As shown in equation no. 1 which is considered by [11], data gathered from digital numbers are transformed to spectral radiance. Data from digital numbers are of the TIR bands of ETM+ and TM5 images for individual years. Then the use of Equation no. (2) thermal infrared images of Landsat 8were transformed considering the standard of USGS.

L λ = L min + L max − L min Q C A L max − Q C A L min ( D N − Q C A L min ) (1)

= M L × Q c a l + A L (2)

In Equations (1) and (2), L_λ denotes spectral radiance in W/(m²srμm) gained by the sensor found from each pixel from the Landsat imagery. Band specific multiplicative is denoted by ML and rescaling factors which are gathered from MTL image file are found to be additive and is denoted using the term by AL. DN of each image and maximum DN is represented by Q_cal and QCAL_max. The value of Qcal and QCAL_max for the 16-bit Landsat 8 is 65535 and for other Landsat missions is 255 respectively. QCAL_min is the least DN (0). Radiances from top of the atmosphere (TOA) are denoted as L_max and L_min that are scaled to QCAL_max and QCAL_min in W/(m²srμm), respectively.

With the use of Equation (3) the radiant images were transformed into temperature detected in blackbody [12].

T b = K 2 ln { ( K 1 L λ ) + 1 } (3)

In the above Equation brightness temperature in Kelvin unit is measured using sensor which is denoted with T_b, spectral radiance in W/(m² srμm) is denoted with L_λ and through image MTL file prelaunch calibration constants in Kelvin unit is obtained. These are represented as K₁ and K₂. Error in measurement of surface temperature can arise during the execution process due to detection of earth as a blackbody because of its brightness [12]. To minimize these error values, emissivity correction is performed to attain land surface temperature (LST) from Tb using Equation (4), [13].

A threshold method was suggested by [14] to achieve land surface emissivity from Normalized Difference Vegetation Index (NDVI) threshold method. If NDVI < 0.2 then pixels are acknowledged as barren land. The emissivity of these barren lands was derived from red spectral region. If NDVI > 0.5 then pixels are acknowledged as lands thoroughly covered with plants or agricultural fields and its emissivity value was estimated to be 0.99 [10]. When NDVI value lies between the range of 0.2 to 0.5 then pixels are acknowledged as partially covered with vegetation. Using Equation (4) the emissivity is obtained.

ε = ε v P v + ε s ( 1 − P v ) + Δ ε (4)

where ε v the emissivity of vegetation coverage, ε s is the soil surface emissivity and, P v is computed to obtain the proportion of vegetation using Equation (5).

[ NDVI − NDVI s NDVI v − NDVI s ] 2 (5)

where NDVI s is the NDVI value of pure soil and NDVI v is the NDVI values of pure vegetation derived from NDVI image. Δ ε in Equation (7) indicates distribution of the land surfaces according to its geometry in addition to its the internal reflection. The value of internal reflection is very low for the ordinary and uniform surfaces [10]. The value of internal reflection is however found to be 2% for rough and heterogeneous surface. Δ ε is calculated using Equation (6)

( 1 − ε s ) ( 1 − P v ) F ε v (6)

where shape factor is denoted by F. The mean value for different land surfaces according to its distributions based on its geometry is assumed as 0.5.

Summarizing Equation (5) and Equation (6), the final equation to calculate emissivity estimation is given as Equation (7).

ε = m P v + n (7)

where m and n co-efficient are calculated as below in Equation (8):

m = ε v − ε s − ( 1 − ε s ) F ε v and n = ε s + ( 1 − ε s ) F ε v (8)

LST = T b 1 + { λ T b ( K ρ ) × ln ε } (9)

In the above Equation (9), wavelength of emitted radiance (11.5 μm) [11] is represented by λ, ρ = hc/σ [15], K is the Stefan–Boltzmann’s constant, h is the Planck’s constant, velocity of light (2.998 × 10⁸ ms⁻¹) by c and surface emissivity by ε. Finally, the derived LST values were converted to the conventional Degree Celsius (˚C) unit by adding the absolute zero which is approximately minus 273.15˚C. Finally retrieve the (LST) as shown in Figure 2.

Table 2 involves urban indices and indices of vegetation. These were estimated through a digital number of measured bands. As recently referenced, the ultimate objective is to investigate the superiority in a relationship's characteristics with the surface temperature. In addition to this to perceive indices with highest grounded capacity for assessing temperature of surfaces of urban areas, a couple of indices were tried. Simulating LST using polynomial curve fitting was shown in Figure 3.

Evaluating temperatures of land surfaces with the use of various factors requires high value of correlation between the predictor variables and temperature measured at land surfaces, with no inter-factor show evaluation of indices for quality assessment of correlation with LST. The indices that exceptionally correlate with LST were chosen and applied to develop a linear regression model in order to analyze future temperatures of land surfaces. To analyze the relation between land cover indices (e.g., UI, MNDWI, NDVI and NDBI) and the LST for each of the periods multiple linear regression models are designed. The investigation outputs demonstrate statistically significant correlations which are valid for an entire year [5]. Land cover indices and LST values were mined from each pixel in the designated area of study for individual point data type in order to achieve this objective. These points were utilized to calibrate the linear regression model [20]. The model offers a general idea about the correlation between LST and LULC indices. This realization is relatively coherent with those narrated by other earlier researches [21].

The Land Use Land Cover Indices for 26.12.1988, 9.3.1998, 3.2.2000, 22.2.2010, and 12.2.2018 for the same seasons were the input variable in the polynomial curve fitting [13] to map future state of the indices for 2010 and 2018, Similarly, the Land Use Land Cover Indices 22.2.2010 and 12.2.2018 was used in polynomial curve fitting analysis to predict the state of the Land Use Land Cover Indices in 2026, as shown below:

[ m ∑ x i ∑ x i 2 ⋯ ∑ x i n ∑ x i ∑ x i 2 ∑ x i 3 ⋯ ∑ x i n + 1 ∑ x i 2 ∑ x i 3 ∑ x i 4 ⋯ ∑ x i n + 2 ⋮ ⋮ ⋮ ⋱ ⋮ ∑ x i n ∑ x i n + 1 ∑ x i n + 2 ⋯ L ∑ x i 2 n ] [ p 1 p 2 p 3 ⋮ p n + 1 ] = [ ∑ y i ∑ x i y i ∑ x i 2 y i ⋮ ∑ x i n y i ]

Curve fitting over m pairs of data ( x 1 , y 1 ) , ( x 2 , y 2 ) , ⋯ , ( x m , y m ) is a process of obtaining a polynomial regression between the two pairs as follows: p ( x ) = P 1 x n + P 2 x n − 1 + ⋯ + P n x + P n + 1 ; where, p ( x ) is a curve fitting of data pairs; p 1 , p 2 , ⋯ , p n + 1 are the model variables, and the x 1 , ⋯ , x n are the n inputs variables. These input variables have a dependency on the possible data available across. Through the current study, the unknown parameters are found using the least square method. At last Land Use Land Cover Indices predictions were converted into land surface temperature distributions for 2010, 2018 and 2026 in the same seasons through multiple linear regression analysis functions.

The proposed method is applied to forecast temperature of land surfaces for the year 2016 and its rate of accuracy in predicting was measured through Mean Absolute Percentage error. [MAPE]—Equation (10)—(Owen, Carlson et al. 1998).

MAPE % = 1 N ∑ i = 1 N | ( T predicted − T observed ) / T observed | ∗ 100 (10)

where: the modeled surface temperature and the actual land surface temperature recorded from Landsat data for the ith pixel is denoted by T_predictedand T_observed. Root Mean Square Error and ration RMSE/std can also be regarded as the measure for calculating accuracy of the prediction model in forecasting temperature. Land surface temperature distribution for the period from 2026 is then found using the prediction model after the accuracy assessment.