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Appendix-C
Weighted aggregation of criteria in GIS-based multi-criteria
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Figure C1: Decision strategy triangle within the continuum of risk and trade-off dimension [4]
WLC involves standardizing continuous criteria (factors) to a common numeric range and then combining them using a weighted average. WLC enables criteria to make trade-offs between their qualities. Very poor quality might be offset by several extremely excellent qualities. This operator is neither AND nor OR—it is in between these two extremes. It is neither risk-averse nor risk-taking. It is proved that vector methods to MCDM are dominated by Boolean strategies, whereas raster systems are dominated by WLC solutions [5]. WLC is defined by complete trade-off and average risk, which is precisely halfway between AND and OR operations, i.e., neither excessive risk aversion nor extreme risk-taking (Figure C1). WLC can be expressed by Equation C1.
𝑆𝑗 = ∑ 𝑤𝑖
𝑛
𝑖=1
𝑥𝑖 𝐶1
Where, 𝑆𝑗 is the composite suitability (of the jth pixel or area); wi =weight of factor i; xi = criterion score of factors i; n = total number of factors. The weight of factor (wi) can be determined using Analytic Network Process (AHP) method as described in Appendix-B.
However, none is superior—they merely offer two opposed perspectives on the decision-making process—what can be called a choice strategy. The WLC approach is not always appropriate for territorial analysis due to the inherent risk of concealing a limiting factor between the high values of other criteria. Thus, the decision-maker has limited control over Boolean and WLC methods to decide the level of risk and trade-off in decision making. To avoid this problem, an Ordered Weighted Average (OWA) was proposed by Yager [6]. The
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third method, OWA, is identical to the WLC procedure, except that OWA requires additional weights, referred to as order weights, and calculates the factor weights' combination. This method provides a comprehensive range of decision methods along the two fundamental dimensions of the degree of trade-off and degree of risk associated with the solution [2] (Figure C1). OWA can be expressed by Equation C2.
𝑆𝑖 = ∑( 𝑢𝑗𝑣𝑗
∑𝑛𝑗=1𝑢𝑗𝑣𝑗)𝑧𝑖 𝑗
𝑛
𝑗=1
𝐶2
Where 𝑆𝑖 is the suitability at ith location, n is the number of indicators, 𝑢𝑗 is the original weight factor of the criteria, 𝑣𝑗 is the ordered weight of the criteria such that 𝑣𝑗ɛ [0, 1] 𝑓𝑜𝑟 𝑗 = 1, 2, … , 𝑛, and ∑𝑛𝑗=1𝑣𝑗 = 1, 𝑧𝑖𝑗 is the ordered value of criteria at ith location.
In the case of OWA, two sets of weights are used. The first weight (𝑢𝑗) of the criteria can be determined using Analytic Network Process (AHP) method as described in Appendix-B. The second weight which is known as ordered weight was assigned on the ordered criteria either based on increasing order or decreasing order depending on the level of acceptable risk and type of criteria. In this study, we have used OWA with different combinations of ordered weights to generate alternatives with different levels of trade-off and risk following the guidelines of Eastman [4]. The combination of ordered weights has been presented in Table C1.
Table C1: Ordered weights used in different strategies.
Decision strategies Ordered weights
Average Level of Risk - Full Tradeoff 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125
Low Risk and No Tradeoff 1 0 0 0 0 0 0 0
High Risk and No Tradeoff 0 0 0 0 0 0 0 1
Low Level of Risk - Some Tradeoff 0.4 0.3 0.12 0.07 0.05 0.03 0.02 0.01 High Level of Risk - Some Tradeoff 0.01 0.02 0.03 0.05 0.07 0.12 0.3 0.4 Average Level of Risk - No Tradeoff 0 0 0 0.5 0.5 0 0 0
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Appendix-D
Method of calculating land surface temperature (LST)
In this study, we have used the TIRS band 10 of the Landsat image to derive LST in the study area. Particulars of Landsat image has been given in Table D1.
Table D1. Particulars of Landsat images used in this study.
OLI = Operational Land Imager, TIRS= Thermal Infrared Sensor
Six step procedure was followed to derive LST [1]. The steps are described below.
Conversion of DN to top of atmospheric spectral radiance: Firstly, the DN values of the image were converted to top of atmospheric (TOA) spectral radiance using Equation D1.
𝐿𝜆 = 𝑀𝐿∗ 𝑄𝑐𝑎𝑙+ 𝐴𝐿 D1
Lλ = TOA spectral radiance (Watts/ (m2 * srad * μm)), ML =Band-specific multiplicative rescaling factor from the metadata, AL=Band-specific additive rescaling factor from the metadata, and Qcal = Quantized and calibrated standard product pixel values [2].
Conversion of Radiance to At-Sensor Temperature: Second, TOA spectral radiance was converted to at-sensor temperature. The at-sensor temperature is also known as brightness temperature (BT). The BT was derived from TOA spectral radiance using Equation D2.
𝐵𝑇 = 𝐾2
ln(𝐾1
𝐿𝜆+1)− 273.15 D2
Lλ =TOA spectral radiance (Watts/( m2 * srad * μm)), K1 =Band-specific thermal conversion constant from the metadata, K2 =Band-specific thermal conversion constant from the metadata [2]. The original Equation of deriving BT was revised by adding -273.15 in order to get the result in Celsius [3].
Landsat Scene ID Acquisition Date Satellite Sensor Path/Row LC81380432021115LGN00 25/04/2021 Landsat 8 OLI/TIRS 138/43
D-2
Calculation Normal Difference Vegetation Index (NDVI): NDVI is important to derive LST because this NDVI would be used to calculate the proportion of vegetation. The NDVI of the study area was calculated using Equation D3.
𝑁𝐷𝑉𝐼 =𝑁𝐼𝑅 (𝑏𝑎𝑛𝑑 5)−𝑅 (𝑏𝑎𝑛𝑑 4)
𝑁𝐼𝑅 (𝑏𝑎𝑛𝑑 5)+𝑅 (𝑏𝑎𝑛𝑑 4) D3
where NIR represents the near-infrared band (Band 5) and 𝑅 represents the red band (Band 4).
The value of NDVI ranges from -1 to +1; the higher value of NDVI refers to healthy and dense vegetation and lower NDVI values represent sparse vegetation.
Calculation of Proportion of Vegetation: The proportion of the vegetation (PV) was calculated using Equation E4 [4]. The minimum and maximum values were used to calculate PV.
𝑃𝑣 = ( 𝑁𝐷𝑉𝐼−𝑁𝐷𝑉𝐼𝑚𝑖𝑛
𝑁𝐷𝑉𝐼𝑚𝑎𝑥−𝑁𝐷𝑉𝐼𝑚𝑖𝑛)2 D4
Calculation of Land Surface Emissivity (LSE): LSE is important to estimate LST. In the next step, LSE was calculated using Equation D5 [5].
𝐿𝑆𝐸 = 0.004 ∗ 𝑃𝑣+ 0.986 D5
Calculation of LST: In the final step, LST was derived from the value of BT and LSE using Equation D6 [1].
𝐿𝑆𝑇 = 𝐵𝑇
{1+[(𝜆∗𝐵𝑇 𝜌⁄ ) ln(𝐿𝑆𝐸)]} D6
𝜆 is the wavelength of emitted radiance. The average value of the limiting wavelength (𝜆
=10.895) [6] was used in the above equation. The value of 𝜌 was calculated using Equation D7.
𝜌 = ℎ𝑐
𝜎 = 1.438𝑋10−2𝑚 𝐾 D7
where 𝜎 is the Boltzmann constant (1.38 × 10−23 J/K), ℎ is Planck’s constant (6.626 × 10−34 J s), and 𝑐 is the velocity of light (2.998 × 108 m/s) [7]
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