Excerpt
TABLE OF CONTENTS
DECLARATION AND RECOMMENDATION
DECLARATION
RECOMMENDATION
COPYRIGHT
ACKNOWLEDGEMENT
DEDICATION
ABSTRACT
LIST OF SYMBOLS
ABBREVIATIONS AND ACRONYMS
LIST OF TABLES
LIST OF FIGURES
CHAPTER ONE
INTRODUCTION
1.1 Background information
1.2 Statement of the problem
1.3 Objectives
1.3.1 Main objective
1.3.2 Specific objectives
1.4 Research questions
1.5 Justification
1.6 Scope
CHAPTER TWO
LITERATURE REVIEW
2.1 Occurrence of droughts
2.1.1 Types of droughts
2.1.2 Drought modelling
2.1.3 Determination of drought threshold level
2.1.4 Selection of drought threshold level
2.2 Climate change and variability
2.2.1 Impact of climate change on water resources
2.2.2 Effect of water balance components on drought
2.2.3 Effect of global warming on droughts
2.2.4 IPCC and IGAD approach
2.3 Major causes of drought in Kenya
2.3.1 Impact of drought in Kenya
2.3.2 Drought monitoring in Kenya
2.4 Drought forecasting
2.5 Drought mitigation
2.6 Drought assessment methods
2.7 Satellite based drought indices
2.7.1 Vegetative condition index
2.7.2 Normalized difference vegetative index
2.7.3 Normalized difference water index
2.7.4 Water supply vegetative index
2.7.5 Normalized difference drought index
2.8 Data driven drought indices
2.8.1 Standardized precipitation index
2.8.2 Palmer drought severity index
2.8.3 Surface water supply index
2.8.4 Aggregated drought index
2.8.5 Deciles index
2.9 Drought forecasting models
2.9.1 Seasonal autoregressive integrated moving average model
2.9.2 Adaptive Neuro-fuzzy inference system model
2.9.3 Markov chain model
2.9.4 Log-linear model
2.9.5 Artificial Neural Network models
2.10 Description of ANN model
2.10.1 Classification of ANN model architectures
2.10.2 Drought forecasting using ANN models
2.10.3 ANN data pre-processing
2.11 ANN learning processes
2.11.1 Supervised learning
2.11.2 Unsupervised learning
2.12 Purpose for ANNs learning process
2.12.1 Learning for classification
2.13 Drought assessment and forecasting in river basins
2.14 Drought assessment and forecasting in the upper Tana River basin
2.15 AquaCrop model
2.16 Kriging interpolation technique
CHAPTER THREE
MATERIALS AND METHODS
3.1 Study area
3.2 Assessment of spatial and temporal drought using selected DIs
3.2.1 Hydro-meteorological data acquisition
3.2.2 Stream flow data
3.2.3 Precipitation data
3.2.4 Consistency test of the hydro-meteorological data
3.2.5 Filling in missing data
3.2.6 Surface Water Supply Index
3.2.7 Stream flow drought index
3.2.8 Standardized precipitation index
3.2.9 Effective drought index
3.2.10 Soil Moisture Deficit Index
3.2.11 Smulation of Soil Water (SW) content using AquaCrop model
3.2.12 Palmer Drought Severity Index
3.2.13 Evaluation of Spatial distribution of drought severity
3.2.14 Mann-Kendall trend test for drought conditions
3.3 Drought forecasting using DIs and ANNs
3.3.1 Drought forecasting
3.3.2 Temporal drought forecasting using DIs
3.3.3 Short-term drought forecasting
3.3.4 Medium-term drought forecasting
3.3.5 Long-term drought forecasting
3.4 Formulation of Nonlinear-Integrated Drought Index (NDI)
3.4.1 Computation of principal components (PC)
3.4.2 Assessment of drought characteristics using the formulated NDI
3.5 Drought forecasting using NDI
3.5.1 Identification of ANN model structure
3.5.2 Drought projection using NDI and Recursive Multi-Step Neural Networks
3.6 Sensitivity analysis of drought indices
3. 7 Time series drought characterization
3.8 Model calibration
3.9 Model validation
3.9.1 The correlation coefficient
3.9.2 Mean absolute error
3.9.3 Mean square error
3.9.4 Nash–Sutcliffe efficiency
3.9.5 Modified index of agreement
CHAPTER FOUR
RESULTS AND DISCUSSIONS
4.1 Temporal and spatial drought conditions
4.1.1 Time series SWSI
4.1.2 Sensitivity of SWSI to weighting parameters
4.1.3 Development and modification of SWSI equation
4.1.4 Spatially distributed drought severity based on SWSI
4.1.5 Time series SDI
4.1.6 Time series SPI
4.1.7 Spatially distributed drought severity based on SPI
4.1.8 Monthly time series EDI
4.1.9 Spatially distributed drougt severity based on EDI
4.1.10 Time series Soil Moisture Deficit Index (SMDI)
4.1.11 Spatially distributed drought severity based on SMDI
4.1.12 Time series Palmer Drought Severity Index (PDSI)
4.1.13 Spatially distributed drought severity based on PDSI
4.1.4 Characteristics of time series drought conditions
4.2 Forecasted drought using DIs and ANNs
4.2.1 Hydrological drought forecasts
4.2.2 Meteorological drought forecasts
4.2.3 Agricultural drought forecasts
4.3 Formulated NDI for the upper Tana River basin
4.3.1 Sensitivity of NDI to the input parameters
4.4 Forecasts of NDI values using ANNs
4.4.1 Drought projections based on NDI and RMSNN
4.4.2 Spatially distributed drought severity based on NDI
CHAPTER FIVE
CONCLUSIONS AND RECOMMENDATIONS
5.1 Conclusions
5.2 Recommendations
APPENDICES
LIST OF SYMBOLS
Abbildung in dieser Leseprobe nicht enthalten
ABBREVIATIONS AND ACRONYMS
Abbildung in dieser Leseprobe nicht enthalten
LIST OF TABLES
Table 3. 1: Stream flow gauge stations
Table 3. 2: Meteorological stations
Table 3. 3: Drought classification based on SWSI
Table 3. 4: Definition of states of drought based on SDI
Table 3. 5: Drought conditions based on SPI
Table 3. 6: Drought severity based on EDI
Table 3. 7: Dominant soils for the upper Tana River basin
Table 3.8: Classification of drought based on PDSI
Table 3. 9: Different categories of drought forecasting
Table 3. 10: Key variables for short-term drought forecasting
Table 3. 11: Variables for medium-term drought forecasting
Table 3. 12: Variables for long-term drought forecasting
Table 3. 13: Principal Component Analysis in MATLAB
Table 3. 14: Classification of drought based on NDI
Table 3. 15: NDI input variables for drought forecasting
Table 4. 1: Weighted Parameters for SWSI at Gura gauging station for 1970
Table 4. 2: Best ANNs for short-term drought forecasting of SWSI and SDI
Table 4. 3: Best ANNs for medium-term drought forecasting of SWSI and SDI
Table 4. 4: Best ANNs for long-term drought forecasting of SWSI and SDI
Table 4. 5: Best ANNs for short-term drought forecasting of SPI and EDI
Table 4. 6: Best ANNs for medium-term drought forecasting of SPI and EDI
Table 4. 7: Best ANNs for long-term drought forecasting of SPI and EDI
Table 4. 8: Best ANNs for short-term drought forecasting of SMDI and PDSI
Table 4. 9: Best ANNs for medium-term drought forecasting of SMDI and PDSI
Table 4. 10: Best ANNs for long-term drought forecasting of SMDI and PDSI
Table 4. 11: Computed NDI for January at Sagana FCF hydrometric station
Table 4. 12: Best ANNs for different months lead time forecasts of NDI
Table 4. 13: Classification of the reported drought at Kamburu hydrometric station
LIST OF FIGURES
Figure 2. 1: Propagation of drought via hydrological cycle
Figure 2. 2: Network of phases of drought modelling
Figure 2. 3: Fundamental structure of a biological neuron
Figure 2. 4: Fundamental parts of a typical neural network
Figure 2. 5 (a-d): Types of ANN activation functions
Figure 2. 6: Classification of ANN model architecture
Figure 2. 7 ( a and b): Feed forward artificial neural networks
Figure 2. 8(a and b): Types of recurrent ANNs
Figure 2. 9: Trend of training error and the point of over-fitting
Figure 3. 1: Map of the upper Tana River basin
Figure 3. 2: Flow chart of the steps used in filling the missing data using ANN
Figure 3.3: Process of evaluating drought using the modified SWSI
Figure 3. 4: Process for computation of the time series SPI
Figure 3.5: Flow chart showing the steps in computation of SMDI
Figure 3. 6: Flow chart of the applied ANN-based drought forecasting model
Figure 3. 7: ANN Architecture used for the forecasting of SWSI
Figure 3. 8: Flow chart showing the process for computation of NDI
Figure 3. 9: Flow chart of ANN-based drought forecasting model for NDI
Figure 3. 10: A three-layer DNN
Figure 3. 11: Three Layer RMSNN used for drought projection
Figure 4. 1: Sensitivity of SWSI to decrease in weighted parameters
Figure 4. 2: Sensitivity of SWSI to increase in weighted parameters
Figure 4. 3: Time series of SWSI for Yatta furrow gauging station
Figure 4. 4: Time series of SWSI for Nyamindi gauging station
Figure 4. 5: Time series of SWSI for Tana sagana gauging station
Figure 4. 6: Time series of SWSI for at Amboni gauging station
Figure 4. 7: Trend of the mean monthly SWSI and precipitation
Figure 4. 8(a-e): Spatially distributed drought severity in the upper Tana River basin
Figure 4. 9(a-e): Spatially distributed drought frequency of severe drought for SWSI
Figure 4. 10: Spatially distributed Mann-Kendall trend of severe drought for SWSI
Figure 4. 11: Mean yearly temperature at Nyeri hydrometric station for 1978-2012
Figure 4. 12: Mean decadal temperature at Nyeri hydrometric station for 1970-2010
Figure 4. 13: Time series of SDI and stream flow for Amboni gauging station
Figure 4. 14: Time series of SDI and stream flow for Tana sagana gauging station
Figure 4. 15: Time series of SDI and stream flow for Nyamindi gauging station
Figure 4. 16: Time series of SDI and stream flow for Kamburu gauging station
Figure 4. 17: Time series SPI and precipitation for Sagana FCF meteorological station
Figure 4. 18: Time series SPI and precipitation at Kerugoya DWO meteorological station
Figure 4. 19: Time series SPI and precipitation for Nyeri meteorological station
Figure 4. 20: Time series SPI and precipitation for Naro-moru meteorological station
Figure 4. 21(a-e): Spatially distributed drought severity based on SPI
Figure 4. 22(a and b): Frequency of severe drought and its trend based on SPI
Figure 4. 23: Time series EDI and precipitation for Nyeri meteorological station
Figure 4. 24: Time series EDI and precipitation forKerugoya DWO meteorological station
Figure 4. 25: Time series EDI and precipitation for Sagana meteorological station
Figure 4. 26: Time series EDI and precipitation for Naro-moru meteorological station
Figure 4. 27(a-e): Spatially distributed drought severity based on EDI
Figure 4. 28: Spatially distributed drought frequency based on SPI
Figure 4. 29: Spatially distributed Mann-Kendall trend test of drought based on SPI
Figure 4. 30: Time series of SMDI for dry season of at MIAD meteorological station
Figure 4. 31: Time series of SMDI for wet season at MIAD meteorological station
Figure 4. 32: Time series of SMDI for dry season at Naro-moru meteorological station
Figure 4. 33: Time series of SMDI for wet season at Naro-moru meteorological station
Figure 4. 34(a-c): Spatially distributed drought severity based on SMDI
Figure 4. 35: Time series of PDSI for dry seasons of at MIAD meteorological station
Figure 4. 36: Time series of PDSI for wet seasons at MIAD meteorological station
Figure 4. 37: Time series of PDSI for dry seasons at Naro-moru meteorological station
Figure 4. 38: Time series of PDSI for wet seasons at Naro-moru meteorological station
Figure 4. 39(a and b): Spatially distributed magnitude of PDSI-based drought in October
Figure 4. 40(a-d): Spatially distributed PDSI-based drought severity
Figure 4. 41: Drought characteristic curves of SWSI for Amboni hydrometric station
Figure 4.42: Severity-Duration-Frequency (SDF) curves at Amboni based on SWSI
Figure 4. 43: Drought-severity contour map for (a) 10 and (b) 50-year return period
Figure 4. 44: Mean drought frequency for entire basin and south-eastern areas
Figure 4. 45: Relation between the Qm, SDI and T for different gauging stations
Figure 4. 46: Plot of Y versus Cv at Kamburu station in the upper Tana River basin
Figure 4. 47: Forecasting efficiency verses trials at Yatta furrow gauge station
Figure 4. 48: Regression of the best ANNs of SWSI at Yatta furrow gauging station
Figure 4. 49: MSE results for L-M algorithm at Yatta furrow station
Figure 4. 50: Observed SWSI and best ANNs forecasts at Amboni gauge station
Figure 4. 51: Comparison of SWSI and SDI forecasts at Amboni gauge station
Figure 4. 52: Comparison of SWSI and SDI forecasts at Yatta furrow gauge station
Figure 4. 53: Comparison of SPI and EDI forecasts at MIAD meteorological station
Figure 4. 54: Comparison of SPI and EDI forecasts at Naro-moru meteorological station
Figure 4. 55: Comparison of SMDI and PDSI forecasts for drought at MIAD station
Figure 4. 56: Comparison of SMDI and PDSI forecasts for drought at Naro-moru station
Figure 4. 57: Absoulute sensitivity for NDI at MIAD meteorological station
Figure 4. 58: Performance of NDI forecasts at MIAD meteorological stations
Figure 4. 59: Performance of NDI forecasts at Naro-moru meteorological stations
Figure 4. 60(a-c): Observed NDI and best ANNs forecast results at Sagana FCF station
Figure 4. 61: Observed and projected drought at Kamburu hydrometric station
Figure 4. 62: Spatially distributed drought severity for the years based on NDI
DEFINITION OF TERMS
Abbildung in dieser Leseprobe nicht enthalten
ACKNOWLEDGEMENT
I express my appreciation to the academic staff in the Faculty of Engineering and Technology, Egerton University who have provided a very conducive environment and opportunity for my mentorship during my theoretical and research training period.
Distinguished gratitude goes to my supervisors; Prof. Dr.-Ing. Benedict M. Mutua and Dr. (Eng.) James M. Raude. They offered scholastic guidance, constant encouragement, valuable suggestions and great support throughout the development of this thesis. Their approach allowed me great freedom to pursue independent work of interest. In addition, they kept my ideas focused in the right direction. They always listened to me and advised on how to get solutions to come up with a worthwhile research thesis. It is a great opportunity and fortune to work with such great supervisors, for they have immensely multiplied my determination in research conceptualization and development. I am thankful to Dr. Kamau, D. N, Prof. Onyando, J. O and Dr. Kundu, P. M. all of the Faculty of Engineering and Technology for their encouragement and advice during the proposal, research and thesis development. I wish to appreciate the support from Dennis Theuri, V.O. Odongo, and C. W. Maina for the guidance they offered in the GIS application.
I thank the African Development Bank (AfDB) through the Ministry of Education, Science and Technology (MoEST) for providing scholarship and financial support for the research project. I recognize Water Resources and Management Authority (WRMA) and Ministry of Environment and Natural Resources, Kenya Meteorological Department, for providing data that was used in this study. I would also like to appreciate all the field assistants that helped me during the data collection.
DEDICATION
This research work is dedicated to my wife Theresa Monthe, who has been a source of encouragement and inspiration during challenging times of my academic programme. I appreciate her prayers, understanding and support during my study and research period. To my children; Anthony Wambua and Daniel Mutuku, for they have always been sources of determination and happiness in the family.
To my parents Mr. Benard Wambua and Mrs. Elizabeth Mutono Wambua who have supported me through out my life; when challenged in various levels of my education, they prayed for God’s blessings on my endeavours. They cared and/or encouraged me to develop good character and positive attitude anchored on persistence which is paramount for any successful journey. In addition, they also sacrificed immensely in numerous ways to ensure that I pursued education to great altitudes of success.
To my brothers; Urbanus Kioko, Stephen Ndolo and all my sisters: for continous support during my education. I also dedicate this thesis to my numerous friends and church members who have supported me throughout the process. I appreciate all what they have done, especially their prayers and advice needed to overcome challenges encountered along the research process.
In a special way, this thesis is dedicated to all my trainers of good will; the primary and secondary school teachers, University lecturers and all those who have immensely contributed to my success in the academic exploration.
Finally this thesis is dedicated to humankind who depends on the limited water resources for life and socio-economic development, yet droughts, climate change and human-induced activities adversely affect the water resource systems.
ABSTRACT
Drought is acritical stochastic natural disaster that adversely affects water resources, ecosystems and people. Drought is a condition characterized by scarcity of precipitation and/or water quantity that negatively affects the global, regional and local land-scales. At both global and regional scales, drought frequency and severity have been increasing leading to direct and indirect decline in water resources. For instance, increase in drought severity and frequency in the upper Tana River basin, Kenya, water resources systems quantity and quality have been adversely affected. Timely detection and forecasting of drought is crucial in planning and management of water resources. The main objective of this forecasting of drought using Indices and Artificial Neural Networks (ANNs) for the basin. Hydro-meteorlogical data for the period 1970-2010 at sixteen hydrometric stations was used to test the performance of the indices in forecasting of the future drought at 1, 3, 6, 9, 12, 18 and 24-months lead times, by constructing ANN models with different time delays. Drought conditions at monthly temporal resolution were The occurrence of drought was investigated using non-parametric Man-kendall trend test. Spatial distribution of drought severity was determined using Kriging interpolation techinique. In addition, a standard Nonlinear-Integrated Drought Index (NDI), for drought forecasting in the basin was developed using hydro-meteoroogical data for the river basin. The performance of the drought forecasting models at the selected lead times were assessed using Mean Absolute Error (MAE), correlation coefficient (R), Nash-Sutcliffe Efficiency (NSE), Ratio of mean square error (RSR) and modified index of agreement (d1). The results of spaial drought show that the south-eastern parts of the basin are more prone to drought risks than the north-western areas. The Mann-Kendall trend test indicates an increasing drought trend in the south-eastern and no trend in north-western areas of the basin at both 90 and 95% significant levels. Another output of this research was the development of Surface Water Supply Index (SWSI) function, NDI and characteristic curves defining the return period and the probability of different drought magnitudes based on Drought Indices (DIs). In addition, drought Severity-Duration-Frequency (SDF) curves were developed. The formulated NDI tool can be adopted for a synchronized assessment and forecasting of all the three operational drought types in the basin. The results can be used in assisting water resources managers for timely detection and forecasting of drought conditions in prioritized planning of drought preparedness and early warning systems.
CHAPTER ONE
INTRODUCTION
1.1 Background information
Drought is one of the critical stochastic natural disasters that adversely affect water resource systems, people and ecosystems (Zargar et al., 2011; Jahangir et al., 2013). Drought is defined as a hydro-meteorological event on land characterized by temporary and recurring water scarcity. According to Morid et al. (2007), the magnitude of the drought is indicated by the extent with which it falls below a defined threshold level over an extended period of time. Drought has been identified as the most complex natural hazards due to difficulty in its detection. Drought preparedness and mitigation depend upon timely information on its onset, and propagation in terms of temporal and spatial extent.Such information can be obtained if effective and continuous drought monitoring indices are used in drought evaluation. The study of spatial and temporal drought conditions has greatly been applied in planning and management of water resource systems such as water supplies, irrigation systems, and hydropower generation (Ceppi et al., 2014; Abad et al., 2013; Alaa, 2014; Okoro et al., 2014). These studies were undertaken in Lombarrdy region of norh Italy, Bashar river basin, Mashtul pilot area of Nile Delta and the river basins in Imo state of Nigeria respectively.
At a global scale, demand for water resources has continued to increase as a result of the population pressure and related socio-economic development needs. As a result, numerous sectors have been affected by water scarcity and thus, effective management of impacts of drought-induced water deficit is required. These drought impacts are more severe on Arid and Semi-Arid Lands (ASALs) than in the humid areas (UNDP, 2012). Therefore, management of drought has become an important issue in most of the countries in the world.The drought characteristics in terms of frequency, duration and severity have been assessed using Drought Indices (DI) in some parts of the world (Mishra and Sign, 2010; Barua, 2010; Belayneh and Adomowski, 2013). However in many regions of the world such as Kenya, drought forecasting is still inadequate and thus the need to develop forecasting techniques for Kenya.
Globally, drought has become more frequent and severe due to climate variability with some regions experiencing droughts at varying scales and times. Therefore, global impacts of drought on environmental, agricultural and socio-economic aspects need to be studied. Drought is classified into four distinct types namely; meteorological, agricultural, hydrological and socio-economic. According to IPCC (2014), these droughts have either direct or indirect impacts on river basins and human lives. Direct impacts include degradation of water resources in terms of quantity and quality, reduced crop productivity, increased livestock and wildlife mortality rates, increased soil erosion and land degradation, and increased plant diseases and insect attacks (UN, 2008; Scheffran et al., 2012). On the other hand, the indirect impacts comprise reduced income, unemployment, and migration of people and animals (Figure 1B; Appendix B). Worldwide, more than eleven million persons have died since 1900 as a result of drought related impacts. In addition, two billion persons have been critically affected by the impacts (FAO, 2013). The main challenges associated with drought are that it causes ill health through water scarcity, malnutrition and famine (UN, 2008).
African countries are among the most vulnerable to adverse impacts of climate variability and drought. This is because the Gross Domestic Product (GDP) of these countries depends heavily on rain-fed agriculture. The impacts in Africa are compounded by numerous factors such as poverty, high population density, and human diseases. According to Mwangi et al. (2013) in East Africa, it has been projected that water availability will decline due to drought. In addition, there is a likelihood of increased drought frequency due to decline in precipitation especially during the dry months. Extreme weather events will cause more frequent and intense droughts in some areas and flooding in others. This may be accompanied by hurricanes and tropical storms especially on coastal areas (IPCC, 2001).
In Kenya, very notable droughts of 2009 and 2011 adversely affected the agricultural sector where crop yields were drastically reduced. During this period, the country’s wheat yield dropped by 45% compared to the 2010 growing season (FAO, 2013). Similarly, between 2002 and 2010, Australia suffered a multi-year drought. The total wheat yield in Australia at the time dropped by 46% compared to the annual average level. In 2010, Russia suffered a long and severe drought which significantly affected the environment, human health and economy. In the US, the southern states experienced a severe drought in 2011while in 2012 more than 6.3 million persons were negatively affected by drought in China. During such drought episodes, people experienced challenges in food access and drinking water (FAO, 2013).
Drought and climate variability have significantly impacted on Kenyan river basins. Human activities have also led to human-induced climate variability in most of the Kenyan river basins. This has aggravated pressure on water resources in these river basins. Climate change is expected to impact on different sectors of Kenya’s economy. For instance, the water sector which is the driver of other sectors of Kenya’s economy will be adversely affected. Kenya is a water scarce country with approximate renewable freshwater of 643 m[3] per capita per year and is projected to drop by a third to 235 m[3] by the year 2025 (GoK, 2009). These values are far below the recommended level of 1000 m[3] per capita per year (WHO, 2010).
The agricultural sector in Kenya is highly vulnerable to climate variability (FAO, 2011). This sector contributes more than 51% of the gross domestic product (GDP) (Mwangi et al., 2013) and has been critically affected by frequent droughts. Over the past 50 years, Kenya has experienced at least one main drought per decade (FAOSTAT, 2000). In addition, there has been a notable increase of drought in terms of frequency, duration and intensity. Any damage caused by drought on agriculture and water resources leads to famine, humanitarian crisis, rationing of water supply and decline in hydropower generation, the main source of power in the country that constitutes approximately 50.7% of the total power (World Bank, 2013). Effective drought forecast allow water resource decision makers to develop drought preparedness plans. Such plans are critical for advance formulation of programmes to mitigate drought-related environmental, social and economic impacts. Therefore, accurate drought assessment and forecasting with an adequate lead time is paramount in formulation of mitigation measures for river basins (Sharda et al., 2012).
With the current trend in climate change, the average temperatures in the highlands of Kenya are predicted to increase by approximately 1.4[0]C whereas the annual rainfall will increase by 20% by 2050. This will lead to an increase in runoff in most parts of Kenya. However, some areas such as the northern and the western regions may have little increase of about 5% of annual rainfall. In addition, potential evapo-transpiration is expected to increase by 10% due to an increase in temperature by 2050 (WWF, 2006). These changes are going to affect drought and its impacts at varying magnitudes and time-scales within basins.
Most of the electrical power in Kenya is generated within the upper Tana River basin mainly from five hydro-power plants. These include Masinga, Kamburu, Kindaruma, Gitaru and Kiambere which have a generating capacity of 40, 44, 225 and 156 MW respectively (Oludhe, 2012). These plants are part of the Seven-Forks dams designed for hydro-power generation. Masinga dam was originally designed as the main water reservoir purposely to regulate water flow to the other plants. The quantity of hydro-power generation depends wholly on the availability of water in the reservoirs at reasonable design levels to drive the turbines. However, in recent times, the availability of water within the Masinga reservoir has greatly been adversely affected by occurrence of droughts. The occurrence of drought is associated with low water levels in the reservoirs as a result of reduced stream flows feeding the major reservoir.
Drought forecasting has received a new approach especially with the development of the Artificial Neural Networks (ANNs). An ANN is a computing system made up of a number of simple and highly interconnected information processing elements. Such a system has performance characteristics that resemble biological neural networks of human brain. ANN has numerous merits when used for data processing. The system processes information based on their dynamic state response to external input (Morid, 2007). ANNs have the capacity to model relationships that are quite dynamic and can capture many kinds of relationships including non-linear functions which are difficult or impossible to determine using other approaches (Mustafa et al., 2012). The ANNs have recently been used in water resources engineering (WRE). WRE comprises the study of hydraulics, hydrology, environment and geological related variables. Such variables are dynamic and exhibit non-linear and stochastic characteristics. These properties make WRE variables complex and difficult to determine due to spatial and temporal variations. Thus due to their advantages, ANNs provide effective analytical techniques in modelling and forecasting non-linear and dynamic time series variables in WRE such as drought (Mustafa et al., 2012).
Agricultural drought can be monitored by assessing soil moisture content levels. However, direct soil moisture data measurement is not available at regional and basin scales at fine resolutions. To estimate soil moisture content, process based models are used where an integration of both the random variables of climate, physical properties of land are considered. One of the advanategs of these models is that they can be used to provide information at different spatial and temporal resolutions. Some of these process based models include the FAO developed AquaCrop model (Casa et al., 2013) and Soil and Water Assessment Tool (SWAT) (Fiseha et al., 2013). The Aquacrop model has been used in modelling crop response to soil water availability. For instance, AquaCrop model was applied in evaluating wheat grain yield and crop biomass in China for several irrigation systems (Du et al., 2011). The model was applied by Iqbal et al. (2014) to assess crop grain yield and biomass response to soil water content, actual evapotranspiration under deficit irrigation conditions.
Most basins in Kenya have limited or lack adequate quantifiable information on drought characteristics such as magnitude, frequency, duration and severity. In addition, there is very limited information on appropriate drought assessment and forecasting methods. Drought assessment for different river basins is essential in understanding the trends in drought frequency, severity, magnitude and associated impacts. The information of such assessment can be adopted in informed decision making by governments and its support programmes on the affected communities. Drought models can be used to estimate and forecast drought conditions on a spatial and temporal domain. To prepare for effective mitigation of drought risks in Kenya, asessment and forecasting of drought conditions is vital. Thus in this research, assessment and formulation of drought forecasting models for upper Tana River basin through the application of Drought Indices and Artificial Neural Networks (ANNs) was accomplished.
1.2 Statement of the problem
There has been a problem of erratic drought occurence that has negatively affected water resource systems and consequently socio-economic development in the upper Tana River basin. The problem of drought in the basin is aggravated by the fact that its frequency and severity has been increasing over the years. Being a stochastic hydrological phenomenon, it is difficult to assess and forecast drought. Despite cascaded adverse impacts of the drought occurrence on decline of quantity and quality of water resources (WRMA, 2009; IFAD, 2012), hydropower generation in the Seven fork cascade dams (Word Bank, 2006; WSI, 2011), limited research has been conducted to assess and forecast its characteristics. Drought occurrence in the basin is attributed to the combined effects of both climate change and land use/cover change that lead to increased evapo-transpiration (ET). According to Nohara et al. (2006), increase in ET is caused by increase in temperature that negatively affect river basin water resource systems such as stream flows, reservoir levels and soil moisture levels at varying magnitudes. There is need to formulate sustainable drought mitigation and copying mechanisms for the basin. However, an appropriate tool(s)/methods for assessing and forecasting drought conditions (severity, duration and frequency) in the basin are limited. To address this challenge, this study therefore formulated effective models and Non-linear Integrated Drought Index (NDI) for assessment and forecasting of drought in the upper Tana River basin.
1.3 Objectives
1.3.1 Main objective
The main objective of this study was to formulate the most appropriate models for assessment and forecasting of drought using Indices and Artificial Neural Networks (ANNs) in the upper Tana River basin for guiding decision making in water resources planning and management.
1.3.2 Specific objectives
The specific objectives of this study were to:
i) Assess temporal and spatial drought using selected Drought Indices (DIs) based on hydro-meteorological data from 1970 to 2010 for the upper Tana River basin
ii) Evaluate the performance of the selected DIs and ANNs in forecasting of short, medium and long-term drought conditions in the upper Tana River basin
iii) Formulate a Nonlinear-Integrated Drought Index (NDI) using principle component analysis based on the basin hydro-meteorological data for 1970-2010
iv) Assess the performance of the formulated NDI and ANNs in forecasting and projecting short, medium and long-term drought conditions in the river basin
1.4 Research questions
i) How can the the spatial and temporal drought condition based on Drought Indices (DIs) assessment using hydro-metrological data of 1970 to 2010 for upper Tana River basin?
ii) How do the DIs and ANNs perform in forecasting drought indices for short, medium and long-term drought characteristics of upper Tana River Basin?
iii) How effective can a Nonlinear-Integrated Drought Index (NDI) be formulated using hydro-metrological data for upper Tana River basin?
iv) How do the formulated NDI and ANNs perform in forecasting NDI values for short medium and long-term drought characteristics for the basin?
1.5 Justification
Improved socio-economic development which is associated with sustainable water resources availability, food security and hydro- power generation and supply in the upper Tana River basin is one of the Kenya’s priorities. To reduce the levels of poverty, hunger, improve sustainable access to safe drinking water, sustainable conservation of resources, as per the Kenya’s Vision 2030 (GoK, 2007), understanding of drought characteristics is paramount (GoK, 2012). Although drought has affected water resources in the basin, quantification of its impacts and characterization of drought is limited. There is need to end poverty, hunger, promote sustainable water management and access to sustainable energy as stated in the UN sustaibale development goals 1, 2 6 and 7 respectively (UN, 2016). To achieve these goals, enhanced drought preparedness through planning of mitigation measures to reduce adverse drought impacts on water resources, food security, hydro-power generation and livelihoods is paramount. This requires data on spatial and temporal characteristics of drought. Such data is useful in identification of drought on-set, its propagation as well as detection of drought risk areas in the basin. However, such data for the upper Tana River basin is scanty and not readily available.
1.6 Scope
Although the Tana River basin covers 126,026 km[2], the current research focused only on the upper Tana River basin with an area of 17,640 km[2]. Assessment and forecasting of drought was based on data-driven drought indices that use hydro-meteorological data from 1970 to 2010 for eight gauged stream flow stations and eight meteorological stations. The data analysis is based on monthly and 90-m temporal and spatial resoulutions respectively.
CHAPTER TWO
LITERATURE REVIEW
2.1 Occurrence of drought s
Drought is a condition on land characterised by scarcity of water that falls below a defined threshold level. The term drought has been defined differently in numerous applications (UNDP, 2012). However, it is a challenge to quantitatively define the term. Droughts may be expressed in terms of indices that depend on precipitation deficit, soil-water deficit, low stream flow, low reservoir levels and low groundwater level. Drought may be defined differently depending on the sector involved. For example, a hydrological-drought occurs whenever river or groundwater levels are relatively low. In addition, water-resources drought occurs when basins experience low stream flow, reduced water reservoir volume and groundwater levels. The water resources drought is influenced by climatic and hydrological parameters within a river basin and drought management practices. The hydrological drought, mainly deals with low stream flows. This drought adversely affects various aspects of human interest such as food security, water supply and hydropower generation (Karamouz et al., 2009; Belayneh and Adamowski, 2013).
A sequence of drought occurrence in a river basin may lead to desertification of vulnerable areas such as arid, semi-arid and sub-humid areas. Within these fragile ecosystems, water resources, soil structure and soil fertility are critically degraded by drought occurrence (El-Jabi et al., 2013). The occurrence of any drought in terms of magnitude, frequency, duration and severity has not been clearly understood for numerous river basins in the world, and this calls for intensified research in drought and such related fields.
2.1.1 Types of droughts
According to Zoljoodi and Didevarasl (2013), there are four main categories of droughts. These include the Hydrological, Meteorological, Agricultural and Socio-economic droughts. The first three types are called the operational droughts and can be integrated into a drought management algorithm. Their relation can then be applied in development of water resource strategy in a river basin (Karamouz et al., 2003). Propagation of hydrological and agricultural drought originates from meteorological droughts which develop from changing phenomena within the hydrological cycle as given in Figure 2.1. The main droughts may further be classified into other types of drought.
Abbildung in dieser Leseprobe nicht enthalten
Figure 2. 1: Propagation of drought via hydrological cycle (Source: Author’s own work)
The hydrological drought is associated with below average quantity of surface and sub-surface water resources resulting from deficit precipitation. Its characteristics which are defined by magnitude, severity, duration and frequency can be studied at a basin scale. Hydrological drought impacts on large areas and large human population and may be triggered by climate change and /or variability (Mondal and Mujumdar, 2015). Like other drought events, hydrological drought is considered to be a ‘creeping hazard’ because it develops slowly, it is not easily noticed, covers extensive areas and it lasts for a long period of time with adverse impact on water resources, ecological systems, and socio-economic development (Liu et al., 2015; Van-loon, 2015).
According to Van-loon and Laaha (2015) and as shown in Table 1A of Appendix A, hydrological drought has the most significant effects across different sectors compared to other types of droughts. Hydrological drought may be categorized into surface and ground water droughts (GD). The Surface Water Drought (SWD) is caused by direct reduction in precipitation that subsequently leads to low surface runoff. The SWD is also caused indirectly by reduced groundwater discharge to surface water resources. This may be attributed to reduced flow of groundwater into surface flow in influent rivers and springs. In some instances, increase in groundwater on specific areas within a basin for an effluent river contributes to the SWD. The common indicators of SWD are reduced river flows, low water levels in reservoirs and lakes. SWD results from a combined interaction of meteorological drought, water resources development infrastructure and operational management.
Groundwater Drought (GD) is a hydrological type of drought caused by significantly low quantity of water in aquifers that may be due to reduced recharge. The recharge normally takes place through permeation and inflow from sub-basins (Adindu et al., 2013). The GD is assessed by measuring the volumetric ground water storage. However, these data are not readily available in most river basins. Thus, aquifer level is considered to be a better indicator than the volumetric ground water storage. The GD is also determined from the evaluation of its secondary effects such as base flow into rivers. Ground water is a vital source of water supply especially in river basins where surface water exhibits a high temporal variability. In some cases, groundwater availability is used as an indicator of relative drought risk.
The meteorological drought which is the most commonly known drought is associated with long time intervals of significantly low or no precipitation and increased air temperature. The deficiency in rainfall leads into low infiltration, decreased runoff and ground water recharge. High air temperatures lead to changes in wind characteristics such as increased wind velocity, low Relative Humidity (RH) and increased evapo-transpiration.
The three operational types of droughts are interconnected. For instance, Agricultural drought links meteorological and/or hydrological drought to agricultural impact. Agricultural droughts impact negatively on farming systems whenever they occur. Their impacts are normally two-fold; environmental and economic impacts. The agricultural drought is a type associated with low agricultural production, increased food insecurity, decline in output from agro-processing industries and unemployment incidents in the agricultural sector. From the environmental perspective, agricultural drought is caused by insufficient precipitation, high temperature that causes elevated rates of evapo-transpiration, increased salt concentration in the crop root zones and soils within irrigation systems (Mishra and Singh, 2010). The term environmental drought is sometimes used to address the adverse effects of extremely low flows on ecosystems, and may be analyzed in the emerging field of eco-hydrology.
The term socio-economic drought relates the supply and demand of economic goods with elements of meteorological, hydrological and agricultural drought. This drought occurs when the demand for an economic good exceeds the supply. It is caused by weather related deficit in water supply within a basin. Some of the noticeable impacts of socio-economic drought include increased unemployment, increased food prices, reduced income, reduced tax revenues and increased migration.
2.1.2 Drought modelling
Drought modelling is a technique of using simple and or complex mathematical, scientific and conceptual representations of drought charactersitics. The purpose of drought modelling is mainly to provide a concise understanding of its occurrence, characteristcics and forecast. The fundamental role of modelling the drought is to offer solutions to the challenges of increasing water scarcity due to population growth, expansion in agriculture, industrial and energy sectors. The scarcity of water in the world is compounded by the droughts that affect both surface and ground water resources. Drought modelling may be categorized into eight aspects such as drought forecasting, temporal assessment, spatial assessment, probability characteristics, management, impacts of climate change on drought, assimilation of land data systems into drought and impacts of drought on different sectors. All the aspects of drought are interconnected in drought modelling and are inter-linked as given in Figure 2.2 which was modified from Mishra and Singh (2011).
Abbildung in dieser Leseprobe nicht enthalten
Figure 2. 2: Network of phases of drought modeling (Source: Author’s own work)
2.1.3 Determination of drought threshold level
The threshold level of any drought is based on the theory of crossing technique where the properties of runs above and below a truncation level are determined. The truncation level may be considered as the long-term mean or median value computed from hydro-meteorological data as shown in Figure 12B, Appendix B. The truncation level for a drought may be smaller than the lowest available value of a particular data set. The truncation level is used to specify some statistics of a drought variable. This is achieved by portioning the time series of the variable deficit and surplus segment. The truncation level may be constant or vary with time. When using a constant threshold level in most cases, the absence of trend should be first checked. Given a time series data such as stream flow, reservoir and ground water levels, a threshold level may be determined using a modified function according to Peters et al. (2003) given as:
Abbildung in dieser Leseprobe nicht enthalten
Abbildung in dieser Leseprobe nicht enthalten
Where;
Q(t) = stream flow (m[3]/s)
QT(c) = threshold value
Qb = minimum value (average supply of average demand) (m[3]/s)
M = period of time in series
c = drought criterion factor
The drought criterion factor c is the ratio of deficit value below the threshold to the deficit below the average value. The drought criterion parameter c determines the height of the threshold level. If the value of c is one, the threshold level is equal to the average of Q. If the value of c is zero, then the threshold level is equal to the minimum of Q. The definition of the threshold also ensures that the total drought deficit decreases with decreasing amplitude of Q(t). The last line of the function is included by considering that deficiency is influenced by supply and demand (Awass and Foerch, 2006a). The QT(c) is an arbitrary value that depends upon the objective of the study of drought based on water scarcity as a relative concept that can occur at any level of supply or demand. The truncation level is normally used to objectively demarcate the on-set and end of a drought event.
2.1.4 Selection of drought threshold level
The truncation level for any drought assessment may be chosen based on mean, median and percentage of exceedance of the available data set and purpose of the drought study. For instance, the truncation levels for stream flow may be taken as a percentile such as Q50, Q70, Q90 (Awass, 2009). The first step in selecting a threshold level Q0 is to define its value, below which a stream flow or precipitation data is considered as drought. Then the method of ‘Crossing Theory’ also called ‘Run-sum Analysis’ is applied to investigate drought characteristics.
When the plotted data for drought falls below the threshold value, a drought event starts, and when it rises above the threshold, the drought event ends. The beginning and ending of drought are defined using the start and end times. For river basins with mainly perennial rivers, relatively low fixed thresholds in the range of Q70 to Q95 may be considered appropriate (Meigh et al., 2002; Sung et al., 2013). However, for basins dominated by intermittent and ephemeral rivers, which have large proportion of zero flow, Q70 or mean flow is recommended. If the trend of the data changes with time, then a variable threshold can be applied for detection of deviations and drought assessment. For instance, monthly flow duration curves could be developed and a variable threshold value of Q75 used for the given river basin (Kjeldsen et al., 2000).
2.2 Climate change and variability
Climate change is explicitly defined as statistically significant and long-term (at least a decade) continuous disparity form mean climatic conditions (IPCC, 2001). Climate variability refers to the deviation in mean state of climate variables on temporal and spatial scales. It is a deviation of climate statistics over a defined period of time such as a month, season or a year compared to long-term staitistics of the same calendar period (WMO, 2015). This change comprises shifts in the magnitude and frequency of erratic weather events and slow and continuous increase in mean temperature. Climate variability dynamics was not well understood in early times on all scales. There has been a great improvement in understanding dynamics of climate variability on a large-scale in recent times. However, more research needs to be done on its cascaded effects to hydrology, quantity and quality of water and sustainable management of water resources at the river basin scale (Peng et al., 2013).
Climate variability has the ability to drive climate and ecosystems across certain thresholds and create a new condition. There may be a shift in mean climatic conditions and extreme events such as more frequent floods and droughts, severe soil erosion and prolonged periods of low stream flow. The climatic change and variability plays a key role in the modification of spatial-temporal patterns of hydrology. For instance, according to Ma et al. (2009), climate variability has led to alteration of the hydrological processes, reduced glaciers and water supply downstream of Himalaya’s catchment. In addition, it influences the human socio-economic activity through land cover and land use within a watershed (Hundecha and Bardossy, 2004). In addition, climate has asignificant adverse effects on all aspects of economy. For instance the 1998-2000 droughts in Kenya were estimated to have economic decline of 2.8 billion United States Dollars. This was due to the losss of crops, livestock damage to fisheries, reduced hydropower generation, reduced industrial production and decline in water supply (Gok, 2012)
Due to climate variability, the natural ecosystems require sustainable management for resilience. The process of resilience involves recovery of ecosystems from significant multi-hazard threats with minimum damage to environment, social and economic wellbeing. However, due to systems going beyond thresholds, managing resources for resilience may not remain a viable strategy. Instead, it might be more viable to manage systems in their new stable states. It is therefore important to conserve, adapt and/or mitigate climatic variability impacts and reduce their risks using integrated approaches to hydrological risks (UNEP, 2015).
The term hydrological risk is used to refer to a combination of the magnitude of the effects of climate variability impacts and the probability that the effects will occur. The effect of climate variability on hydrological extreme events has a great challenge in water resources and basin management. There is the need to formulate strategies to efficiently manage the challenges at river basin level (WRMA, 2010).
The threat of climate change on basins degradation in the past decade has been of great concern as far as conservation of biodiversity is concerned (Ayyad, 2003; Chen and Rao, 2008). Based on the period in years of occurrence, climate variability may be categorized as short-term and long-term (IPCC, 1995). Climate change has been attributed to global warming. For instance, global surface air temperature has been noted to change by 0.74Abbildung in dieser Leseprobe nicht enthalten0.18oC from 1906 to 2005 (IPCC, 2007). The climatic variables which impact immensely on water resources include the precipitation and temperature. One of the resulting adverse effects on water resource systems is the climatic-induced water scarcity. The impact of climate variability on the hydrological response of a basin may be investigated using accurate field data and modelling techniques.
Some of the hydrologic variables affected by the climatic variability include surface runoff, sub-surface runoff, stream flow, sediment yield and soil erosion (Kamga, 2001; Tong and Cheng, 2002). The climatic variables which impact immensely on water resources include the precipitation and temperature. One of the resulting adverse effects on water resource systems is the climatic-induced water scarcity that needs detailed investigation. The impact of climate variability on the hydrological response of a basin may be investigated using accurate field data and modelling technique.
Prediction of hydrological response associated with climate variability requires the use of simple or complex distributed hydrological models. However, the use of complex models may require large datasets for both the pre-change and post-change basin conditions. Alternative approaches that combine spatial modelling and qualitative techniques may therefore be applied (Bassey and Akinkunmi, 2013). A number of approaches have been used to study changes within different basins. Hydro-meteorological measurements and the application of Remote Sensing (RS) and Geographical Information Systems (GIS) techniques have been widely used to detect effects of climate change on basins.
Some of the efforts made to monitor climatic change are measurements of stream flow, water balance components, levels of lakes, tree ring growth rates and drought risks. The regional trends in climate change may be reflected in the projections for countries such as Kenya. It is expected that the mean annual temperatures in Kenya will increase from 1 to 2.8°C by the 2060s, and 1.3 to 4.5°C by the 2090s (IPCC, 2007). On the other hand, the mean annual rainfall will increase by up to 48%. It is anticipated that there will be a general increase in precipitation due to climate change in most arid and semi-arid lands (ASALs) of Kenya (IPCC, 2007).
2.2.1 Impact of climate change on water resources
Climate change and land-cover change within a basin may greatly impact on the quantity and quality of water resources. Reservoir water availability is influenced by drivers of climate variability especially the ones that alter runoff within a basin. To manage water resources under climate variability, the capacity of water reservoirs should be increased for offsetting the impacts of climate variation and maintaining existing water yields.
Climate change influences the patterns of hydrology, droughts and water resources systems (Ma et al., 2010). Although hydrology and water resources are interrelated, the two terms differ in their definition. Hydrology is the study of the water on, above and below the earth in terms of its occurrence, circulation, distribution, chemical and physical properties, and the reaction of the water with the environment, including the interaction with ecosystem within a basin. On the other hand, water resources refer to the quantity and quality of water based on a defined temporal and/or spatial resolution.
Due to climate change and land-cover change, hydrological processes in basins have significantly been altered especially by extreme climate events (Jones, 2010). Extreme climate events such as droughts are series of occurrences that happen with greater intensity or frequency than common events. The climate variability has led to increased climatic uncertainty with variation in the weather pattern, mainly between the seasons and years. The impact of climate change on water resources and agriculture is considered as both regionally distinct and spatially heterogeneous. One of the challenges in monitoring trends in climate change in river basins is the paucity of meteorological data both in terms of period of record and the distribution of stations. For instance, in many countries, precipitation, temperature, wind and other hydro-meteorological data recording began only after the Second World War. This makes it difficult to statistically determine the extent of variability of an event in terms of magnitude and frequency (Awass, 2009). However, in some areas there are enough data to study annual trends.
2.2.2 Effect of water balance components on drought
Different components of the hydrological water balance equation are influenced by the drought. The water balance equation components for a river basin are summarized in the expression:
Abbildung in dieser Leseprobe nicht enthalten
Where;
P = Precipitation (mm)
ET = Evapo-transpiration (mm)
ΔS = Change in soil water storage (mm)
G = The ground water recharge (mm)
From Equation (2.3), it implies that precipitation is a vital contributor to the runoff and thus it greatly influences stream flow. During a drought event, the quantity of precipation is significantly reduced below the normal average while temperature in a river basin may increase above long-term mean value. As a result, there is an increase in evapo-transpiration, and thus alteration of the magnitude of water balance equation components. When evapo-transpiration (ET) on the soil surface is high, then the quantity of water stored in the top soil horizon or the root zone is low. This condition contributes to agricultural drought. The ET, Δ S and G components of the water balance are greatly influenced by physiological charatceristics of the river basin which may cause accelerated or delay in hydrological response.
2.2.3 Effect of global warming on droughts
Global warming is due to firstly, the climate change which slows down the global circulation of ocean currents due to moderated differences in temperature between tropical and temperate sea water bodies. Secondly, due to the ice melting in the Polar regions implying cold water entering the oceans and drifting into the tropics affect global warming. The ice melting flowing leads into cooling of tropical oceans whose effect is picking significantly of low moisture by the prevailing winds (IPCC, 2011). The wind carries with it the little moisture picked along its course.
Global warming influences the rate and timing of evapo-transpiration. Due to global warming, some regions in the world are likely to get wetter while those that are already under dry conditions are likely to get drier. Thus, global warming is likely to increase drought occurrence and expansion of the dry areas (Dai, 2011). For instance, the regions in southern Africa, the Sahel region of Africa, southern Asia, south west of United States of America have generally been getting drier over the years (FAO, 2013).
In addition, the quantity of water resources is expected to decline by up to 30% in the areas affected by climate change. These notable changes will occur partly because of an expanding atmospheric circulation pattern. This pattern is called Hadley cell in which warm air in the tropics rises, loses moisture, and descends in the sub-tropics as dry air. During this process, jet streams shift to the higher latitudes, and storm patterns shift along with them leading to expansion of Arid and Semi-arid Lands (ASALs) (Dai, 2011).
2.2.4 IPCC and IGAD approach
The inter-governmental panel for climate change (IPCC) projects changes in climate system using a hierarchy of climate and earth system models. The models simulate changes in climtate based on a set of scenarios of anthropogenic forcing. One of the new sets of the scenarios such as Representative Concentration Pathways (RCP) is used for climate monitoring. This scenario projects atmospheric carbon concentration to be higher in the year 2100 than at present.
The IPCC monitors global surface temperature data sets for the purpose of projecting global warming using integrated climatic models such as simple, atmospheric chemistry and global carbon cycle models (IPCC, 2013; IPCC, 2014). The Intergovenmental Authority on Development (IGAD) also applies a number of climatic models to predict ten-day, monthly and other periodical climate data. It uses climate data bank comprising decadal precipitation and temperature for respective regions. These data sets are then used to develop regional precipitation and temperature risk maps (IGAD, 2007). However, the IPCC and IGAD provide information based on global scales and thus does not give a direct and effective indicator at basin scale scenarios. Most of the information at a basin scale is obtained through downscaling of the global data.
2.3 Major causes of drought in Kenya
Climate variability and global warming that affect atmospheric circulation play a fundamental role in influencing drought occurrences in Kenya. When the Indian Ocean surface water temperature is abnormally low, it leads to the cooling of South-East and North-East trade winds. These two air masses move across the land and converge near the equator within a region called Inter-Tropical Convergence Zone (ITCZ). When the winds are cool, they do not pick up enough moisture from the ocean water surface and thus lead to erratic rainfall patterns in eastern parts of Kenya (GoK, 2009). On the western parts of Kenya, the Atlantic and Congo prevailing winds bring the same drought conditions in case they are abnormally too cold to pick sufficient moisture.
Other factors leading to increase in drought frequency and severity are human-induced such as poor land use practices, deforestation and destruction of catchment areas. Generally the forest cover has decreased by 72% between 2000 and 2007 (GoK, 2009) and further to 6.1% of the total land mass in 2011 (World Bank, 2014). Among the five main water towers in Kenya, the Mau Complex lost the highest with 70% of the forest cover destroyed within the stated period. The major causes of deforestation have been the need to expand agricultural land, uncontrolled exploitation of forest resources, overgrazing and establishment of new settlements on forest land as accelerated by increasing population growth.
2.3.1 Impact of drought in Kenya
Kenya has experienced approximately 30 major droughts during the last 100 years according to UNDP (2012). Over 70% of the natural disasters in Kenya are associated with droughts and extreme weather conditions. The severity and frequency of droughts in the country have been increasing over the years (Arnell, 2004). Some of the recognizable droughts include the 1952-1955, 1973-1974, 1983-1984, 1992-1993, 1999-2000 and 2009-2011 (UNDP, 2012). The occurrence of droughts in Kenya has impacted negatively on people’s livelihoods in the sense that, huge resources which would have otherwise been used for other socio-economic activities are normally diverted to cater for food shortages and water scarcity. The 2008-2009 droughts for instance, caused adverse effect on 3.8 million pastoralists and agro-pastoralists in Kenya in terms of food insecurity. Approximately 1.5 million children in primary schools needed to be fed due to famine (KFSSG, 2009). Many people within the arid and semi-arid lands (ASALs) suffered malnutrition due to lack of food and water.
The drought of 2011 was also severe in terms of food and water scarcity in Kenya where over 3.2 million persons needed urgent care (KFSSG, 2011). In addition, over half a million persons were forced to migrate from their settlements in search of forage, food and water. The cattle were grazed through wildlife protected areas bringing competition and conflict between people, livestock and wildlife on pasture and water resources. Additionally, the tourism sector which is a major foreign exchange earner for Kenya was greatly affected. This is because drought affects the environmental resources that are crucial attraction for tourists including the wildlife, biodiversity, water quantity and quality.
Drought has impacted negatively on numerous river basins in Kenya. For instance, the upper Tana River basin has been negatively affected by notable droughts such as the La Niña of 1999 to 2000, and 2008 to 2009 (Oludhe, 2012). These led to severe water scarcity in the region and a significant reduction in hydro-power generation that was characterized with rationing of power. To address the problem, there is need to have information for planning ahead. Therefore, this led to a detailed assessment evaluation and characterization of drought occurrences within the basin and finally the development of a prediction system that could be used for early warning on drought occurrence in the upper Tana River basin.
2.3.2 Drought monitoring in Kenya
To minimize the impacts of drought in Kenya, an effective and timely monitoring system is necessary. Such monitoring activities are meant to provide critical information in the development of early warning systems. Since drought has become a recurrent phenomenon in Kenya, different organizations are involved in coming up with methods to address drought-induced challenges.
The organizations involved in data collection for early drought warning systems in Kenya include; the Kenya Meteorological Department (KMD), Ministry of Agriculture, livestock and fisheries, Department of Resource Survey and Remote Sensing (DRSRS), Kenya National Bureau of Statistics (KNBS), Inter-governmental Authority for Development (IGAD) Climate Prediction and Application Centre (ICPAC), World Food Programme (WFP) Kenya Office, Regional Centre for Mapping of Resources for Development (RCMRD), Livestock Network and Knowledge System (LINK), Famine Early Warning system-Network (FEWS-NET) and the Arid Lands Resource Management Project (ALRMP) (WFP, 2011; UNDP, 2012). To prepare adequately for mitigation of the drought impacts, a thorough assessment, evaluation and forecasting of drought conditions is very critical. However, the main challenges or gaps with drought preparedness and mitigation in Kenya include the fact that:
i) Drought assessment has been based on past and present drought indices (DIs) developed for specific regions in other countries and their suitability in Kenya has not been sufficiently tested
ii) Efficiency of the performance of DIs in drought forecasting for different lead times is not well explored in most basins in Kenya
iii) Spatial and temporal drought assessment and forecasting of drought using hydro-meteorological variables is limited for most basins in Kenya
Due to these challenges, there was need to test the applicability of the Drought Indices (Dis) in Kenya using data forms available in concerned river basins. To effectively define the droughts in Kenya, a combination of hydro-meteorological data should be used as input variables in drought assessment and forecasting models.
2.4 Drought forecasting
The terms short, medium and long-term forecasting have been used in drought studies as indicators of lead time in months of future drought. In most of the drought forecasting research, 1 to 3 months lead time is considered as short-term forecast. The medium to long-term drought forecast is lumped into one category of 4 to 12 months lead time (Mishra et al., 2006; Cassiamani et al., 2007; Belayneh, 2012). Forecasting of short-term drought conditions is useful for monitoring the effect of drought on agricultural systems. Under the short-term drought forecasting soil moisture and crop water stress may be defined especially during growing seasons. On the other hand, forecasting medium and long-term droughts helps to understand the overall effect of drought on water resources at basin and regional scales. The medium to long-term forecasting is critical in water resources management. It may be used for drought risk management as emerging early warning systems in Kenya. The three categories of drought forecasting can be used to formulate long-term plans for sustainable management of water resources and agricultural systems. Droughts are likely to persist in river basins such as upper Tana, with varied projections. Some models have projected an intensification of drought events in some areas, although other models indicate a reduction in drought severity (GoK, 2012).
2.5 Drought mitigation
Numerous methods are used to alleviate, protect or reduce the impacts of drought on people’s livelihoods. The methods are directed to alter the drought effect on water resources, agriculture, livestock and other basin resources. Some of the commonly used methods at the basin scale for drought mitigation include:
i) Drought monitoring: This involves gathering of information about drought in terms of water and rainfall levels. Such information is used in formulation of early warning systems and design of other mitigation strategies
ii) Rainwater harvesting: It involves collection, conveyance and storage of water during wet conditions for use during the drought. The water may be stored in dams reservoirs or in storage tanks
iii) Desalination: This is a process of treating and removing dissolved salts in sea water or irrigation water within a river basin
iv) Recycling: It is the treatment and purification of wastewater through recycling, recovery and reuse in community water supply and irrigation systems
v) River engineering: It is the design and construction of river training channels, water canals or to divert and re-direct water to drought prone areas for water supply or irrigation
vi) Cloud seeding: A rainfall inducing substance is sprayed in the atmosphere to act as nuclei on which clouds form and generate precipitation
vii) Development of Crop varieties: This is the development of new plant varieties that can tolerate drought with significant crop yields
viii) Development of farming technologies that cope with drought: It is the development and use of farming technologies that have high water use efficiency and crop production and animal husbandry such as drip and greenhouse systems
ix) Reduction in outdoor water use and wastage: This involves the regulation and control of water wastage via the use of sprinklers, hoses, water containers, pipes and other usage and maintenance activities
2.6 Drought assessment methods
Drought indices or models are used for assessment of occurrence and severity of droughts. The Drought Indices (DIs) were developed for specific regions using specific structures and forms of data input. There is limited information in the application of drought indices that combines both temporal and spatial drought evaluation at river basin scales. Drought has been assessed in terms of temporal and spatial domain using evapotranspiration mapping as illustrated by Eden (2012). There are two broad categories of drought indices; satellite based and the data driven drought indices (Belayneh and Adamowski, 2013).
2.7 Satellite based drought indices
The satellite Remote Sensing (RS) may be defined as the science and art of obtaining information of points, objects, areas or phenomena through analysis of data acquired by a sensor, which is not in direct physical contact with the target of investigation (Sayanjali and Nabdel, 2013). The RS provides an aerial view of land, water resources and vegetation cover. This technique gives a spatial and temporal context of assessing drought and has the ability to monitor vegetation dynamics over large surface areas. Currently, there is a considerable interest in collecting remote sensing data at multiple time scales. Such data is used to conduct a near real time information management (Mulla, 2013). Examples of satellite drought indices are the Vegetation Condition Index (VCI), Normalized Difference Vegetation Index (NDVI), Normalized Difference Water Index (NDWI), Water Supply Vegetative Index (WSVI) and Normalized Difference Drought Index (NDDI).
2.7.1 Vegetative condition index
The Vegetative Condition Index (VCI) is computed from an advanced accurate and high resolution radiometer radiance data. This data is usually adjusted to match land conditions, climate, and ecology and weather conditions. The index is used for drought detection and trend tracking. It can be used to determine the time of on-set of drought, intensity, duration and associated impacts on vegetation (Mishra and Singh, 2010). The main challenge with the use of VCI is that it is used during dry seasons and the areas under consideration should have significant vegetation cover. The VCI for a month j is computed from the relation:
Abbildung in dieser Leseprobe nicht enthalten
Where;
VCIj = Vegetation condition index for month j
NDVIj = the NDVI for a month j
NDVImin = minimum NDVI for the period under consideration
NDVImax = maximum NDVI for the period
2.7.2 Normalized difference vegetative index
The Normalized Difference Vegetative Index (NDVI) is a satellite data driven index that is used to monitor ground vegetation which could be linked to drought conditions. The index can filter out green vegetation using Landsat Multispectral Scanner (MSS) digital data (Musaningabe, 2012). It is normally expressed as a function of the near-infrared and red bands given as:
Abbildung in dieser Leseprobe nicht enthalten
Where;
NIR = near-infra red band
R = the red band
The NDVI is the most commonly used satellite based index. One advantage of the NDVI is that it has distinct values ranging from -1 to 1 with zero taken as an approximate value denoting absence of vegetation. The negative NDVI values indicate a non-vegetative surface while values closer to 1 represent dense vegetation.
2.7.3 Normalized difference water index
The Normalized Difference Water Index (NDWI) is determined based on leaf water content and vegetative type. Its value ranges from -1 to +1. The higher the NDWI value, the higher the vegetative water content and the higher the proportion of vegetative cover.The values of NDWI are computed by processing the satellite data in which green and near infra red bands are used as per the relation:
Abbildung in dieser Leseprobe nicht enthalten
Where;
NDWI = normalized difference water index
G = green band
NIR = near infra red band
The NDWI is very sensitive to soil moisture content, vegetation cover and leaf moisture content (Tychon et al., 2007). Although NDWI is used for drought detection, it is sometimes affected by land cover and pests and diseases on vegetation. However, it has an advantage of detecting drought more effectively as compared to the NDVI (Gu et al., 2007).
2.7.4 Water supply vegetative index
The Water Supply Vegetative Index (WSVI) is a drought indicator based on relationship between the NDVI and the land surface temperature. The higher the values of WSVI, the higher the moisture levels, canopy temperature and the lower the NDVI. On the other hand, lower values of WSVI give an indication of extreme drought. The WSVI values range from -4 for extreme drought to +4 for highly moist conditions (Luke et al., 2001). The values of WSVI are obtained by analyzing the effect of vegetation on the reflection of red, near infra red and thermal bands. This index is more effective in drought detection under the conditions when the NDVI is greater than 0.3. Combining the WSVI and the NDVI in drought detection provides a more sensitive approach and better results (Jain et al., 2010).
2.7.5 Normalized difference drought index
The Normalized Difference Drought Index (NDDI) is used for detection of drought by combining the outputs of NDWI and NDVI derived from satellite data. The values of the two indices decrease with decrease in slope gradient of the cumulative precipitation. However, NDDI values decrease more abruptly during dry period than the NDVI. Thus, NDDI is more sensitive to water content and it is a better index for drought detection than NDVI. The NDDI has been noted to detect drought conditions on grassland than NDVI (Gu et al., 2007). The NDDI values can be computed from the following relation:
Abbildung in dieser Leseprobe nicht enthalten
Where;
NDDI = Normalized difference drought index
NDVI = normalized difference vegetative index
NDWI = normalized difference water index
While the satellite based drought indices can be used to detect the on-set, intensity and duration of drought, their limitations and advantages need to be examined. These indices are limited to the areas with significant vegetation cover. Thus, in case the vegetation is infested by the pests and diseases, such indices can give misleading results. In addition, the indices are difficult to harmonise their drought characterization in terms of magnitude as they result from analyzing different bands of the satellite imageries.
2.8 Data driven drought indices
The Data Driven Drought Indices (DDDI) use a single or a combination of hydro-meteorological variables as input parameters to assess drought intensity, duration, severity and magnitude. Some of the data driven indices as reported by Belayneh and Adamowski (2013) include; the Standardized Precipitation Index (SPI), Palmer Drought Severity Index (PDSI), Surface Water Supply Index (SWSI), Aggregated Drought Index (ADI), Effective Drought Index (EDI), Reclamation Drought Index (RDI), Crop Moisture Index (CMI) and Murger Index (MuI). These indices use different input data such as rainfall, temperature, catchment soil moisture content, snow water content, stream flow, storage reservoir volume, and potential evapo-transpiration (Zoljoodi and Didevarasl, 2013). However, the suitability of the indices and their testing for Kenyan conditions has not been adequate. Therefore, Kenya does not have generic indices for drought forecasting. Due to scanty and lack of drought assessment indices that can be used for defining critical drought conditions in Kenya, this research assessed selected suitable indices.
Some of the most critical elements of drought which are used for the design of water storage systems to cope with drought impacts include; longest duration, largest severity, highest intensity and spatial and temporal variation of droughts (Sharma and Panu, 1997; Manikandan and Tamilmani, 2013). Drought duration, severity and intensity are fundamental characteristics of any drought event. Drought duration refers to any continuous period of the sequence with deficit, while intensity is the magnitude below a truncation level. Severity is the cumulative deficit below a truncation level during drought period and may mathematically be defined as the product of the drought intensity and duration.
For better understanding of drought characteristics, assessment of the influencing variables is paramount. The statistical analysis for stream flow and precipitation as drought variables include parameters such as: the mean, coefficient of variation, log-1 serial correlation coefficient and the probability distribution function of the sequence under study. The extreme values of drought which include; duration, severity and intensity may be modelled with reference to a certain truncation level. The truncation level is usually taken as the long-term mean of the drought variable (Dracup et al., 1980a; Bonacci, 1993).
2.8.1 Standardized precipitation index
The Standard Precipitation Index (SPI) was developed by Mckee et al. (1993) to quantify the rainfall deficit and monitor drought conditions within Colorado, USA. For calculation of SPI, long-term historical precipitation record of at least 30 years is integrated into a probability distribution function which is then transformed into a normal distribution function. The SPI requires less input data compared to most other drought indices and this makes it flexible for wide applications (Mckee and Edwards, 1997; Bacanli et al., 2008).
The SPI has several advantages which make it more applicable in many river basins. First, it requires only the precipitation as the input data.This makes it ideal for river basins that do not have extensive hydrological data records. Secondly, its evaluation is relatively easy since it uses precipitation data set only. Thirdly, it is a standardized index and this makes it independent of geographical location as it is based on average precipitation values derived from the area of interest. In addition, the SPI exhibits statistical consistency, and has the ability to present both short-term and long-term droughts over time scales of precipitation variation (Belayneh and Adamowski, 2012). However, the SPI has some disadvantages in its use as a drought assessment tool. First, it is not always easy to find a probability distribution function to fit and model the raw precipitation data. Secondly, most river basins do not have reliable time-series data to generate the best estimate of the distribution parameters. In addition, application of SPI in arid and semi-arid lands of time-series of less than three months may give inaccurate values.
To overcome the challenge of simulating and modelling the data for SPI outputs, application of different probability distribution functions may be employed. These include the Gamma, Pearson type III, Lognormal, Extreme Value and Exponential distribution functions (Cacciamani et al., 2007). However, the Gamma probability distribution function is preferred in hydrological studies. In hydrology, it has an advantage of fitting only positive and zero values since hydrological variables such as rainfall, and runoff are always positive or equal to zero as lower limit values (Markovic, 1965; Aksoy, 2000). The Gumbel and Weighbull distribution functions are used for study of extreme hydrological variables. The Gumbel distribution function is used for frequency analysis of floods, while the Weibull distribution function is used to analyse low flow values observed in rivers (Bulu an Aksoy, 1998).
Although the SPI can be used to present significant drought conditions within a river basin, identification of key dry periods requires an analysis of data for time scales greater than 6 months. This is because the high frequency of SPI values at shorter time scales conceal the critical dry periods. For time scales shorter than 6 months, there is insignificant autocorrelation while for time scales greater than 6 months, the autocorrelation increases significantly (Awass, 2009).
2.8.2 Palmer drought severity index
The Palmer Drought Severity Index (PDSI) was developed based on a criterion for determining the beginning and end of drought or wet period spell (Palmer, 1965; Wang, 2010). It is a simple monthly water balance model which requires rainfall, temperature and catchment soil moisture content as input parameters. This index applies a concept of supply and demand over a two-layer model. In this concept, the difference between the quantity of precipitation needed to maintain a natural water balance level and the actual precipitation is determined. The index does not consider stream flow, reservoir water balance, and other hydro-meteorological variables that influence the drought (Karl and Knight, 1985; Yan et al., 2013).
Several coefficients which are calculated to define local hydrological characteristics influenced by precipitation and temperature are calculated for use in PDSI. These coefficients depend on soil water capacity of the principal layers. The original PDSI has been modified to yield Palmer Hydrological drought Index (PHDI). The original PDSI does not take into account the human-induced impacts on water balance such as irrigation. However, the new version is a model mainly for evaluation and monitoring of water supply. The model has been applied on a number of catchments for detecting and planning of drought relief programmes (Loucks and Van Beek, 2005).
The PDSI has some limitations or disadvantages as a drought index. In some regions, the PDSI assumes that all the precipitation is rain. This may sometimes give misleading results in regions which experience winter season and also on high elevation areas. In addition, it under-estimates runoff since it assumes that overland flow occurs after all soil layers have been saturated. The other disadvantage is that the PSDI responds slowly to developing or ending of a drought event (Mishra and Sigh, 2010). Lastly, the original model is more suitable for agricultural drought than hydrological drought based on the applied time series.The original PDSI has some advantages and disadvantages (Narasimhan and Srinivasan, 2005). The major advantages of the original PDSI include:
i) The two indices provide decision makers with measurement of abnormality of recent weather condition for a basin or region
ii) It provides an opportunity to place current drought condition on a historical perspective
iii) It has the capacity to express historical drought conditions on spatial and temporal domain
The disadvantages of the PDSI are that:
i) The index uses two-layers in water balance computation and this is an over-simplification which leads to inaccurate values
ii) Potential evapo-transpiration (PE) in PDSI is computed based on Thornthwaite method which is a poor method of estimating the PE
iii) The original PDSI considered coarse resolution of land use and land cover parameters of 700-100,000 km[2] yet the land use changes within such a large area may be great
2.8.3 Surface water supply index
The Surface Water Supply Index (SWSI) was developed in Colorado USA, as an indicator of surface water or moisture levels (Shafer and Dezman, 1982). The index requires input variables which include; snow water content, stream flow, rainfall and storage reservoir volume (Castano, 2012). Normally the snow water content, rainfall and storage reservoir volume are used for computing the SWSI values for winter season. However, during the summer season, stream flow substitutes snow water content. At a basin scale, the SWSI values are determined from monthly catchment average values of rainfall, reservoir volumes, snow water content and stream flows measured at stations within the catchment. One of the advantages of the SWSI is that it gives a representative measurement of surface water supplies across the river basin.
The SWSI is unique in specific basins or regions. It requires long term record data for its calibration and thus may be limited in basins that lack sufficient data. Another limitation of the SWSI is that any additional change in the water management within a basin calls for modification of its algorithm. The change may be due to an addition of new water reservoirs and flow diversions that based on their weights, require to be accommodated in the algorithm (Barua, 2010). Thus, it is difficult to have a homogeneous time series of the index for several basins.
2.8.4 Aggregated drought index
The Aggregated Drought Index (ADI) is used for determination of three categories of drought; hydrological, agricultural and meteorological droughts. In the use of each category, the specific drought is determined by selectively inserting input variables required into the model. This index uses rainfall, stream flow, potential evapo-transpiration, soil moisture content, snow water content and reservoir storage volume as input data (Keyantash and Dracup, 2004).
The Principal Component Analysis (PCA) is used as a numerical method for construction of ADI using appropriate input data sets. The PCA is used to transform spatially correlated data series from a basin into two sets of orthogonal and uncorrelated functions. The principle components are used to express the original p -variable data set in terms of uncorrelated component Zj, where 1<j>p. The p -model is used where the analysis explains temporal fluctuations of the input variables (such as precipitation, steram flow, reservoir levels, soil moisture content, temperature and evapotranspiration) of a basin. The calculation of the principle components involves the construction of a pxp symmetric correlation matrix Cx. The matrix gives the correlation between the original data where p is the number of variables. This matrix is expressed using the relation:
Abbildung in dieser Leseprobe nicht enthalten Abbildung in dieser Leseprobe nicht enthalten
Where;
Cx = correlation matrix
x = vector observation data
ux = mean value of x
T = the transpose matrix
The correlation matrices developed undergo the PCA through the application of Eigenvectors. The Eigenvectors are unit vectors that establish the relationship between the principle components and standardized data. A unit vector may be derived from the relation:
Abbildung in dieser Leseprobe nicht enthalten
Where;
Z = n x p matrix of principle components
X = n x p matrix standardized observation data
E = p x p matrix of eigenvectors
The first PC to represent ADI is determined and normalized by use of its standard deviation function defined by:
Abbildung in dieser Leseprobe nicht enthalten
Where;
ADIi,k = ADI value for month k in year i
Zi,l,k = the first PC for month k in year i
σk = the sample standard deviation over all years for month k
To determine ADI thresholds, the empirical cumulative distribution of the ADI values given in Equation (2.10) are constructed. The ADI thresholds are then calculated using empirical cumulative distribution function and used to classify drought conditions based on the specified thresholds as summarized in Table 2A (Appendix A).
2.8.5 Deciles index
The deciles index was developed by Gibbs and Maher (1967) and has found a wide application in some regions such as Australia (Morid et al., 2007). The Deciles Index uses long term monthly rainfall records where the records are ranked from the highest to the lowest and then a cumulative frequency distribution constructed. This distribution is then partitioned into ten sections called deciles. One major limitation in using the Deciles Index approach is that it requires long-term rainfall records of 30-50 years if accurate calculations are to be realised (Hayes, 2003). The first decile is the precipitation value not exceeded by the lowest 10% of all precipitation values within the period under study. This is followed by the second decile that falls between the lowest 10 and 20% in that order. By comparing the amount of precipitation in a certain period with a long term cumulative distribution of precipitation amount in the mentioned period, the severity of the drought can be assessed. Although the data driven indices have been used in other basins in the world, they cannot directly be applied in Kenyan river basins before their calibration and validation are done.
2.9 Drought forecasting models
Development in forecasting and early warning of the drought phenomena is increasingly being applied in many regions in the world. For instance, drought forecasting in Kenya has previously been based on rainfall prediction. To forecast rainfall, the prevailing and expected sea surface temperature anomalies (SSTAs) over the Pacific, Indian and Atlantic oceans are used. The factors of the SSTAs are assessed by applying various tools such as ocean atmosphere models, statistical models, satellite derived information and expert interpretation. The onset, cessation and distribution of rainfall are derived from statistical analysis of previous years that exhibit same characteristics as that of the year under consideration (GoK, 2014). Drought forecasting is being done to help mitigate consequences of drought on vulnerable river basins. Different drought modelling and forecasting techniques are in use today. Some of the commonly used drought forecasting models include; Seasonal autoregressive integrated moving average model (SARIMA), Adaptive Neuro-fuzzy inference system, Markov chain model, Log-linear model and Artificial Neural Network (ANN) model.
2.9.1 Seasonal autoregressive integrated moving average model
The Seasonal Autoregressive Integrated Moving Average model (SARIMA) model is a time series tool. Time series events reoccur in every given number of observations (Chatfield, 2003). For monthly measurements, the recurrence over a year of twelve months, it is expected that the recurring value (xt) will depend on values that are based on annual lags. These lags are defined by xt-12 or xt-24. It may be influenced by recent non-seasonal values. The model has been generalized to deal with seasonality as defined by the relation given as:
Abbildung in dieser Leseprobe nicht enthalten
Where;
ut = seasonal value representing seasonality
w= the period for monthly series, typically of multiples of 12
xt = recurring value
To achieve stationery conditions, the seasonal difference can be repeated many times defined by D. For instance, if D = 2 and D =12, then the following function which is called SARIMA model results to:
Abbildung in dieser Leseprobe nicht enthalten
Where;
xt = recurring value
Abbildung in dieser Leseprobe nicht enthalten= annual lag
ut = seasonal value representing seasonality
2.9.2 Adaptive Neuro-fuzzy inference system model
The Adaptive Neuro- fuzzy logic approach was developed by Zadeh (1965). It is a linguistic uncertainty function that applies the fuzzy inference system (FIS). The adaptive Neuro-fuzzy inference system uses a combination of Artificial Neural Networks (ANNs) and Fuzzy Inference System (FIS), and has greatly been used to come up with engineering solutions. The term fuzzy influence system is a principle that comprises three conventions (Firat and Gungor, 2008) given as:
i) A Rule base that consist of fuzzy ‘if-then rules’ incorporated in their algorithms
ii) A data-base defining the membership function that converts input value into a value between 0 and 1
iii) An influence system combining fuzzy rules to generate system results
2.9.3 Markov chain model
The Markov chains have greatly been used in stochastic characterization of drought (Cancelliene and Salas, 2004). For instance, an early warning system using Markov chain model in conjunction with PDSI based on probabilistic severity, duration and return period of drought may be developed (Shatanawi et al., 2013). Drought has also been characterized in terms of probabilistic occurrence by combining Markov chain model with SPI for short term drought prediction within a period of 1 to 3 months lead time (Paulo et al., 2005; Paulo and Pereira, 2007; 2008). The Markov chain model has two main applications; modelling stochastic characteristics of drought and forecasting future series of drought using historical data sets.
The Markov chain model exhibits a discrete stochastic process where a drought state (x) at a future time step (t+1) is dependent upon the present state xt and independent of previous states Xt-1, Xt-2,..., Xt-n. If a system of n states is considered, then the relation applies:
Abbildung in dieser Leseprobe nicht enthalten
Such a system can be transformed from S to S[2], S[3]..., Sn according to specified transitional probabilities P12, P13, ..., P1n or remain at state S1 with a transitional probability of P11 (Shatanawi et al., 2013). Therefore, the Pij may be used to denote transitional probabilities from Si to Sj. The Pij can be represented in form of nxn matrix. Entries of such a matrix defined as P may be computed from a number of transitions nij from state i to the next state j using the relation:
Abbildung in dieser Leseprobe nicht enthalten
Where;
pij = the nxn matrix
nij = the entries of the P matrix
The following summation holds true for the matrix
Abbildung in dieser Leseprobe nicht enthalten
The transitional matrix at any given time step is calculated using the function:
Abbildung in dieser Leseprobe nicht enthalten
Where;
Pt+[1] = transition matrix at any given time
Pt = transition matrix at initial time
Pt+n-[1] = transition matrix of previous time step
The Markov chain attains a steady state after several time steps. It is thus possible to define a stationery matrix Abbildung in dieser Leseprobe nicht enthalten as Eigenvector of Pt using the relation:
Abbildung in dieser Leseprobe nicht enthalten
Since Abbildung in dieser Leseprobe nicht enthaltenis a stationery probability for state j, thus
Abbildung in dieser Leseprobe nicht enthalten
The persistence and recurrence time can be presented using two main terms of the Markov chain. The first is the probability that the system will retain the same state in a subsequent time step. Thisis called persistence. The persistence probability Pr is defined using the relation:
Abbildung in dieser Leseprobe nicht enthalten
On the other hand, the recurrent time is defined as the average time for a system to transit from a certain state j and then back to the same state j and is computed using the function:
Abbildung in dieser Leseprobe nicht enthalten
The time required for a system to be transformed for the first time from state i to j is called first passage time (tij) and is computed using the relation:
Abbildung in dieser Leseprobe nicht enthalten
2.9.4 Log-linear model
The log-linear model was developed in 1990 and can be used on Poisson-distributed data sets. It is a linear model that can fit in Poisson distribution function. The model is an extension of two dimensional contingency tables where the correlation between two or more discrete and categorical variables is determined by getting the natural logarithm of frequency entries in a contingency table. A contingency table is a type of table in a matrix format that is used to display a frequency distribution of variables. The table can provide basic information on interrelationship between two variables and the interactions between them.The model has been used in forecasting drought in various regions in the world including catchments in Portugal where a twelve month data within SPI was modelled (Moreira et al., 2008).
2.9.5 Artificial Neural Network models
The Artificial Neural Network (ANN) model is an information processing system developed with a structure and operation similar to that of a human brain (Maier et al., 2010). The model has been improved over time by use of different calibration techniques. With sufficient amount of data and complexity, the ANN model can be adapted to establish any correlation between series of independent and dependent variables (Luk et al., 2000). The ANNs have some advantages (Morid et al., 2007; Tran et al., 2009; Mustafa et al., 2012; Beale et al., 2014) which include:
i) the ability to process information based on their dynamic response to external input
ii) the ability to capture numerous kinds of relationships including non-linear functions which are not usually detected by other techniques
iii) the ability to provide effective analytical techniques in modelling and forecasting systems
iv) the ability to model dynamic/stochastic time series variables in Water Resources Engineering
v) the ANNs to processes large and complex data sets, including that of drought forecasting
2.10 Description of ANN model
The ANN model processes information through an elaborate network of neurons that are linked together. It simulates outputs based on certain inputs by a working principle resembling that of human brain where in the human brain; the neuron receives a set of input signals and generates outputs. The nervous system of human beings is represented by a number of architectural structures that range from simple to complex structures. Whether the structures are simple or complex, the systems consist of neurons or neural cells as the chief building blocks as shown in Figure 2.3.
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