Wavelet Analysis of Hydroclimatic Teleconnections and Precipitation Variability over Ethiopia

Introduction

Data Source and Method of Analysis

Description of Study Area

Ethiopia is located between latitudes 3.3°N and 15°N, and longitudes 33°E and 48°E, covering an approximate area of 1.02 million square kilometers (Korecha & Barnston, 2007). This study focuses on various regions within Ethiopia, including the eastern, western, northern, southern, and central parts, as depicted in Figure 1.

The country's natural landscape is characterized by high plateaus and mountain ranges, with elevations ranging from approximately 4,500 meters above sea level in the northern and central regions to about 100 meters below sea level in the northeastern lowlands. This diverse topography contributes to significant climatic variations across Ethiopia, influencing different rainfall patterns as described by the National Meteorology Service Agency.

The northwestern regions predominantly experience a single rainy season, known as Kiremt, which extends from June to September (Tsidu, 2012). This season is influenced by the movement of the Intertropical Convergence Zone (ITCZ), the South Atlantic anticyclone, and the moist southwesterly monsoon flow originating from the Southern Hemisphere. The timing and spatial distribution of rainfall during Kiremt are closely linked to ITCZ oscillations and the strength of the southern hemispheric anticyclones (Segele & Lamb, 2005).

In contrast, the eastern regions of Ethiopia exhibit a bimodal rainfall pattern, consisting of a primary rainy season in summer and a secondary, shorter rainy season during the spring months (Belg: February-May). These rainfall patterns are primarily driven by moist easterly winds from the northwestern Indian Ocean, influenced by high-pressure systems over the Arabian Sea. Similarly, the southern regions follow a bimodal rainfall distribution, with peak rainfall occurring in both spring and autumn (October-December). This seasonal pattern aligns with that observed in equatorial East Africa and is attributed to the north-south migration of the ITCZ.

Figure 1. The study area.

Figure 1. The study area.

Method of Analysis

Standardized Precipitation Ratio (Ip)

To identify the alternating dry and wet periods of the country in this study the standardized pre- capitation ratio was first performed on the annual scale. Then, analytical and spectral tools (continuous wavelet Transform analysis) were used to identify the variability modes and determine the origin of this variability.

The standardized precipitation ratio (Ip) was used to determine wet and dry periods and their alternations consistent with positive (wet) and negative (dry) values(Jemai et al., 2017). It’s a dimensionless parameter, based on the ratio of the precipitation difference from the mean to standard deviation (Equation (1). It is defined as follows:

$$I_p={\displaystyle \frac{y_i-\overline{y}}{\sigma }}$$

(1)

Where Ip: standardized precipitation ratio, y_i: rainfall for a given study year,$\overline{y}$: average rainfall during a given study period, σ: standard deviation of rainfall for the same period

Wavelet Transform

Various methods have been introduced for signal processing and time series analysis. Most of these methods are mathematical transforms that convert vectors or functions from one space to another. Some of the transformations (e.g. Fourier transforms) are only appropriate for the analysis of stationary time series exhibiting a uniform oscillation pattern over time. These transformations are not robust enough for the analysis of non-stationary time series, since they cannot identify all frequencies within the time series(Olkkonen, 2011; Percival & Walden, 2000).

Wavelet Transform is a powerful tool used for time-frequency analysis, allowing the decomposition of signals into components that vary in both time and frequency domains. This method is particularly useful for identifying localized variations in power and understanding the temporal evolution of different frequency components. The current study employs the Continuous Wavelet Transform (CWT) to analyze the variation of power across multiple timescales.

Previous studies, such as (Sarker & Sarker, 2022) have demonstrated the effectiveness of wavelet-based approaches in studying the spectral properties of complex systems, including the dynamics of water hammer waves. Similarly, (Sarker et al., 2023) highlighted the utility of time-frequency analysis in examining morphodynamic properties and their temporal evolution. These works provide valuable insights into the practical applications of Wavelet Transform in analyzing both natural and engineered systems, reinforcing its suitability for the current analysis.

Wavelet analysis is a common method for analyzing the power of local variation within a time series by decomposing a time series into time versus frequency space and hence it describes the variability of precipitation in terms of pattern and representing the contribution of each pe- riod/scale for the overall. The wavelet transforms breaking up a signal into scaled versions of a wavelet function; where the size of the wavelet (the window) varies with frequency. The wavelet transform are often used to analyze time series that contain non-stationary power at many various frequencies. Localization during a signal analysis by decomposing or transforming a one- dimensional time Series into a diffuse two-dimensional time-frequency image, at an equivalent time. Then, it is Possible to get information on both the amplitude of any periodic signals within the series, and how this amplitude varies with time. Continuous wavelet transform was chosen because it is widely used complex wavelet having good time-frequency localization than other real wavelets (Addison, 2017).

The theory behind wavelet transform is relatively similar to that of Fourier transforms, but offers much greater flexibility allowing a highly accurate capture of all frequencies present in a given time series ((Percival & Walden, 2000; Sang, 2013). Continuous wavelet transform (CWT) is recognized as one of the robust tools for the investigation of processes with high temporal variability ((Labat, 2005). It is appropriate in particular for examining non-stationary processes, like climate variables.

In this study, it is performed on gauge precipitation data and ocean indices to examine their oscillation. The CWT of the signal x(t) generating a wavelet spectrum is expressed as Equation (3.3) (Grossmann and Morlet, 1984).

$$W\left(s,\tau \right)={\displaystyle \frac{1}{\sqrt{s}}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }} x\left(t\right){\psi }^{\mathrm{*}}\left({\displaystyle \frac{t-\tau }{s}}\right)dt$$

(2)

Every wavelet length is finite and ever precisely localized in time. The mother wavelet includes two parameters: scaling s and temporal location τ

$${\psi }_{s,\tau }\left(t\right)={\displaystyle \frac{1}{\sqrt{s}}}\psi \left({\displaystyle \frac{t-\tau }{s}}\right)$$

(3)

With ψs,τ (t): Wavelet daughter; s: scale parameter ; τ : time-localization parameter. and ψ and ψ ∗ represent the wavelet function and complex conjugate, respectively. With τ = 0 and s = 1 denotes the basic or mother wavelet. The flexibility in the wavelet transform comes due to the variations in the scale, which enable to capture the long and short frequency in the time series. Parameterization of scale and wavelet daughters admits the finding of various frequencies that compose signal. The actual computation of the wavelet transform described(Santos et al., 2016). Finally we used mat lab and origin software to analysis the data.

Figure 2. Flowchart illustrating the methodology used in the study, highlighting the sequential steps from data collection and preprocessing to analysis and interpretation of results using wavelet transforms and other techniques.

Figure 2. Flowchart illustrating the methodology used in the study, highlighting the sequential steps from data collection and preprocessing to analysis and interpretation of results using wavelet transforms and other techniques.

Result and Discussion

Standardized Precipitation Ratio (Ip)

Precipitation is the result of several atmospheric systems (such as topography and ITCZ) that behave at various scales and periodicity. Further, the irregularity and seasonality exhibited by rainfall, can be assessed by means of the climatic index is crucial. Standardized precipitation ratio(Ip) developed by (McKee et al., 1993) is used to classify the wet and dry rainfall anomalies. Based on the irregularity of precipitation distribution, standardized precipitation was calculated to separate wet and dry periods over 31 years of the study area. For this investigation, we used standardized precipitation ratio equation (1) mentioned in the method of data analysis. The calculated rainfall irregularity (standardized precipitation ratio) was calculated to indicate the estimated year to year variation of inter annual rainfall expressed in terms of standardized precipitation ratio. Negative standardized rainfall indicates dry (below normal rainfall) period and positive anomalies indicate wet (above normal rainfall) period.

Figure 3a) depicts the Ip for the northern part of the country, where positive standardized precipitation values indicate wet periods and negative values represent dry periods. The northern region experienced a generally dry period from 1987 to 1997, with a peak negative value of -1.25 in 1990, followed by wet periods from 1998 to 2001, peaking at 2.76 in 2001. From 2002 to 2017, more dry periods were observed, with only a few slightly humid years. The standard precipitation ratio for this location shows a pattern consistent with observations by (Ayalew et al., 2012; Wolde-Georgis, 1997) who also reported positive standardized anomalies in the northern region during 1999–2001.

Figure 3b) illustrates the standard precipitation ratio for the eastern part of the country. In this region, a predominantly dry period was observed from 1987 to 1991, except for 1987. This was followed by generally wet periods from 1992 to 1998, with a maximum value of 2.1 in 1996. The final period, from 1999 to 2017, indicates more dry periods compared to wet periods, with a maximum positive peak value of 2.1 in 2010 and a maximum negative peak value of -1.5 in 2015.

Figure 3. Standardized precipitation ratio for a) northern b) eastern regions of Ethiopia.

Figure 3. Standardized precipitation ratio for a) northern b) eastern regions of Ethiopia.

Figure 4a illustrates the standard precipitation ratio for the western parts of the country. From 1987 to 1996, alternating slightly dry and wet periods were observed. Wet periods were prominent between 1997 and 2001, with the highest positive peak value of 2.1 recorded in 1997. From 2002 to 2012, the region experienced a predominantly dry period, interspersed with occasional humid years. The driest year during this period was 2003, with a negative peak value of -1.8. From 2013 to 2017, the region experienced a predominantly wet period, interrupted by a few dry years

Figure 4. Standardized precipitation ratio for a) western b) southern regions of Ethiopia.

Figure 4. Standardized precipitation ratio for a) western b) southern regions of Ethiopia.

Figure 4b presents the standard precipitation ratio (Ip) for the southern parts of the country. From 1987 to 1994, the region experienced predominantly dry years, with only a few wet periods. Between 1995 and 1998, positive Ip values indicate wet periods, followed by a dry period from 1999 to 2005. The period from 2005 to 2017 was characterized by an increase in both wet years and the frequency of dry years compared to the earlier study periods. During this time, a notable negative peak value of -2 was recorded in 2015, while a positive peak value of 1.75 was observed at the end of the study period.

As shown in Figure 5a, the temporal variability of the standard precipitation ratio (Ip) reveals distinct patterns across four periods. The first period, spanning 1987–1993, exhibited predominantly wet conditions, with more wet periods than dry periods. The second period, from 1994 to 2004, was characterized by generally dry conditions, interrupted by a slightly wet year in 1996, which recorded a maximum Ip value of 3.2. The third period, from 2005 to 2011, was marked by predominantly wet conditions, except for 2008 and 2009. Finally, the fourth period, from 2012 to 2017, was a generally dry period, with the most pronounced negative peak value of -1.5 observed in 2014.

The occurrence of extreme positive and negative peak values during these periods indicates episodes of heavy rainfall and drought, respectively, highlighting the impact of extreme climatic events on rainfall variability and water availability in the study area.

Figure 1. Standardized precipitation ratio for a) central regions of Ethiopia.

Figure 1. Standardized precipitation ratio for a) central regions of Ethiopia.

Mode of Rainfall Variability

To investigate precipitation variability and identify oscillation modes, the wavelet transform technique was applied to 31 years (1987–2018) of annual precipitation data across five selected regions of Ethiopia: Northern, Eastern, Southern, Western, and Central. The analysis, presented in Figure 6, Figure 7 and Figure 8, reveals several dominant frequency modes and power spectra on an annual scale. These findings emphasize significant oscillation periods and their temporal evolution, offering insights into region-specific and temporal rainfall variability. The wavelet transform effectively captures key trends and oscillatory behaviors, providing a detailed understanding of precipitation dynamics across the study regions.

Figure 6a illustrates the wavelet analysis of periodicity in a year versus the time series of rainfall for the northern part of Ethiopia, revealing two dominant oscillations or periodicities. The first is a periodicity of 6 –10 years, corresponding to a frequency range of 0.1– 0.17 cycles per year, with a power of 55.4 W. This oscillation prevailed for 11 years, spanning from 1992 to 2003. The second is a 20-year periodicity, corresponding to a frequency of 0.05 cycles per year, lasting from 1992 to 2005. These findings indicate that the northern region is characterized by the dominance of a long-term oscillation of 6 –10 years during 1992–2003 and a 20-year periodicity from 1992 to 2005.

Figure 6. Wavelet analysis of precipitation time series of a) northern b) eastern part of the Ethiopia.

Figure 6. Wavelet analysis of precipitation time series of a) northern b) eastern part of the Ethiopia.

Figure 6b describes a wavelet analysis of the periodicity in year and time series of rainfall in the eastern part of Ethiopia. From this graph, we observe three dominant oscillations, or periodicities. The first strong oscillation or periodicity of 2-4 years (frequency of 0.25-0.45 with power of 25.4 w) prevailed for 4 years from 2008–2012. The second oscillation or periodicity of 5–6.5 years (in terms of frequency 0.15–0.2 with a power of 7.87 w) lasts from 1995–2000, and the third period periodicity of 10–20 years (frequency of 0.05–0.1 is continued from 1992–2010. So the eastern part of the country dominates the long-term oscillation of 2–4, 5–6.5, and 10–20 years.

Figure 7a presents the wavelet analysis of periodicity in years versus the time series of rainfall for the western part of Ethiopia, revealing three dominant oscillations or periodicities. The first is a strong periodicity of 3–5 years, corresponding to a frequency range of 0.2–0.3 cycles per year, with a maximum power of 39 W. This oscillation dominates for 13 years, spanning from 1992 to 2005. The second is a moderate periodicity of 8 years, corresponding to a frequency of 0.12 cycles per year, with a power of13.9, lasting from 1995 to 2000. The third is a weak periodicity of 27 years, corresponding to a frequency of 0.037 cycles per year, occurring briefly from 1992 to 1993. These findings indicate that the long-term oscillation of 3–5 years is the most significant, prevailing for 13 years (1992–2005) and strongly influencing rainfall patterns in the region.

Figure 7. Wavelet analysis of precipitation time series of a) western b) southern part of the Ethiopia.

Figure 7. Wavelet analysis of precipitation time series of a) western b) southern part of the Ethiopia.

Figure 7b presents the wavelet analysis of periodicity in years versus the time series of rainfall for the southern part of Ethiopia, identifying two dominant oscillations or periodicities. The first is a strong periodicity of 2.5–4 years, corresponding to a frequency range of 0.25–0.4 cycles per year, with a power of 24 W. This oscillation prevailed for 7 years, spanning from 2009 to 2016. The second is a 10-year periodicity, corresponding to a frequency of 0.1 cycles per year, with a power of 7.66 W, lasting from 1995 to 2009. These findings suggest that the southern region is characterized by the dominance of a short-term oscillation of 2.5–4 years during 2009–2016 and a long-term 10-year periodicity from 1995 to 2009.

Figure 8. Wavelet analysis of precipitation time series of a) central part of the Ethiopia.

Figure 8. Wavelet analysis of precipitation time series of a) central part of the Ethiopia.

Figure 8a presents the wavelet analysis of periodicity in years versus the time series of rainfall for the central part of Ethiopia, highlighting two dominant oscillations or periodicities. The first is a strong oscillation with a periodicity of 2–3 years, corresponding to a frequency range of 0.35–0.45 cycles per year, and a maximum power of 27.6 W. This oscillation prevailed for 8 years, spanning from 1992 to 2000. The second is a 10-year periodicity, corresponding to a frequency of approximately 0.1 cycles per year, with a power of 6.63 W, lasting from 2002 to 2010. These findings indicate that the central part of Ethiopia is characterized by a dominant short-term oscillation of 2–3 years from 1992 to 2000, followed by a long-term 10-year periodicity from 2002 to 2010

Modes of Oceanic Induced Oscillations

To identify the modes of variability associated with different oceanic indices, the Continuous Wavelet Transform (CWT) was applied. Figure 9 and Figure 10 present contour plots illustrating the periodicities of oceanic indices such as Nino-4, North Atlantic Oscillation (NAO), Mediterranean Oscillation Index (MOI), and Southern Oscillation Index (SOI).

Figure 9a depicts the wavelet analysis of the periodicity in years versus time series for the Nino-4 index. The analysis reveals a dominant oscillation of 8–13 years (corresponding to a frequency range of 0.067–0.125 cycles per year) with a maximum power of 1.36 W, prevailing for 20 years from 1992 to 2012. This Nino-4 oscillation clearly aligns with the periodicities observed in rainfall data for the northern, eastern, and central regions of Ethiopia.

In the northern region, the long-term rainfall oscillation of 6–10 years (1992–2003) coincides with the Nino-4 oscillation (1992–2012), showing a strong negative correlation (r = -0.84). During this period, the standardized precipitation ratio indicated predominantly wet conditions. For the eastern region, the 10–20-year rainfall periodicity (1992–2010) overlaps with the Nino-4 oscillation, albeit with a weak positive correlation (r = 0.05). In the central region, the 10-year rainfall periodicity (2002–2010) coincides with the Nino-4 oscillation, exhibiting a moderate negative correlation (r = -0.5).

These findings suggest a significant teleconnection between the Nino-4 index and rainfall variability in Ethiopia, particularly influencing the northern (1992–2003) and central (2002–2010) regions. Previous studies (Viste & Sorteberg, 2013; Wolde-Georgis, 1997) support these observations, reporting drought events during 1987, 1991, 1994, and 2009 across many parts of Ethiopia, which correspond to variations in the Nino-4 index.

Figure 9b presents the wavelet analysis of the periodicity in years versus-time series for the North Atlantic Oscillation (NAO) index. The analysis reveals two distinct oscillations, or periodicities. The first oscillation, lasting 2–3 years, corresponds to a frequency range of 0.3–0.5 cycles per year with a power of 1.55 W and occurred between 2007 and 2012. The second oscillation, spanning 10–12 years, corresponds to a frequency range of 0.083–0.1 cycles per year, and was observed during two time periods: 1987–1997 and 2007–2015.

Figure 9. Wavelet analysis of periodicity in year’s verses time series of the a) Nino-4 index value b) NAO index value.

Figure 9. Wavelet analysis of periodicity in year’s verses time series of the a) Nino-4 index value b) NAO index value.

Among these oscillations, the 2–3-year NAO oscillation coincides with rainfall periodicities in the southern part of Ethiopia. Specifically, the short-term rainfall oscillation of 2.5–4 years (2009–2016) aligns with the NAO oscillation from 2007–2012, showing a weak negative correlation (r = -0.15). During this time, the standardized precipitation ratio indicated alternating wet and dry periods.

These findings suggest that rainfall variability in the southern part of Ethiopia is influenced by the NAO index during the 2007–2012 period. This conclusion aligns with previous studies by (Viste & Sorteberg, 2013; Wolde-Georgis, 1997) reported that 1987, 1991, 1994, and 2009 were drought years in Ethiopia including 1987, 1991, 1994, and 2009, corresponding to variations in the NAO index.

Figure 10 a presents the wavelet analysis of the periodicity in years versus-time series for the Mediterranean Oscillation Index (MOI). The analysis reveals a dominant oscillation with a period of 6–10 years, corresponding to a frequency range of 0.1–0.15 cycles per year and a power of 0.923 W, occurring from 1995 to 2005. This 6–10-year MOI oscillation aligns with rainfall periodicities observed in the northern, eastern, and southern parts of Ethiopia. Specifically, in the northern part of Ethiopia, the long-term oscillation of 6–10 years from 1992 to 2003 coincides with the MOI oscillation from 1995 to 2005, showing a moderate positive correlation of 0.53. During this period, the rainfall in the northern part exhibited wet conditions. In the eastern part, a 5–6.5-year oscillation observed from 1995 to 2000 correlates negatively with the MOI oscillation (correlation coefficient of -0.75), with wet conditions observed in the region. For the southern part, the 10-year periodicity from 1995 to 2009 correlates weakly with the MOI oscillation (correlation coefficient of -0.12). These results suggest that the teleconnection of the MOI significantly affects rainfall variability in the northern (1992–2003) and eastern (1995–2000) parts of Ethiopia.

Figure 10. Wavelet analysis of periodicity in year’s verses time series of the a) Nino-4 index value b) NAO index value.

Figure 10. Wavelet analysis of periodicity in year’s verses time series of the a) Nino-4 index value b) NAO index value.

Figure 10b shows the wavelet analysis of the periodicity in a year-versus-time series for the Southern Oscillation Index (SOI). The analysis identifies two dominant oscillations: a 3–5-year oscillation (frequency of 0.2–0.3 cycles per year, power of 0.15 W) that prevailed from 1995 to 2000, and an 11-year oscillation (frequency of 0.09 cycles per year, power of 1.9 W) from 1995 to 2012. The 3–5-year SOI oscillation coincides with rainfall periodicities in the western part of Ethiopia, where a dominant 3–5-year oscillation from 1992 to 2005 correlates negatively with the SOI (correlation coefficient of -0.4) during the 1995–2000 period. This period was marked by wet conditions in the western part, indicating that the SOI influenced rainfall variability in the region. Thus, we conclude that the Southern Oscillation Index significantly impacted the western part of Ethiopia during 1992–2005.

Conclusion

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