**Consequences of Land Use Changes on Hydrological Functioning**

Luc Descroix and Okechukwu Amogu *IRD / UJF-Grenoble 1 / CNRS / G-INP, LTHE UMR 5564, LTHE, Laboratoire d'études des Transferts en Hydrologie et Environnement, France* 

#### **1. Introduction**

86 Water Resources Management and Modeling

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experimental and modelling approach. Dissertation, Vrije Universiteit Amsterdam,

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for the tropics: Influence of sea surface temperature anomalies on the West African

Land use and land cover changes are well known to have a strong influence on the water cycle and its evolutions. Vegetation cover is the main component of the land surface that explains the behaviour of rainwater transforming into runoff, infiltration or evapotranspiration. Land use changes such as afforestation or deforestation, increase in cropping or grazing areas, likely modify this water balance and land surface behaviour. However, urbanisation and sometimes some geopolitical events, could significantly influence the water cycle.

Human activities that change the land cover can considerably change the water cycle within a basin; for example, the present-day situation of the Aral Sea basin in Central Asia where intensive irrigation resulted in a near cessation of the water inflow from the Syr Daria and the Amu Darya rivers. In the Mediterranean basin and the Sahel, the successive removal of vegetation by forest clearing and overexploitation increased drought risks. Inversely, the increase in frequency of flooding in the Ganges basin can be explained by deforestation (Kuchment, 2008). Furthermore, dams and reservoirs dedicated to irrigation increase evaporation and decrease the transport of sediments.

In the following sections, we will describe the main changes observed in the water cycle in different areas. Firstly, we will compare the hydrodynamic changes in two different mountainous areas subject to very different socio-economic changes in the recent decades and centuries. As "water towers", mountains are the natural reservoir and providing areas for most of the semi-arid and arid regions in the World with the land use and land cover changes in these areas causing changes in water supply in the downstream plains. Secondly, we will compare two different areas in West Africa in the current period in order to highlight the role of soil hydrological functioning from the primary scale to the regional scale. The non-linearity of hydrological processes in this area gives rise to the "Sahelian paradox" as the anthropic impact exceeds the influence of the climatic inter decadal signal. LUC/LCC (land use change/land cover change) not only lead to consequences in hydrological balances, but also in sedimentary balance. Land cover has a direct influence on soil erosion and conservation. Both erosion and runoff determine the river transport capacity, whose evolution appears as geomorphic evidences of the historical behaviour of the basins. In a third section, we will describe the consequence of land use changes in erosion, transport and sedimentation processes; finally, other causal factors of water cycle modifications will be discussed in the last part of this chapter.

### **2. Consequences**

Mountainous areas are commonly characterized by forests, whose removal reduces infiltration and improves the conditions of overland flow (Kuchment, 2008). As a consequence, flood runoff and peak discharges may significantly increase. On the contrary, the main effect of afforestation on the water cycle is an increase in transpiration and interception of precipitation, which in turn results in a decrease of the volume of total discharge. Simultaneously, the higher infiltration capacity of forest soils increases the opportunity for groundwater recharge, and the flow of small rivers tends to be more sustained.

In this section, the hydrological behaviours of two different mountains are compared.

**The French Southern Alps** constitute a mountain area in stage of resilience. This region suffered up till the end of the 19th century from severe land degradation due to deforestation, crops extension and overgrazing. Large areas and hillslopes were degraded and numerous "torrents" appeared. Floods and inundations in the alpine valleys and in the downstream flood plains were frequent and devastative. Flood flows increased and on the contrary, base flows were reduced, making water management very difficult. A strong increase in soil losses in the upstream part of the basins led to river beds widening and rising. As in other mountainous areas of Southern Europe, a hard policy of land reclamation was initiated in the second half of the 19th century (French 2nd Empire and 3rd Republic). Reforestation and torrent correction were strongly developed (Table 1); then, socioeconomic evolution and three wars (against Prussia in 1870, and the 1st and 2nd World Wars) depleted these areas of their inhabitants (Descroix & Gautier, 2002; Descroix & Mathys, 2003).


Table 1. Evolution of the forested area in the three French Southern Alps districts since 1770


Table 2. Measured erosion depth in the main fragile outcrops in Southern French Alps

describe the consequence of land use changes in erosion, transport and sedimentation processes; finally, other causal factors of water cycle modifications will be discussed in the last

Mountainous areas are commonly characterized by forests, whose removal reduces infiltration and improves the conditions of overland flow (Kuchment, 2008). As a consequence, flood runoff and peak discharges may significantly increase. On the contrary, the main effect of afforestation on the water cycle is an increase in transpiration and interception of precipitation, which in turn results in a decrease of the volume of total discharge. Simultaneously, the higher infiltration capacity of forest soils increases the opportunity for groundwater recharge, and the flow of small rivers tends to be more

In this section, the hydrological behaviours of two different mountains are compared.

Alpes Hte Provence 6954 10 18 26 31 36 2500 Hautes Alpes 5534 11 19 27 29 34 1860 Drôme 6560 15 23 29 31 43 2800

Black marls 9.4 13 Blue marls 8.6 4 Grey marls 8 1 Molasses 1.4 1 Glacial fields 16 1 Oligocene clays 30 1

Table 2. Measured erosion depth in the main fragile outcrops in Southern French Alps

Table 1. Evolution of the forested area in the three French Southern Alps districts since 1770

Bed rock outcrop soil losses in mm.yr-1 number of sites documented

**The French Southern Alps** constitute a mountain area in stage of resilience. This region suffered up till the end of the 19th century from severe land degradation due to deforestation, crops extension and overgrazing. Large areas and hillslopes were degraded and numerous "torrents" appeared. Floods and inundations in the alpine valleys and in the downstream flood plains were frequent and devastative. Flood flows increased and on the contrary, base flows were reduced, making water management very difficult. A strong increase in soil losses in the upstream part of the basins led to river beds widening and rising. As in other mountainous areas of Southern Europe, a hard policy of land reclamation was initiated in the second half of the 19th century (French 2nd Empire and 3rd Republic). Reforestation and torrent correction were strongly developed (Table 1); then, socioeconomic evolution and three wars (against Prussia in 1870, and the 1st and 2nd World Wars) depleted these areas of their inhabitants (Descroix & Gautier, 2002; Descroix & Mathys,

Forested area (%) 1992 forested area

in 1992 (km2 1770-1780 1878 1914 1948 1992 )

part of this chapter.

**2. Consequences** 

sustained.

2003).

Department total area

(km2)

The most important sediment providing material in the French Southern Alps is the black marls (Table 2), which outcrops exceed 1000 km2; but some other fields are also very sediment productive, as indicated in Table 2. For centuries, these very degraded marly terrains were providing stream flow and suspended sediment load to the Alpine rivers.

Nowadays, the forested area is more than three times that of the mid 19th century, and erosion is no more a concern for the population neither is the flood, and inundation has become very uncommon in the last few decades. Flooding and sedimentation are now very sparse and they are no more a constraint to development either for agriculture or for other socio-economic activities. The Durance river valley was completely developed during the 20th century and a series of dams were built along the river in order to provide electric power as well as water to the irrigated district of the lower Provence and the Comtat Venaissin and for the cities of Marseille and Toulon.

Inversely, in Northern Mexico, the hillslopes of the **Western Sierra Madre** have for some decades been going through a great erosion stage. This is due to a generalised overexploitation of the environment. This area has been characterized at least since the 1970s by a general overgrazing and forest clearing in the upper part of the basins (Descroix et al., 2002, Viramontes & Descroix, 2003). Both human originated processes cause an increase in bare soils areas, leading to soil compaction and a significant decrease in soil hydraulic conductivity and water infiltration. Other consequences are a decrease in soil water holding capacity and an increase in runoff. Instead of gullies, this area is characterised overall by sheet erosion and sheet runoff, all the soil surface being degraded: fine soil particles are removed by drop detachment and transported by runoff. This decrease in soil permeability is demonstrated by data collected in Table 3 and Table 4. At the end of the 20th century, deforestation was severe and more than 50% of the extended pine and oak forests of the upper parts of the Sierra had already been cleared, with the exploited wood volume being commonly three times higher than the authorized volume (Viramontes & Descroix, 2003). Measurements made at the 50 m2 plot scale show that runoff was increased by one order of magnitude and erosion by two orders of magnitude after the tree cutting (Table 3). In the first years following clearing, the remaining litter partially protects the soil.


Table 3. Influence of clearing on runoff and erosion measured at the plot scale (50 m2) in a forest environment in the Western Sierra Madre (Northern Mexico)

In the same upper part of the Sierra Madre, soil physical characteristics were measured on grassland areas, in order to compare their behaviour in non grazed or normally grazed areas on the one hand, and in spread overgrazed sectors on the other hand. Obviously, overgrazing causes a strong increase in runoff (five times higher) and erosion (one order of magnitude higher), and a significant reduction in hydraulic conductivity and soil porosity and inversely a rise in soil bulk density (Table 4). The three latter parameters are part of the explaining factors of the two former ones. They are due to cattle trampling which provokes soil compaction and a closing of the pores, most of them becoming inactive (Descroix et al., 2002).


Table 4. Soil characteristics on non-grazed and overgrazed areas in a grassland environment. Runoff and erosion measured on 50 m2 and 1 m2 plots (10 repetitions per class); hydraulic conductivity measured with disk infiltrometer, bulk density by the "pool method", porosity using a mercury porosimeter.

Thus, we observed in the last 3 decades the following features at the basin scale (those of Rio Ramos and Rio Sextin, two tributaries of the Rio Nazas, whose catchment areas are respectively 7100 and 4700 km²) (Fig.1):


Fig. 1. Observed hydrological changes linked to land use changes by the evolution of hydrographs in sub-basins of some km² into the Ramos and Sextin Rivers basins

All these facts are the results of a reduction of the water holding capacity of the soils and the basins.

These consequences make water management more difficult, because there is less water in the rivers during the dry season, at the moment where water is most required for agriculture, grazing and other needs. Simultaneously, the increase of the discharge, the peak flow and the flooding risk during the rainy season also makes water management more difficult due to both the necessity to increase the volume of reservoirs to ensure the same volume of supplied water and the severe increase in silting up which could cause other problems as dam sedimentation, and local difficulties for irrigation and navigation.

This will also have consequences in the sediment balance (see part 4).

Land cover highly impacts the water cycle; this is even more marked in mountainous areas where there is commonly more rainfall, and steep slopes favourable to runoff and erosion. Mountains being natural water towers, land use and land cover changes could have there consequences in the water balance of extended areas. Processes described in fig.1 applied at the scale of great water providing areas could lead considerable changes in water supply and oblige to great water management modification. In sub-Saharan Africa, two main water towers (the Guinean mountains and the Ethiopian Highlands) provide water to hundreds of millions inhabitants, with significant proportion of them living in arid and semi-arid areas. An increase in both flood and drought hazards is observed from some decades and this is a threat for the downstream located societies. This is also a geopolitical challenge, and water scarcity could be in the future the cause of severe conflicts.

#### **3. The sahelian paradox**

90 Water Resources Management and Modeling

Non grazed 8 7 77 1.21 49 Overgrazed 43 90 24 1.55 35 Table 4. Soil characteristics on non-grazed and overgrazed areas in a grassland environment. Runoff and erosion measured on 50 m2 and 1 m2 plots (10 repetitions per class); hydraulic conductivity measured with disk infiltrometer, bulk density by the "pool method", porosity

Thus, we observed in the last 3 decades the following features at the basin scale (those of Rio Ramos and Rio Sextin, two tributaries of the Rio Nazas, whose catchment areas are

Fig. 1. Observed hydrological changes linked to land use changes by the evolution of hydrographs in sub-basins of some km² into the Ramos and Sextin Rivers basins

hydraulic conductivity mm.h-1 bulk density g.cm-3

Porosity %

soil losses g.m-2

Soil

characteristics

using a mercury porosimeter.

respectively 7100 and 4700 km²) (Fig.1):




runoff coefficient %

> In the West African Sahel, it was observed in recent decades that during the Great Drought that occured in this region (this has not yet come to an end, although rainfall amounts are partially increasing again; see Ali and Lebel, 2009 and Lebel and Ali, 2009), Sahelian rivers showed a significant increase in runoff in spite of the reduction of the annual rainfall amount. Since this cannot be due to rainfall, there is a consensus to estimate that this is a consequence of land use change (Albergel, 1987; Descroix et al., 2009; Amogu et al., 2010). West Africa suffered a great drought from 1968 to 1995. The Sahelian area (semi-arid part of West Africa, with annual rainfall lower than 700 mm) was more particularly affected with a decrease in rainfall ranging from 20 to 35% during the 1970s and the 1980s compared with the 1950-1968 period. The drought has not really ended, only attenuated in the eastern half of the Sahel, and rainfall remains very low in its western part (Ali and Lebel, 2009; Lebel and Ali, 2009).

> The large rivers of West Africa (Senegal, Gambia, Niger, Chari among others) experienced an expected decrease in discharge since the beginning of the Drought (1968); for the period of 1970–2000, the decrease in the mean annual discharge of the region's largest rivers (fig.2), namely the Senegal and Niger rivers, was in proportion almost twice as much as the decrease in rainfall. Similar trends have been observed for smaller river systems while, in contrast, other studies have indicated a runoff increase in some Sahelian catchments. Albergel (1987) remarked a paradoxical fact: although the rainfall was decreasing from 1968, runoff increased in experimental Sahelian catchments in Burkina Faso; in the Sudanian catchments of the same country, more classically, the discharges were decreasing. This is even more paradoxical because rainfall decreased stronger northward, in the Sahelian area,

than in the Sudanian one (Fig. 3). Amani and Nguetora (2002) highlighted an increase in runoff of some right bank tributaries of the Niger River, and they observed that their flood began one month earlier during the 1980s than during the 1960s; they also remarked that in the Niger River at Niamey station, sometimes in the recent years (at the moment) the first flood peak was higher than the "guinean flood" peak, a process that never occurred before 1984. Mahé et al. (2003) observed an increase in discharge of the Nakambé River, one of the upper Sahelian tributaries of the Volta River; Mahé and Paturel (2009) noticed an increase in runoff coefficients and in discharges in the Black and White Gorgol basins in the western Sahel (Mauritania). This is also observed in the south Sahelian Sokoto basin in Nigeria (Mahé et al., 2011). Most of the Sahel seems to be concerned by this "Sahelian paradox". Amogu et al (2010) proposed a regionalisation of hydrological processes, with a big difference between Sudanian areas where runoff decreased strongly and Sahelian (and, in recent years, north-Sudanian) areas where runoff increased significantly. They also determined that this increase in runoff was observed at all the spatial scales.

Fig. 2. The effect of the Great Drought on the discharge of the main West African rivers: a general expected decrease in runoff after 1968

Until the end of the 1990s, all the water supplied to the Niger River downstream from Gao came from the right bank tributaries. On the left bank, endorheism is the main hydrological mode, and the observed increase in runoff led to an increase in the number, the volume and the duration of ponds. As the ponds are the main groundwater recharge points, this caused a significant increase in groundwater recharge and a rise in water table level in the western part of the Iullemeden sedimentary basin, during the drought; this was named the "Niamey's paradox" by Leduc et al (1997) and described by Séguis et al. (2004). Also in this case, the increase in runoff is supposed to be due to land use change, particularly land clearing (Leblanc et al., 2008). Finally, if the drought is the most cited process in the Sahel, "floods in drylands" are not to be considered as an oxymoron, and they are unfortunately not well known, in spite of their current rise in number and strength (Tschakert and al., 2010).

than in the Sudanian one (Fig. 3). Amani and Nguetora (2002) highlighted an increase in runoff of some right bank tributaries of the Niger River, and they observed that their flood began one month earlier during the 1980s than during the 1960s; they also remarked that in the Niger River at Niamey station, sometimes in the recent years (at the moment) the first flood peak was higher than the "guinean flood" peak, a process that never occurred before 1984. Mahé et al. (2003) observed an increase in discharge of the Nakambé River, one of the upper Sahelian tributaries of the Volta River; Mahé and Paturel (2009) noticed an increase in runoff coefficients and in discharges in the Black and White Gorgol basins in the western Sahel (Mauritania). This is also observed in the south Sahelian Sokoto basin in Nigeria (Mahé et al., 2011). Most of the Sahel seems to be concerned by this "Sahelian paradox". Amogu et al (2010) proposed a regionalisation of hydrological processes, with a big difference between Sudanian areas where runoff decreased strongly and Sahelian (and, in recent years, north-Sudanian) areas where runoff increased significantly. They also

determined that this increase in runoff was observed at all the spatial scales.

Fig. 2. The effect of the Great Drought on the discharge of the main West African rivers: a

Until the end of the 1990s, all the water supplied to the Niger River downstream from Gao came from the right bank tributaries. On the left bank, endorheism is the main hydrological mode, and the observed increase in runoff led to an increase in the number, the volume and the duration of ponds. As the ponds are the main groundwater recharge points, this caused a significant increase in groundwater recharge and a rise in water table level in the western part of the Iullemeden sedimentary basin, during the drought; this was named the "Niamey's paradox" by Leduc et al (1997) and described by Séguis et al. (2004). Also in this case, the increase in runoff is supposed to be due to land use change, particularly land clearing (Leblanc et al., 2008). Finally, if the drought is the most cited process in the Sahel, "floods in drylands" are not to be considered as an oxymoron, and they are unfortunately not well known, in spite of their current rise in number and strength (Tschakert and al.,

general expected decrease in runoff after 1968

2010).

Fig. 3. The unexpected increase in discharge of gauged Sahelian rivers during the Great Drought

Most of the changes in the water cycle characteristics in West Africa result from the long drought period (observed since 1968) and the reduction of the soil water holding capacity. However, recent studies (Mahé et Paturel, 2009, Descroix et al., 2009) have shown that anthropic factors better explain the spatial and temporal variability of the processes than the climatic factors.

As a matter of fact, as in all geo-climatic environments, a combination of natural and human parameters leads to changes in processes, which explains the resource evolution and possibly, as a feed back effect, the climatic evolution; but the last hypothesis remains to be founded on tangible evidence. The combined action of the very clear climatic and anthropic trends led to the following hydrological consequences (Albergel, 1987, Mahé, 1993, Amani et Nguetora, 2002, Mahé *et al,* 2003, Mahé *et al,* 2005, Andersen *et al*., 2005, Mahé, 2009, Mahé *et al*, 2009, Amogu, 2009, Descroix *et al*., 2009, Amogu *et al*., 2010; Descroix *et al*. 2012):


In the Sudanian region, where the rainfall deficit was lower, an inverse trend has long been observed, and runoff and discharge were logically and expectedly decreasing with the reduction in rainfall. Hydrological processes remain there mostly "hewlettian" (Hewlett, 1961, Cappus, 1960), i.e. runoff occurs only when the soil is saturated; as soils are mainly deep, the reduction of rainfall firstly affected the part of water which was dedicated to runoff (fig.4). The increase in runoff is not observed in the Sudanian area (where rainfall amount ranges from 700 to 1300 mm per year), where the reduction in rainfall was not so strong (10-15%) due to lower land use changes and a reduced impact of these changes. With soils being deeper in this area, the important role of soil and groundwater in runoff yield is highlighted. In this "hewlettian" context, a decrease in rainfall firstly reduces the part dedicated to runoff.


Fig. 4. The expected decrease in discharge of gauged Sudanian rivers during the Great Drought

During the end of summer 2010, the surroundings of the Niger River in the Niamey area suffered a severe flood. We show here, that although there was a partial recovery after the

amount ranges from 700 to 1300 mm per year), where the reduction in rainfall was not so strong (10-15%) due to lower land use changes and a reduced impact of these changes. With soils being deeper in this area, the important role of soil and groundwater in runoff yield is highlighted. In this "hewlettian" context, a decrease in rainfall firstly reduces the part




Fig. 4. The expected decrease in discharge of gauged Sudanian rivers during the Great

During the end of summer 2010, the surroundings of the Niger River in the Niamey area suffered a severe flood. We show here, that although there was a partial recovery after the

inhabitants for a long time (due to sanitary and historical reasons).

of 20th century (Olivry, 2002; Andersen et al., 2005).

dedicated to runoff.

Drought

severe droughts of the 80's and 90's, rainfall was not the main factor of this exceptional flood: 2010 annual rainfall was below the 50's 60's values, and neither rainfall distribution during the year, nor the size of the rainy events changed significantly during the last decades. Rather, the hydrological behavior of the right bank tributaries of the Niger River shows a severe increase in runoff since the 80's (Fig.3), which is still ongoing for two of them, and resulted in changes in the hydrograph of the Niger River during the rainy season, lasting from June to August in Niamey (Fig. 6). Otherwise, during the last decade (2000- 2009), the flood occurred 2 or 3 weeks earlier at the Niamey station than in previous decades. During the last two decades, the end of the main flood has come relatively early in spite of the increase in discharge, continuing to occur 2 months earlier than in the 1950s and 1960s (before the drought) (Fig. 6). We show that this is the result of an increase in bare and crusted soils as a consequence of human pressure and/or of non reversible effects of the drought. This questions the idea of the re-greening Sahel arising from recent remote sensing studies. Of particular interest to policymakers is the effect of human pressure (and particularly intensive cropping and grazing) on hydrological regimes of Sahelian rivers, which could be partly mitigated by either the use of updated urbanization rules of by promoting re-greening experiments which had already some success in this area.

The hydrograph of Niger River downstream from the Niger Inner Delta (northern Mali) is bi-modal, being constituted by (Fig. 6):


Fig. 5. The intermediate behaviour of northern Sudanian basins: towards a Hortonian process ?

Fig. 6. Evolution of the hydrograph of Niger River at Niamey station representing the evolution of discharge and regime

Fig. 7. Inundation of the right bank part of the city of Niamey (Niger) by the Niger River in August 2010


During the red flood of 2010, the water reached its highest recorded level, the flooded area was 3.1 km² in the city of Niamey, some 5 000 people lost their homes and crops and the food shortage resulting from drought of the two previous years provoked an increased havoc on local population life (Fig. 7) (Descroix *et al*., 2011).

### **4. Influence on sedimentation**

96 Water Resources Management and Modeling

Fig. 6. Evolution of the hydrograph of Niger River at Niamey station representing the

Fig. 7. Inundation of the right bank part of the city of Niamey (Niger) by the Niger River in

evolution of discharge and regime

August 2010

In this section, the consequence of land use changes on erosion, transport and deposition of sediment will be analysed through the pattern of river beds evolution. As it was shown in section 2, a major land management policy was driven in the Southern French mountains at the end of the 19th century (as in other European Mediterranean areas, and, later, in French North African dependencies). The aim was land reclamation in order to control soil degradation as an origin of flooding and sedimentation in downstream floodplains. At the small basin scale, the results were very effective and efficient, since even very degraded hillslopes were reforested and protected against erosion (see fig. 8).

Fig. 8. The Brusquet catchment (French Southern Alps) in 1880 (left) and in 1990 (right) (*photos IRSTEA (formerly CEMAGREF))*

This led to important changes in the downstream hydrography, the decrease in bed load and suspended load linked to the afforestation and flood hazard reduction causing a strong evolution of river beds in most of the major rivers in the southern European mountains (Descroix and Gautier, 2002). The river beds were affected by:


These processes appeared some decades after the beginning of reforestation which was initiated in the second half of the 19th century and the rural abandonment strongly accelerated the increase in forested areas during the 20th century. During the 1970s, the first evidence of the sediment load deficit appeared with the beginning of the river bed entrenchment, accelerated by gravel extraction in most of the French Alps rivers. Although this activity was forbidden at the beginning of the 1980s, the river entrenchment remains severe due to the remaining sediment transport deficit linked to the natural afforestation, preventing soil material from being detached and transported. In many cases, the river bed entrenchment reached several meters, up to 3 meters in the Buech river valley (Descroix and Gautier, 2002), locally more than 10 m in northern French alpine rivers (Peiry, 1988). In the last decades, 90% of the total length of French Alpine rivers have been affected by this sedimentary déficit (Peiry et al., 1994). This could lead to the destruction of bridges and embankments, and led the authorities to drive new land management policies applied to river beds: new embankment, protection of bridges, etc (Fig. 9).

In Northern Mexico, the strong overgrazing and deforestation cited in section 2 are linked to speculation; it is easy to earn much money without taking into consideration the land conservation; selling wood to sawmills, and calves to the US market became good businesses during the last decades. In the 1980s and the 1990s, Durango state (Northern Mexico) produced 5 millions m3 of wood per year instead of the 1.7 million m3 authorized by federal forestry institutions (Descroix et al., 2001). During the 1990s and the beginning of the 2000s, a general trend of overgrazing was observed in the whole Northwestern part of Mexico (Chihuahua, Sonora, Sinaloa and Durango States) (Descroix, 2004).

Fig. 9. The Céans river bed in 2004 (left) and in 2011 (right) (Southern French Alps). The bed of the Céans was entrenching strongly in the last decades, due to a great decrease of erosion and sediment transport. The river banks were falling regularly in the entrenched bed (left), obliging the authorities in building new embankments to protect surrounding areas from destruction (right).

This led to the on-going land degradation described in section 2. This causes a severe increase in suspended and bed load sediment in river beds, which in turn leads to the

evidence of the sediment load deficit appeared with the beginning of the river bed entrenchment, accelerated by gravel extraction in most of the French Alps rivers. Although this activity was forbidden at the beginning of the 1980s, the river entrenchment remains severe due to the remaining sediment transport deficit linked to the natural afforestation, preventing soil material from being detached and transported. In many cases, the river bed entrenchment reached several meters, up to 3 meters in the Buech river valley (Descroix and Gautier, 2002), locally more than 10 m in northern French alpine rivers (Peiry, 1988). In the last decades, 90% of the total length of French Alpine rivers have been affected by this sedimentary déficit (Peiry et al., 1994). This could lead to the destruction of bridges and embankments, and led the authorities to drive new land management policies applied to

In Northern Mexico, the strong overgrazing and deforestation cited in section 2 are linked to speculation; it is easy to earn much money without taking into consideration the land conservation; selling wood to sawmills, and calves to the US market became good businesses during the last decades. In the 1980s and the 1990s, Durango state (Northern Mexico) produced 5 millions m3 of wood per year instead of the 1.7 million m3 authorized by federal forestry institutions (Descroix et al., 2001). During the 1990s and the beginning of the 2000s, a general trend of overgrazing was observed in the whole Northwestern part of

 Fig. 9. The Céans river bed in 2004 (left) and in 2011 (right) (Southern French Alps). The bed of the Céans was entrenching strongly in the last decades, due to a great decrease of erosion and sediment transport. The river banks were falling regularly in the entrenched bed (left), obliging the authorities in building new embankments to protect surrounding areas from

This led to the on-going land degradation described in section 2. This causes a severe increase in suspended and bed load sediment in river beds, which in turn leads to the

destruction (right).

river beds: new embankment, protection of bridges, etc (Fig. 9).

Mexico (Chihuahua, Sonora, Sinaloa and Durango States) (Descroix, 2004).

widening and enhancement of these beds (fig. 10a): downstream, this provokes the silting up of numerous dams located at the boundary of the coastal Sinaloa and Sonora plain (Fig. 10b); the latter being arid or semi arid, it needs water from the Western Sierra Madre to supply the irrigated districts. Clearly, land overexploitation in the upper part of the basins is threatening the agricultural and economic development of the relatively rich coastal plain, as well as the endorheic basin of the Nazas River and the Conchos basin, the main Mexican tributary of the Rio Bravo/Rio Grande River.

Fig. 10. Evidence of silting up in the Matapé River bed: 10a (left) a general widening and enhancement of river bed is observed; 10b (right) downstream, sedimentation is filling the dams reservoirs, threatening the irrigation downstream in the Coastal, semi-arid, plain.

In section 3, we saw that West Africa is suffering a strong land use change due to both a 40 year long drought and a severe demographic pressure (the highest demographic growth in the World since the beginning of the 1990s).

What are the consequences of this pressure on the regional sedimentary balance?

There is a strong opposition in the sedimentary balance of the Niger River upstream vs downstream from the Niger Inner Delta.


The sedimentary deficit of the Niger riverbed upstream from Bamako is also evidence of the impact of urbanisation on the hydrological behaviour of river basins.

Like the Niger River, some other great river basins in the world are affected by changes in sedimentary balance; the following examples show the importance of taking into account this balance in developing land management policies:

Fig. 11. Evidence of the silting up of Niger River in its middle reach: at left, sandy deposit in the river bed at Niamey (Niger); at right: the new Kourtéré (formed in 1998) alluvial cone invading the Niger River bed in its right bank, just upstream from Niamey (*photo E. Gautier*)


In these three cases, an increase in the silting up in the cited dams is observed.

#### **5. Specific impacts of agriculture and urbanisation**

In this section, we will study other human factors that influence the modification of hydrological behaviour, such as agriculture, irrigation, drainage, urbanisation and socioeconomics, in several parts of the World (mainly in tropical areas).

 Fig. 11. Evidence of the silting up of Niger River in its middle reach: at left, sandy deposit in the river bed at Niamey (Niger); at right: the new Kourtéré (formed in 1998) alluvial cone invading the Niger River bed in its right bank, just upstream from Niamey (*photo E.* 




In this section, we will study other human factors that influence the modification of hydrological behaviour, such as agriculture, irrigation, drainage, urbanisation and socio-

In these three cases, an increase in the silting up in the cited dams is observed.

**5. Specific impacts of agriculture and urbanisation** 

economics, in several parts of the World (mainly in tropical areas).

*Gautier*)

river bed.

villages back;

Descroix, 2011).

As has been shown in section 4, rural activities (mostly cropping, livestock grazing and forest clearing) are commonly the cause of an increase in soil losses in the hillslopes, due to the consequent decrease in vegetation cover which causes a soil direct exposure to wind and water erosion. Even if the crops can reach or exceed the same biomass as the removed natural vegetation, this is the case only at the end of the cropping season: evapotranspiration, soil coverage, and protection against wind and water erosion and against runoff remains low during the first weeks and months of the crop cycle. Seasonal influence also impacts deforestation, and clearing has little effect on flow under moderate climates during winter and early spring, because in these seasons, evapotranspiration is very low (Kuchment, 2008).

The impact of irrigation on the water cycle is particularly significant in arid areas, but it is also considerable in moderate climates; it is common for runoff and evaporation from irrigated areas to increase significantly (Kuchment, 2008). In many dry areas, a rise in water table can occur because of water filtration from reservoirs, leakage from water distributing systems or inappropriate practices; this may cause waterlogging and soil salinisation (Kuprianov and Shiklomanov, 1973; Kuchment, 2008). In order to avoid water logging, drainage is applied in many regions of the world; the first effect of drainage is the lowering of water table; as a result, evapotranspiration may drop considerably; it increases flow velocities and runoff can rise by 20-30%. The effect of agriculture and forestry on the water cycle depends on physiographic and climatic conditions. Ploughing usually breaks up overland flow and increases infiltration (Kuchment, 2008). Tillage and the activity of plant roots system modify the structure of the upper soil layer and change the vertical permeability and the soil water retention capacity. Extension of vegetation cover increases the interception of precipitation and increases evapotranspiration.

Urbanisation is well known to disturb natural hydrology, mostly due to the reduction in infiltration capabilities and in the obstruction of stream flows. The increasing urbanisation of watersheds increases the amount of impervious areas; thereby increasing the volumes of stream flow and shortening the time of rain water concentration. The consequence is an increase in maximum flow discharges thereby increasing the flooding risk (Kuprianov and Shiklomanov, 1973). Natural land that used to soak up runoff is replaced by roads and large areas of pavement. Due to the increased pavement area, there is more storm runoff and erosion because there is less vegetation to impede the flow of water as it runs down hills; more land erosion and more sediment is washed into streams, less water will soak into the ground. This means that the water table will have less water to recharge it; it can cause subsidence of the urban area, which is commonly exaggerated by the overuse of ground water for urban needs. Mexico City pumps almost 1.5 billion m3.yr-1 from its water table, provoking a subsidence of several cm per year. Otherwise, large areas of Bangkok became liable to flooding in the recent decades due to the subsidence caused by the overuse of groundwater.

Thus, urbanisation causes an increase in both the runoff coefficient and the stream velocity due to the increasing proportion of impervious surfaces (factories, roofs, roads, car parks, pavements, etc). It also causes strong modifications in water quality, rain water draining these impervious areas and loading urban and industrial dust and wastes through the rivers and the water tables.

As has been shown in section 3, urbanisation needs gravel and sand for construction and river beds have long been considered as abundant supplying areas for these materials. However, these reserves are not inexhaustible; uncontrolled extraction of material may affect the sediment balance and the hydraulic equilibrium of the river, and sometimes that of surrounding lands, bridges and embankments. These consequences are amplified in mountainous areas due to the steep slopes.

A case study of consequences of mountain urbanisation: the La Ravoire torrent (Northen French Alps) appearance in March 1981.

At the end of March 1981 a major ecological catastrophe occurred in the French northern Alps. At the snowmelt, the 31st March, the Ravoire brook (Upper Isère valley, in Savoie region) suddenly reawakened (Périnet, 1982) (Fig.12).

Fig. 12. a) Situation of the Les Arcs hillslope before the 31/3/1981; b) Situation after the 31/3/1981: damages caused by the rainy event; c) Main restoration measures; d) the series of dams built to prevent further deepening of the torrent bed (photo F. Périnet)

The banks were destabilized and collapsed in great bundles and the brook became a torrent, carrying enormous volumes of mud mixed with snow, trees and rocks. Within five days, its bed had deepened by more than 10 meters; 300 000 m3 of rocks and material hurtled down from the mountain, covering more than 5 hectares of grassland at foothill, blocking the Isère River and pushing its course towards its right bank, in the industrial area of Bourg Saint Maurice. Significant damages were recorded in this area with some buildings destroyed; on the left bank of Isère River, where the Ravoire brook flows, the railway and five roads (including the access road to Les Arcs ski station) were cut, five houses partially or completely destroyed. In a first assistance operation, a single way road was established to provide access to the 15 000 stranded tourists trying to leave the ski station, the torrent water was diverted towards the penstock pipe of the Malgovert hydropower station (located near Bourg St Maurice in the bottom of the Isère valley). Then works for the torrent course reclamation, the housing protection and the building of new bridges were undertaken as well as a gallery for the railroad (Fig.12).

#### **6. Geopolitics**

102 Water Resources Management and Modeling

As has been shown in section 3, urbanisation needs gravel and sand for construction and river beds have long been considered as abundant supplying areas for these materials. However, these reserves are not inexhaustible; uncontrolled extraction of material may affect the sediment balance and the hydraulic equilibrium of the river, and sometimes that of surrounding lands, bridges and embankments. These consequences are amplified in

A case study of consequences of mountain urbanisation: the La Ravoire torrent (Northen

At the end of March 1981 a major ecological catastrophe occurred in the French northern Alps. At the snowmelt, the 31st March, the Ravoire brook (Upper Isère valley, in Savoie

Fig. 12. a) Situation of the Les Arcs hillslope before the 31/3/1981; b) Situation after the 31/3/1981: damages caused by the rainy event; c) Main restoration measures; d) the series

of dams built to prevent further deepening of the torrent bed (photo F. Périnet)

mountainous areas due to the steep slopes.

French Alps) appearance in March 1981.

region) suddenly reawakened (Périnet, 1982) (Fig.12).

(a) (b)

(c) (d)

Geopolitical events can significantly impact the water cycle. The main concern is the strong land use changes caused by massive migrations due to wars or local conflicts.

"The Indochinese section of the Mekong Basin has been subjected to major environmental disturbances over the last half century" (Lacombe et al., 2010). The Vietnam War is invoked as a central explanation for the extensive deforestation in specific areas while conflict induced exoduses caused the abandonment of cultivated lands, followed by forest regeneration. Although the socio-economic consequences of these episodes have been analysed, their hydrological impacts remained unknown until the study of Lacombe et al. (2010). They investigated hydrological changes in two catchments of the lower Mekong Basin that were either heavily bombed (in southern Laos) or depopulated (in northern Laos) (Fig.13). The first one is located on the Ho Chi Minh trail; one third of the latitudinal extent of the Ho Chi Minh trail in Laos is included in this southern catchment and it was a target which the United States Air Forces (USAF) made a concerted effort to neutralize. During this conflict, the USAF practised deliberate massive removal of vegetation as a military tactic to deny cover and land to opposition forces. A total of between 690 000 and 2 948 000 tons of high explosive ordnance were exploded over this southern catchment between October 1965 and September 1973 (Lacombe et al., 2010). In the northern basin, while air operations of the Vietnam War caused losses of forest, continuing ground battles in northern Laos over the period 1953–1975 induced significant emigration. Between 730 000 and 1 million people (one fourth of the country's population as per 1970) were displaced over this period (Taillard, 1989; Goudineau, 1997; cited by Lacombe et al., 2010). This migration flow was sustained after the end of the war. As an immediate consequence of the communist takeover more than 300 000 people fled to Thailand between 1975 and 1985, escaping the new regime. This massive exodus particularly affected the northern provinces of Laos, traditionally inhabited by Hmong ethnic groups (Central Intelligence Agency, 1970, cited by Lacombe et al., 2010) which served the US army during the Vietnam War and were forced to flee the new communist regime after the war ended. The abandonment of large mountainous areas, which were traditionally subject to shifting or permanent cultivation, most likely induced a shift toward forest regeneration (Lacombe et al., 2010). The analysis (Lacombe et al., 2010) is based on the widely and independently recognized fact that vegetation, via evapotranspiration, is a central driver of groundwater recharge and stream flow production. The analysis of the most complete Vietnam War air mission database and of available hydrometeorological data over the period 1960–2004 reveals a sharp runoff increase in the southern catchment when bombing climaxed in the early 1970s while no hydrological change is observed in the northern catchment over the same period (Lacombe et al., 2010). From 1995 onwards, the northern and southern catchment's runoff productions are significantly lower and higher than in the pre-war conditions, respectively. Although causalities could not be ascertained because of data limitations, these short- and long-term hydrological shifts were found to be consistent, in terms of occurrence, spatial distribution and magnitude, with the expected changes in the vegetation cover, either denser in the north (in response to abandonment of cultivated lands) or sparser in the south (as a result of bomb-induced deforestation and soil degradations) (Lacombe et al., 2010). Furthermore, as a consequence, some areas near Vientiane (the capital of Laos) and overall near Luang Prabang (a city located 200 km north of Vientiane) were cleared in past years including on steep slopes (Chaplot et al., 2007), causing the appearance of gullies (Fig.14).

Fig. 13. Location of the two sub basins studied by Lacombe et al. 2010

In other areas in the world, geopolitics could lead to severe land degradation: in the Republic of South Africa, the apartheid regime created some Bantustans, also known as black African homelands or simply homelands), which were territories set aside for black inhabitants, not recognized by the international community. Due to population density and land management, soil erosion was strong in some of these areas such as the Kwazulu. The same kind of process occurred in northern Pakistan during the USSR-Afghanistan war; a great number of Afghans took refuge in Pakistan. Finally, in Hispaniola Island (West Indies), which includes two countries, Haïti on the western side and the Dominican Republic to the east, the rural areas of the Republic of Haïti, are much more densely populated than the ones of the Dominican Republic resulting in severe land degradation (deforestation to harvest crops, grass and wood). The fact that the population is much poorer in Haiti should also be noted. In these three examples, there is a strong gradient in land exploitation between two areas, noticeable in aerial pictures and satellite scenes. Overall, this provokes severe damages in the overexploited side due to cyclones in Haïti and to monsoon rains in Pakistan and in South Africa.

Fig. 14. Gullies appearing in rain fed rice fields in central Laos, 10 km eastward from Luang Prabang. These areas were settled on fields recently cleared by peasants coming from the eastern Laotian mountains near the Vietnam boundary, judged insecure by the government which encouraged people to migrate within easily controlled areas (*photo NAFRI*) .

### **7. Conclusion**

104 Water Resources Management and Modeling

based on the widely and independently recognized fact that vegetation, via evapotranspiration, is a central driver of groundwater recharge and stream flow production. The analysis of the most complete Vietnam War air mission database and of available hydrometeorological data over the period 1960–2004 reveals a sharp runoff increase in the southern catchment when bombing climaxed in the early 1970s while no hydrological change is observed in the northern catchment over the same period (Lacombe et al., 2010). From 1995 onwards, the northern and southern catchment's runoff productions are significantly lower and higher than in the pre-war conditions, respectively. Although causalities could not be ascertained because of data limitations, these short- and long-term hydrological shifts were found to be consistent, in terms of occurrence, spatial distribution and magnitude, with the expected changes in the vegetation cover, either denser in the north (in response to abandonment of cultivated lands) or sparser in the south (as a result of bomb-induced deforestation and soil degradations) (Lacombe et al., 2010). Furthermore, as a consequence, some areas near Vientiane (the capital of Laos) and overall near Luang Prabang (a city located 200 km north of Vientiane) were cleared in past years including on

steep slopes (Chaplot et al., 2007), causing the appearance of gullies (Fig.14).

Fig. 13. Location of the two sub basins studied by Lacombe et al. 2010

In other areas in the world, geopolitics could lead to severe land degradation: in the Republic of South Africa, the apartheid regime created some Bantustans, also known as black African homelands or simply homelands), which were territories set aside for black inhabitants, not recognized by the international community. Due to population density and land management, soil erosion was strong in some of these areas such as the Kwazulu. The same kind of process occurred in northern Pakistan during the USSR-Afghanistan war; a great number of Afghans took refuge in Pakistan. Finally, in Hispaniola Island (West These last two types of human impacts (urbanisation and geopolitics) demonstrate the great influence of societies, socio-economic conditions, the strategic context and the migration processes, on the land cover and thus on the water cycle. The poor understanding of the influence of these factors could lead to exaggerations recorded in history, such as the sentencing of goats as "the main enemy of Mediterranean forests and soils" by the French governments during the 19th century, while most of the land degradation problem could result from the lack of forest policy during the period just after the French Revolution (from 1797 to 1804) and the forest exploitation during previous centuries in order to build the ships of the French Royal Marine. The Little Ice Age (from the end of Middle Age to the end of 19th century) and its low winter temperatures (mostly during the 19th century) was also a natural constraint for the vegetation. It is always more appropriate to link these severe crises of water cycle to a combination of natural and human factors.

#### **8. Acknowledgements**

We are grateful to the French research Institute for Development (IRD), the Niger Basin Authority (NBA), and the AMMA program which funded the field experiments in Mexico, in France and in Niger. This study was also partially funded by the French ANR projects ECLIS (Contribution of livestock to the reduction of rural population vulnerability and to the promotion of their adaptability to climate and society changes in Sub-Saharan Africa) and ESCAPE (Environmental and Social Changes in Past, present and future)

### **9. References**


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## **Fuzzy Nonlinear Function Approximation (FNLLA) Model for River Flow Forecasting**

P.C. Nayak1, K.P. Sudheer2 and S.K. Jain3 *1Deltaic Regional Centre, National Institute of Hydrology, Kakinada 2Dept of Civil Engineering, Indian Institute of Technology Madras, 3NEEPCO, Department of Water Resources Development and Management, Indian Institute of Technology, Roorkee, India* 

#### **1. Introduction**

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Mahé, G. (2009) Surface/groundwater interactions in the Bani and Nakambe rivers, tributaries of the Niger and Volta basins. West Africa. *Hydrol. Sci. J.*, *54*, 704-712. Mahé, G. & Paturel, J-E. (2009) 1896-2006 Sahelian annual rainfall variability and runoff

Mahé, G., Lienou, G., Bamba, F., paturel, J-E., Adeaga, O., Descroix, L., Mariko, A., Olivry, J-

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Peiry, J-L., Salvador, P.G. & Nouguier, F. (1994). L'incision des rivières dans les Alpes du nord: état de la question. Revue de Géographie de Lyon, 69 (1), 47-56. Périnet, F. (1982). Stations de sport d'hiver ; réflexionc à propos d'un accident. Revue

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northern Mexico : the hydrological consequences of environment degradation.

It is well understood that the limitations of hydrological measurement techniques warrants for modeling of hydrological processes in a basin. However, most hydrologic systems are extremely complex and modeling them with the available limited measurements is a difficult task. The basic purpose of a model is to simulate and predict the operation of the system that is unduly complex, and also to predict the effect of changes on this operation. It is well known, of various hydrological processes, the rainfall-runoff process is the most complex hydrologic phenomenon to comprehend due to tremendous spatial and temporal variability of basin characteristics and rainfall patterns, as well as a number of other variables associated in modeling the physical processes (Tokar and Markus, 2000). The transformation from rainfall to basin runoff involves many hydrologic components that are believed to be highly nonlinear, time varying, spatially distributed, and not easily described by simple models. The artificial neural network (ANN) and Fuzzy Inference System (FIS) approaches are becoming increasingly popular in the context rainfall-runoff modeling due to their various advantages. This Chapter discusses an effective integration of these two models in a different manner.

It is a common belief that the ANN (and FIS models to an extent) models of the rainfall runoff process are purely black models as they do not explain the process being modeled, but for a few recent studies (Wilby et al., 2003; Jain et al., 2004; Sudheer, 2005). However, it must be realized that the hydro-meteorological data that are employed in developing rainfall runoff models (ANN, FIS or conceptual) contain important information about the physical process being modeled, and this information gets embedded or captured inside the model. For instance, a flow hydrograph, which is normally used as the output variable in an ANN rainfall runoff model, consists of various components that result from different physical processes in a watershed.

For example, the rising limb of a runoff hydrograph is the result of the gradual release of water from various storage elements of a watershed due to gradual repletion of the storage due to the rainfall input. The rising limb of the hydrograph is influenced by varying infiltration capacities, watershed storage characteristics, and the nature of the input *i.e*. intensity and duration of the rainfall, and not so much by the climatic factors such as temperature and evapotranspiration etc. (Zhang and Govindaraju, 2000). On the other hand, the falling limb of a watershed is the result of the gradual release of water from various storages of the watershed after the rainfall input has stopped, and is influenced more by the storage characteristics of the watershed and climatic characteristics to some extent. Further, the falling limb of a flow hydrograph can be divided into three parts: initial portion just after the peak, middle portion, and the final portion. The initial portion of the falling limb of a flow hydrograph is influenced more by the quick-flow (or interflow), the middle portion of the falling limb is more dominated by the delayed surface flow, and the final portion of the falling limb (of smaller magnitudes) is dominated by the base flow. Hence it is apparent that a local approximation technique, which maps the changing dynamics using different functions, would be an effective way to model the rainfall runoff process. Local approximation refers to the concept of breaking up the domain into several small neighbourhood regions and analysing these separately. This argument is supported by the results of Sudheer (2005), wherein he proposes a procedure to extract knowledge from trained ANN river flow models.

Farmer and Sidorowitch (1987) have found that chaotic time series prediction using local approximation techniques is several orders of magnitude better. In their approach, the time series is first embedded in a state space using delay coordinates, and the underlying nonlinear mapping is inferred by a local approximation using only nearby states. This approach can be easily extended to higher order local polynomial approximations. Singer et al. (1992) derived the local approximation as a state dependant autoregressive modelling. However, this becomes complex with large data sets as the inefficient computation of nearby state search makes the implementation much harder. In order to overcome this limitation we proposed to simplify the signal representation (input) with a vector clustering procedure. That is, the local model fitting is based on statistically averaged prototypes instead of the original state vector samples. Also, the nearby state search can be significantly simplified with all prototypes organized according to a certain metric such as pattern similarity.

Most of the applications of local approximation technique (e.g. FIS) employ linear relationship as an effective local approximation. However, the rainfall runoff process being highly nonlinear, a nonlinear local approximation would be a better approach. The objective of this Chapter is to illustrate that a nonlinear local approximation approach in modelling the rainfall-runoff process may offer better accuracy in the context of river flow forecasting. More specifically, the chapter discusses about subdividing the data into subspaces and evaluates the limitations of linear and nonlinear local approximations. The proposed approach is illustrated through a real world case study on two river basins. Both the applications are developed for river flow forecasting, one on a daily time step and the other on an hourly time step.

#### **2. Theoretical considerations of local approximation**

Consider modeling a river flow time series such that *<sup>t</sup>* ( ,) *x <sup>t</sup> y f* , where it is required to forecast the value of flow (*yt*) and **x**t is the input vector to the model at time *t*. Generally, the modeler uses a set of *n* 'candidate' examples of the form (**x***i, yi*), *i=1,2,…,n*, and finds an optimal set of parameter vector ( ) by calibrating an appropriate model. The inputs to the model typically are the previous values of the time series and the output will be the forecast value. The model is normally trained and tested on training and testing sets extracted from the historical time series. In addition to previous time series values, one can utilize the values or forecasts of other time series (or external variables) that have a correlation or causal relationship with the series to be forecasted as inputs. For a river flow forecasting problem, such exogenous time series could be the rainfall or evaporation over the basin. Each additional input unit in a model adds another dimension to the space in which the data cases reside, thereby making the function to be mapped more complex. The model attempts to fit a response surface to these data.

For many applications in data driven modeling (regression, classification etc.) an estimate of expected response (output) is desired at or close to one fixed predictor (input) vector. This estimate should depend heavily on predictor vectors in the sample which are close to the given fixed predictor vector. The predictive relationship between the current state **xt** and the next value of the time series can then be expressed as:

$$y(t+T) = f\_T^0(\mathbf{x}(t))\tag{1}$$

where *T* is the prediction time horizon. The problem of one step ahead (*T*=1) predictive modeling is to find the mapping 0 1 : *<sup>N</sup> Tf R to R* . In local approximation, a local predictor is constructed based on the nearby neighbors of **x***t*, that is, fitting a polynomial to the pairs (**x**(*t*i), y(*t*i+T)) with **x**(*t*i) being the nearest neighbors of **x***(t)* for *ti<t*. The original signal can also be viewed as an evolution of the state **x**(*t*) of a dynamical system in **R**N. The signal history (input samples) compose the map from state space of dimension *N* to a scalar space, the parameters of the mapping can be estimated by interpolating *f(***x***)* from noisy signal samples. The local modeling is superior and simpler under the condition that the given dynamics is locally smooth and a long enough signal history is available (Singer et al., 1992). Under this condition, *f(***x***)* can be approximated by the first few terms of multi dimensional Taylor series expansion, resulting in,

$$f(\mathbf{x}) = \mathbf{a}^{\mathsf{T}} \mathbf{x} + \mathsf{b} \tag{2}$$

Based on the fuzzy modeling approach originally developed by Takagi and Sugeno (1985), the global operation of a nonlinear process is divided into several local operating regions. Within each local region, *Ri*, a reduced order linear model is used to represent the processes behavior. Fuzzy sets are used to define the process operating conditions such that the dynamic model of a nonlinear p`rocess can be described in the following way:

Ri: IF operating condition *i*

110 Water Resources Management and Modeling

temperature and evapotranspiration etc. (Zhang and Govindaraju, 2000). On the other hand, the falling limb of a watershed is the result of the gradual release of water from various storages of the watershed after the rainfall input has stopped, and is influenced more by the storage characteristics of the watershed and climatic characteristics to some extent. Further, the falling limb of a flow hydrograph can be divided into three parts: initial portion just after the peak, middle portion, and the final portion. The initial portion of the falling limb of a flow hydrograph is influenced more by the quick-flow (or interflow), the middle portion of the falling limb is more dominated by the delayed surface flow, and the final portion of the falling limb (of smaller magnitudes) is dominated by the base flow. Hence it is apparent that a local approximation technique, which maps the changing dynamics using different functions, would be an effective way to model the rainfall runoff process. Local approximation refers to the concept of breaking up the domain into several small neighbourhood regions and analysing these separately. This argument is supported by the results of Sudheer (2005), wherein he proposes a procedure to extract knowledge from

Farmer and Sidorowitch (1987) have found that chaotic time series prediction using local approximation techniques is several orders of magnitude better. In their approach, the time series is first embedded in a state space using delay coordinates, and the underlying nonlinear mapping is inferred by a local approximation using only nearby states. This approach can be easily extended to higher order local polynomial approximations. Singer et al. (1992) derived the local approximation as a state dependant autoregressive modelling. However, this becomes complex with large data sets as the inefficient computation of nearby state search makes the implementation much harder. In order to overcome this limitation we proposed to simplify the signal representation (input) with a vector clustering procedure. That is, the local model fitting is based on statistically averaged prototypes instead of the original state vector samples. Also, the nearby state search can be significantly simplified with all prototypes

Most of the applications of local approximation technique (e.g. FIS) employ linear relationship as an effective local approximation. However, the rainfall runoff process being highly nonlinear, a nonlinear local approximation would be a better approach. The objective of this Chapter is to illustrate that a nonlinear local approximation approach in modelling the rainfall-runoff process may offer better accuracy in the context of river flow forecasting. More specifically, the chapter discusses about subdividing the data into subspaces and evaluates the limitations of linear and nonlinear local approximations. The proposed approach is illustrated through a real world case study on two river basins. Both the applications are developed for river flow forecasting, one on a daily time step and the other

forecast the value of flow (*yt*) and **x**t is the input vector to the model at time *t*. Generally, the modeler uses a set of *n* 'candidate' examples of the form (**x***i, yi*), *i=1,2,…,n*, and finds an

model typically are the previous values of the time series and the output will be the forecast

) by calibrating an appropriate model. The inputs to the

, where it is required to

organized according to a certain metric such as pattern similarity.

**2. Theoretical considerations of local approximation** 

Consider modeling a river flow time series such that *<sup>t</sup>* ( ,) *x <sup>t</sup> y f*

trained ANN river flow models.

on an hourly time step.

optimal set of parameter vector (

$$\text{THEN } \hat{y}\_i(t) = \sum\_{j=1}^{m} a\_{ij}y(t-j) + \sum\_{j=1}^{m} b\_{ij}u(t-j) \text{ (i=1, 2, 3, ..., } nr\text{)}\tag{3}$$

The final model output is obtained by firing strength

$$\hat{y}(t) = \left[\sum\_{i=1}^{nr} \mu\_i \hat{y}\_i(t)\right] / \left(\sum\_{i=1}^{nr} \mu\_i\right) \tag{4}$$

where *y* is the process output, *<sup>y</sup>* is the model prediction, u is the process exogenous input, *<sup>i</sup> y* is the prediction of the processes output in the *i*th operating region, *nr* is the number of fuzzy operating regions, *ni* and *no* are the time lags in the input and the output respectively, *<sup>i</sup>* is the membership function for the *i*th model, *aij* and *bij* are the local linear parameters, and *t* represents the discrete time.

Operating regions of a process can be defined by one or several process variables. A number of fuzzy sets, such as 'low, medium and high' are defined for each of these process variables. An operating condition is constructed through logical combinations of these variables which are used to define the process operating regions and assigned the fuzzy sets: 'low, medium and high'.

Fig. 1. Takagi-Sugeno fuzzy model computational procedure

The graphical representation for computational procedure for TS fuzzy model is presented for sine curve (Fig 1). From the figure it can be observed that if MFs can be developed for a given data set, then using the membership grade of each predictor variable, local linear models can be developed and their parameters can be estimated. The overlapping of different MFs and different local fuzzy regions are graphically presented in Fig 2. From the figure it is seen that different local linear models are developed from different fuzzy regions and fuzzy reasoning is applied to estimate the model output. Fuzzy regions are represented with different membership grade used in the fuzzy-if- then rules which is the crux of fuzzy modeling approach. Therefore, classification of different local regions in an input/output data set is very important in fuzzy modeling approach and fuzzy clustering technique is widely used for such purposes. In the clustering approach, classification is carried out using different distance measures. The degree of similarity can be calculated by using a suitable distance measure. Based on the similarity, data vectors are clustered such that the data within a cluster are as similar as possible, and data from different clusters are as dissimilar as possible.

Fig 3 gives an example of two clusters in <sup>2</sup> with prototypes **v**1 and **v**2. The partitioning of the data is expressed in the *fuzzy partition matrix* **U** = [µ*ij*] whose elements are the

Fig. 2. Fuzzy local liner model developments

*<sup>y</sup>* is the model prediction, u is the process exogenous input, *<sup>i</sup> y* is the prediction of the processes output in the *i*th operating region, *nr* is the number of fuzzy operating regions, *ni* and *no* are the time lags in the input and the output respectively,

*<sup>i</sup>* is the membership function for the *i*th model, *aij* and *bij* are the local linear parameters,

Operating regions of a process can be defined by one or several process variables. A number of fuzzy sets, such as 'low, medium and high' are defined for each of these process variables. An operating condition is constructed through logical combinations of these variables which are used to define the process operating regions and assigned the fuzzy sets: 'low, medium

The graphical representation for computational procedure for TS fuzzy model is presented for sine curve (Fig 1). From the figure it can be observed that if MFs can be developed for a given data set, then using the membership grade of each predictor variable, local linear models can be developed and their parameters can be estimated. The overlapping of different MFs and different local fuzzy regions are graphically presented in Fig 2. From the figure it is seen that different local linear models are developed from different fuzzy regions and fuzzy reasoning is applied to estimate the model output. Fuzzy regions are represented with different membership grade used in the fuzzy-if- then rules which is the crux of fuzzy modeling approach. Therefore, classification of different local regions in an input/output data set is very important in fuzzy modeling approach and fuzzy clustering technique is widely used for such purposes. In the clustering approach, classification is carried out using different distance measures. The degree of similarity can be calculated by using a suitable distance measure. Based on the similarity, data vectors are clustered such that the data within a cluster are as similar as possible, and data from different clusters are as dissimilar

Fig 3 gives an example of two clusters in <sup>2</sup> with prototypes **v**1 and **v**2. The partitioning of the data is expressed in the *fuzzy partition matrix* **U** = [µ*ij*] whose elements are the

where *y* is the process output,

and *t* represents the discrete time.

Fig. 1. Takagi-Sugeno fuzzy model computational procedure

and high'.

as possible.

Fig. 3. Identification of membership functions through fuzzy clustering.

membership degrees of the data vectors **x***<sup>k</sup>* in the fuzzy clusters with prototypes **v***j*. The antecedent membership functions are then extracted by projecting the clusters onto the individual variables. The performance of models of this kind depends heavily on the definition of neighboring state space. Euclidian distance is the most commonly employed measure of closeness to find out the neighboring state space. However, it is not usually an appropriate measure of closeness due to the curse of dimensionality. For instance, consider that the predictor distribution is uniform on a ball of radius unity in *D* dimensional space. For an expected response at or close to origin of the ball, it would be reasonable to use only sample vectors inside a ball of radius < 1, assuming Euclidean distance as a measure of closeness. But, since the probability of a sample vector lying in the smaller ball is *<sup>D</sup>* , it is necessary that sample sizes are exponential in *D* to get enough close vectors for accurate estimation. To avoid this curse, we might assume that *fi(***x***)* has a ridge approximation (*i*  corresponds to the fuzzy region),

$$f\_i(\mathbf{x}) \equiv \sum\_{n=1}^{l} c\_n \mathbf{g}\_n(a^T \mathbf{x}) \tag{5}$$

in which *gn* is a transformation function of the linear combiner of *x; <sup>T</sup> a* , *nc* are the parameters of the ridge function; *l* is the number of sub domains. Also note that the linear local approximation is valid under the condition that the dynamics of *f(***x***)* is locally smooth. However, equation 5 typically accounts for the nonlinear local dynamics of *f(***x***)*. A close examination of equation 5 reveals that the ridge function is an ANN with single hidden layer having *l* nodes, and a linear transfer function on the output layer.

Hence it is apparent that if the state space is classified into sub-domains, and each of these domains is modeled independently by a neural network approach, which when combined together, the resulting model may provide a better global modeling of the nonlinear dynamics in the state space. It appears that this heuristic has not been addressed or confirmed by empirical trials. The present study illustrates this heuristic by comparing the performance of models developed using the proposed approach, local linear approximation and a global nonlinear approach.

#### **3. Methodology**

In general, the above discussed nonlinear local model fitting is composed of two steps: a set of nearby state searches over the signal history and model parameter fitting. For a given signal, this procedure results in a set of local model parameters which when combined together provide a single function over the entire space. Since the neighborhood search is performed over the whole signal history a lot of redundant computation results which in turn hinders effective implementation of this approach. These redundant computations can be avoided by classifying the state spaces into homogenous subspaces by means of an appropriate vector clustering technique.

#### **4. Clustering for classification**

The objective of cluster analysis is the classification of objects according to similarities among them, and organizing of data into groups. Clustering techniques are among the *unsupervised* methods, they do not use prior class identifiers. The main potential of clustering is to detect the underlying structure in data, not only for classification and pattern

measure of closeness to find out the neighboring state space. However, it is not usually an appropriate measure of closeness due to the curse of dimensionality. For instance, consider that the predictor distribution is uniform on a ball of radius unity in *D* dimensional space. For an expected response at or close to origin of the ball, it would be reasonable to use only

necessary that sample sizes are exponential in *D* to get enough close vectors for accurate estimation. To avoid this curse, we might assume that *fi(***x***)* has a ridge approximation (*i* 

> 1 () ( ) *l*

parameters of the ridge function; *l* is the number of sub domains. Also note that the linear local approximation is valid under the condition that the dynamics of *f(***x***)* is locally smooth. However, equation 5 typically accounts for the nonlinear local dynamics of *f(***x***)*. A close examination of equation 5 reveals that the ridge function is an ANN with single hidden

Hence it is apparent that if the state space is classified into sub-domains, and each of these domains is modeled independently by a neural network approach, which when combined together, the resulting model may provide a better global modeling of the nonlinear dynamics in the state space. It appears that this heuristic has not been addressed or confirmed by empirical trials. The present study illustrates this heuristic by comparing the performance of models developed using the proposed approach, local linear approximation

In general, the above discussed nonlinear local model fitting is composed of two steps: a set of nearby state searches over the signal history and model parameter fitting. For a given signal, this procedure results in a set of local model parameters which when combined together provide a single function over the entire space. Since the neighborhood search is performed over the whole signal history a lot of redundant computation results which in turn hinders effective implementation of this approach. These redundant computations can be avoided by classifying the state spaces into homogenous subspaces by means of an

The objective of cluster analysis is the classification of objects according to similarities among them, and organizing of data into groups. Clustering techniques are among the *unsupervised* methods, they do not use prior class identifiers. The main potential of clustering is to detect the underlying structure in data, not only for classification and pattern

*i n n n f cg ax* 

*T*

< 1, assuming Euclidean distance as a measure of

(5)

, it is

*a* , *nc* are the

*x*

in which *gn* is a transformation function of the linear combiner of *x; <sup>T</sup>*

layer having *l* nodes, and a linear transfer function on the output layer.

closeness. But, since the probability of a sample vector lying in the smaller ball is *<sup>D</sup>*

sample vectors inside a ball of radius

corresponds to the fuzzy region),

and a global nonlinear approach.

appropriate vector clustering technique.

**4. Clustering for classification** 

**3. Methodology** 

recognition, but for model reduction and optimization. Various definitions of a cluster can be formulated, depending on the objective of clustering. Generally, one may accept the view that a cluster is a group of objects that are more similar to one another than to members of other clusters. The term "similarity" should be understood as mathematical similarity, measured in some well-defined sense. In metric spaces, similarity is often defined by means of a *distance norm*. Distance can be measured among the data vectors themselves, or as a distance from a data vector to some prototypical object of the cluster.

Since clusters can formally be seen as subsets of the data set, one possible classification of clustering methods can be according to whether the subsets are fuzzy or crisp (hard). Hard clustering methods are based on classical set theory, and require that an object either does or does not belong to a cluster. Hard clustering in a data set **X** means partitioning the data into a specified number of mutually exclusive subsets of **X**. The number of subsets (clusters) is denoted by *c*. Fuzzy clustering methods allow objects to belong to several clusters simultaneously, with different degrees of membership. The data set **X** is thus partitioned into *c* fuzzy subsets. In many real situations, fuzzy clustering is more natural than hard clustering, as objects on the boundaries between several classes are not forced to fully belong to one of the classes, but rather are assigned membership degrees between 0 and 1 indicating their partial memberships. Fuzzy clustering can be used to obtain a partitioning the data where the transitions between the subsets are gradual rather than abrupt. Most analytical fuzzy clustering algorithm is fuzzy c-means (FCM) clustering. The FCM clustering algorithm is based on the minimization of an objective function called *C-means functional.* In the current study subtractive clustering algorithm is used for classification. Subtractive clustering method (Chiu, 1994) is an extension of the FCM and mountain clustering method (Yager and Filev, 1994), where the potential is calculated for the data rather than the grid points defined on the data space. As a result, clusters are elected from the system training data according to their potential. Subtractive clustering compared to mountain clustering has an advantage that there is no need to estimate a resolution for the grid.

#### **5. Fuzzy non linear local approximation model**

As discussed earlier a novel hybrid model is proposed herein which performs independent nonlinear local approximation, and combines each of them using the fuzzy framework. The architecture of the proposed model is presented in Fig 4. The method is based on the concept that the input space is divided into sub regions of similar dynamics using an appropriate clustering algorithm (subtractive clustering algorithm in this study), and modeling of each of these regions is carried out using nonlinear local function approximation (ANN in this study). The proposed model is termed as Fuzzy Non Linear Local Approximation (FNLLA) model, which is based on fuzzy concept and neural technique is applied for nonlinear local function approximation.

In Fig 4, the gating is done to identify the membership grade associated with any given input vector for each of the clusters. This input vector will be passed to each of the developed ANN, and the output from each ANN is combined by computing the weighted mean with the membership grade in each cluster.

Fig. 4. Schematic representation of the proposed FNLLA model

Fig. 5. Computational architecture of the proposed FNLLA model

The architecture of FNLLA is depicted in Fig 5, in which the fuzzy computing scheme is represented in an adaptive neural network structure. The consequent part of each of the fuzzy rule is a nonlinear function (ANN in this case). The computations are performed in 4 layers in FNLLA. In the layer 1, (L1), the incoming input vector (**x**) is passed to different sub regions (C1, C2 and C3) and the associated MFs are computed as:

$$\mu\_j = e^{-\alpha \left\| \mathbf{x}\_j - \mathbf{x}\_i \right\|^2} \tag{6}$$

in which xi is the ith input vector, x is the cluster centre and *j* the sub-region number, and is a function of the cluster radius. In layer 2 (L2), the weights of the hidden nodes of the ANN consequent models are estimated, and the output from each ANN consequent model is arrived at layer 3 (L3). At layer 4 (L4), each of these consequent outputs are combined to arrive at the final output.

#### **6. Demonstrative case examples**

116 Water Resources Management and Modeling

Fig. 4. Schematic representation of the proposed FNLLA model

Fig. 5. Computational architecture of the proposed FNLLA model

regions (C1, C2 and C3) and the associated MFs are computed as:

The architecture of FNLLA is depicted in Fig 5, in which the fuzzy computing scheme is represented in an adaptive neural network structure. The consequent part of each of the fuzzy rule is a nonlinear function (ANN in this case). The computations are performed in 4 layers in FNLLA. In the layer 1, (L1), the incoming input vector (**x**) is passed to different sub The proposed model is illustrated through two examples by developing rainfall-runoff models: (a) for Kolar basin up to the Satrana gauging site in India, (b) the Kentucky basin, USA. The Kolar River is a tributary of the river Narmada that drains an area of about 1350 sq km before its confluence with Narmada near Neelkanth. In the present study the catchment area up to the Satrana gauging site is considered, which constitutes an area of 903.87 sq km (Fig 6). The 75.3 km long river course lies between north latitude 21009' to 23017' and east longitude 77001' to 77029'. For the study, rainfall and runoff data on an hourly interval during the monsoon season (July, August, and September) for three years (1987–1989) are used. The rainfall data available were in the form of areal average values in the basin. The total available data has been divided into two sets, calibration set (data during the years 1987–1988) and validation set (data during the year 1989). Different models for lead times of up to 6 hours have been developed in the study.

Fig. 6. Map of Kolar River basin, India

Fig. 7. Study area map of Kentucky River basin in USA

The Kentucky River basin covers over 4.4 million acres of the state of Kentucky. Forty separate counties lie either completely or partially within the boundaries of the river basin. The Kentucky River is the sole water source for several water supply companies of the state. There is a series of fourteen locks and dams on the Kentucky River, which are owned and operated by the US Army Corps of Engineers. The drainage area of the Kentucky River at Lock and Dam 10 (LD10) near Winchester, Kentucky is approximately 6,300 km2 (Fig 7).

The data used in the study presented in this paper include average daily streamflow (m3/s) from the Kentucky River at LD10, and daily average rainfall (mm) from five rain gauges (Manchester, Hyden, Jackson, Heidelberg, and Lexington Airport) scattered throughout the Kentucky River Basin. The total length of the available rainfall runoff data was 26 years (1960-1989 with data in some years missing).

The input vector identified, according to Sudheer et al. (2002) for modeling the river flow in Kolar, included a total number of 4 variables. Accordingly, the functional form of the model, in the case of Kolar, for rainfall runoff modeling is given by:

$$Q(t) = f[R(t-9), R(t-8), R(t-7), Q(t-1)]\tag{7}$$

where Q(t) and R(t) are river flow and rainfall respectively at any time t in hour.

The functional form for rainfall-runoff dynamic for Kentucky River basin is given by:

$$Q(t) = f\left[R(t), R(t-1), R(t-2), Q(t-2), Q(t-1)\right] \tag{8}$$

Different statistical indices that are employed to estimate the model performance include coefficient of correlation (CORR), efficiency (EFF), root mean square error (RMSE) and noise to signal ration (NS).

### **7. Results and discussions**

118 Water Resources Management and Modeling

The Kentucky River basin covers over 4.4 million acres of the state of Kentucky. Forty separate counties lie either completely or partially within the boundaries of the river basin. The Kentucky River is the sole water source for several water supply companies of the state. There is a series of fourteen locks and dams on the Kentucky River, which are owned and operated by the US Army Corps of Engineers. The drainage area of the Kentucky River at Lock and Dam 10 (LD10) near Winchester, Kentucky is approximately 6,300 km2 (Fig 7).

The data used in the study presented in this paper include average daily streamflow (m3/s) from the Kentucky River at LD10, and daily average rainfall (mm) from five rain gauges (Manchester, Hyden, Jackson, Heidelberg, and Lexington Airport) scattered throughout the Kentucky River Basin. The total length of the available rainfall runoff data was 26 years

The input vector identified, according to Sudheer et al. (2002) for modeling the river flow in Kolar, included a total number of 4 variables. Accordingly, the functional form of the model,

where Q(t) and R(t) are river flow and rainfall respectively at any time t in hour.

The functional form for rainfall-runoff dynamic for Kentucky River basin is given by:

*Qt f Rt Rt Rt Qt* ( ) [ ( 9), ( 8), ( 7), ( 1)] (7)

*Qt f Rt Rt Rt Qt Qt* ( ) [ ( ), ( 1), ( 2), ( 2), ( 1)] (8)

Fig. 7. Study area map of Kentucky River basin in USA

(1960-1989 with data in some years missing).

in the case of Kolar, for rainfall runoff modeling is given by:

#### **7.1 Parameter Estimation in FNLLA**

The optimal number of clusters in FNLLA has been obtained by varying cluster radius in the subtractive clustering algorithm, combined with the ANN model development. Different ANN models are developed for different clusters of input space changing hidden neuron from 2 to 10. Single hidden layer with sigmoid function nodes is used in the ANN. The sigmoid activation function is considered in the output layer also. A standard back propagation algorithm with adaptive learning rate and momentum factor has been employed to estimate the network parameters for different clusters. In order to have a true evaluation of the proposed nonlinear local approximation in fuzzy models, the result obtained for both the basins from FNLLA was compared with FIS, which performs a linear local approximation. In FIS model, subtractive clustering has been used for fuzzy model identification which includes optimal number of if-then-rule generation and consequent parameters are optimized using least square error (LSE) technique.


Table 1. Partition of input space for Kolar and Kentucky basin

In the analysis, the radius of influence (ra) of the cluster centre is fixed by various trials, which is the foremost interest for the current study. The value of ra is varied from 0.001 to 1.0 with a step size of 0.01; at each stage number of clusters is estimated. From the current data set 1 to 7 clusters are found while changing the cluster radius from 0.05 to 1 for Kolar river basin and maximum 4 clusters are observed by changing radius from 0.001 to 1 for Kentucky basin. The number of clusters identified corresponding to various radiuses for both the basins are presented in Table 1. It is evident from Table 1 that as the cluster radius increases, the number of cluster decreases. Different ANN models are developed for data belonging to different clusters (effectively representing different ranges of flow), and as discussed earlier hidden neurons are by trial and error procedure. The stopping criteria for ANN model building was maximum efficiency.

Fig. 8. Variation of Efficiency Plot for three clusters with respect to hidden neurons

It was observed that model performance was good when input space is classified into 3 clusters for both the basins. The efficiency plots with respect to different hidden neurons for 3 ranges of flows are presented in Fig.8 for Kolar basin. From the Fig 8 it is observed that the maximum efficiency is for the models having hidden neurons are 4, 4 and 8 for individual clusters for Kolar River basin. Similar procedure has been followed for optimization of model parameter for Kentucky River basin. The optimum hidden neuron obtained 3, 3 and 3 for Kentucky basin. The identified optimal clustering radius with number of if-then rules for Kolar and Kentucky basin are furnished in Table 2.


Table 2. Optimal model structure for three sub-domains for Kolar and Kentucky basin

The summary statistics of the flow data belonging to each of the identified clusters are presented in Table 3. It is clear from Table 3 that the clustering based on the input vector clearly identifies distinct clusters that have different nonlinear dynamics. This is evident from overlapping clusters; the cluster C1 contain flow range from 4.67 m3/s to 240.55 m3/s, and the cluster C2 contain flow range from 3.63 m3/s to 240.24 m3/s, in the case of Kentucky river basin data. A similar observation is found in the case of Kolar basin data too. It is worth mentioning that the consequent ANN models for each of these clusters preserve the summary statistics much effectively in both basins (see Table 3). Also, it is evident that the classification of the data into different clusters is according to the range of flow, though not exclusively forced by the FNLLA.

 Cluster 1 Cluster 2 Cluster 3

2 4 6 810

Number of hidden neuron

It was observed that model performance was good when input space is classified into 3 clusters for both the basins. The efficiency plots with respect to different hidden neurons for 3 ranges of flows are presented in Fig.8 for Kolar basin. From the Fig 8 it is observed that the maximum efficiency is for the models having hidden neurons are 4, 4 and 8 for individual clusters for Kolar River basin. Similar procedure has been followed for optimization of model parameter for Kentucky River basin. The optimum hidden neuron obtained 3, 3 and 3 for Kentucky basin. The identified optimal clustering radius with number of if-then rules for

Fig. 8. Variation of Efficiency Plot for three clusters with respect to hidden neurons

Model Kolar River basin Kentucky River basin

FNLLA 0.12 [3 ranges/rules] 0.010 [3 ranges/rules] Table 2. Optimal model structure for three sub-domains for Kolar and Kentucky basin

The summary statistics of the flow data belonging to each of the identified clusters are presented in Table 3. It is clear from Table 3 that the clustering based on the input vector clearly identifies distinct clusters that have different nonlinear dynamics. This is evident from overlapping clusters; the cluster C1 contain flow range from 4.67 m3/s to 240.55 m3/s, and the cluster C2 contain flow range from 3.63 m3/s to 240.24 m3/s, in the case of Kentucky river basin data. A similar observation is found in the case of Kolar basin data too. It is worth mentioning that the consequent ANN models for each of these clusters preserve the summary statistics much effectively in both basins (see Table 3). Also, it is evident that the classification of the data into different clusters is according to the range of flow, though not

FIS 0.20 [3 rules] 0.008 [4 rules]

20

exclusively forced by the FNLLA.

Kolar and Kentucky basin are furnished in Table 2.

40

60

Efficiency [%]

80

100


Table 3. Summary statistics of the river flow in the identified sub-domain by FNLLA Model


Table 4. Cluster wise FNLLA performance for Kolar Basin

The results of the FNLLA were first analyzed for its effectiveness in capturing the nonlinear dynamics at local level. In order to achieve this, the performance indices were computed for each sub-domain for Kolar basin for the calibration and the validation period, and are presented in the Table 4. Note that in the study the FNLLA model classified 1975, 1782 and 593 patterns as low, medium and high flow respectively using the subtractive clustering algorithm. It is evident from the Table 4 that the nonlinear consequent models of the FNLLA are effective in capturing the nonlinear dynamics in each sub regions.

#### **7.2 Performance of FNLLA at 1 step-ahead forecast**

The values of various evaluation measures during calibration and validation period for FNLLA and FIS for both the basins for 1-hour lead forecast are summarized in the Table 5, from which it can be observed that both the models possess high value of correlation (0.96 and more) between the forecasted and the observed river flow at 1-hour ahead. The high value for the efficiency index indicates a very satisfactory model performance in capturing the nonlinear dynamics involved in the rainfall-runoff processes. Note that the FNLLA performs with higher efficiency in the case of Kolar basin compared to the FIS model. The


value of RMSE varies from 24 m3/sec to 42 m3/sec for Kolar basin indicating a very good performance by both the model; the RMSE values are relatively less in Kentucky basin for FNLLA model. It is also noted that the NS ratio for the FNLLA model is less than that corresponding to the FIS model.

Table 5. Statistical Indices for 1-hour and 1- day lead forecast for Kolar and Kentucky basin

The scatter plots of flows for 1-hour lead forecast validation period for both the models are presented in Fig 9 for Kolar basin (Fig 10 for Kentucky basin for 1 day lead). These plots give clear indication of the simulation ability of the developed model across the full range of flows. It is noted that most of the flows tend to fall close to the 450 line (reduced scattering), showing a good agreement between observed and forecasted flows. From the plot it is observed that both the models are quite competent in forecasting river flow at 1 hour/day lead time. In general it is noted that the mapping of the low flow region is relatively better compared to high flow region.

Fig. 9. Scatter plots for observed and computed flows by both models at 1 hour lead time for Kolar basin (a) FIS (b) FNLLA

value of RMSE varies from 24 m3/sec to 42 m3/sec for Kolar basin indicating a very good performance by both the model; the RMSE values are relatively less in Kentucky basin for FNLLA model. It is also noted that the NS ratio for the FNLLA model is less than that

Correlation 0.98 0.99 0.98 0.98 Efficiency 95.84 96.72 95.91 96.61 RMSE (m3/s) 42.39 23.29 42.06 24.31 Noise to Signal ratio 0.20 0.11 0.34 0.12

Correlation 0.98 0.97 0.96 0.96 Efficiency 95.68 93.87 92.17 91.68 RMSE (m3/s) 50.12 51.87 67.45 60.41 Noise to Signal ratio 0.21 0.25 0.28 0.29

Table 5. Statistical Indices for 1-hour and 1- day lead forecast for Kolar and Kentucky basin

The scatter plots of flows for 1-hour lead forecast validation period for both the models are presented in Fig 9 for Kolar basin (Fig 10 for Kentucky basin for 1 day lead). These plots give clear indication of the simulation ability of the developed model across the full range of flows. It is noted that most of the flows tend to fall close to the 450 line (reduced scattering), showing a good agreement between observed and forecasted flows. From the plot it is observed that both the models are quite competent in forecasting river flow at 1 hour/day lead time. In general it is noted that the mapping of the low flow region is relatively better

<sup>2000</sup> [a]

0

1000

Computed flow (m3/sec]

Fig. 9. Scatter plots for observed and computed flows by both models at 1 hour lead time for

Calibration Validation Calibration Validation

0 1000 2000

/sec]

Observed flow [m3

[b]

Statistical Indices FNLLA FIS

corresponding to the FIS model.

**Kolar River Basin**

**Kentucky River Basin**

compared to high flow region.

0

Kolar basin (a) FIS (b) FNLLA

1000

Computed flow [m3/sec]

2000

0 1000 2000

/sec]

Observed flow [m3

Fig. 10. Scatter plots for observed and computed flows (a) FIS (b) FNLLA model for 1 day lead forecast for Kentucky basin

Fig. 11. Comparison plot between FIS and FNLLA model for a typical storm event for (a) Kolar basin (b) Kentucky basin

The forecasted hydrograph for a typical flood event (during validation period) for both the basins by both the models are presented along with its observed counterpart in Fig 11. It can be observed from Fig 11 that FNLLA is preserving the peak flows effectively than the FIS for 1 hour/day lead forecast for Kolar and Kentucky basin, while in low and medium ranges of flow the performance of both models is similar. This observation effectively brings out the capturing of nonlinear dynamics at the local regions of the input space.

#### **7.3 Performance of FNLLA at higher lead time forecasts**

Both the models are further evaluated for their effectiveness to forecast flows at higher lead times. The statistical indices for higher lead forecasting are presented in Table 6 for both the basins. The superior performance of the FNLLA compared to FIS is clearly visible at higher lead times from the results presented in Table 6. It is observed that even though the performance of FNLLA deteriorates as the lead time increases, it falls down to only 79.15% efficiency at 6 hours ahead, while FIS shows only 51.00% efficiency at the same lead time. A similar behavior is exhibited by FNLLA in the case of Kentucky basin also, as the efficiency is 77.23% at 3 days ahead compared to 46.92% in the case of FIS. A similar argument holds well in the case of other performance indices too.


Table 6. Performance indices for FIS and FNLLA model at higher forecast lead times

#### **8. Summary and conclusions**

The objective of this chapter was to present and illustrate a nonlinear local approximation approach in modelling the rainfall-runoff process which offers better accuracy in the context of river flow forecasting. Based on the theoretical considerations of the fuzzy modelling in the state space (input-output), it is clear that if the state space is classified into sub-domains and each of these domains is modeled independently by a neural network approach which are combined together, it may provide a better global modeling of the nonlinear dynamics in the state space. In general, the proposed nonlinear local model fitting is composed of two steps: a set of nearby state searches over the signal history and model parameter fitting. For a given signal, this procedure results in a set of local model parameters which when combined together provide a single function over the entire space. Since the neighborhood search is performed over the whole signal history, a lot of redundant computations result which in turn hinders effective implementation of this approach. These redundant computations can be avoided by classifying the state spaces into homogenous subspaces by means of an appropriate vector clustering technique. The proposed model is termed as Fuzzy Non Linear Local Approximation (FNLLA) model, which is based on fuzzy concept and neural technique is applied for nonlinear local function approximation. The partition of the state space is achieved by subtractive clustering algorithm and nonlinear local function approximation is by ANN in the proposed method. The antecedent parameters of the model are simultaneously estimated during clustering, and the standard back propagation algorithm is employed for ANN parameter estimation.

The potential of FNLLA is illustrated using two case examples: (i) data pertaining to Kolar River basin, and (ii) data corresponding to Kentucky River basin. The optimal architecture of the FNLLA model, which is defined by the number of sub-regions in the data and the structure of each ANN, is arrived after a trial and error procedure. When the performance of the FNLLA is compared with that of a pure FIS, it is observed that the FNLLA certainly possesses the advantages of nonlinear mapping. Though both the models perform similar at 1 step-ahead forecasts, the FNLLA performs much better than FIS at higher lead times. Overall, the results of the study confirm the heuristic that a nonlinear local approximation is a better approach in fuzzy modeling especially when complex nonlinear functions are being mapped.

#### **9. References**

124 Water Resources Management and Modeling

1 hour/day lead forecast for Kolar and Kentucky basin, while in low and medium ranges of flow the performance of both models is similar. This observation effectively brings out the

Both the models are further evaluated for their effectiveness to forecast flows at higher lead times. The statistical indices for higher lead forecasting are presented in Table 6 for both the basins. The superior performance of the FNLLA compared to FIS is clearly visible at higher lead times from the results presented in Table 6. It is observed that even though the performance of FNLLA deteriorates as the lead time increases, it falls down to only 79.15% efficiency at 6 hours ahead, while FIS shows only 51.00% efficiency at the same lead time. A similar behavior is exhibited by FNLLA in the case of Kentucky basin also, as the efficiency is 77.23% at 3 days ahead compared to 46.92% in the case of FIS. A similar argument holds

FNLLA FIS

Forecast Lead time 1 hour 3 hour 6 hour 1 hour 3 hour 6 hour

Efficiency (%) 95.84 81.52 79.15 95.91 57.79 51.00 RMSE (m3/s) 42.39 89.40 94.95 42.06 135.11 145.70 Noise to Signal ratio 0.20 0.43 0.923 0.34 0.65 1.95

Efficiency (%) 96.72 79.95 77.73 96.61 50.54 46.92 RMSE (m3/s) 23.29 60.45 62.37 24.31 92.12 96.48 Noise to Signal ratio 0.11 0.35 0.39 0.12 0.77 0.83

Forecast Lead time 1 day 2 day 3 day 1 day 2 day 3 day

Efficiency (%) 95.68 84.21 64.88 92.17 73.58 52.50 RMSE (m3/s) 50.12 95.82 142.9 67.45 123.93 166.20 Noise to Signal ratio 0.21 0.40 0.59 0.28 0.51 0.69

Efficiency (%) 93.87 78.68 58.66 91.68 73.14 50.56 RMSE (m3/s) 51.87 96.68 134.63 60.41 108.53 147.24 Noise to Signal ratio 0.25 0.46 0.64 0.29 0.52 0.70

Table 6. Performance indices for FIS and FNLLA model at higher forecast lead times

The objective of this chapter was to present and illustrate a nonlinear local approximation approach in modelling the rainfall-runoff process which offers better accuracy in the context of river flow forecasting. Based on the theoretical considerations of the fuzzy modelling in the state space (input-output), it is clear that if the state space is classified into sub-domains

capturing of nonlinear dynamics at the local regions of the input space.

**7.3 Performance of FNLLA at higher lead time forecasts** 

well in the case of other performance indices too.

**Kolar River Basin** 

**Calibration** 

**Validation** 

**Calibration** 

**Validation** 

**Kentucky River Basin** 

**8. Summary and conclusions** 

