SOI, WeMO and MO (Fig. 2b). The goal of this analysis is to study and characterise the distribution of precipitation and to check the random nature. The simple analysis is done first in the short and long term for a one-month sampling step and the truncation was chosen as 120 months, in order to take into account the detail and highlighting of annual and multiannual cycles (Mangin 1984). In this study, we opted the Turkey filter in spectral analysis.

Rainfall analysis clearly highlights the annual cycles (Fig. 3a), and this is by relatively high correlation peaks every 12 months. This result is also emphasised by a spectacular peak on the variance density spectrum for the frequency equal to 0.0833 corresponding to 12 months (Fig. 3b). Furthermore, we note the presence of other weakly presented peaks at a frequency of approximately 0.16 corresponding to 6 months. As such, it appears that the climate of the Sebaou river basin is characterised by two seasons of equal periods quite distinct: a relatively rainy season and a dry season. The rainfall of Sebaou river basin could be considered as a short-term structured process.

Fig. 3

Rainfall (a) correlogram and (b) spectral density, (c) correlogram and (d) spectral density of indices

However, the correlograms of the indices NAO, SOI, MO and WeMO show the annual cycles but of a lower magnitude in comparison with those of rainfall (Fig. 3c). This is probably due to the short-term non-stationarity of these phenomena. According to Meurisse-Fort (2009), their periodicity is still difficult to determine, each phase lasting indifferently for one or several years. In contrast, the variance density spectrum (Fig. 3d) is well presented by characteristic peaks particularly for the annual cycle, as well as other short- and long-term peaks. The SOI indicates a peak of quasi- annual periodicity of 9-month corresponding to frequency f = 0.1167 and a peak that expresses the decadal cycle corresponding to the frequency f of 0.0083. The WeMO reveals a single relatively sharp peak corresponding to the annual component. Spectrum analysis of NAO and MOi indices shows several peaks of high-energy intensity well distributed approximately on the same frequency bands. It is also possible to observe moderately high peaks on the same frequency bands corresponding to low frequencies of multi-annual periodicity. Characteristic peaks of high frequency corresponding to 6 months and 3 to 4 months relative to semiannual and seasonal periodicity. These observations require cross-analyses between these phenomena in order to understand the interactions and the coherences between them. According to correlation and spectral analyses, we can say that climate indices and the rainfall of the study area can be considered sufficiently well-structured process.

Continuous wavelet analysis

Continuous wavelet analysis (CWT) has been applied to the monthly rainfall and climatic indexes time series presented in Fig. 2 to identify the main modes of variability, stationary, non-stationary, periodic events, as well as the wet and the dry periods. The cone of influence covers only from 12 to 16 years of data on the 39 years taken for analysis. It is therefore easy to know if the high variance power at the scale, monthly, seasonal, annual, multi-annual and decennial, is stable over the period of study or not.

Figure 4 shows the local wavelets power spectrum of rainfall and climatic indexes, precipitation power spectrum indicates the existence of several fluctuations on the difference time scales. They highlight the existence of short-term non-stationary structures (Fig. 5), where the seasonal cycle weakly structured shows a very low contribution that does not exceed 12% on total variance of rainfall (Fig. 6). Despite that, the wavelet power spectrum provides clear evidence of the annual fluctuations marked at level of all stations. This annual cycle is also well expressed by the global wavelet spectrum (GWS) (Fig. 5), explains for between 30 and 51% of the total variance (Fig. 7). However, the interannual mode of (1–3 year) is considered as the second dominant mode in the intra-annual rainfall variability because it explains approximately between 25 and 28% of the variance (Fig. 6).

Fig. 4

The continuous wavelet power spectrum for monthly rainfall and indices time series used for the analysis. The Dark red is the significant level at 5%. The dark line denote the “cone of influence,” where edge effects become important

Fig. 5

The global wavelet powers spectrum of rainfall and indices (the red line is the 95% confidence line for a red-noise background)

Fig. 6

Contribution of time scale bands on total variance of rainfall and indices in percentage

Fig. 7

The scale-average wavelet power for 8–16 month band of rainfall and indices (The red line marks the 95% confidence level)

Oued Sebaou river watershed is also characterised by other modes of variability on a larger scale. At the interannual scale, fluctuations (2–4 years) are temporally localised, appear quite visible for all four stations. According to Ozger et al. (2011): the significance of these large-scale phenomena indicates the existence of different rainfall generating mechanisms, but, they are marked by a rupture and blue colour in the spectrum (Fig. 4) between the late 1970s until the mid-1990s more marked at the station Dra’a El Mizen, which seems disrupted their continuity. At the multiannual scale (3–6 years), the fluctuations appear discontinuously separated by the rupture 1980–1993, which corresponds to the relatively dry period. These different interannuals up to long-term, which are more or less localised in time, represent a part of the cumulative total variance comprised between 36 and 69% across the basin (Fig. 6). These observations probably can be explain that cyclic components are responsible for the appearance of wet periods in the watershed.

For indexes series, the wavelet spectrum (Fig. 4) shows the existence of structures more or less continuous for medium and high frequencies. The seasonal and annual cycles are slightly structured showing a lower contribution not exceeding 16% of the total variance of the indices; for NAO, these structures constituting singularities in a discontinuous manner between 1988 and 2000, whereas the SOI, they are only locally presented between 1980–1984 and 1999–2002. For the WeMO and MO, the annual component presents in a sparsely manner between 1975 and 2000. At the interannual scale, the fluctuations (1–3 years, 2–4 years and 3–6 years) have not clearly expressed; they are randomly organised over time and it explains that 7–18% of total variance (Fig. 6). The large-scale circulation is characterised by other modes of variability on a larger scale. At the interannual scale, a multi-year component (6–7 years) is particularly visible for the NAO and SOI, which gradually passes towards decadal fluctuation more or less continuous, significant within the Cone of influence (Fig. 4) represents 40 to 60% of the total variance especially for NAO, SOI and WeMO (Fig. 6). Nevertheless, as an observation on MO index, a clear rupture at about the year 2000, this rupture is spreading deeply in the ranges of scales inducing a low variability for almost all components (Fig. 4). The decadal component (8–16 year) appears to be more attenuated but they are always visible in the GWS (Fig. 5). These different interannual fluctuations up to long-term, which are more or less localised in time, represent a part of the cumulative total variance comprised between 60 and 90%, further confirming the long-term processes of these phenomena.

From the continuous wavelet spectrum (Fig. 4), the scale average wavelet power of the annual component (8–16 months) has been isolated and extracted to examine its variance in the space time and analyse its climatic behaviour. Figure 7 represents the evolution of rainfall and climate indices variance. According to Santos and Morais (2013), an increase in the power or amplification of the curve corresponding to wet periods and a decrease in the power or attenuation of the curve corresponding to the dry periods compared to the significance level of 95%. On the annual scale, for 8–16 month band, the exceptionally rainy years exceeding the 95% confidence level: 1973, 1981, 1984, 1986, 1994, 1998, 2001, 2002, 2003, 2004 and 2005. In contrast, the dry period is marked with low variability of rainfall variance below the threshold significance level between 1976–1985 and 1987–2001, with the exception, Beni Yenni station which indicated a decrease of the variance from the year 1974 with some non-significant amplifications. These observations can confirm the deficit or a significant negative rainfall trend affecting the Sebaou river basin and particularly its northern part from the mid-1970s and early 1980s that has also been observed in Algeria in the example of the work of Elouissi et al. (2017), Elouissi et al. (2016), Taibi et al. (2017), Zeroual et al. (2017), Megnounif and Ghenim (2016), Achite and Ouillon (2016), Elmeddahi et al. (2014) and Meddi et al. (2014). Furthermore, the scale average wavelet power analysis of spectral band 8–16 months of climatic indices (Fig. 7) shows an increase in NAO variance between the 1980s and 1990s and a decrease from the year 2000, followed by a net and significant increase between 2006 and 2008. According to Hamlaoui-Moulai et al. 2013 and Ouachani et al. 2013, the NAO entered into positive phase from the middle of 1970 to 2000. The SOI index shows strong variability with a clear increase towards the end of the 1990s. The values of the MO index present a slight increase from 1980 to 1996 and a clear decrease in the variance from 1997. In contrast, the WeMO index indicates more or less regular fluctuations during the study period.

Multiresolution wavelet analysis

Multiresolution wavelet analysis (MRWA) allows us to highlight the various modes of variability present in the non-linear signal, from the modes of small time scales until the long-term modes of variability (Labat et al. 2002). During the last decade, several researches have documented and proven the effectiveness of the coupling between the discrete wavelet transform and the nonparametric statistical methods and the artificial intelligence systems in the analysis and forecasting of hydrological series (Djerbouai and Souag-Gamane 2016). In this part, a MRWA has been used as a complementary tool to continuous wavelet analysis to identify and extract more information about non-linear trends and climate responses present in the precipitation and the climate indices and make comparisons between them scale by scale.

The original series of precipitation and indices with 468 observations correspond to appropriate dyadic lengths of 2⁷ (7 scales). They have been decomposed into a series of detail D1 (mode 2 month), D2 (mode 4 month), D3 (mode 8 month), D4 (mode 16 month), D5 (mode 32 month), D6 (mode 64 month), D7 (mode 128 month) and into an approximation (A7) (Fig. 8). This decomposition was carried out in several consecutive steps of filtering based principally on Mallat algorithm (Mallat 1989) using mother wavelet of Daubechies (db).

Fig. 8

Multiresolution wavelet decomposition of (a) Aghrib, (b) Dra’a El Mizen, (c) Nezlioua, (d) Beni Yenni rainfall series and (e) NAO, (f) SOI, (g) MO and (h) WeMO indices series

The first observations from the analysis by periodic component or mode of variability (Fig. 8) indicate an inverse relationship between the level of decomposition and the component variability, when the scale increases variability decreases. In addition, the evolution and the distribution of the wavelets coefficients of rainfall over time appears as not uniformly distributed across the scales. The analysis allows to capture the variations of the signal at the different scales, consequently, a large and great variations involve high values for the coefficients at small scales (Labat et al. 2000; Chettih and Mesbah 2010); they are more pronounced for seasonal and semiannual variability modes from 2 to 8 months between D1 and D3. The distributions of the annual and multiannual component of 8–16 to 32–64 month from D4 to D6 present some periodic phenomena in time. The decadal components (D7) of 128 months indicate a quasi-alternation of wet and dry periods during the study period, this is clearly visible for all the series. The approximation component (A7) shows an interesting and strong trend towards a drier trend well illustrated at Dra’a El Mizen and Beni Yenni stations, this negative trend began in the mid-1980s. This analysis clearly revealed the alternation of three periods: the first wet (increasing) between the years 1970–1980, the second dry (decreasing) from the year 1980 to the middle of 1990s. The third, a quasi-humid period (increasing) compared to the previous beginning in the mid-1990s to the year 2005 (Fig. 8). This observation has already been observed using CWT (Fig. 7). On the other hand, the components (A7) are significant to show the signs of change in trend at multiannual and decadal periodicity (Partal 2017) and can explain the climate change hypothesis and its impacts on rainfall decreases around the world and particularly the Mediterranean basin, which remains a region more vulnerable to climate variability as already mentioned. Furthermore, the approximation series carries the greater part of the trend component (Joshi et al. 2016). According to Joshi et al. (2016), the interannual periodic components resulting from the multiresolution wavelet analysis play an important role in the detection and monitoring of climate trends. Our observations corresponding to the periodic components D7 and A7 are the best components that clearly explain the wet and dry trends who have had to influence our study area.

Concerning wavelet coefficients of climate indices (Fig. 8), they appear in an irregularly distributed manner and present a high variability of variance at short-term (D1, D2 and D3) compared with those of rainfall. In the medium term for the components D4, D5 and D6 can be detected some cyclical fluctuations more or less regular roughly comparable to those of the rainfall fluctuations, but, it is difficult to discern any relationship between rainfall and indices. In the long term, for (D7) component, the climatic trends are clearly highlighted corresponding globally for SOI to a relatively dry trend, absolutely in accordance with long-term rainfall trends particularly the period 1972 to 1991. However, the NAO, MO and WeMO behave in the long term in a reverse way in this period, probably indicating a distant influence. In general, on a large scale (D7 and A7), we observe a significant trend in phase and dephasing according to the scale of resolution selected for the indices and rainfall (Fig. 8). This finding will be further substantiated by CWMRA at different scales.

Cross spectrum analysis

The aim of cross-analysis in our study is seeking to highlight the causal relationship in time-frequency space between climatic indices and rainfall in the watershed of Oued Sebaou. In this section, the indices NAO, SOI, MO and WeMO are considered as an input variable representing the impulse and the rainfall as output variable representing impulse response.

Cross correlation

The cross correlogram highlights the possible causal relationship between climate indices and rainfall. It can provide an approximate vision of the impulse response of the system. Cross correlograms between NAO, SOI, WeMO and MO indices and rainfall (Fig. 9a) show overall a low correlation in its positive and negative parts with quasi-cyclic impulse responses particularly clearly distinguished for the NAO and rainfall.

Fig. 9

Effective index-rainfall (a) cross-correlation, (b) cross-amplitude function, (c) Phase function, (d) gain function and (d) coherence function

Cross-amplitude function

The cross-amplitude function for each pair gave the same behaviour for all stations of the basin used in this study. For the Aghrib station as an example, the cross-amplitude function between NAO, SOI, WeMO and MO indices and rainfall (Fig. 9b) shows a particularly important peak at a frequency of 0.083 for NAO and SOI, thus materialising the annual periodic phenomenon. As well, this frequency is also fairly well marked for the MO and WeMO but with relatively smaller amplitude. Moreover, we have noticed relatively low peaks for frequencies ranging from 0.133 to 0.166 representatives of the semiannual phenomena for NAO, SOI, MO and WeMO. However, the low frequencies phenomena are also timidly manifested for WeMO and SOI explaining some pluriannual phenomena respectively for frequencies 0.0042 and 0.0083.

Phase function

The phase shift of the relationship rainfall-index representative of the circulation time of information is calculated for the cut-off frequency (Fig. 9c). Phase functions indicate unstructured distributions on the frequency axes, with values oscillate between 2 and − 2. This may be explained by the complexity of the system behaviour.

Gain function

The gain function that reflects the way in which the large circulation phenomena influence the rainfall, shows well-illustrated amplifications by peaks for high frequencies for NAO and SOI (Fig. 9d). However, it shows a pretty bad attenuation for WeMO and MO. Furthermore, a characteristic peak highlights the amplification of low frequencies for the NAO and SOI at a frequency of 0.083 corresponding to the annual periodicity. We note also some interesting values for WeMO and rainfall at frequencies 0.15, 0.125 and 0.05.

Coherence function

The coherence values of each pair rainfall-index present a high variability and oscillate between 0.02 and 0.80 with an average of 0.43 relatively stable for the three modes of frequency (Fig. 9e). This might probably already indicate a non-linear causality relationship between index-rainfall. According to cross-correlation and cross-spectral analysis, there is a low correlation between indices and rainfall, nevertheless, for characteristic frequencies, the large-scale climatic fluctuations seem to influence the rainfall, but the causal relationship may remain possibly nonlinear. The alternative, time-scale analysis techniques of cross-wavelet transform can highlight this possible relationship.

Cross analysis in space time-scale

The large-scale climatic fluctuations such as SOI, NAO, MO and WEMO indicate their strong influence on rainfall of Europe and the Mediterranean basin. The subjects of this context are currently debated worldwide by the scientific community and the researchers, may be mentioned for example, the work of: Taibi et al. (2017); Beranová and Kyselý (2016); Turki et al. (2016); Filahi et al. (2016); De Lima et al. (2015); Ozger et al. (2009); Martín Vide et al. (2008) and Martin-Vide and Lopez-Bustins (2006). In this part, XWT, WCT and CWMRA have been applied in order to investigate and to assess the relationships that may exist between large-scale climatic fluctuations and precipitation in space time-frequency for the period 1972–2010.

Cross wavelet analysis

Figure 10 shows the cross wavelet spectrum (XWT) between the monthly indices and rainfall series. The XWT spectrums between the indices and the rainfall reveal an important significant common power for a significance level of 5% expressed by the dark red colour. Cross-spectrum (XWT) does not highlight the seasonal components of 4–8 month, except for a few isolated events temporally constituting singularities (Fig. 10). At the annual scale, in the spectral band of 8–16 months, presents a fairly continuous component over time with strong coefficients for all XWT spectrum. It should be noted, however, a brief discontinuity in the time approximately from 1976 to 1980 for NAO and SOI (Fig. 10). At the multiannual scale, in the spectral band of 64–128 months and higher than 128 month of 5 to 10 year between 1992–2005 and 1986–2000 respectively, there are strong coefficients well illustrated and in a continuous manner particularly for the SOI, but in a sparse and local manner with lower coefficients for NAO, MO and WeMO.

Fig. 10

Cross wavelet spectra (XWT) between monthly indices and rainfall time series. The thick black is significance level of 5%. Cone of Influence (COI) in which edge effects cannot be ignored

As in the case of a univariate analysis, a time average taken from the cross spectrum (Fig. 10) may give more immediate and more visible information on the different scales characteristics of significant correlations (Fig. 11). In addition, the percentage of contribution of each mode of variability was calculated from the global cross wavelet spectrum (GCWS) (Fig. 12).

• The contribution of seasonal and semi-annual modes of 2–8 months shows a small contribution not exceeding 16% of the total variance (Fig. 12), expressed by low amplification in the GCWS

• Annual processes (8–16 months) are very visible on the GCWS and seem the most dominant between each pair rainfall-index, this mode explains the largest part of the variance with a contribution of more than 50% for NAO, MO and WeMO and less than 30% for SOI (Fig. 12)

• The modes of multiannual variability of 16–32 and 32–64 months describing 1–5 y mode show less importance amplifications in comparison with those of the annual mode with a cumulative contribution range between 8 to 26% compared to the total variance (Fig. 12).

• The multiannual up to decennial components of 64–128 months characterising 5–10 y mode also shows obvious amplifications in the GCWS particularly for SOI-rainfall with a contribution exceeding 40% compared to the total variance (Fig. 12).

• The modes of variability more than 128 month (10 y) appear fairly important, the relationships express a maximum contribution of 20 and 50% compared to the total variance respectively for SOI and WeMO (Fig. 12).

Fig. 11

Global cross wavelet spectrum between (a) NAO, (b) SOI, (c) MO, (d) WeMO index and rainfall series in frequency space

Fig. 12

Contribution of cross time scale bands on total variance between indices and rainfall in percentage

As part of the cross wavelet analysis, the variability at temporal scale of the 12 month component has been also isolated and analysed (Fig. 13). The amplitude of the cross spectrum between NAO and rainfall, presents generally very considerably and significant fluctuations for the last 25 years. For the SOI and rainfall, coefficients with high magnitude are recorded in the period 1972–1976, followed by periods of high coefficients from 1980 to 1990 and 1995–2004. Similar pics more amplified are also observed between MO and rainfall. For the WeMO and rainfall, we notice primarily very high coefficients fluctuations at all stations in the periods 1972–1976, 1984–1988 and 2000–2004. These different periods characterised by high coefficients seem to show the possible links between the indices and the rainfall at least concerning the annual component.

Fig. 13

Cross wavelet spectrum between monthly (a) NAO, (b) SOI, (c) MO, (d) WeMO index and rainfalls series in time space for 12-month isolated component

Wavelet coherence analysis

Wavelet coherence analysis (WCT) was carried out between two cross wavelet spectrum index rainfall- (Fig. 14), a value equal to ‘1’ for a selected frequency band indicates that the response of the rainfall signal (output) is 100% linear with respect to the input signal (index).

Fig. 14

Wavelet coherence spectra (WCT) between monthly (a) NAO, (b) SOI, (c) MO, (d) WeMO index and rainfalls series. The thick black is significance level of 5%. Cone of Influence (COI) in which edge effects cannot be ignored

The WCT spectrum between NAO and rainfall shows a significant covariance observed on the seasonal spectral band of 4–8 months, which is discontinuous more or less localised in time. At the annual scale, from the spectral band between 8 and 16 months, the most significant coherence between the NAO and rainfall is evident in the second half of the observation period, and more specifically for the periods: 1983–1984, 1988–1996 and 1998–2003. At the multiannual scale, another period of strong and significant relationship was highlighted in the spectral band from 16 to 32 month with a transition to 32–64 month observed at all stations during the period 1995–2005. This result is consistent with the result of Turki et al. (2016) that have found a strong coherence between the NAO index and the rainfall of Marrakesh (Morocco) in particular to low frequency events. According to Ozger et al. (2009), the multiannual scales of 6–8 year are the better for drought occurrences in relationship with low-frequency phenomena.

On a small scale, the WCT indicates a strong covariance between SOI and rainfall (Fig. 14) but in a discontinuously and scattered manner observed in 1982, 1984, 1992, 1994, 2003 and 2006. At the annual scale, the coherence is less visible, it exists in a very discontinuous way and it is localised temporally by two small singularities after the year 2000. At the multiannual scale of 1–3 years between 16- and 32-month bands, the SOI affects rainfall only locally in the periods 1983–1987 and 2007–2008. For the modes 3–5 years (32–64 months), note a very high covariance in the period 1993–2004 with a transition of the covariance from the first scale to the second scale. The WCT between SOI and rainfall indicates a high coherence in the period 1972–2010 for the scales higher than 128 month (10 years), it is more significant within the cone of influence in the period 1986–2000.

The coherence between the MO and rainfall (Fig. 14) is well presented at small scale (4–8 month) and well materialised in the period 1976–1984, but elsewhere, it exists in a discontinuous way. The annual spectral band (8–16 months) is well illustrated after the year 2000, with high coefficients indicating the good linear relationship between the MO and rainfall, this is for all stations of the study. At the multiannual scale of 32–64 month, the coherence is less representative than at the annual scale observed between the years 1980 and 2000, it is well illustrated for Aghrib and Nezlioua stations. For the phenomena higher than 128 month, a fairly marked coherence within the cone of influence between 1985 and 1995.

The WCT spectrum between the WeMO and rainfall (Fig. 14) shows significant values of coherence but more or less localised in the time-frequency space. At the annual and multiannual scale of 8–16 and 16–32 month, there is a high coherence more marked between 1984 to 1992 and 2000 to 2006. The mode 3–5 years of the spectral band of 32–64 month shows good coherence observed around 1978–1984. The events higher than 128 month show a significant influence of WeMO on rainfall in the period 1986–2000 only for Aghrib and Nezlioua stations.

Similarly, the component of 12-month for the annual process was also isolated in this study using the concept of wavelet coherence, where Fig. 15a, b represents the square of the wavelet coherence (real part) and the phase of the coherence function respectively. For the selected mode, the analysis showed a moderately medium coherence about 0.4 (Fig. 15a). Figure 15a indicates an increase in the coherence between NAO-rainfall in the early 1980s, with maximum values within the range 0.8 to 1 for the periods 1992–1996, 1998–2004 and 2008. The highest values of the coherence correspond to consecutive passages of direct disturbance of the positive phase of NAO on rainfall, a “good” agreement is found, while during wet periods less agreement is highlighted. This observation has been also reported by Hamlaoui-Moulai et al. (2013); Ouachani et al. (2013). Mühlbauer et al. (2016) found that the positive phase of NAO recorded after 1975, corresponds to the period of the rainfall deficit for all plains of the Iberian Peninsula. Vergni et al. (2016) have observed in the region of Abruzzo (Italy) that the positive NAO is responsible for the main cases of negative trends, while the negative phase of NAO plays the opposite role. Sun et al. (2014) have shown that the southern part of Europe has become drier since 1980, and this is due to the positive phase of NAO. Coherence greater than 0.6 is observed between the input signal SOI and the rainfall signal more marked in the years: 1974, 1984, 1988, 2002 to 2004. The MO shows a high average coherence with rainfall of study area observed in the years: 1974, 1984 to 1992 and 2008 to 2010. In addition, the WeMO and rainfall revealed a discontinuity in the coherence spectrum with a greater consistency of 0.6 more pronounced in 1975, 1986 to 1988 and 2002 to 2004. The WeMO and rainfall showed a discontinuity in the coherence spectrum with a coherence exceeding 0.6 more pronounced in 1975, 1986 to 1988 and 2002 to 2004.

Fig. 15

(a) Coherence real part spectrum and (b) phase function spectrum between monthly NAO, SOI, MO, WeMO index and Beni Yenni rainfall in time space for 12-month isolated component

Overall, the phase function shows very high fluctuations (Fig. 15b). Furthermore, when the real part of the coherence increases, the phase of the coherence function decreases reflecting an inverse relationship. For example, the NAO-precipitation relationship oscillated around zero, between 0 and − 1.5 after the 1980s to the year 2006, this indicates a quasi-linear relationship between NAO-rainfall, after the year 2006, the phase begins to increase, which corresponds to a decrease in coherence, consequently, the relationship tends to non-linearity. The behaviour of the phase for MO and rainfall is quite similar to that of the NAO and rainfall. However, for the SOI and WeMO indices, the phase decreases during periods of good coherence shown in Fig. 15a. In a general way, the rainfall variability in the Sebaou river basin seems to show a locally significant correlation with large-scale phenomena that influence rainfall in the short and medium term and monitoring long-term trends. Cross-multiresolution wavelet analysis using Daubechies wavelets can identify the possible relationships scale by scale of resolution to confirm the existence of significant links between indices and rainfall.

Cross-multiresolution wavelet analysis

In this part, cross-multiresolution wavelet analysis (CMRWA) has been used to determine the correlations and the degree of causality between each detail-index and detail-rainfall, scale by scale up to the approximation for different wavelets of Daubechies. Labat et al. (2000) have successfully applied this technique in order to characterise the degree of linearity of Pyrenees karstic systems.

In the framework of this analysis, scales have been decomposed into details and approximations are aggregated into several sub-series describing the short, medium and long-term processes in order to present several combinations of correlations for Beni Yenni station data (Fig. 16). The combinations retained in this analysis are: the overall correlation between original indices and rainfall signals, the short-term cross-correlations from D1 to D4 of index and rainfall details, the long-term cross-correlation from D5 to D7 details of indices and rainfall and approximation A7 of indices and rainfall (Fig. 16a). We also retained the combinations D3-D4 (Fig. 16b) for annual process, D4-D5 (Fig. 16c), D5-D6 (Fig. 16d) for multiannual process, D6-D7 (Fig. 16e) and D7-A7 (Fig. 16f) to visualise the long-term evolution.

Fig. 16

Cross-multiresolution wavelet analysis between indices and rainfall signals at the same multiresolution level for short, medium and long-term process

As first observations, cross-multiresolution wavelet spectrum between the original signals of indices and rainfall signals (overall correlation) shows a very low correlation, explained by coefficients which oscillates between − 0.15 to 0.15 (Fig. 16). In the short term, the correlation is quite similar to that of the global correlation. In the long term, the cross-correlations are not characterised by typical impulse responses, but the correlation coefficients are relatively higher (Fig. 16a), which indicates that even low correlation, the relationship index-rainfall is dominated by the short term and the short-term process masks distant long-term phenomena. As such, in order to show the evolution of these correlations, we have taken scale-by-scale D3-D4, D4-D5, D5-D6, D6-D7and D7-A7 and represent their cross-correlations (Fig. 16b–f).

The CWMRA between NAO-rainfall for the components 8–16 and 16–32 months shows a higher correlation than the overall correlation with quite remarkable cyclical impulse responses oscillate between − 0.4 and 0.4 (Fig. 16b, c). The analysis indicates that the best relations are obtained from the scales between D5-D6 and D6-D7 characterising the multi-year variability modes of 3–5 and 5–10 years with remarkable impulse response coefficients between − 0.52 and 0.37 for the first mode and between − 0.4 to 0.63 for the second mode (Figs. 16d, e, and 17). In addition, the analysis shows that the Daubechies wavelet 20 (db20) is the best to represent the good correlations between NAO and rainfall (Fig. 17).

Fig. 17

Maximums positive and negative correlations obtained by cross-multiresolution wavelet analysis observed between each detail index / detail rainfall and approximation index / approximation rainfall, scale per scale at monthly level for the different wavelets of Daubechies (12, 20, 30 and 40)

The CWMRA between SOI and rainfall indicates an improvement and increase in the correlation coefficients from the annual correlation scale (D3-D4) (Fig. 16b) to the scales of low frequency events corresponding to decadal phenomena 10 years (128 months), with a spectacular impulse response in its negative part represented by a maximum coefficient of − 0.71 (Fig. 16f) explain by Daubechies wavelets 12 and 20 (Fig. 17). Zeroual et al. (2017) had also noted this negative correlation in northern Algeria using canonical correlation analysis.

For the MO and rainfall, the CWMRA shows correlations quite similar to the correlations between NAO, SOI and rainfall, but these correlation coefficients probably remain medium ranges between − 0.34 and 0.45 for the Beni Yenni station (Fig. 16e) and ranges between − 0.52 to 0.53 at Aghrib station (Fig. 17). These significant relationships between MO and rainfall are observed for scales (D6-D7) describing 5–10 years mode of variability using Daubechies wavelet 12 (db12) (Fig. 17).

In addition, we note that the cross-correlations are evolving from one sense to another across the range of long-term scales. This is well illustrated by the decompositions of the details of WeMO index and rainfall showing fairly characteristic impulse responses based on the Daubechies wavelet 20 (db20) for the scales between D6-D7 explaining also the mode variability 5–10 years and that gives coefficients ranging from 0.33 to −0.59 (Fig. 16e, 17). In addition, by way of indication, the decompositions of the WeMO, MO and rainfall details and their approximations become positive in the long term (for the mode higher than 128 month). Ouachani et al. (2013) have also found a significant links between MO, WeMO and rainfall of the Upper Medjerda basin of Tunisia. in addition, Martín Vide et al. (2008) have shown that there are clear links between the negative WeMO values and the torrential rainfall in the northern part of the Iberian Peninsula.

Another representation of the results of the multi-scale cross analysis (Fig. 17) explains in detail the maximums of positive and negative correlations observed between each detail index / detail rainfall and approximation index / approximation rainfall, scale per scale at monthly level for the different wavelets of Daubechies (12, 20, 30 and 40). It allows characterising and identifying the most significant scales explaining the possible relationships existing between the indices and the rainfall according to the Daubechies wavelet properties. Finally, the CMRWA allowed showing that the large-scale atmospheric circulation has a very low impact on the precipitation of the study area in the short term, but they significantly influence the rainfall in the long-term and thus control its climate trends.

Conclusion

The numerical and graphical observations obtained from CSA, CWT, MRWA, XWT, WCT and CMRWA have made it possible to assess the possible existing relationships between NAO, SOI, MO, WeMO indices and rainfall of Sebaou river basin. The following points summarised the main results obtained:

• 1. The rainfall analysis have been clearly demonstrates the dominance of 1 y and 1–3 y mode, which they explain 30 to 51% and 25 to 28% of the total variance respectively. However, the indices have shown that inter-annual fluctuations up to long-term represent the biggest part of the cumulative total variance ranging between 60 and 90%.

• 2. The analysis of 8–16 months component of CWT indicated significant fluctuations materialising a dry period between 1976–1985 and 1987–2001. In addition, The MRWA has shown an alternation of dry and wet periods with strong trend towards drier conditions starting from 1980s, where the periodic components D7 of decadal fluctuation (10 y) and the approximation A7 were the best components that could explain rainfall trends and drought in the study area.

• 3. In general, the cross correlograms showed a low correlation between indices and rainfall. However, cross-amplitude and gain functions have highlighted the peculiarity of annual periodic phenomena well distinguished for NAO, SOI and rainfall. Moreover, the values of the coherence function obtained could probably indicate a non-linear causal relationship.

• 4. The XWT spectrum have shown at the 8–16 month scale a fairly continuous component over time presenting the highest coefficients. As well, for the modes of 5–10 years and higher than 10 years characterising the multiannual to decadal variability, the spectrum reveals strong coefficients for the SOI between 1992–2005 and 1986–2000 respectively and less intense for NAO.

• 5. The WCT indicated the most significant relationship between NAO and rainfall at the mode 1 year, 1–3 years and 3–5 years, approximately from the early 1980s corresponding to the period with tendency to dry, which explains that the NAO is one of the main factors responsible for the appearance of drought in northern Algeria and the Mediterranean basin. However, the SOI affects rainfall only locally, but this influence could be significant for decadal variability mode of low frequency (higher than 10 years). Likewise, the coherence between the MO, WeMO and rainfall is also fairly well presented with significant values more or less localised in the time-frequency space.

• 6. According to this study, CMRWA is an efficient and complementary tool to XWT and WCT techniques. It allows extraction of hidden correlations between scales, the example of the negative spectacular impulse response between WeMO-rainfall for 5–10 years mode.

• 7. The CMRWA spectrums have shown that the short-term correlation is quite similar to that of the global correlation, proving that the short-term processes dominate the relationship index-rainfall, which masks the long-term phenomena, thus explaining the complexity of the relationship in space time-frequency scales, whose influence on the rainfall process can sometimes be very distant.

• 8. The CWMRA shows that the scales between D6-D7 and D7-A7 characterising multiannual and decadal variability patterns of 5–10 years and higher than 10 years are the most effective to represent climate index -rainfall relationships. Where, Changing Daubechies wavelet properties for the CWMRA could enhance the correlations between index-rainfall scales.

The prospects of this work are inexhaustible and constituting a new field to exploited. The results obtained from this study on the possible relationships between large-scale fluctuations and rainfall are important, however, these results remain spatially local at the level of the Sebaou basin. As such, our prospects would be to extend to cover the completely Northern Algeria, likewise North Maghreb using long hydroclimatic series as well as other multi-scale techniques.