Introduction

Land and water are vital resources towards ensuring food security, economic and social progress and ultimately for sustenance of life. These resources are hampered day by day by various environmental hazards. Besides environmental hazards, soil erosion plays a vital role in deteriorating the natural resources. Soil erosion also threatens the global food security by reducing the productivity of soil, reducing the reservoir capacity etc. The dynamics of soil erosion and sediment yield are affected by spatial and temporal characteristics of the catchment like climate, soil type, land use pattern, topography and anthropology activities. Since these factors bear temporal and spatial variability, they can be isolated by discretizing the catchment into smaller homogeneous units and eventually adopting feasible soil erosion models for sub-watersheds. However, the major problem with execution of these models is the input data which are too spatial and rare. GIS tool along with different soil erosion models is very useful for estimation of soil erosion from un-gauged watershed. For this purpose, watershed is treated as unit for sustainable development of natural resources (Patel et al. 2012). Watershed management is defined as judicious use of the natural resources in a watershed to ensure optimum and sustained productivity (Yadav et al. 2014). Watershed is a natural hydrological entity which allows surface run-off to a defined channel, drain, stream or river at a particular point. It is the basic unit of water supply which evolves over time. Also it is describes as an area of land that contains common set of streams and rivers that all drain into a single large body of water (Black 2005). However, Chow (1964) defined watershed as the separating boundary of a drainage basin and termed it as a catchment. For sustainable management of watersheds, soil erosion is a major factor, which accelerates the rate of land degradation and hence influences agricultural productivity, run-off movement and sometimes leads to flood in the lower basin (Essiet 1990). Scientific planning of soil conservation requires knowledge of the relations among those factors that cause loss of soil and those that help to reduce such losses. Recent studies (Jaiswal et al. 2012; Pandey et al. 2007; Yoshino and Ishioka 2005; Sharma et al. 2001; Khan et al. 2001; Sidhu et al. 1998 and several others) discovered that Geographic Information System and remote sensing are great use in characterization and prioritization of watershed areas because of spatial computation of soil loss and other affecting parameters in erosion. Shrimali et al. (2001) presented mapping, monitoring and prioritizing the areas based on their susceptibility to degradation using remote sensing and GIS. Geomorphology parameters such as stream length, stream order, drainage density, stream frequency, farm factor, texture ratio, circulatory ratio, bifurcation ratio, compactness ratio and elongation ratio have been extensively used across the world for prioritization of sub-watersheds (Mishra and Nagarajan 2010; Sharma et al. 2010; Javed et al. 2009; Hlaing et al. 2008; Chopra et al. 2005 and Vittala et al. 2004).

Analytical hierarchy process (AHP) approach is a multi-criteria decision method that uses hierarchical structures to represent a problem and then to develop priorities for the alternatives based on the judgment of the user (Saaty 1980). It involves building a hierarchy (ranking) of decision elements and then making comparisons between each possible pair in each cluster (as a matrix) to compute weight for each element within a cluster (or level of the hierarchy) with a consistency ratio (useful for checking the consistency of the data) of decision. Yahaya et al. (2010) has studied the causative factors for flooding in watershed using AHP and found the weightage for the different factors as 33.9% for rainfall, 25.5% for drainage network, 19.7% for slope, 15.2% for soil type and 5.7% for land cover. Oyatoye et al. (2010) used AHP method in decision-making on investment portfolio selection in the banking sector of the Nigerian capital market. Chowdary et al. (2013) has conducted a study for prioritization of micro-watersheds using multi-criteria decision approach of AHP and sediment yield index model (AHPSYI) under GIS environment.

In spite of popularity of AHP, this method is often criticized for its inability to adequately handle the inherent uncertainty and imprecision associated with the mapping of the decision-maker’s perception to exact numbers (Deng 1999). Since fuzziness and vagueness are common characteristics in many decision-making problems, a fuzzy-AHP (FAHP) method should be able to tolerate vagueness or ambiguity (Mikhailov and Tsvetinov 2004). In other words, the conventional AHP approach may not fully reflect a style of human thinking because the decision makers usually feel more confident to give interval judgments rather than expressing their judgments in the form of single numeric values and so FAHP is capable of capturing a human’s appraisal of ambiguity when complex multi-attribute decision-making problems are considered (Erensal et al. 2006). This ability comes to exist when the crisp judgments are transformed into fuzzy judgments. Zadeh (1965) published his work fuzzy fets, which described the mathematics of fuzzy set theory. The main characteristic of fuzziness is the grouping of individuals into classes that do not have sharply defined boundaries (Hansen 2005). The uncertain comparison judgment can be represented by the fuzzy number. Jaiswal et al. (2015) gives the flow chart for proposed FAHP-based decision support, as shown in Fig. 1.

Fig. 1
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Flow chart of FAHP-based decision support model for prioritization of sub-watershed

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The present study aims at for prioritization of sub-watershed using different erosion hazard parameters by fuzzy analytical hierarchical process (FAHP) decision support system. This study is mainly helpful for planning and management of natural resource on sustainable basis. Tam et al. (2002) proposed a nonstructural fuzzy decision support system (NSFDSS) to integrate both expert’s judgment and computer decision model for selecting site layout plan, and advocated the superiority of non NSFDSS over simple AHP because of its automatic consistency checking and correction, simplified scale of comparison and elimination of consistency deviation.

Methodology for Prioritization

Description of the Study Area

Study area, i.e. Nagwan watershed (92.33 km2) situated in upper part of Damodar Valley, district Hazaribagh, Jharkhand, India, is characterized as seriously eroded area (ranked second) in the world (El-Swaify et al. 1982). The latitudes of Nagwan watershed vary from 23°59′33″N to 24°5′37″N and longitudes vary from 85°16′41″E to 85°23′50″E as shown in Fig. 2. The soils of the study area are clay loam and silty loam type. The topography of the watershed is undulating, and maximum and the minimum elevations of the area are 667 m and 560 m (Fig. 3), respectively. The area experiences sub-humid subtropical monsoon type of climate, characterized by hot summers (40 °C) and mild winters (4 °C). The annual rainfall of the study area is about 1272.5 mm, out of which 80% received in monsoon season. The distribution of rainfall is uneven and most of the storms are intense storms of 10 cm/h while considering the duration of rainfall is 30 min. The most of the agriculture of the Nagwan watershed is rain fed (only 20% area under irrigation) and mono-cropped. Farmers grow paddy and maize in kharif or wheat, gram and mustard in rabi season (Source: Directorate of Census, DVC, Hazaribagh).

Fig. 2
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Location map of the study area

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Fig. 3
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Contour map of the study area (Nagwanw atershed)

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Fig. 4
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Topographic map of the study area (Nagwan watershed)

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Fig. 5
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False colour composite image of the Nagwan watershed

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Fig. 6
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Soil texture map of the study area (Nagwan watershed)

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Data acquisition

The acquired data used in the present study, their source and purpose of acquisition are presented in summarized form in Table 1.

Table 1 Acquired data and their source used for study
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Methodology

The methodology used includes selection and computation of EHPs for sub-watersheds, calculation of weights of EHPs using FAHP-based decision support and finally prioritization of sub-watersheds for identification of environmentally stressed areas.

Erosion Hazard Parameters (EHPs)

EHPs analysis is carried out through measurement of spatial, linear, aerial and relief aspects of the basin and slope contribution (Nag and Chakraborty 2003) to understand the run-off characteristics of the area and potentiality of watershed deterioration. The linear morphometric parameters are mean bifurcation ratio (Rb), length of overland flow (Lo), and aerial or shape morphometric parameters are drainage density (Dd), drainage texture (Dt), stream frequency (Fs), drainage intensity (Di), drainage factor (Df), circularity ratio (Rc), elongated ratio (Re), shape factor (Bs), compactness coefficient (Cc); relief aspects are relative relief, and other three parameters are soil loss, sediment yield and sediment production rate calculated as suggested by Jaiswal et al. (2014) and Smith (1950).

Computation of Weights

The principle of AHP theory lies on uncertainty in decision that creates basis of applying fuzziness in AHP. The AHP for MCDS constructs a matrix of pair-wise comparisons (ratios) between the factors affecting the decision. A numerical value between 0 and 9 has been suggested by Saaty to indicate how one criterion is important than other as stated in Jaiswal et al. (2014). The degree of importance between two factors in the matrix is filled on the basis of field experience, survey results, judgment of decision makers and result reported by researchers (Jaiswal et al. 2014). The size of comparison matrix in AHP may be a square matrix with size equal to number of parameters (n) considered for decision. In FAHP method, the comparison matrix can be fuzzified by triangular or trapezoidal membership functions (Kordi 2008) and that can be expressed as \( \bar{A} = [\bar{a}_{i,j} ] \) in the following matrices form:

$$ \left[ {\bar{a}_{i,j} } \right] = \left| {\sigma_{i,j} ,\varepsilon_{i,j} ,\tau_{i,j} ,\mu_{i,j} } \right|\quad {\text{where}}\;i\;{\text{and}}\;j\;{\text{may}}\;{\text{vary}}\;{\text{from}}\;1\;{\text{to}}\;n $$
(1)

The members of fuzzy matrix can be described by \( \sigma \), \( \varepsilon \), \( \tau \) and \( \mu \) which are four parts of the fuzzy membership function in such a way that \( 0 < \sigma \le \varepsilon \le \mu \). Buckley (1985) proved, if \( \bar{A} \) is consistent then fuzzy matrix \( = [\bar{a}_{i,j} ] \) will also be consistent. The weights from fuzzy matrix can be computed either by geometric mean method or \( \lambda_{\hbox{max} } \) methods, in which both provide nearly same results for small consistency ratio. The normalized principal eigenvector, which is called priority vector, is used in \( \lambda_{\hbox{max} } \) to assign the weights for different parameters. In the present case, geometric mean method is used to compute fuzzy membership function for the weights using following equations:

$$ \sigma_{i} = \left[ {\prod\limits_{j = 1}^{n} {\sigma_{ij} } } \right]^{1/n} \quad {\text{and}}\quad \sigma = \left[ {\sum\limits_{i = 1}^{n} {\sigma_{i} } } \right] $$
(2)
$$ \varepsilon_{i} = \left[ {\prod\limits_{j = 1}^{n} {\varepsilon_{ij} } } \right]^{1/n} \quad {\text{and}}\quad \varepsilon = \left[ {\sum\limits_{i = 1}^{n} {\varepsilon_{i} } } \right] $$
(3)
$$ \tau_{i} = \left[ {\prod\limits_{j = 1}^{n} {\tau_{ij} } } \right]^{1/n} \quad {\text{and}}\quad \tau = \left[ {\sum\limits_{i = 1}^{n} {\tau_{i} } } \right] $$
(4)
$$ \mu_{i} = \left[ {\prod\limits_{j = 1}^{n} {\mu_{ij} } \mu_{ij} } \right]^{1/n} \quad {\text{and}}\quad \mu = \left[ {\sum\limits_{i = 1}^{n} {\mu_{i} } } \right] $$
(5)

The fuzzy membership function describing the weights of different parameters is defined by the following equation:

$$ \bar{x}_{i} = \varphi \left( {x_{i} } \right) = \left[ {\frac{{\sigma_{i} }}{\mu },\frac{{\varepsilon_{i} }}{\tau },\frac{{\tau_{i} }}{\varepsilon },\frac{{\mu_{i} }}{\sigma }} \right] $$
(6)

The centroid method is employed to de-fuzzify the membership function, which gives crisp weights for all the parameters used in the analysis. The formula used to compute crisp weights xi of membership function \( \varphi (x_{i} ) \) between the limits \( \sigma \) and \( \mu \) (Zimmermann 1934) is

$$ x_{i} = \frac{{\int_{{x_{i} = \sigma }}^{\mu } {\varphi \left( {x_{i} } \right)x_{i} {\text{d}}x} }}{{\int_{{x_{i} = \sigma }}^{\mu } {\varphi \left( {x_{i} } \right){\text{d}}x} }} $$
(7)

Consistency Check

The comparative importance between the members in fuzzy-AHP is subjective that depends on personal ability or understanding about the subjects and feedback from different sources. Therefore, a consistency check is employed to judge the consistency of decisions in FAHP analysis. The consistency of judgment is checked by estimating consistency ratio (CR) computed using the following equation:

$$ {\text{CR}} = \frac{{\text{CI}}}{{\text{RI}}}\times 100 $$
(8)

where CI is the consistency index and RI is the random consistency index. The consistency index depends on size of matrix (number of parameters), and the consistency in decisions is estimated using the following equation:

$$ {\text{CI}} = \frac{{\lambda_{\hbox{max} } - n}}{n - 1} $$
(9)

where \( \lambda_{\hbox{max} } \) is the principal eigen value (Han and Tsay 1998; Malczewski 1999) that can be computed approximately by calculating the product of the pair-wise comparison matrix and the weight vectors, de-fuzzifying this matrix and adding all elements of the resulting vector. A unitless random consistency index (RI) depends on matrix size (n), after generating reciprocal matrix of various sizes (Saaty 1980). The average RI for different sizes of matrix is given in Table 2.

Table 2 Random consistency index for different matrix size
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If CR is less than 10%, the subjective evaluation about decision is considered as consistent.

Priority assessment

The EHPs considered may vary in the diverse ranges, and therefore, there is a need to bring down those on same scale. Normalization approach to restrict the variation in the range from 0 to 1 is applied using Eq. (10) as follows:

$$ W_{ij} = \frac{{{\text{EHP}}_{ij} - {\text{OLB}}_{i} }}{{{\text{OUB}}_{i} - {\text{OLB}}_{i} }} $$
(10)

where Wij is the normalized value of ith EHP of jth watershed; OUBi and OLBi are the original upper and lower bound for ith EHP. EHPij is the original value of ith EHP for jth sub-watershed. The final priority (Fj) of a watershed in the present FAHP-based MCDS is computed by summing the product of normalized value of EHP and its corresponding weights obtained from FAHP analysis, as follows:

$$ F_{j} = \sum\limits_{i = 1}^{n} {x_{i} *W_{ij} } $$
(11)

After getting the final priorities of all sub-watersheds in the study area, they are grouped in different priority classes for deciding the intensity and urgency of soil conservation measures.

Results

Nagwan watershed was divided into 21 sub-watersheds named as SW-1 to SW-21 (Fig. 7). An analysis of the configuration of stream pattern revealed that the complex pattern is found in Fig. 8. The land uses and land cover were classified into eight classes, namely agriculture land (with crops—7.32 km2 and without crops—45.99 km2), barren land—1.25 km2, built-up land—2.43 km2, dense forest—1.64 km2, grass land—24.34 km2, open forest—5.50 km2, shrubs—2.15 km2 and water body—1.48 km2 (Fig. 9). For evolution of FAHP-based MCDS, weights of each erosion hazard parameter (EHP) using FAHP method, priority calculation using weights of EHP and their corresponding normalized intensity and finally categorization of different sub-watersheds in different groups for soil conservation measures are determined.

Fig. 7
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Sub-watersheds of Nagwan watershed

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Fig. 8
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Drainage map of study area (Nagwan watershed)

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Fig. 9
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Land use/land cover map of the study area (Nagwan watershed)

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EHPs computation

Thirteen EHPs, viz. USLE model (SL), Sy, SPR, Rhp, Di, Dd, Df, Fs, Dt, Rc, Lo, Cc and Rb, have been computed as suggested in Jaiswal et al. (2014) and Smith (1950), and result is shown in Table 3. For computation of soil loss from sub-watersheds, the R-factor, K-factor, LS-factor, C-factor and P-factor of the watershed were found as 540.92 MJ mm ha−1 h−1 year−1, 0.19 to 0.34 Mg h MJ−1 mm−1, 0.03 to 19.54, 0.009 to 1.000 and 0.80 to 0.1, respectively. These factors are combined in a number of formulas in USLE, which returns a single number, the computed soil loss per unit area, equivalent to predicted erosion in t ha−1 year−1 (Wischmeier and Smith 1978). Once all erosive factor maps were generated, these maps were included in a Raster operation of multiplication in ArcGIS; therefore, a soil loss map was obtained (Fig. 10). The mean annual soil loss was predicted as 26.81 t/ha/year.

Table 3 Variation of different EHPs in sub-watersheds of Nagwan watershed
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Fig. 10
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Soil erosion map of the watershed

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For calculation of sediment production rates from different sub-watersheds, the empirical model proposed by Jose and Das (1982) and further elaborated by Mishra et al. (1984) that uses geomorphological parameters was used. The sediment yield was estimated by a simple regression model (Rao and Mahabaleswara 1990) using rainfall, slope, land use and some other geomorphological parameters. The variation of EHPs for different sub-watersheds is given in Table 3.

FAHP-Based Prioritization

Considering the huge investment in the watershed development program, it is important to plan the activities on priority basis for achieving relevant fruitful results. Such an effort also facilitates the problematic areas to arrive at optimal solutions. The resource-based approach is found to be realistic for watershed prioritization, as it involves an integrated approach. For prioritization of sub-watersheds, fuzzy analytical hierarchy process (FAHP) along with different number of erosion hazard parameters (EHP’s) from 9 to 15 was used to prioritize the watershed under reference. The fuzzy matrix representing triangular, narrow trapezoidal, medium trapezoidal and wide trapezoidal membership functions was used to assess the uncertainty in the decision-making. The geometric mean method (Eqs. 2–6) was used for computing and fuzzy assigning weightage to the matrix. The crisp weightage to all EHPs was determined by centroid method de-fuzzification of matrix weightage. The fuzzy matrix of geometric mean \( [\sigma_{i} ,\varepsilon_{i} ,\tau_{i} ,\mu_{i} ] \) and fuzzy membership function of the final weightage towards triangular and trapezoidal with narrow, medium and wide trapezoidal shape functions were computed in respect of from 9 to 15. The consistency ratios of fuzzy-AHP decision matrix for triangular, narrow rectangle and wide rectangle functions were computed for different sets of EHP’s from 9 to 15 and are present in Table 4.

Table 4 Variation of consistency ratio with increasing parameters
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It has been observed that when the number of parameters exceeds 10, no significant difference is noticed in consistency ratio. The best weightage of the EHPs was found to be assorted with number 13 parameters having narrow trapezoidal shape with consistency ratio or as 9.44, it has also the smallest consistency ratio. As such this was taken as base for determining the final priority scores for sub-watershed. The fuzzy matrixes for pair-wise comparison of narrow trapezoidal function of 13 EHP’s are present in Table 5.

Table 5 Fuzzy matrix for pair-wise comparison of 13 EHP’s for narrow trapezoidal function
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Fuzzy matrix of geometric mean and fuzzy matrix of the final weightage of 13 EHP’s in respect of narrow trapezoidal fuzzy function are shown in Table 6. The computed weights for thirteen erosion hazard parameters are present in Table 7.

Table 6 Fuzzy matrix of geometric mean and fuzzy matrix of the final weight for narrow trapezoidal fuzzy function
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Table 7 Weightage values for thirteen erosion hazard parameters
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Having the weightage assigned to 13 erosion hazard parameters (EHPs) using FAHP with narrow trapezoidal shape, the final priority of each sub-watershed was computed using normalized values of EHPs and their respective weightage. The computed values of priority assessment for different sub-watersheds are given in Table 8, which varied from 0.278 to 0.627. Based on the cluster of the distribution, scatter graph in Fig. 11 was divided into five different ranges of prioritization, i.e. very high, high, medium, low and very low, as present in Table 9.

Table 8 Normalization of EHPs and priority assessment of sub-watersheds
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Fig. 11
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Scattered graph for priority classes

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Table 9 Details of sub-watersheds as per priority classes derived using FAHP
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The analysed results of sub-watersheds’ prioritization using FAHP (Table 9 and Fig. 12) depicted that of 21 sub-watersheds, 5 sub-watersheds (SW-18, SW-3, SW-17, SW-19 and SW-20) covering a total area of 19.02 km2 and 4 sub-watersheds (SW-21, SW-16, SW-6 and SW-14) of total area 16.89 km2 could be put under very high and high prioritization category in immediate execution of catchment area treatment plan. The remaining 11 sub-watersheds categorized of medium, low and very low priorities can have implementation of soil conservation measures at later stage. However, the agronomical and/or biological treatment is recommended to pursue and continue for such prioritized sub-watersheds.

Fig. 12
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Priority map of Nagwan watershed

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The analysis revealed that the combination of soil loss from USLE model (SL), sediment yield (SY), sediment production rate (SPR) and relative relief has contributed nearly 66% weight in decision-making and those can be considered as the super parameters in prioritization work.

Conclusions

Prioritization of sub-watersheds for planning and management of soil conservation measures can significantly economize the cost of catchment area treatment. A fuzzy analytical hierarchical process (FAHP)-based multi-criteria decision support model (MCDSM), as an extension of Saaty’s AHP by overcoming its uncertainty in selection of environmentally degraded sub-watersheds, has been developed for prioritizing the areas in a catchment/watershed for soil conservation measures. Instead of incorporating large number of parameters, it is advisable to go for selective and relevant parameter while applying MCDS method for prioritizing watersheds. The model considered the weights of thirteen erosion hazards parameters (EHPs), namely soil loss by USLE, mean bifurcation ratio (Rb), drainage density (Dd), drainage texture (Dt), stream frequency (Fs), drainage intensity (Di), sediment production rate (SPR), sediment yield (Vs), circularity ratio (Rc), compactness constant (Cc), elongation ratio (Re), length of overland flow (Lo), drainage factor (Df) and relative relief (Rhp) with triangular, narrow rectangular, medium rectangular and wide rectangular fuzzy membership functions. The geometric mean method for weight matrix and centroid method for crisping of weights have been considered for determining weights of EHPs. The model can successfully be used in prioritizing the areas in any watershed for soil conservation measures.