The Open Access Journal of Science and Technology

Vol. 3 (2015), Article ID 101120, 7 pages

doi:10.11131/2015/101120

## Modelling West African Total Precipitation Depth: A Statistical Approach

#### Abstract

Even though several reports over the past few decades indicate an increasing aridity over West Africa, attempts to establish the controlling factor(s) have not been successful. The traditional belief of the position of the Inter-tropical Convergence Zone (ITCZ) as the predominant factor over the region has been refuted by recent findings. Changes in major atmospheric circulations such as African Easterly Jet (AEJ) and Tropical Easterly Jet (TEJ) are being cited as major precipitation driving forces over the region. Thus, any attempt to predict long term precipitation events over the region using Global Circulation or Local Circulation Models could be flawed as the controlling factors are not fully elucidated yet. Successful prediction effort may require models which depend on past events as their inputs as in the case of time series models such as Autoregressive Integrated Moving Average (ARIMA) model. In this study, historical precipitation data was imported as time series data structure into an *R *programming language and was used to build appropriate Seasonal Multiplicative Autoregressive Integrated Moving Average model, ARIMA (*p*, *d*, *q*)*(*P*, *D*, *Q*). The model was then used to predict long term precipitation events over the Ghanaian segment of the Volta Basin which could be used in planning and implementation of development policies.

#### 1. Introduction

##### 1.1. Background

Climate change has resulted in extreme drought condition in some parts of the world and flooding in other parts [14]. Environmental changes in Africa have mostly been directly related to rainfall [8,15]. Precipitation over most regions of the continent traditionally has been associated with the seasonal excursion of the Inter Tropical Convergence Zone (ITCZ). Recent studies have however revealed that the seasonal development of the tropical rain belt over Africa is driven by several factors of the general atmospheric circulation which in turn control the location and characteristics of the ITCZ [15,8,10]. This atmospheric circulation is believed to generate and maintain wave disturbances that modulate the rainfall field. [9] found the Tropical Easterly Jet (TEJ) as one of the most intense circulation features over Africa and concluded that the TEJ may be a critical factor in the development of the rainy season and the overall climate in West Africa as opposed to the traditional belief of seasonal movements of the inter-tropical convergence zone (ITZC). It is obvious that long term prediction of rainfall over West Africa using Global Circulation Model (GCMs) could be flawed as factors controlling precipitation over the region are still not well understood [14]. Successful prediction effort will require models which depend on past events as their inputs as in the case of Autoregressive Integrated Moving Average (ARIMA) model. The overarching goal of this research is to develop a model that can reliably forecast the catchment's long term rainfall event for effective and sustainable water resources management.

#### 2. Materials and Methods

##### 2.1. The study area

The study focused on the Ghanaian segment of the Volta River Basin. The area stretches from the mid section of the country (7˚30′20.35″N, 2˚59′47.22″W to 7˚10′24.80″N, 1˚09′10.09″E) to the northern section bordering three West African countries namely Togo in the east, Burkina Faso in the north and La Cote d'Ivoire in the west. Below the mid section it tapers towards south eastern and ended at 5˚46′11.60″N, 0˚43′02.74E″, where the Volta River empties into the Atlantic Ocean. The study area covers a total of 167, 692 square kilometres (70% of the country's total land area) constituting 40.18% of the total area of the Volta River Basin (417, 382 square kilometres) [1,3,11,2].

##### 2.2. Selection of appropriate ARIMA model

Rainfall over West Africa may not be modelled adequately using Global Circulation Model (GCMs) since the factors controlling precipitation over the region are still not well understood [14,12]. Successful prediction effort will require statistical modelling approach which uses historical records to estimate the model parameters as in the case of Autoregressive Integrated Moving Average (ARIMA) model. The general forms of these models are [13]:

${Y}_{t}={\Phi}_{0}+{\Phi}_{1}{Y}_{t-1}+\cdots +{\Phi}_{p}{Y}_{t-p}+{\u03f5}_{t}+\mu -{\theta}_{1}{\u03f5}_{t-1}-\cdots -{\theta}_{q}{\u03f5}_{t-q},$ | (1) |

where: *Y*_{t} = response variable at time *t*, *Y*_{t−1}, ⋯, *Y*_{t−p} = response variables at time lags at *t* − 1, ⋯, *t* − *p*, respectively, *Φ*_{0}, *Φ*_{1}, ⋯, *Φ*_{p} = coefficients to be estimated for the AR(p) model, *θ*_{1}, ⋯, *θ*_{p} coefficients to be estimated, *μ* = constant mean of the process, *ϵ*_{t} = error term at time *t*.

Rearranging the ARMA (*p*, *q*) model and introducing a differencing order gives the general Autoregressive Integrated Moving Average, ARIMA (*p*, *d*, *q*) model as shown below [13]:

$\begin{array}{cc}\hfill {Y}_{t}-{\Phi}_{0}-{\Phi}_{1}{Y}_{t-1}-\cdots -{\Phi}_{p}{Y}_{t-p}& =\mu +{\u03f5}_{t}-{\theta}_{1}{\u03f5}_{t-1}-\cdots -{\theta}_{q}{\u03f5}_{t-q}\hfill \\ \hfill \left(1-{\Phi}_{0}-{\Phi}_{1}B-{\Phi}_{2}{B}^{2}-\cdots -{\Phi}_{p}{B}^{p}\right){Y}_{t}& =\mu +\left(1-{\theta}_{1}B-\cdots -{\theta}_{q}{B}^{q}\right){\u03f5}_{t}\hfill \\ \hfill \Phi \left(B\right){\left(1-B\right)}^{d}{Y}_{t}& =\mu +\theta \left(B\right){\u03f5}_{t}\hfill \\ \hfill {Y}_{t}& ={\Phi}^{-1}\left(B\right){\left(1-B\right)}^{-d}\left(\mu +\theta \left(B\right){\u03f5}_{t}\right),\hfill \end{array}$ | (2) |

where *Φ*
(*B*) = *Φ*_{0} + *Φ*_{1}*Y*_{t−1} + ⋯+*Φ*_{p}*Y*_{t−p} + *ϵ*_{t}*a**n**d**θ*
(*B*) = *ϵ*_{t} − *θ*_{1}*ϵ*_{t}1 − ⋯−*θ*_{q}*ϵ*_{t−q}.

##### 2.3. Modelling of spatial component of the rainfall total

The ARIMA model considers the temporal variations only since the observed data are collected at some locations within the study area, for the spatial dimension to be accounted for in the time series analysis, a dedicated spatial analysis package is required to model the spatial component in the observations. In this study, the researcher used Geographic Information System (GIS) to model historical spatial variation of the precipitation over nearly forty year (40) record. 1967–2006 monthly precipitation data of the study area spanning roughly forty years were obtained from Ghana Meteorological Agency. A total of 33 rainfall stations data were analyzed. Though the available number (stations) of rainfall data was few it covered the entire study area thus giving a good spatial representation (Figure 1).

The monthly rainfall for each year was interpolated over a grid using moving average interpolation method. Data gaps in the monthly recordings were filled using predicted values from the interpolation. The complete monthly data as obtained above were converted into a vector layer. The vector data was rasterized and a distance between each station point was calculated. A Thiessen polygon map depicting area coverage of a particular rainfall station was created for each month. The Thiessen polygon map was crossed with sub-basins map of the study area (Figure 2). Monthly rainfall total for each sub-catchment was generated from the cross operation by using aggregate function of GIS attribute data, the sub-catchments monthly rainfall were summed to monthly total precipitation within the study area. The monthly total rainfall data were then imported into a statistical programming environment *R*. Using *R*time series analysis function, the data was decomposed into seasonal, trend and irregular components.

Box-Jenkins approach was used to select the appropriate Seasonal Multiplicative Autoregressive Integrated Moving Average ARIMA (*p*, *d*, *q*)*(*P*, *D*, *Q*) model which fits the time series data. The dataset was divided into two, comprising 1967–2000 (33 years) record for model building and 2000–2006 (6 years) for model validation. The researcher used graphical methods to identify normality, trend, cyclical and seasonal variations in the data. A second order differencing was then used to render the data stationary. Six different options of the ARIMA models were fitted and for each model the researcher computed the Box-Jenkins test and also compared histogram of the residuals, residuals quantile versus theoretical quantile plot, autocorrelation function and the p-values of the Ljung-Box statistics. The model with the least Akaike Information Criterion (AIC) was parsimoniously selected among the rest. The fitted model was used to predict a six-year (2000–2006) time step ahead. The predicted values were plotted with 95% confidence interval with the observed data (2000–2006 validation record) superimposed on the plot of the predicted data. The model prediction was very good. The validated model was therefore used to predict forty-year time step ahead of precipitation events over the study area.

#### 3. Results

##### 3.1. Rainfall pattern

Different model parameter estimates were obtained during the search stage (Table 1). The Aike Information Criterion (AIC) for each model was compared. Based on the AIC and the principle of parsimony in model fitting, second order seasonal autoregressive integrated moving average model ARIMA (2,1,1) (1,1,1) was selected.

Substituting the estimated parameters of the selected seasonal ARIMA (2,1,1)*(1,1,1)_{12} model that predicts the average monthly rainfall depth *Y*_{t} − *Y*_{t−1} = *Δ**Y*_{t}, in the general formula gave the Seasonal Autoregressive Integrated Moving Average Model as below:

$\begin{array}{cc}\hfill {\displaystyle}& \Phi \left({B}^{h}\right)\Phi \left(B\right){\Delta}_{h}^{D}{\Delta}^{d}{Y}_{t}=\alpha +\Theta \left({B}^{h}\right)\theta \left(B\right){\u03f5}_{t}\hfill \\ & \left(1-{\Phi}_{1}{B}^{h}-\cdots -{\Phi}_{p}{B}^{ph}\right)\left(1-{\Phi}_{1}B-\cdots -{\Phi}_{p}{B}^{p}\right){\left(1-{B}^{h}\right)}^{D}\Delta {Y}_{t}\hfill \\ \hfill {\displaystyle}& =\alpha +\left(1-{\Theta}_{1}{B}^{h}-\cdots -{\Theta}_{Q}{B}^{Qh}\right)\left(1-{\theta}_{1}B-\cdots -{\theta}_{q}{B}^{q}\right){\u03f5}_{t}\hfill \\ \hfill {\displaystyle}& \left(1-{\Phi}_{1}{B}^{12}\right)\left(1-{\Phi}_{1}B-{\Phi}_{2}{B}^{2}\right)\left(1-{B}^{2}\right)\left(1-{B}^{12}\right)\left(1-B\right)\Delta {Y}_{t}\hfill \\ \hfill {\displaystyle}& =\left(1-{\Theta}_{1}{B}^{12}\right)\left(1-{\Theta}_{1}{B}^{12}\right)\left(1-{\theta}_{1}B\right){\u03f5}_{t}\hfill \\ \hfill {\displaystyle}& \left(1+0.0468{B}^{12}\right)(1-0.1413B-0.0860{B}^{2})(1-{B}^{12})\left(1-B\right)\Delta {Y}_{t}\hfill \\ & =\left(1+0.9356{B}^{2}\right)\left(1+0.9813B\right){\u03f5}_{t}\hfill \\ & (1+0.0468\Delta {Y}_{t-12}\left)\right(1-0.1413\Delta {Y}_{t-1}-0.0860\Delta {Y}_{t-2}\left)\right(1-\Delta {Y}_{t-12}\left)\right(1-\Delta {Y}_{t-1})\Delta {Y}_{t}\hfill \\ & =\left(1+0.9356{\u03f5}_{t-12}\right)\left(1+0.9813{\u03f5}_{t-1}\right){\u03f5}_{t}\hfill \\ & {\displaystyle \Delta {Y}_{t}=\frac{\left(1+0.9356{\u03f5}_{t-12}\right)\left(1+0.9813{\u03f5}_{t-1}\right){\u03f5}_{t}}{\left[\right(1+0.0468\Delta {Y}_{t-12}\left)(1-0.1413\Delta {Y}_{t-1}-0.0860\Delta {Y}_{t-2}\left)\right(1-\Delta {Y}_{t-12}\right)(1-\Delta {Y}_{t-1}\left)\right]}}\hfill \end{array}$ | (3) |

After applying a non-seasonal and seasonal differencing of the time series data it was rendered stationary. The stationarity effect is shown in autocorrelation graph as spike cuts off quickly. Plot of the residuals, theoretical quantile plot and histogram plot after applying non-seasonal and seasonal autoregressive (AR) and moving average (MA) models to the transformed data indicate good fit of the ARIMA model. The Box-Jenkins statistics also clearly indicates the goodness of fit of the model as the p values are highly significant from zero (0). A plot of the six-year model prediction (from 2000 to 2006) against observed recordings over the same period at 95% confidence interval clearly shows an excellent performance of the model (Figure 3).

##### 3.2. Prediction result

A forty-year time step ahead (2007 to 2047) prediction of rainfall over the study area by the

model at 95% confidence interval indicates a steady decline in precipitation. It is observed from the decomposed plot that precipitation over the study area will decline within the next forty year period by approximately 9%.

#### 4. Discussion

##### 4.1. Rainfall pattern

The result of the analysis clearly showed that precipitation over the Volta catchment area is on the steady decline. The total rainfall within the catchment over the last forty years has declined by approximately seven and half percent (7.5%). The model prediction for the next forty years indicated a further decline of approximately nine percent (9%). The model performed very well as the predicted values corresponded well with the observed values. The results of the analysis corroborated other earlier reports of the decline of the catchment's precipitation [3,4]. Similar downward trend was reported by [5] in the Sahel region of West Africa. [6] reported that the decade of 1980s was the driest of the twentieth century for the continent as a whole and for West Africa in particular. Rainfall fluctuations in the continent are similar as observed in the 1980s. There were periods when rainfall within West Africa was high as opposed to the situation in East and Southern Africa. For instance, in the 1990s and 1950s when rainfall throughout East Africa was low, the reverse occurred in West Africa [7].

#### 5. Conclusion

Integrating Geographic Information System (GIS) with Autoregressive Integrated Moving

Average (ARIMA) model for time series phenomena could account for the spatial component. In this study, GIS assessment of spatio-temporal distribution of monthly total precipitation depth was carried out, and the resultant time series data was used to build ARIMA model. From the model prediction, it was observed that the catchment's rainfall will decrease further approximately by 9% within the next forty years. This indicates that the catchment's water resources would be significantly threatened and calls for prudent and pragmatic water resources management strategies. The study counteracted the findings that the end as well as the total amount of rainfall will remain more or less fixed.

#### 6. Recommendations

- 1. It is recommended that the government should continuously review the catchment's precipitation to assess any variability which might not be accounted for in the current model prediction.
- 2. Similar models of the catchment's surface temperature, evapotranspirtation and insolation should be built so that relationship between rainfall and these variables could be developed through regression modelling.
- 3. The Ghanaian Government should further promote and intensify storage of surface run-off in small reservoirs in the northern part of the country as part of climate change adaptation strategies as the study showed further decrease in total precipitation depth in the coming years.

#### Acknowledgments

The author is completely indebted to International Foundation for Science (IFS), Swedish International Development Cooperation Agency (SIDA), NATUR, and OIC's Standing Committee on Scientific and Technological Cooperation for the Promotion and Cooperation of Science and Technology (COMSTECH) for the funding support for this project (Grant no. W/4300-1).

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The Open Access Journal of Science and Technology

Vol. 3 (2015), Article ID 101120, 7 pages

doi:10.11131/2015/101120

## Modelling West African Total Precipitation Depth: A Statistical Approach

#### Abstract

Even though several reports over the past few decades indicate an increasing aridity over West Africa, attempts to establish the controlling factor(s) have not been successful. The traditional belief of the position of the Inter-tropical Convergence Zone (ITCZ) as the predominant factor over the region has been refuted by recent findings. Changes in major atmospheric circulations such as African Easterly Jet (AEJ) and Tropical Easterly Jet (TEJ) are being cited as major precipitation driving forces over the region. Thus, any attempt to predict long term precipitation events over the region using Global Circulation or Local Circulation Models could be flawed as the controlling factors are not fully elucidated yet. Successful prediction effort may require models which depend on past events as their inputs as in the case of time series models such as Autoregressive Integrated Moving Average (ARIMA) model. In this study, historical precipitation data was imported as time series data structure into an *R *programming language and was used to build appropriate Seasonal Multiplicative Autoregressive Integrated Moving Average model, ARIMA (*p*, *d*, *q*)*(*P*, *D*, *Q*). The model was then used to predict long term precipitation events over the Ghanaian segment of the Volta Basin which could be used in planning and implementation of development policies.