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A specific statistical analysis is applied on Sahel precipitation anomaly measurements, during the period 1900 - 2010. The conclusions obtained show persistent long-range power-law self-similarity for the time series of the rainfall anomalies and a semi-log distribution for the anomalies located over the mode of the data. It should be emphasized that these findings could substantially contribute to the forecasting of the precipitation index which is of crucial importance for the simulation of the global climate variability.
Marengo (2004) studied the decadal and long-term patterns of rainfall based on a combination of rain gauge and gridded rainfall datasets, for the entire Amazon basin and for its northern and southern sub-basins, during 1929-1998. Concentrating on the period 1950-1998, Marengo (2004) analyzed the rainfall variability, the variations in circulation and the sea surface temperature fields. Negative rainfall trends were detected for the entire Amazon basin, while in the northern (southern) sub-basin a negative (positive) trend seemed to be present. Spectral analyses exhibited decadal time scale variations for southern Amazonia and both inter-annual and decadal scale variations for northern Amazonia. Finally, in both sub-basins shifts in the rainfall regime occurred in the mid-1940s and 1970s, while a decreasing rainfall was observed in the northern Amazonia after 1975.
V. Krishnamurthy and J. Shukla (2007) used the 70-yr-long high-resolution gridded daily rainfall data over India in order to analyze the lagged composites of rainfall anomalies. They showed that the ~16 days active (break) cycle, starts with positive (negative) rainfall anomalies over the Western Ghats and eastern part of central India and intensifies and expands to a region covering central India and parts of north India during the peak phase, while negative (positive) anomalies cover the sub-Himalayan region and southeast India. Moreover, V. Krishnamurthy and J. Shukla (2007) detected 45- and 20-day oscillations and seasonally persisting components in the seasonal monsoon rainfall. The two dominant intra-seasonal oscillations seemed to have a poor correlation with the seasonal mean rainfall, while the inter-annual persisting components with anomalies of the same sign indicated a high contribution to the total seasonal mean rainfall.
Ip et al. (2011) studied the multi-scale variability and trends of an historical series of precipitation data available spanning more than a century and a long-term historical flood/dryness grade dataset, during the period of 1470 - 2000 in North China. Obvious seasonal changes, quasi biennial oscillations, inter-annual 4-7 year component and inter-decadal 19-year periodicity were detected in the studied precipitation time series. Moreover, 4~5 year ENSO mode inter-annual oscillation, quasi-10 year inter-decadal variability, quasi-24 year component and 50-80 year centurial periodicity were detected in the historical flood/dryness grade time series.
Machado et al. (2011) analyzed the decadal to centennial scale hydrological response using a combined documentary-sedimentary-instrumental 500 years comprehensive register of climatic data (rainfall and flooding). Wet years were seen to be closely linked to the presence of autumn positive anomalies (e.g. early 18th century). It was also derived that continuous, decadal wet periods coincided with both autumn and spring positive rainfall anomaly years (e.g. 1570/90, 1830/40, 1870/1900). During the late Medieval Warm Period (950-1200) and during some decades of the Little Ice Age (1440 - 1490, 1520 - 1570, 1600 - 1740, 1770 - 1800, 1820 - 1840, 1870 - 1900) high frequencies of large floods were detected. Flood frequency seemed to decrease in the 20th Century (1945 - 1973) with an average of 0.14 floods/year. It was also established that during wet phases (e.g. late 19th century) large floods occurred during all seasons, whereas a predominantly autumn extreme flooding (>70%) was attributed to higher inter-annual rainfall variability (e.g. 1945 - 1973). Finally, Machado et al. (2011) confirmed a shift from autumn rainfall maxima towards winter since the early 1990's.
The principal aim of this study is to examine the temporal fluctuations of Sahel precipitation anomalies over 20 - 10N and 20W - 10E, during the period 1900 - 2010, in order to define whether they exhibit persistent long-range correlations. From the other hand, this work attempts to investigate into the distribution of the rainfall anomaly data, concentrating on the Sahel geographical area.
2 Data and analysis
In the current study, we have used mean monthly values of Sahel precipitation anomaly (SPA), over 20 - 10N, 20W - 10E during the period 1900 - 2010. These measurements were taken of the the NOAA Global Historical Climatology Network (GHCN) V2 monthly data set, which contains gridded precipitation anomalies. 2064 homogeneity adjusted precipitation stations (from the U.S., Canada, and Former Soviet Union) have been combined with a data set containing 20590 raw precipitation stations throughout the world to create these gridded fields. In grid boxes with no homogeneity adjusted data, GHCN raw data was used to provide the greatest possible global coverage. Each month of data consists of 2592 gridded data points produced on a 5 x 5 degree basis for the entire globe (72 longitude x 36 latitude grid boxes). Gridded data for every month from January 1900 to the most recent month is available. These anomalies have been estimated with respect to the period 1950 - 1979 using the traditional anomaly method. Anomalies were calculated on a monthly basis for all adjusted stations having at least 25 years of data in the 1950 - 1979 base period. Station anomalies were then averaged within each 5 x 5 degree grid box to obtain the gridded anomalies. For those grid boxes without adjusted data, anomalies were calculated from the raw station data using the same technique. Measurements for Sahel in this data set are raw precipitation stations and have not undergone any homogeneity adjustments. The averaging region was chosen from a rotated principal component analysis of African precipitation by Janowiak (1988).
Time series of the mean monthly SPA values was characterized by strong seasonality and a weak long-term trend which were both removed (detrending and deseasonalization, respectively) by using specific statistical tools. The detrending was simply achieved by applying linear best fit to the whole time series, while the deseasonalisation was implemented by applying the classical Wiener method (e.g., filtering out all the detected seasonal changes, which were the semiannual and annual oscillation, the quasi 38 year component and the 50 - 100 year centurial periodicity) (see figure 1a).
A modern statistical method, called Detrending Fluctuation Analysis (DFA-method) was applied on the detrended and deseasonalised mean monthly SPA time series, so as to study their intrinsic properties. DFA method emanates from random walk theory, allowing the detection of intrinsic self-similarity in non-stationary time-series of observations often collected in a variety of research fields (Varotsos et al., 2005, 2006a). The sequential steps of DFA, which has proved useful in analyzing a large variety of complex systems with self-organizing behavior (Peng et al., 1994; Weber and Talkner, 2001; Varotsos et al., 2006b), are explicitly described in Varotsos et al. (2007).
3 Results and discussion
In order to study the SPA distribution, their mean monthly values were grouped into classes of equal length. Figure 1b illustrates the percentage relative frequency histogram of the mean monthly SPA values and the smoothed line of the Gaussian distribution. However, using the statistical best-fit tests Kolmogorov Smirnov, Chi-square and Anderson Darling the hypothesis whether SPA values fit the Normal distribution was rejected, at the 95% confidence level.
Inspection of figure 1b shows that the mode of the values seem to be mainly responsible for the poor-flexibility of the Normal distribution on SPA (Efstathiou, 2006).
The data set of SPA was also tested for meeting the requirements of an exponential distribution, a geometric distribution, a simple power law distribution, a generalized power law distribution (the Zipf-Mandelbrot distribution) and a lognormal distribution, respectively. The observed data didn't show any statistically significant fit to the above mentioned distributions.
In order to define the distribution of Sahel rainfall anomalies, the empirical probability P(X>x) of exceeding a fixed SPA value x, was calculated and then plotted in semi-logarithmic graph, against the value x (see figure 2a).
In the following, applying linear regression analysis between the logarithm of the probability P(X > x) and the SPA value x, the slope derived α for values located over the mode of the data equalled -0.26, with coefficient of determination R2 = 0.99. The latter denotes that the slope derived above is obviously statistically significant, at 95% confidence level yielding to the relationship below:
P(X>x) ~ 10αx (1)
According to the equation (1), SPA values exceeding the mode of the data the distribution of the Gutenberg-Richter law (Goldstein et al., 2004; Rundle, 1989). This result was confirmed by using the statistical best-fit test Kolmogorov-Smirnov, at 95% confidence level. Moreover, figure 2b establishes an obvious correlation (r2 = 0.99) between the cumulative function of the empirical and the semi-log distribution.
In the following, DFA-method applied on the detrended and deseasonalised mean monthly SPA values, during the period 1900 - 2010, revealed persistent long-range power-law self-similarity, with scaling exponent α = 0.59 ± 0.01 for all the time lags between 4 months and 28 years (see figure 3a,b). In other words, the fluctuations of the SPA in small time-intervals were found to be positively correlated to those in longer time intervals in a power-law fashion. More precisely, the averaged square of the detrended fluctuation function F(τ) over the N/τ intervals with length τ seemed to obey a power-law, notably:
<F2(τ)> ~ τ2α (2)
and the power spectrum function scales with 1/fβ, where β = 2a -1 (Kantelhardt et al., 2002). This is very different from the spectral properties of other meteorological quantities and indicates SPA as a manifestation of self-organised criticality (Peters and Neelin, 2006).
Vatay and Harnos (1994) showed scaling behavior in daily air humidity fluctuations, proposing that the exponent β can be related to the scaling of the weighted average of the avalanches (Christiensen et al., 1991). Matsoukas et al. (2000) investigated the rainfall time series of 9 states in America by using DFA method, and pointed out the persistent long-range power-law correlation. The scaling exponents are divided into two parts, presenting from 0.92 to 1.06 and 0.62 to 0.89, corresponding to the scaling in temporal intervals from 75 min to 5 days and 10.5 days to 16 months respectively.
It is worthwhile to clarify at this point that the persistence found above provides, in principle, a rainfall index forecast, which assumes that the SPA value of the ''following time interval'' (up to 28 years) will be the same as for the corresponding ''current time interval''. It obviously has a different meaning from the conventional forecast in climatology, which assumes that the SPA value in the ''following'' e.g. 28 years will be the same as the ''overall climatological'' SPA mean.
The above-mentioned findings provide power-law relationship that is practically simple simulating models for precipitation index fluctuations and they might also be helpful for the development of new models improving prediction of future precipitation index under different scenarios.
The first result of the present paper was that Sahel rainfall anomalies don't fit the Gaussian distribution but instead anomalies located over the mode of the data seemed to obey the Gutenberg-Richter law. Furthermore, a 'long memory' within the time series of the mean monthly SPA values was derived. More precisely, the application of the DFA method to the above mentioned detrended and deseasonalised time series revealed persistent long-range power-law correlations. That scaling behaviour of SPA denotes that there is a tendency an increase in the Sahel rainfall anomaly to be followed by another increase at a different time in a power-law fashion. It should be noted that these findings could substantially contribute to the forecasting of the rainfall index which is of crucial importance for the simulation of the global climate variability.