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The analysis of time series is essential for building mathematical models to generate synthetic hydrologic records, to forecast hydrologic events, to detect intrinsic stochastic characteristics of hydrologic variables as well to fill missing and extend records. To this end, this paper examined the stochastic characteristics of the monthly rainfall series of Ilorin, Nigeria vis-à-vis modelling of same using four modelling schemes. The Decomposition, Square root transformation-deseasonalisation, Composite, and Periodic Autoregressive (T-F) modelling schemes were adopted. Results of basic analysis of the stochastic characteristics revealed that the monthly series does not show any discernible presence of long-term trend, though there is a seeming inter-decadal annual variation. The series exhibits strong seasonality throughout its length, both in the moments and autocorrelation and significantly intermittent. Based on assessment of the respective models, the performance of the different modelling schemes can be expressed in this order: T-F > Composite > Square root transformation-Deseasonalised > Decomposition. Considering the results obtained, modelling of monthly rainfall series in the presence of serial correlation between months should be based on the establishment of conditional probability framework. On the other hand, in view of the inadequacy of these modelling schemes, because of the autoregressive model components in the coupling protocol, nonlinear deterministic methods such as Artificial Neural Network, Wavelet models could be viable complements to the linear stochastic framework.

The assessment of the dynamics and regime of a particular hydrologic phenomenon is imperative; especially the time-based characteristics. Time-based characteristics of hydrological data are of great significance in the planning, designing and operation of water systems. This significance is informed more largely due to the variability and oscillatory behaviour of hydrological sequences. Against this backdrop therefore, as noted by Kottegoda [

Like any other aspect of science and engineering developments, there has been a tremendous introduction of new concepts and ideas in rainfall cum precipitation study in general. Notable of such are researches in various directions including space-time structure and variability of rainfall. In this regard, there has been a significant shift from point process models to models based on concepts of scale invariance [

Though generally, hydrologic processes such as precipitation and runoff evolve on a continuous time scale and their estimation correspondingly unduly difficult, in particular, rainfall modelling and its quantitative estimation or forecasting are important considering the fact that it is a critical weather parameter in the estimation of crop water requirement, and development of long lead time flood and flash-flood warning systems. However, it suffices to note that despite substantial progress, several modelling issues still remained unresolved [

The study location is Ilorin (North central Nigeria) at longitude 4˚35' and latitude 8˚30'. It has elevation of between 273 to 333 m and a mean annual temperature of about 27˚C and is characterised by a distinct bi-seasonal weather pattern; i.e., wet and dry. The wet season starts in April and ends in October, while the dry season starts in November and ends in March. The mean annual rainfall is 1150mm, while the relative humidity ranges from 65% - 80%.

Map of Nigeria showing the study location

characteristics like moments and dependence structure of the data series was done to be able to evaluate randomness and trend pattern. In this regard, the time series plot was examined to establish whether it does exhibit intermittency or otherwise as well as seasonal characteristics like trend and moments. The objective here is to evaluate seasonality in the moments. Analysis of dependence structure was done in time and frequency domains; basically through autocorrelation and spectral density, respectively.

In this study, four (4) different modelling schemes were employed; these are a) decomposition, b) square root transformation-deseasonalisation strategy, c) composite modelling and d) Periodic modelling (Thomas-Fier- ing).

1) Decomposition strategy

Here, the data series was de-trended, deseasonalised and further smoothen with a moving average (MA) of order 6 based on the autocorrelation structure of the original raw data. To this end, an additive model of the form in equation (1) was employed.

where,

This procedure requires that the data series be decomposed into seasonal components; the deseasonalisation after the removal of the long-term trend was done by using the seasonal adjustment factors (SAF). These values (SAF) indicate the effect of each period on the level of the series.

After the decomposition process and smoothening, an ARIMA model was fitted into the random or stochastic component left. Based on the analysis of the autocorrelation functions of the random component, a multiplicative ARIMA model was fitted; in this regard,

2) Square root transformation-deseasonalisation scheme

Based on the suggestion of Delleur and Kavvas [

. Seasonal indices (factors).

Month | April | May | June | July | Aug | Sept | Oct | Nov | Dec | Jan | Feb | Mar |
---|---|---|---|---|---|---|---|---|---|---|---|---|

SAF | 0.99 | 64.42 | 85.80 | 37.32 | 35.76 | 145.77 | 43.26 | −88.68 | −91.22 | −94.95 | −88.34 | −50.13 |

Seasonal analysis of the original mean rainfall series (RF) before (a) and after (b) detrending

rescaled (deseasonalised) by subtracting from each term of the series by the corresponding seasonal mean and dividing same by the corresponding standard deviation. The deseasonalisation process is according as:

where,

Correlogram of ARIMA (1, 0, 0) × (1, 0, 1)_{12} model residuals fitted to the stochastic component

Using the autocorrelation functions of the square root transformed and deseasonalised series, a seasonal

ARIMA model of the form:

transformed series with its seasonal component, a reversed rescaling procedure was done; that is,

where, j is the month in a 12-month annual cycle and

monthly rainfall series.

3) Composite modelling

The composite modelling entails decomposing the original data series into its various components; i.e., deterministic and a stochastic component which accounts for the random effects (dependent and independent parts) [_{, }was represented by a decomposition model of the additive type according as equation (4).

where, T(t) is the trend component, P(t) the periodic component and ε(t), the stochastic component.

For the identification of trend, annual rainfall series was used. The annual series was obtained by aggregating the 43 years annual series. In the actual trend detection procedure, a hypothesis of no trend was made and the value of the test statistic (Z) was calculated by using 1) Turning Point Test, 2) Kendall’s Rank Correlation Test and 3) Mann-Kendall Trend Test. The computed values of the test statistic in all instances were −0.852, −0.429, and 0.195, respectively. Considering the values of the computed test statistic (Z), at 5% level of significance, the Z values do not provide reason to suspect the presence of any discernible long-term trend. Thus, the observed rainfall series may be treated as trend free. Hence the composite model, i.e., equation (4) reduces to:

To confirm the presence of periodic component in the monthly rainfall series, a correlogram of the series was drawn.

The parameters of the periodic component of the composite model were evaluated by using the classical harmonic analysis method. To this end, the Cumulative Periodogram (CP) approach was adopted. In this case, the point of intersection of the fast increase in the Periodogram (CP_{i}) and the slow increase is considered and the corresponding harmonics taken as significant and the remaining treated as errors and passed on to the random component; i.e., insignificant. From figure 5, the first four harmonics are considered significant. The periodic component can be expressed as in equation (6a).

Autocorrelation function of the original rainfall series based on water year regime

Cumulative periodogram of the mean monthly rainfall series

where, k is the maximum harmonics,

Based on figure 5, the resulting periodic component can be expressed according as equation (6b).

Based on the autocorrelation of the residual series left after the periodic component was removed from the

original series,

model of the monthly rainfall series, i.e., equation (5) then becomes

4) Periodic Autoregressive modelling Scheme

Modelling of the monthly rainfall series using periodic autoregressive model was done by adopting the Thomas-Fiering (T-F) model. The T-F model is a linear stochastic model for stimulating synthetic series of seasonal hydrologic process. The schema for the rainfall modelling using this framework takes the form

. Harmonic coefficients.

Harmonics | Coefficients | |
---|---|---|

j | α | β |

1 | −0.630724 | 4.410554 |

2 | 0.512422 | 0.773266 |

3 | 4.514855 | 6.113874 |

4 | −2.476685 | −3.618778 |

This model uses a linear regression relationship to relate the storm rf_{t+1} in the (t+1) month to storm rf_{t} in the t(th) month. Here, _{j} is the regression coefficient and

In all the instances, for the respective modelling strategy, split sampling procedure was adopted; i.e., one segment of the monthly rainfall series (40 years’ time period) was used for modelling while the remaining three years data was used for model validation. For model validation/forecasting, forecast functions corresponding to the respective ARIMA modelling scheme was adopted using the difference equation form. In this regard, recalling that Z_{t}(L) = [_{Zt+L}], using square brackets to signify conditional expectations and noting that

the following forecast functions were employed, viz:-

a) Decomposition modelling scheme:

b) Square root transformation-deseasonalisation strategy:

c) Composite modelling scheme (i.e., the stochastic component):

Hydrologic processes such as precipitation and runoff evolve on a continuous time scale. The implication(s) of this is simple; as shown by

Monthly rainfall time series plot

Seasonal correlogram showing periodic stochasticity/intermittency

the fact the series takes on nonzero and zero values throughout the entire length of the record (

In the same context, figure 8 shows inter-annual decadal variation in the rainfall series; long-term trend pattern is seemingly not evident. However, there is large variability among the monthly values of rainfall of different years, with the period 1995-2009 showing slight increases in the storm event during the peak seasons. On the other hand, figure 9, figure 10 shows the presence of seasonality in the moments, meaning that monthly statistics for dry season are significantly different from those of the wet season period. Unlike intermittent stream flow process, the seasonal means have higher values than the seasonal deviations throughout the year. As noted in

To assess this, analysis of dependence structure in time series via spectral density is critical; figure 11 shows the dependence structure of the monthly rainfall in a frequency domain. The spectral density exhibits a discrete spectral component at the frequency of 1/12 cycle per month. This periodicity is seen in figure 11(a). Similarly, the periodogram exhibits quite a corresponding pattern in terms of the periodicity. However, as noted by Kottegoda [

Inter-annual mean monthly rainfall variation pattern

Variation in seasonal moments

. Seasonal coefficients of skewness (g).

S/No. | Months (Seasons) | Coefficients of Skewness (g) |
---|---|---|

1 | April | 0.8960 |

2 | May | 0.5088 |

3 | June | −0.0808 |

4 | July | 0.6821 |

5 | August | 0.9659 |

6 | September | −0.2462 |

7 | October | 0.2905 |

8 | November | 1.7530 |

9 | December | 4.8412 |

10 | January | 3.5736 |

11 | February | 3.0213 |

12 | March | 0.5701 |

(a) (b)

to varying degrees of accuracy whereas the decomposition strategy failed completely. In above, the positive attribute of the T-F model ahead of others reinforces its suitability for adoption in rainfall modelling.

. Seasonal moments for both the observed series and the models in the simulation phase.

Months | Observed data | Decomposition | Sqrt-deseasonlised | Composite | T-F | |||||
---|---|---|---|---|---|---|---|---|---|---|

Mean | Std. dev | Mean | Std. dev | Mean | Std.dev | Mean | Std.dev | Mean | Std.dev | |

April | 131.84 | 35.01 | 93.61 | 0.67 | 102.34 | 2.01 | 94.39 | 4.19 | 71.45 | 49.02 |

May | 152.43 | 115.29 | 94.10 | 0.08 | 146.51 | 0.63 | 147.30 | 8.66 | 159.95 | 84.11 |

June | 177.01 | 20.52 | 92.72 | 0.06 | 174.59 | 0.21 | 185.62 | 3.56 | 174.21 | 86.32 |

July | 191.46 | 8.70 | 108.53 | 0.07 | 155.32 | 2.36 | 140.53 | 1.74 | 147.60 | 60.72 |

Aug | 155.99 | 21.41 | 99.58 | 0.07 | 134.72 | 1.08 | 145.54 | 5.13 | 96.01 | 33.33 |

Sept | 203.93 | 33.83 | 89.96 | 0.06 | 206.15 | 3.31 | 235.12 | 2.33 | 193.85 | 106.50 |

Oct | 168.46 | 64.67 | 96.97 | 0.06 | 137.54 | 1.17 | 125.88 | 3.62 | 114.12 | 39.18 |

Nov | 19.50 | 16.63 | 97.78 | 0.06 | 6.51 | 0.33 | 13.58 | 0.83 | 7.38 | 6.77 |

Dec | 2.20 | 3.81 | 96.12 | 0.06 | 1.23 | 0.01 | 6.44 | 0.38 | 3.55 | 4.42 |

Jan | 5.36 | 9.29 | 98.56 | 0.06 | 1.86 | 0.12 | 6.54 | 0.59 | 8.91 | 15.43 |

Feb | 0.00 | 0.00 | 96.46 | 0.06 | 5.12 | 0.10 | 10.32 | 0.61 | 0.53 | 0.91 |

March | 25.93 | 23.62 | 94.68 | 0.06 | 34.72 | 0.28 | 43.65 | 2.26 | 43.79 | 28.16 |

Considering the performance of the models adopted, it is imperative to look at the implications of the data pre-processing strategy. In all the models, except the Thomas-Fiering (T-F) model, ARIMA models were used to model the supposedly stationary stochastic component. To achieve stationarity, seasonal differencing (12-lag) and seasonal standardisation (deseasonalisation) were respectively applied but not without its associated problems. For instance, the deseasonalisation process is a misnomer since it implies that the deseasonalised series is free of seasonality; however, other seasonality may still be present [

For purposes of identifying a more realistic modelling scheme for the rainfall series, assessment of the stochastic characteristics was done to be able to understand the dynamics of the monthly series. Sequel to this, four different modelling schemes: Decomposition, Square root transformation-deseasonalisation, Composite, and Periodic autoregressive modelling (T-F), were adopted. Results of basic analysis of the stochastic characteristics revealed that the monthly series does not show any discernible presence of long-term trend, though there is a seeming inter-decadal annual variation. It is evident that the series exhibits strong seasonality throughout its length, both in the moments and autocorrelation. This gives rise to significant correlation which is attributable to the serial dependence of the same month on several years; this serial dependence is same for all 12 months. The strong seasonal autocorrelation structure connotes intermittency considering the fact that the series assumes nonzero and zero values throughout its length for the period considered.

Resulting from the analysis and the modelling exercise, the Thomas-Fiering (T-F) model can be used for monthly rainfall modelling and short-term forecast. In addition, both the composite and square root transformation-deseasonalisation schemes may also be employed but not without caution. Because of the ARIMA model component of these models in the coupling, their forecast abilities were impaired considering the inadequacy of their respective forecast errors to preserve the observed standard deviations of the rainfall series. This primarily might have arisen from the second-order stationarity assumptions requirement of the autoregressive models. In the same vein, whole decomposition of any trend-free series requiring de-trending, deseasonalisation followed by moving average smoothing, and fitting of ARIMA model might be too excessive as it distorts the entire spectrum in the overall and not encouraged. The results obtained suggest that modelling of monthly rainfall series in the presence of serial correlation between months should be based on the establishment of conditional probability framework; in this case, two conditional probabilities: probability that month t has zero rainfall given that month t-1 had non-zero rainfall and probability that month t has zero rainfall, given that month t-1 had zero rainfall. On the other hand, considering the inadequacy of these modelling schemes because of the autoregressive model components, nonlinear deterministic methods such as Artificial Neural Network, Wavelet models could be viable complement to the linear stochastic framework.