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Spectral analysis is a modern tool of estimating periodicities and a modification of Fourier analysis (Davis, 1973; Chatfield, 1989). Its importance lies in the fact that periodicities contribute significantly to the determination of cyclic behaviour of weather (Ayoade, 1973; Davis, 1973; Kendall and Ord, 1990; Burroughs, 1992).
Studies on periodicities have been carried out in Africa (Rhode and Virji, 1976; Dyer and Tyson, 1977; Ogallo, 1979; Peterbaugh, 1983; Nicholson, 1985), North America (Armstrong and Vines, 1975; Walsh et al., 1982), India (Bhalme and Mooley, 1981) and Europe (Wigley et al., 1982; Logue, 1983). Other studies have associated periodicities with lunar cycle. These studies include Campbell et al. (1983), Currie (1984) and Vines (1986). The study of Currie (1984) postulated that lunar tidal forcing of 18.6-year period is a determinant of Indian floods and droughts. A number of studies have attempted to relate periodic tendencies in rainfall to atmospheric phenomena of El-Nino/Southern Oscillation, quasi-biennial oscillation and ENSO-like quasi-periodicities in Atlantic sea surface temperature (Farmer and Wigley, 1985) and long-term quasi-periodic fluctuations (Tyson and Dyer, 1978; Bhalme and Jadhav, 1984; Mooley and Partharasethy, 1984). Gribblin (1973) and Wood and Lovett (1974) linked periodicities with atmospheric circulation in connection with sunspot cycle, while Campbell et al. (1983) associated it with gravitational interactions in the earth moon–sun system. Klaus (1977) studied spatial distribution and periodicity of mean annual precipitation south of Sahara. His studies reveal that the Sahel had witnessed moist conditions from 1926 to 1931, 1947 to 1952, 1954 to 1959, 1964 to 1967 and in 1969. However, in the same climatic zone, in 1931–1933, 1941–1943, 1948–1949 and 1968–1972 excluding 1969, dry conditions have prevailed.
Many scholars including Flohn (1960), Dalby and Harrison-Church (1973), Bunting et al. (1976) and Lusigi and Glaser (1984) have found little or no evidence of apparent trends and periodicities in annual rainfall series of West Africa. Bunting et al. (1976) using 30 years (1936–1965) rainfall record of West Africa examined annual total series for linear trends and concluded that the search for periodicities in West African rainfall is a fruitless effort. However, periodicities have been detected in the study of rainfall periodicities in West Africa and Nigeria, in particular (Ayoade, 1973; Adejuwon et al., 1990; Ologunorisa and Adejuwon, 2003). Despite these studies, no detailed research work has been carried out on the spectral analysis of rainfall in Edo and Delta States of Nigeria. However, in this article, annual rainfall is spectrally analysed and the significant spectral peaks are identified. The filtered time series applied to the rainfall record examines in detail the amplitude of the periodicities found.
2. The study area
Edo and Delta States are located in the southern part of Nigeria. It lies between longitude 5°00′E and 6°45′E and latitude 5°00′N and 7°34′N (Figure 1). It has a total land area of 34 840 km2.
Edo and Delta States experience tropical climate. It has distinct dry and wet season. Between 8 and 10 months in the year, the climate of the region is dominated by tropical maritime (mT) air mass, while the remaining 2 to 4 months of the year are under the influence of the dry tropical continental (cT) air mass. The mean annual rainfall is about 3470 mm in the coastal areas and decreases inland both in amount and duration to less than 1800 mm in the northern area. The mean monthly temperature is 27 °C. Humidity decreases from the coast inland. It ranges from about 85% in the coastal areas while it is less than 80% in the north.
The relief of the region includes coastal lowland, the Esan Plateau, Orle valley and the dissected uplands of Akoko-Edo (Adejuwon, 2000; Segynola, 2000). The soil types are made up of ferrosols to the north, sandy and clay soil to the west, alluvial and hydromorphic soil close to Niger and Benue rivers and alluvial soil along the coastal areas (Imoroa, 2000). The vegetation varies from the mangrove swamp along the coast to the rainforest in the middle and the savanna in the north.
The drainage patterns of some of the rivers are dendritic. The River Niger drains the eastern flank of the region along with River Ase and discharges into the sea through its several distributaries such as Forcados, Escravos and Warri rivers and creeks such as the Bomadi creeks and others. Rivers Osse, Orihionwon and Ikpoba drain the western area. Rivers Jemiesson and Ethiope along with Osse from the north and northeast join River Benin which eventually discharges into the sea.
Daily, monthly and annual rainfall data for the period 1931–1997 were collected from the archive of Nigerian Meteorological Agency, Lagos. The data were collected from the synoptic stations of Benin (06°20′N; 05°38′E), Sapele (05°55′N; 05°41′E), Warri (05°31′N; 05°44′E) and Forcados (05°55′N; 05°26′E). The above data at the stated time intervals or dimensions were chosen based on the rationale that the period 1931–1997 is considered to have records which are reliable and comprehensive enough for the analysis. Also large fluctuations are best examined over a long period in which large cycles can be carried out.
The preliminary treatment of the data involved basic statistical techniques like computation of the totals and means for rainfall analysis in each station. The statistical analysis employed for this study is spectral analysis which is a modification of Fourier analysis (Chatfield, 1989).
3.2. Techniques in spectral analysis
Spectral analysis estimates periodicities. In the pre-computer era, harmonic analysis (the direct method) was used and the results were usually displayed as a plot of amplitude against frequency known as the periodogram (Tabony, 1979). Blackman and Tukey (1959) later developed indirect approach that was computer based, while Fast Fourier transform for direct method was derived by Cooley and Tukey (1965) in which the ordinary ‘direct’ formulae is replaced by more computationally efficient ones. However, the two most important spectral analyses are the maximum entropy spectral analysis and the power spectral analysis.
The maximum entropy spectral analysis, an indirect method, is a method of analysing time series that employs autoregressive method to extract the maximum amount of information from the available data (Burroughs, 1992). Its success as a method of assessing periodicities in a time series depends mainly on the ‘signal-to-noise’ ratio in the time series. As observed by Burroughs (1992), meteorological series rarely meet the signal-to-noise criteria that can exploit the advantages of maximum entropy spectral analysis which is that of not adding or subtracting information from the data. The maximum entropy method of spectral analysis is not universally accepted (Tabony, 1979). This method, therefore, will not be employed in this study for assessing periodicities.
Power spectral analysis involves the presentation of the square of the amplitude of the harmonics of time series as a function of the frequency of the harmonics (Burroughs, 1992). Spectral analysis is essentially a modification of Fourier analysis so as to make it suitable for stochastic rather than deterministic function of time (Chatfield, 1989). It is a non-parametric procedure in which a finite set of observations is used to estimate a function defined over the range (O, Π). Fourier transform spectral analysis is the mathematical determination of the amplitude of the harmonic component of a time series and the presentation of these in the form of a power spectrum. The plotting of the power of the variance power against the frequency is known as power spectrum. The area under the curve in a power spectrum is proportional to the variance. The transformation from the time to the frequency domain is made by taking the Fourier transform of the time series. Therefore, Fourier analysis method was adopted for the assessment of periodicities in this study.
The main objective of spectral analysis is to estimate the contribution of a particular band of frequencies to the overall variance in the terms of a time series (Ayoade, 1973). The contributions of oscillations of various wavelengths to the variable of a time series are shown by spectrum of a time series. Standard tests covered the mathematics of the techniques (Panofsky and Brier, 1958; Davis, 1973). The serial correlations of the forms in a given time series can be calculated using the formula:
where rt is the serial correlation coefficient between x1 and x1 + t separated by time interval t, cov the covariance and Var the variance.
The degree of likeness between two terms of a time series separated by a given time interval is measured by the serial correlation. The auto-correlation function tends towards zero as the interval between observations increases. The serial correlations between the terms which are 1, 2, …, n places apart are called the first, second, …, nth serial correlations, respectively. The correlogram is given by the graphical plot of the number of serial correlations on the x-coordinate against the value on the y-coordinate. Spectral analysis follows from this idea of the correlogram and is usually done by applying a type of harmonic analysis to the auto-correlation function (Ologunorisa and Adejuwon, 2003).
3.3. Calculation of variance
The power or variance was plotted arithmetically rather than logarithmically after necessary transformation by the computer package. The computer package employed in this study is the Fourier program proposed by Davis (1973). The program was upgraded. The method is applied because spectrum estimates plotted on a logarithmic scale exaggerate the visual effects of variation where the spectrum is small (Chatfield, 1989). Thus, it is often easier to integrate a spectrum plotted on an arithmetic scale as the area under the graph corresponds to power and one can more readily assess the importance of different peaks. Also, arithmetic scale makes interpretation of the final result easier. In addition, it is generally easier to interpret a spectrum if the frequency scale is measured in cycles per unit time rather than radians per unit time (Chatfield, 1989).
3.4. Determination of spectral peaks
The issue of determination of peaks that are significant is of paramount importance. Spectral peaks have arbitrarily been taken to lie between those frequencies where the power is one-third of its peak value (Tabony, 1979). The variance attributed to a given peak is limited to the area contained between one-third power points as shown in Figure 2. The estimation of the statistical significance of a peak is based on the variance ratio, i.e. ratio of the variance observed to that expected from a random series. If the width of the peak defined by the one-third power points is Δf, the mean power over the frequency range Δf is hp and the mean power over the whole of the spectrum is hA, then the variance ratio R is given by hp/hA.
If the series contain an underlying harmonic component with angular frequency β, there will be a peak in the unsmoothed spectrum not only at either side of it determined by local maxima of the sine function. This is known as the ‘side bands’. In general, smoothing with a variety of truncation points will avoid wrong interpretations. However, it may happen that a known (i.e. seasonal) harmonic could distort the estimates and we may prefer to remove the harmonic before developing the sample spectrum. Also, if there is latent harmonic in the series with angular frequency and given a spectral peak at α, we may also expect to see peaks at frequency 2α, 3α and so on. This is known as ‘echo effects’ (Kendall and Ord, 1990).
3.5. Filter method
When the graph of power is plotted, successive values of the power or variance do vary greatly. This is known as ‘raw power spectral’ which is a result of the finite sample length of the time series from which they are derived (Davis, 1973). The (unsmoothed) intensity function plotted against frequency is known as ‘periodogram’. Periodogram in itself is an inconsistent estimate. A true periodic structure of the time series can be obtained from a smoothed power spectrum. There are widely used filters for the purpose of smoothing the power spectral known as spectral windows. Considerable variety of the lag and spectral windows include Barttlete windows, periodogram windows, Daniel windows, Tukey windows, Parzen windows, etc. Barttlete window is no longer in use and is inferior to Tukey and Parzen windows. Periodogram window produces results not attainable in practice. Tukey window and Parzen window are described as the two best known lag windows (Kendall and Ord, 1990).
Tukey window is also called the Tukey–Hanning or Blackman–Tuckey window. The empirical evidence suggests that it is reasonable to use either the Tukey or Pazen window (Kendall and Ord, 1990). However, for the purpose of smoothing the power spectral in this study, the ‘Hanning filter’ by Davis (1973) is employed. This filter is a triangular function of the form:
where Ŝ2 is the smoothed power or variance, S2 the power or variance and n the harmonic number.
The graph of the spectral analysis in Edo and Delta States (formerly Mid-Western Nigeria) over 67 years (1931–1997) was shown in Figure 3(a)–(d). The smoothed power spectrum of the detrended annual rainfall was performed for Benin, Sapele, Warri and Forcados.
The result in this study revealed that the contribution of the total variance by various harmonics in the annual rainfall series of these stations are 234 872 962, 15 146 101 000, 101 474 843 and 6 489 305 284, respectively. In these stations, the critical values of variance above which any spectral peak is highly significant at 95% confidence level for Benin, Sapele, Warri and Forcados are 4175.22, 4785.20, 5669.90 and 6938.52, respectively.
The spectral analysis performed for Benin synoptic station show that significant spectral peaks are prominent at 6.7, 4.6 and 3.7 years periodicities. The most pronounced peak at the station is 3.7 years periodicity (Figure 3(a)). In Sapele, the most pronounced periodicity of 5 years was observed (Figure 3(b)). Although, the spectral peaks are significant at 4.6 and 3.7 years, respectively, at Warri, the most pronounced of these peaks is 3.7 years (Figure 3(c)). However, in the case of Forcados, a single significant spectral peak of 3.6 years cycle is prominent (Figure 3(d)).
Periodic tendencies have been observed in African rainfall (Wood and Lovett, 1974; Dyer and Tyson, 1977). The spectral results from the previous studies show prominent cycles of 1.8–2.6 years (Klaus, 1977), 2.7–3.3 years (Ogalo, 1979) and 2.5–8 years (Farmer and Wigley, 1985). Perterbough (1983) studied normalized rainfall departure series on various regions of Africa and found no apparent statistically significant oscillations in most cases. Notwithstanding, Nicholson (1985) studied 84 rainfall regions in Africa and discovered statistically significant spectral peaks of approximately 2.3, 2.8, 3.6 and 5–6 years.
The evidence in this study shows that significant cyclical patterns are observed in the rainfall of Edo and Delta States (formerly Mid-Western Region) in Nigeria. The range between 3 and 6 years periodicities were detected as significant periodicities in the rainfall series. Ayoade (1973) and Adejuwon et al. (1990) had earlier noted such short-term periodicities in the precipitation pattern of Southern Nigeria. Making use of spectral analysis, 2–6 years periodicities were detected by Ayoade (1973) while Adejuwon et al. (1990) detected prominent periodicities of between 2 and 8 years cycles and less prominent periodicities of 16, 18 and 21 years cycles. This is in contradiction to Bunting et al. (1976), who established that there is no apparent evidence of periodicities. Both Ayoade (1973) and Adejuwon et al. (1990) also disagree with the observation made in Nigerian meteorological note 2 (1952) that the 5-year moving average of many stations shows some slight evidence of a cycle of about 13 years but in some cases that the cycle is ill-defined.
The observed short-term periodic behaviour has been linked with the factor of the ocean–atmosphere interactions (Ayoade, 1973; World Meteorological Organization, 1988; Adejuwon et al., 1990; Burroughs, 1992). The ocean–atmosphere interaction is believed to be a dominant factor in the weather processes in Edo and Delta States of Nigeria in the wet season and contributes the largest share of the total annual rainfall. During this time of the year, the factor takes over the atmospheric processes dominantly (Adedokun, 1978) and breaks down as it is farthest from the South. Based on this fact, the factor of ocean–atmosphere interaction is therefore needed to be understood, while the extent to which they affect cyclical pattern in the distribution of rainfall is also needed to be assessed. The nature and timing of ocean–atmosphere factor may be of significant importance in the explanation of pattern of rainfall in Edo and Delta States because there appears to be similarity in the nature of rainfall in Southern Oscillation.
Dominant cycles with periodicities exist in the annual rainfall of 67 years in Edo and Delta States (formerly Mid-Western Nigeria) of Nigeria. Dominant cycles with periodicities were found in the annual rainfall of 67 years in the region. The frequency of occurrence shows irregularities in their cycles. The periodic tendencies exhibited in the rainfall of the region are closely related with the factors of ocean–atmosphere interactions. The stations examined for the annual rainfall series indicate significant periodic tendencies. The observed significant spectral peaks range between 3 and 6 years. The periodicities observed are basically short term.
The author is grateful to Prof. O. Ekanade, the former Dean of the Faculty of Social Sciences, Obafemi Awolowo University, Ile-Ife, Nigeria for his comments and suggestions which have been very helpful in the preparation of the final draft.