Deriving the weights using Analytical Hierarchy Process (AHP)

CHAPTER 2: LITERATURE REVIEW

4.2 Integration of various thematic layers for groundwater potential zones

4.2.1 Deriving the weights using Analytical Hierarchy Process (AHP)

A pair-wise comparison for the identified themes controlling groundwater was carried out using Saaty’s scale of relative importance (table 4.1). On the scale, the thematic layers can be categorized as less important, equally important and more important. On the scale below, a score of 1 indicates equal importance between two themes, and a score of 9 indicates extreme importance of one layer relative to the other (Saaty, 1980).

Table 4. 1: Saaty’s scale of relative importance (Saaty, 1980).

Less Important Equally

Important

More Important Extremely Very

strongly

Strongly Moderately Equal Moderately Strongly Very Strongly

Extremely

0.111 0.143 0.200 0.333 1 3 5 7 9

A pairwise comparison matrix was based on the nine-point scale of relative importance of the different themes on controlling groundwater occurrence, namely geology, lineaments, land use/land cover and slope. Since the study used four thematic layers, a 4 by 4 matrix was created (table 4.2). In table 4.2, the element matrix values of the different weights for the four themes are presented. In the table, the theme in the row is compared to the theme in the column. For example, in table 4.2, geology is equally important to geology (since it is 1); geology is twice as important to lineament density (since the value is 2); geology is 3 times important to slope and land cover (since the value is 3) etc.

Table 4. 2: A Pair-wise comparison matrix

Theme Geology Lineament Density Land-use/ Land cover Slope

Geology 1 2 3 3

Lineament density 0.5 1 3 3

Land-use/ Land cover 0.33 0.33 1 1

Slope 0.33 0.33 1 1

∑ of column matrix 2.16 3.66 8 8

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To reduce possible bias in the results, usually pair-wise comparison matrix is normalized by dividing each element of the matrix by the sum of its column, as suggested by Saaty (1980). Consequently, the total of the rows were determined, and the principal eigenvector /normalized weights is determined by averaging across all the rows as shown in table 4.3. This average/ normalized weights were then converted into percentage to give the overall percentage influence of each parameter. From the results, it is observed that geology has a total influence on the hydrological properties of 44%, and lineament density 32%, while slope and land cover have 12% each.

Table 4. 3: Normalized matrix (Saaty, 1980).

Geology Lineament Density

Land use/

Land cover

Slope ∑ (Total)

Average/

Normalized (Principal Eigenvector)

%

Influence

Geology

0.46 0.55 0.38 0.38 1.76 0.44 44%

Lineament density

0.23 0.27 0.38 0.38 1.25 0.32 32%

Land use/Land cover

0.15 0.09 0.13 0.13 0.49 0.12 12%

Slope

0.15 0.09 0.13 0.13 0.49 0.12 12%

Analytical Hierarchy Process (AHP) captures the idea of uncertainty in judgments through the principal eigenvalue and the consistency index (Saaty 2004). Principal Eigen value (𝜆𝑚𝑎𝑥) is calculated by summing up the products between each element of Eigen vector and the sum of columns of the pairwise matrix (table 4.4).

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Table 4. 4: Computation of principal Eigenvalue (𝛌_𝐦𝐚𝐱).

Thematic Layer ∑ of Column matrix (1)

Eigen Vector (2)

Products of column (1) and (2) [(1) x (2)]

Geology 2.16 0.44 0.95

Lineament density

3.66 0.32 1.17

Land use/ Land cover

8.00 0.12 0.96

Slope 8.00 0.12 0.96

Principal Eigenvalue (𝜆_𝑚𝑎𝑥) 𝛌_𝐦𝐚𝐱= (0.95+1.17+0.96+0.96) = 4.04

Saaty also provided a measure of Consistency, which is called a Consistency Index (CI).

Consistency Index is a measure of the degree of consistency in the pairwise comparison matrix, and is calculated using equation 4.1 as follows:

CI =

^λmax−n

n−1

(4.1) Where 𝜆_𝑚𝑎𝑥 is the principal Eigenvalue of the pairwise matrix and was computed in table

4.4 above and n is the number of thematic layers being compared. Four thematic layers were compared in this study, therefore n = 4.

The consistency index was therefore computed using equation 4.2 as follows:

𝐶𝐼 =

^λmax−n

n−1

=

^4.04−4

4−1

= 0.013

^(4.2)

Since the elements of the pairwise matrix are obtained from an individual’s preferences and judgments, some inconsistencies are expected and allowed in Analytical Hierarchy Process. In order to determine whether the degree of inconsistencies are within the acceptable range, Saaty (1980) recommended the calculation of the consistency ratio (CR).

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The consistency ratio (CR) is determined by comparing the consistency of the matrix in question (the one with our judgments), versus the consistency index of a random-like matrix. A random matrix is one where the judgments have been entered randomly and therefore it is expected to be highly inconsistent (Brunelli, 2015). Saaty (2012) provides the calculated random index (RI) values for random matrices of different sizes as shown in Table 2.5.

Table 4. 5: Saaty’s table of random index for different (n) values

N 1 2 3 4 5 6 7 8 9 10

RI 0 0 0.58 0.89 1.12 1.24 1.32 1.41 1.45 1.49

It should be noted that the value of the Random Index depends on the number (n) of thematic layers being compared. Since four layers were compared in this study, the random index RI = 0.89. Therefore, the consistency ratio was calculated using equation 4.3 follows:

CR= CI/RI (4.3)

= (0.013)/ (0.89)

= 0.015 or 1.5%

Saaty (2012) has shown that a consistency ratio (CR) of 0.10 or less is acceptable to continue the analysis of the thematic layers. If the consistency ratio is greater than 0.10, it is necessary to revise the judgments to locate the cause of the inconsistency and correct it.

The calculated Consistency Ratio of the analyst in this study was 0.015 (i.e. 1.5 %), which is within the acceptable range of 0 – 0.10, hence it is acceptable.

Having confirmed that the consistency ratio was within the acceptable limits, the different classes generated were then ranked from 1 to 5 based on their significance in controlling groundwater potentiality, where a rank of 1 represents lower potentiality, and 5 represents higher groundwater prospects (table 4.6). The table 4.6 shows the thematic layers, the weights, classes, rank and groundwater potential for each thematic layer.

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In table 4.6, weights represent the percentage of influence on controlling groundwater occurrence and storage as derived in table 4.3. The classes represent the defined classes whereas the rank represents the assigned rank on the scale of 1 to 5 for each class. A scale of 1 being very poor and 5 being very good groundwater potential.

Table 4. 6: Ranks and weights for different parameters influencing groundwater potentiality. (Adopted from: Tessema et al., 2014; Palaka and Sankar, 2015; Shekar and Pandey, 2015).

Thematic Layer Weights Classes Rank Groundwater Potential Geology/

Lithology 44%

Carbonates/Marbles

Sandstones/ Granulestone Mudrocks

Basalt/Gneiss

Granites/Dolerite/Diabase dykes/Metaquartzites

5 4 3 2 1

Very Good Good Moderate Poor Very Poor Lineament Density

32%

Very high density High density Moderate Low density

Very low/ No lineaments 5 4 3 2 1

Very Good Good Moderate Poor Very poor Slope

12%

Flat Gentle Moderate Steep Very steep

5 4 3 2 1

Very Good Good Moderate Poor Very Poor Land Cover/ Land

use data 12%

Cultivated land

Wetlands/ water Body Vegetated areas Settlements Barren land

5 4 3 2 1

Very Good Good Moderate Poor Very Poor

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