Korean Journal of Remote Sensing (대한원격탐사학회지)

Korean Society of Remote Sensing (대한원격탐사학회)
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Comparison of Lambertian Model on Multi-Channel Algorithm for Estimating Land Surface Temperature Based on Remote Sensing Imagery

  • A Sediyo Adi Nugraha (Major in Geographical Sciences, Faculty of Geography, Universitas Gadjah Mada) ;
  • Muhammad Kamal (Department of Geographic Information Science, Faculty of Geography, Universitas Gadjah Mada) ;
  • Sigit Heru Murti (Department of Geographic Information Science, Faculty of Geography, Universitas Gadjah Mada) ;
  • Wirastuti Widyatmanti (Department of Geographic Information Science, Faculty of Geography, Universitas Gadjah Mada)
  • Received : 2024.06.30
  • Accepted : 2024.08.19
  • Published : 2024.08.31
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Abstract

The Land Surface Temperature (LST) is a crucial parameter in identifying drought. It is essential to identify how LST can increase its accuracy, particularly in mountainous and hill areas. Increasing the LST accuracy can be achieved by applying early data processing in the correction phase, specifically in the context of topographic correction on the Lambertian model. Empirical evidence has demonstrated that this particular stage effectively enhances the process of identifying objects, especially within areas that lack direct illumination. Therefore, this research aims to examine the application of the Lambertian model in estimating LST using the Multi-Channel Method (MCM) across various physiographic regions. Lambertian model is a method that utilizes Lambertian reflectance and specifically addresses the radiance value obtained from Sun-Canopy-Sensor(SCS) and Cosine Correction measurements. Applying topographical adjustment to the LST outcome results in a notable augmentation in the dispersion of LST values. Nevertheless, the area physiography is also significant as the plains terrain tends to have an extreme LST value of ≥ 350 K. In mountainous and hilly terrains, the LST value often falls within the range of 310-325 K. The absence of topographic correction in LST results in varying values: 22 K for the plains area, 12-21 K for hilly and mountainous terrain, and 7-9 K for both plains and mountainous terrains. Furthermore, validation results indicate that employing the Lambertian model with SCS and Cosine Correction methods yields superior outcomes compared to processing without the Lambertian model, particularly in hilly and mountainous terrain. Conversely, in plain areas, the Lambertian model's application proves suboptimal. Additionally, the relationship between physiography and LST derived using the Lambertian model shows a high average R2 value of 0.99. The lowest errors(K) and root mean square error values, approximately ±2 K and 0.54, respectively, were achieved using the Lambertian model with the SCS method. Based on the findings, this research concluded that the Lambertian model could increase LST values. These corrected values are often higher than the LST values obtained without the Lambertian model.

Keywords

1. Introduction

Human activities, including the change in land cover, have influenced climate change, which in turn has contributed to the occurrence of severe weather events worldwide. From 2000 to 2020, these changes massively decreased the water supply on land. The intensification of the Earth’s water supply decline has led to a severe drought, which is further aggravated by the lack of replenishment of water resources following droughts. The United Nations Convention to Combat Desertification (2022) report indicates that droughts account for 15% of all global natural disasters, causing massive death casualties with a death toll of 650,000 people between 1970 and 2019. In 2022, around 2.3 billion people worldwide faced water scarcity, while almost 160 million children were affected by the consequences of prolonged drought. Droughts have a global impact on humans, the environment, the economy, society, and culture, as well as the health sectors(National Integrated Drought Information System, 2022). Hence, it is imperative to implement drought mitigation strategies that are in line with the Indonesian Sustainable Development Goals, specifically numbers 2, 13, and 15(BAPPENAS, 2020).

To date, the task of identifying droughts is challenging due to the need for well-equipped technologies that are widely available in different places and provide precise measurements (World Meteorological Organization and Global Water Partnership, 2016). It is crucial to determine that various forms of remote sensing data are necessary for different studies, especially those focusing on droughts that require specific data characteristics. Land Surface Temperature (LST) is a highly correlated indicator of drought. The indicator has become crucial in various indices and formulas used to determine droughts through remote sensing. These include the Evaporative Stress Index (ESI) (Anderson et al., 2007), Temperature Condition Index (TCI) (Kogan, 1997), Vegetation Drought Response Index (VegDRI) (Brown et al., 2008),Crop Water Stress Index (CWSI)(Jackson et al., 1981;Idso et al., 1981), and Temperature Vegetation Dryness Index (TVDI) (Nugraha et al., 2022; Nugraha et al., 2023; Sandholt et al., 2002). In addition, the presence of distinct physical features, especially in mountainous and hilly regions, has led to a diverse terrain structure and LSTs (Albrich et al., 2020; Meybeck et al., 2001; Zhao and Li, 2015).

Since the 1980s, many algorithms have been developed to detect information related to LST from remote sensing data, particularly Thermal Infrared (TIR), including a single-channel algorithm (Jimenez-Munoz et al., 2009; Nugraha, 2019a; Qin et al., 2001) and split-window algorithm (Nugraha and Atmaja, 2021; Nugraha et al., 2019b, 2019c; Sobrino et al., 1994; Wan and Dozier, 1996). However, due to the neglect of physiographic features, specifically topography, the utilization of these two methods does not allow for the possible achievement of LST information through the topographic correction procedure. Furthermore, it is crucial to carefully evaluate many approaches used in the development of the LST method so as to accurately determine LST. Using three different methods, the LST estimation can be achieved through remote sensing data. The first method is the Single-Channel Method (SCM), which was developed by Artis and Carnahan (1982). This method specifically utilizes one thermal band, distinguishing between the low and high thermal bands for obtaining the surface temperature (Nugraha et al., 2024; Sekertekin and Bonafoni, 2020).The second method is the Multi-Channel Method (MCM), which was developed by combining two thermal bands to determine surface temperature more accurately than the SCM (Nugraha et al., 2024; Nugraha and Kurniawan, 2024).The Multi-Angle Method (MAM)is a method utilizing both front and rear angles in remote sensing photographs. It is not commonly used and has only been used in a limited number of cases (Sobrino et al., 2004; Sòria and Sobrino, 2007).

Nugraha et al. (2024) conducted research that demonstrated the suitability of the MCM or Split Windows Algorithm (SWA) for use in regions with diverse land cover. Skokovic et al. (2014) devised the MCM technique in their research, which integrated emissivity and water vapor to enhance the precision of the LST measurement. However, the method employed in the measurement did not consider the varying physical features of the terrain. Consequently, the diverse land surfaces led to more intricate thermal radiation, causing a decrease in the downward radiation in the atmosphere and ultimately amplifying the radiation in the surrounding area (Zhao and Li, 2015). Lipton (1992) suggested that the lack of a topographic effect can lead to measurement bias in LST estimation, particularly when using remote sensing data in mountainous regions. The impact is caused by variations and changes in land slope and tree growth direction.

Hence, it is important to note that not all radiometric corrections can minimize the information on the change (Dymond and Shepherd, 1999; Ekstrand, 1996; Meyer et al., 1993; Teillet et al., 1982). The extensive range of the area and the challenge of establishing model reliability are the primary factors contributing to this situation. On the other hand, the current non-parametric method still necessitates intricate data and an extensive field survey (Schaaf et al., 1994). Given this information, it is crucial to compare topographic corrections to assess their impact on surface temperature estimates using radiance channels.

Topography correction is a method that has been developed to compensate for the reduction in topographic features in the analysis of remote sensing data (Soenen et al., 2008; Sola et al., 2014; Sola et al., 2016). Furthermore, it is performed to account for variations in signal strength that occur when measuring different types of terrain (such as flat plains or mountainous areas). The goal isto produce a consistent reflectance value across all terrains (Chen et al., 2023; Sola et al., 2014; Sola et al., 2016). Topographic correction methods can be categorized into three types: empiric, semi-physical, and physical. These methods have constraints derived from the assumption of Lambertian and non-Lambertian models(Chen et al., 2023; Riano et al., 2003; Vincini and Frazzi, 2003). The empirical method emphasizes the use of statistical correlations between surface reflectance values and topographic conditions, which can be acquired efficiently and applied for topographic correction. However, it is specifically tailored to the specific spatio-temporal conditions. Therefore, applying this method requires a longer extended period in several stages (Qiu et al., 2019; Reese and Olsson, 2011; Richter, 1997). The empirical method incorporating topographic corrections includes C-correction, Sun-Canopy-Sensor+C (SCS+C), and Minnaert.

A semi-physical method aims to simplify and reduce the number of empirical parameters used in radiation transfer, recreating vegetation reflectance (Gu and Gillespie, 1998; Yin et al., 2020a). The physical method simultaneously considers solar radiation throughout the atmosphere, its interaction with the Earth’s surface, and its detection by sensors while excluding the surface (Li et al., 2012; Wen et al., 2009; Zhang and Gao, 2011; Zhang et al., 2015). Hence, the physical method is the most intricate among the other two and presents considerable difficulties, especially when combined with remote sensing data (Li et al., 2012).

The topographic correction has the potential to enhance the precision of remote sensing by focusing on specific information that is not influenced by the overall ranking of different approaches (Richter et al., 2009; Umarhadi and Danoedoro, 2020). However, the constraint is specifically linked to the use of spectral bands in remote sensing. Currently, the topographic correction method uses the visible band from top-of-atmosphere (TOA)reflectance as its input due to the application of non-Lambertian reflectance (Riano et al., 2003; Smith et al., 1980; Teillet et al., 1982).

The Lambertian model requires optimization due to its lack of a radiance band, which limits its ability to accurately capture changes in directly visible reflectance. One of the adjustments that can be applied to the Lambertian model is the Cosine Correction (Teillet et al., 1982) and the SCS correction (Gu and Gillespie, 1998). To date, researchers have used topographic correction mainly for visible and infrared bands in their studies (Ediriweera et al., 2013; Himayah et al., 2016; Umarhadi and Danoedoro, 2019; Umarhadi and Danoedoro, 2020; Yin et al., 2020b; Zhang et al., 2015). However, not all researchers apply this method to their specific research due to the level of brightness. Although a dedicated topographic correction for the thermal band has not been created, Zhu et al.(2021) successfully employed a method to change the LST data by using land height information from a Digital Elevation Model (DEM). This modification resulted in a noticeable variation of up to 1 K in the LST values. Hence, it is imperative to employ a topographic correction method.

Previous research has demonstrated that applying topographic correction can enhance the precision of various analyses, including land coverage classification (Moreira and Valeriano, 2014; Vanonckelen et al., 2013), detection of forest coverage (Vanonckelen et al., 2015), and estimation of biophysical parameters (Yinet al., 2020). In addition, the process of topographic correction can eliminate the influence of numerous elements, such as atmospheric effects and variations in the visual field of different terrains (Chi et al., 2022; Sola et al., 2014). Thus, by combining several methodologies, it is possible to obtain a qualitative-comparative outcome, especially when estimating the LST. Hence, it is crucial to analyze the integration of topographic correction and radian band with LST. Currently, the effect of topography on LST is assessed by establishing a regression correlation between LST and terrain factors, including land elevation, slope, and slope direction (aspect) (Hais and Kučera, 2009; He et al., 2019).

Based on this explanation, the research aims to analyze variations in LST values using the MCM method compared to the Lambertian model. Additionally, understanding the impact of physiographic differences is crucial for interpreting LST results. This activity is to determine the LST in mountainous and hilly areas with drought potential, which is crucial for observing the ecological environment and studying climate change. Moreover, it also seeks to provide distinct LSTs, specifically in hilly and mountainous regions in contrast to flat plains.

2. Materials

2.1. Study Area

The research was conducted in four regencies located in East Java Province, Indonesia (Fig. 1). Each of the four sites represented distinct physiography. The first location, Situbondo, is situated at an elevation of 135 meters relative to the mean sea level. Baluran National Park encompasses a diverse range of topography, including plains, hills, and mountains, providing comprehensive coverage of difficult areas. Bondowoso, the second location, is characterized by its largely hilly and mountainous terrain, with an elevation of 1,533 meters above sea level. The third site comprises some areas of Probolinggo regency, featuring the prominents and sea of Bromo, which is a tourist destination due to its volcanic eruption effects. It is situated at an elevation of 2,144 meters above sea level. Moreover, the fourth location is Tuban regency, characterized by karst topography situated at an elevation of six meters above sea level. It is characterized by a predominance of flat terrain extending from its central to the coastline area.

Fig. 1. Research location: (a) Situbondo regency, (b) Bondowoso regency, (c) Probolinggo regency, and (d) Tuban regency.

2.2. Remote Sensing Data

The remote sensing data used in this research are Landsat images, which include a thermal band that enables the extraction of LST information (bands 10 and 11).In addition to thermal bands, the red and infrared bands are utilized to derive vegetation indices for estimating emissivity. Furthermore, this study employs two levels of Landsat imagery: Level 1 (L1) serves as raw data for the Lambertian and LST models, while Level 2 (L2)is used to validate the LST results. The Landsat images used were acquired during the dry season from August to November 2023. Meanwhile, the topography correction process utilizes DEM data sourced from Shuttle Radar Topography Mission (SRTM) images, which have a spatial resolution of 30 meters. This data is adjusted to match the spatial resolution of the Landsat image’s thermal band, also set at 30 meters. The Landsat 8 and 9 OLI/TIRS imagine data, as well as SRTM, can be accessed for free at https://earthexplorer.usgs.gov/. Additional information regarding the Landsat data is provided in Table 1.

Table 1. Remote sensing data used in this research

3. Methods

3.1. Remote Sensing Data Processing

The pre-processing stage, which involved the radiometry correction, began by converting the Digital Number (DN) to reflectance values for the visible band, and DN to radiance for the thermal band. The processing stages for radiometric correction in the visible bands and brightness temperature in the thermal bands adhere to the formulas outlined in the Landsat imagery handbook published by the U.S. Geological Survey (USGS), Department of the Interior. The atmospheric correction was performed using the Dark Object Subtraction (DOS) method developed by Chavez (1988). This method considers the darkest pixel value on water reflectance to accurately correct for atmospheric effects. The formula used for atmospheric correction is shown as follows:

DNcorrected = DN – bias (minimum pixel value)       (1)

Where, DNcorrected is the result of a Landsat image that has been atmospherically corrected, DN is the reflectance value of the Landsat image, and bias is the minimum value of the image band that is sensitive to the water object, in this case, the band 2 of Landsat 8. This atmospheric correction is carried out only on the visible band to optimize the sensitivity of the vegetation index in estimating emissivity values.

3.2. Model Lambertian

The topographic method applied in this research is part of the Lambertian model that focuses on the semi-physical method. It assumes that all wavelengths with two-way reflectance have a constant value; therefore, it aligns with the change of direct reflectance (Vincini and Frazzi, 2003). Additionally, in the Lambertian model, attention must be given to the illumination value (IL), which calculates the angle of incoming sunlight that strikes the Earth’s surface (Civco, 1989; Colby, 1991; Hantson and Chuvieco, 2011). The illumination value is computed using the following formula:

IL = cos α * cos θZ + sin α * sin θZ * cos (øα – ø0)       (2)

Where, α represents the slope angle; θz is the value of the solar zenith angle from the image Landsat in the metadata; øα is the azimuth angle solar of the Landsat image obtained from the image metadata while ø0 is the aspect angle. The IL value will result in the range 1 to –1 with the relative pixel orientation towards the sunlight(Riano et al., 2003; Umarhadi and Danoedoro, 2020; Vincini and Frazzi, 2003).

3.2.1. Cosine Correction

The Cosine Correction is a simple topographic correction that merely focuses on the surface of Lambertian and reflectance without considering the angle of the incoming light(Civco, 1989; Teillet et al., 1982). The Cosine Correction method is as follows:

\(\begin{align}\rho_{h}=\rho_{t}\left(\frac{\cos \theta_{z}}{I L}\right)\end{align}\)       (3)

Where, ρh is the result of topographic correction using the Cosine Correction method while ρt is the radiance surface of the thermal band from the Landsat image.

3.2.2. Sun-Canopy-Sensor (SCS)

The SCS, developed from the Cosine correction, focuses on areas of dominant vegetation appearance with various terrains (Dymond and Shepherd, 1999; Gu and Gillespie, 1998). The SCS method equates the DN value on the slope area to a plain terrain and aligns with the illumination direction (Soenen et al., 2005). The SCS method applied in the research follows the Eq. (4).

\(\begin{align}\rho_{h}=\rho_{t}\left(\frac{\cos \alpha * \cos \theta_{z}}{I L}\right)\end{align}\)       (4)

Where, ρh is the topographic correction derived from the result of SCS and the method while α represents the slope angle.

3.3. Land Surface Emissivity (LSE)

The LSE for bands 10 and 11 of the Landsat image refers to research by Nugraha et al.(2024), applying the emissivity method developed by Skokovic et al. (2014) whose result proved that the use of emissivity value had the slightest difference. The emissivity method used is the Normalized Difference Vegetation Index (NDVI)-based emissivity method (NBEM)(Sobrino et al., 2003). This method was chosen for its potential to differentiate objects based on vegetation greenness levels to provide emissivity values in heterogeneous geographical conditions(Nugraha et al., 2024). The NBEM method derives from NDVI calculations to obtain fractional vegetation values(PV)(Carlson and Ripley, 1997; Sobrino et al., 2003; Sobrino and Raissouni, 2000; Valor and Caselles, 1996). LSE is computed using Eqs. (5, 6), while the PV is using Eq. (7).

LSEband10 = 0.987 * Pv + 0.971 * (1 – Pv) + dƐλ       (5)

LSEband11 = 0.989 * Pv + 0.977 * (1 – Pv) + dƐλ       (6)

\(\begin{align}\rho_{v}=\left(\frac{N D V I-N D V I_{\min }}{N D V I_{\max }-N D V I_{\min }}\right)^{2}\end{align}\)       (7)

Where, dƐλ represents the surface roughness, where 0 shows the predominant plain, and 0.55 shows a variation of the surface.

3.4. Water Vapor

The determination of water vapor values(w), refers to research by Nugraha et al.(2024) as a result of image extraction from MODIS Terra (MOD021KM) and water vapor of MODIS (MOD05_L2) (Kaufman and Gao, 1992; Moradizadeh et al., 2007; Nugraha, 2019a; Sobrino et al., 2003; Zhao et al., 2009). The W value is obtained through the equation 8.

W = f17 W17 + f18 W18 + f19 W19       (8)

fi is the weighting factor resulting from the water vapor MODIS calculation, and Wi is the calculation in the MODIS Terra bands 17, 18, and 19.

3.5. Land Surface Temperature (LST)

The LST method applied in this research is the MCM, which combines two thermal bands. Despite that combination, this method functions at its maximum extent for Landsat 8, 9 since the Landsat 5, 7 ETM+ only focus on a single thermal band, which is unsuitable for the LST. The MCM method developed by Skokovic et al. (2014) considers the emissivity and water vapor values to determine the LST value. The MCM Skokovic (MCMSko) is presented in the Eq. (9).

Ts = Tb10 + C1 (Tb10 – Tb11) + C2 (Tb10 – Tb11)2 + C0 + (C3 + C4 W) (1 – ε) + (C5 + C6 W) ∆ε       (9)

Where, Tbi represents the brightest temperature of the Landsat image, ci is the coefficient determined by Skokovic et al. (2014) (Table 2).

Table 2. Skokovic coefficient value

ε is the average emissivity value; ∆ε is the emissivity difference of the Landsat image thermal band. The calculation for ε and ∆ε is shown in the Eqs. 10 and 11.

\(\begin{align}L S E(\varepsilon)=\frac{L S E_{10}+L S E_{11}}{2}\end{align}\)       (10)

Difference of LSE (∆ε) = LSE10 – LSE11       (11)

Where, LSE is obtained from Eqs. 5 and 6.

3.6. Validation Data

The validation process utilizes Landsat L2 imagery to assess the distribution of LST derived from the MCMSko method with a Lambertian model. Validation is conducted at 1 × 1 plots to determine the spatial distribution of LST and to compute the coefficient of determination (R2) and Root Mean Square Error (RMSE). The coefficient of determination calculation will focus on the MCMSko method’s results concerning the physiographic distribution within the study area, while RMSE will compare the LST results from Landsat L2 with those from the MCMSko method using the Lambertian model and without the Lambertian model. The formulas used for calculating the coefficient of determination and RMSE follow the equations below:

\(\begin{align}R M S E=\sqrt{\frac{\sum_{i=1}^{N}\left|Z_{i}^{*}-\bar{Z}_{i}\right|^{2}}{\mathrm{~N}}}\end{align}\)       (12)

\(\begin{align}R^{2}=\frac{\sum_{i=1}^{N}\left|Z_{i}^{*}-\bar{Z}_{i}\right|^{2}}{\sum_{i=1}^{N}\left|Z_{i}-\bar{Z}_{i}\right|^{2}}\end{align}\)       (13)

Where, Z*i is a correspondence of LST model, Zi is the LST observation result, \(\begin{align}\bar{Z}_{i}\end{align}\) is the average LST observation values, and N is the number of sample.

4. Results

4.1. Proccessing Data

Topographic correction is integrated into the Lambertian model and constitutes a preprocessing step in remote sensing image data processing. This stage is part of the overall correction process in remote sensing data processing. In topographic correction, it is crucial to consider using elevation data from SRTM imagery. This choice affects the azimuthal direction of slopes (aspect) and the steepness of the study area. These conditions indirectly influence the illumination results in the implementation of the Lambertian model. Additionally, the entire process involves cloud masking, as cloud cover can significantly affect the identification of LST when applying the Lambertian model.

Fig. 2 illustrates the illumination results across the entire study area derived from aspect and slope. Baluran National Park area indicates that only mountainous regions with slopes greater than 15° and aspects facing away from the sun are scattered minimally in the northwest, potentially leading to more shadow appearances compared to other areas. Conversely, in the Wurung crater area, slopes predominantly exceed 15° with aspects facing away from the sun predominantly in azimuth slopes > 150°. In contrast, Mount Bromo features flatland areas centered around the Bromo crater with extensive sandy soil structures and slopes ranging from 1 to 8°. Additionally, the dominant aspect directions of these slopes range from 150 to 250°. Along the coastal region of Tuban Regency, there are notable differences compared to other areas, with slopes reaching up to 24° and aspects spreading to the north and west, characterized by undulating hilly terrain.

Fig. 2. Processing data of Lambertian model: (1) Aspect data, (2) Slope data, and (3) Illumination data (IL); (a) Situbondo regency, (b) Bondowoso regency, (c) Probolinggo regency, and (d) Tuban regency.

4.2. Lambertian Model

The application of the Lambertian model focuses on topographic correction using the SCS and Cosine Correction methods. Topographic correction is performed on the thermal band, red band, and near-infrared bands of Landsat 8 and Landsat 9 OLI/TIRS imagery. Additionally, this study will compare these results with those obtained without applying the Lambertian model to assess the extent of differences in LST estimates.

In Fig. 3, the thermal band without the Lambertian model exhibits unclear visibility trends, particularly on slopes and edges of hills or mountains. In contrast, the thermal band employing the Lambertian model through the SCS and Cosine Correction methods distinctly differentiates mountainous and hilly regions, clearly delineating their slopes and edges. However, the results for flatland areas remain relatively consistent whether the Lambertian model is applied or not. The application of the Lambertian model to the red and near-infrared bands does not visually alter or sharpen the imagery but is evident in statistical metrics. Understanding these conditions demonstrates that the Lambertian model effectively impacts processing outcomes for NDVI and Brightness Temperature (BT).

Fig. 3. Lambertian model result: (1) Without Lambertian model, (2) Lambertian model of cosine correction, and (3) Lambertian model of SCS; (a) Situbondo regency, (b) Bondowoso regency, (c) Probolinggo regency, and (d) Tuban regency.

The implementation of the Lambertian model on NDVI and BT using the SCS and Cosine Correction methods results in differences in image processing outcomes compared to not applying the Lambertian model. The NDVI values were adjusted to account forthe influence of the Lambertian model, namely in hilly and mountainous regions (Situbondo, Bondowoso, and Probolinggo regencies) (Fig. 4). The corrected results predominantly showed hill ridges and valleys, since the vegetation in these areas typically undergoes leaf loss during the dry season. Conversely, the flat landscape in Tuban regency has a strong reflection intensity on its land surface. As a result, areas without clustered vegetation would be identified as having lower vegetation coverage. The result differed from the without Lambertian model of NDVI, indicating that there were distinct coverage regions in nearly all areas, ranging from plains to mountainous areas. Moreover, the difference between SCS and Cosine Correction became noticeable on the elevated areas such as hills and mountains, where the vegetation coverage was more noticeable with Cosine Correction compared to SCS.

Fig. 4. NDVI result: (1) Without Lambertian model, (2) Lambertian model of cosine correction, and (3) Lambertian model of SCS; (a) Situbondo regency, (b) Bondowoso regency, (c) Probolinggo regency, and (d) Tuban regency.

The NDVI was used as the input data for estimating the emissivity value in measuring the LST. The Pv was derived from the NDVI, which quantifies the extent of plant cover on the land. The pixel value used was based on the research conducted by Sobrino et al. (2003, 2004) and Sobrino and Raissouni (2000). They assigned a value of 0.5 to represent vegetation coverage and a value of 0.2 to represent non-vegetation or land reflection. The LSE values for bands 10 and 11 exhibited a negligible difference of 0.001 across all conditions, whether corrected or uncorrected for topography. The dominant LSE value for all research sites was 0.98 for emissivity, which aligns with the value reported by Kuenzer and Dech (2013) and Sabins (1996). This value was observed for both vegetation and water.

In contrast to NDVI, the application of the Lambertian model to BT is illustrated in Table 3, where results from thermal channels band 10 and band 11 tend to show increases(overestimate). These increases in BT values are uneven across the entire region but vary depending on the location, from plains to mountains. The BT error average (K) in Situbondo and Tuban regencies was the lowest, with a value of ±5 K, in comparison to Bondowoso and Probolinggo, which had an error average of ±10 K. This comparison was based on the Lambertian model topographic correction results between SCS and Cosine Correction methods.

Table 3. Brightness temperature comparison between Lambertian model and without Lambertian model

Furthermore, the comparison of without Lambertian model and Lambertian model results between SCS and Cosine Correction revealed that Cosine Correction exhibited an average BT error (K) that was twice as large as that of SCS for the without Lambertian model BT.

4.3. Lambertian Model on LST

The comparison between results from the Lambertian model and those without the Lambertian model in LST shows significant differences. Fig. 5 presents the spatial LST distribution using the algorithm from Skokovic et al. (2014). The without Lambertian model of LST result showed minimal variation in the physiography between the hilly and mountainous terrain (Situbondo, Bondowoso, and Probolinggo) and the plains terrain (Tuban). However, the application of Lambertian model LST using SCS and Cosine Correction methods was particularly effective in hilly and mountainous areas, such as Situbondo, Bondowoso, and Probolinggo, for accurately distinguishing between high and low areas. Furthermore, the analysis of LST values in all Lambertian model areas using SCS and Cosine correction methods proved that the correction for areas directly exposed to sunlight led to a greater increase in temperature compared to areas covered by shade, as shown in Table 4. The LST results with the Lambertian model exhibit an average difference of 6 K, except for the lowland areas (Tuban and Situbondo), which show the lowest difference of approximately ±2 K. While, the comparison of the Lambertian model of LST values using SCS and Cosine Correction revealed that Tuban had the highest average error of 22 K, followed by Situbondo with an error range of 15–21 K, Bondowoso with an error range of 12–16 K, and finally, Situbondo with the lowest average error of 7–9 K. These values were compared to the without Lambertian model values.

Fig. 5. LST of MCMSko result: (1) Without Lambertian model, (2) Lambertian model of cosine correction, and (3) Lambertian model of SCS; (a) Situbondo regency, (b) Bondowoso regency, (c) Probolinggo regency, and (d) Tuban regency.

Table 4. Accuracy of LST comparison with LST L2 product of Landsat

The various LST values depicted in Fig. 5 showed a considerable occurrence in Tuban and Probolinggo regencies, particularly in the area consisting of sea sand. These LST values had an average temperature exceeding 350K. Meanwhile, Situbondo and Bondowoso regencies exhibited an LST distributionin predominantly shaded areas with an average temperature below 300 K, whereas the areas immediately exposed to the sun had temperatures ranging from 310 to 325 K. Hence, it is imperative to extensively validate the results of Lambertian model and without Lambertian model LST using SCS and Cosine Correction. This validation should be based on the Level 2 data of Landsat LST in order to ascertain the difference and error of the obtained results.

4.4. Physiographic Differences in LST

Fig. 6 illustrates the relationship between the various heights of each area and the corresponding LST value. The data illustrates an inverse relationship between elevation and temperature, indicating that as the terrain increases in height, the temperature decreases. This condition applied to both Lambertian model and without Lambertian model data. This relationship exhibits a high value with an average R2 of 0.992. However, the Lambertian model of LST displayed distinct LST values at various altitudes in all areas. The Lambertian model of LST result in Tuban regency (Fig. 6d)revealed a comparable variation. In Bondowoso (Fig. 6b), a similar situation occurred where the lowest temperature was observed at elevations above 1,000 m, while the without Lambertian model of LST in Tuban regency had the highest temperature. In areas with an elevation greater than 1,500 m, such as Probolinggo (Fig. 6c), the LST encountered a notable reduction. In contrast, Situbondo (Fig. 6a) showed LST values within a similar range for both corrected and uncorrected values across all locations with a relatively reasonable height. These findings demonstrated that the topographic correction had been fully implemented in the area with a physiography similar to that of the Situbondo regency. Consequently, the error average was largest in the plains areas when implementing topographic correction, especially in areas with fewer diverse physiographic surfaces.

Fig. 6. Coefficient of determination (R2) between LST and elevation: (a) Situbondo regency, (b) Bondowoso regency, (c) Probolinggo regency, and (d) Tuban regency.

4.5. Validation

The validation of the LST results, depicted in Table 4, showed the correlation (R2) between the MCMSko method and the LST values obtained from Landsat L2. These results demonstrated a strong correlation, with an average R2 value of 0.99, between the Lambertian model and without Lambertian model data. However, the LST in Probolinggo, without Lambertian model, had the lowest RSME of 1.07. The lowest RSME in Situbondo was 0.54 and 0.72. Both Lambertian model and without Lambertian model also indicated that Tuban had the largest RSME, with values of 2.67 and 3.53. For all temperature range samples, the without Lambertian model of LST tended to underestimate the value compared to the LST of the Landsat L2 product, based on the 1 × 1 plot (Fig. 7). Additionally, a comparison was made between field measurements and the results of LST processing (see Table 4). The results indicate a discrepancy between LST derived using the Lambertian model and that obtained without the Lambertian model. In areas predominantly characterized by hills and mountains (such as Bondowoso and Probolinggo Districts), the difference between the LST measured in the field using the Lambertian model and the field measurements was the smallest, with an error (K) of less than 6 K. Conversely, in regions with a physiographic condition of plains, the discrepancy was the highest, exceeding 26 K with the Lambertian model. However, in the plains areas, the smallest discrepancy between field measurements and LST without the Lambertian model was less than 5.5 K.

Fig. 7. Plot 1 x 1 validation: (1)Without Lambertian model, (2)Lambertian model of cosine correction, and (3)Lambertian model of SCS; (a) Situbondo regency, (b) Bondowoso regency, (c) Probolinggo regency, and (d) Tuban regency.

The discrepancy between the processed LST and field measurements of LST may be attributed to the limitations of measurement areas, which are typically confined. This contrasts with imagery where the surface temperature represents the entire scene. Nugraha et al. (2024) indicate that field temperature measurements are influenced by both the method and location of surface temperature acquisition. Furthermore, the heterogeneity and homogeneity of the sample objects used in surface temperature measurements contribute to variations, resulting in either an increase or decrease in field temperature measurements.

The findings demonstrated that the Lambertian model of LST values in Probolinggo and Tuban, obtained using Cosine Correction, were consistent with the validation results. However, the results varied when comparing the corrected LST validation of SCS and Cosine Correction in Bondowoso and Situbondo. The average temperature below300–310 tended to be overestimated, while temperatures over 310 K tend to be underestimated. In contrast to Tuban and Situbondo regencies, the SCS results showed inconsistencies, with the lower temperature range being consistently underestimated and the higher range consistently overstated.

5. Discussion

The SCS and Cosine Correction methods for thermal bands generated different results when applied to areas with close and shaded slopes. The SCS algorithm yielded more accurate results for areas with shadow coverage than those with steep terrain, as demonstrated in Fig. 3. In addition, areas directly exposed to sunlight exhibited distinct variations, and the grayish gradation showed that SCS was capable of producing gradation more effectively than cosine correction. Hence, the SCS result aligns with the findings reported by Kane et al. (2008), Soenen et al. (2005), Sola et al. (2016), Vanonckelen et al. (2013), and Yin et al. (2018), which specifically examined the combination of the slope and incident ray, instead of illumination as in Cosine Correction.

The Lambertian model has a pronounced statistical impact on the thermal band, resulting in a change in radiant value. This effect is visually demonstrated in Fig. 3. However, implementing a Lambertian model for the red and infrared bands only impacts the statistical analysis rather than the visual representation. The impacted area was limited to areas with shadow coverage, such as hilltops, slopes, or hill ridges. Hence, the use of the NDVI is crucial for assessing vegetation in diverse regions. In this case, the use of the Lambertian model supports the research carried out by Adhikari et al. (2016), Buchner et al. (2020), Jasrotia et al. (2022), and Moreira and Valeriano (2014), who demonstrated that topographic correction could enhance the visibility of vegetation in shadowed areas. This process has implications for obtaining fractional vegetation and emissivity. So as to generate fractional vegetation based on the heterogeneous and homogeneous conditions of the research areas, it is necessary to modify the lowest standard for land visibility and the maximum standard for vegetation visibility in the NDVI calculation. Nugraha et al. (2024) and Sobrino et al. (2004, 2008) argue that restricting a diverse region will result in a consistent level of emissivity for items that are not as prominent. Hence, further research is necessary to investigate the fluctuation in vegetation index and establishing a threshold value for achieving optimal emissivity to maximize the existing method combination. This suggestion is crucial for taking into account the significant impact of emissivity on the LST value.

Lambertian model to LST can lead to substantial variations across all physiographic areas (Fig. 5). In addition, the homogeneity and heterogeneity of land covers also have an impact on the LST value. The LST measurement in Tuban showed a significant difference when compared to other areas. These findings demonstratedthattheuseoftopographiccorrectioninpredominantly plains terrains will greatly enhance the LST value. However, places with visible vegetation did not exhibit a substantial impact. Similar conditions were present in Probolinggo, specifically in the sea sand of Bromo and Wurung crater. The temperature rise was substantial at elevations of 1000 meters or above. The variation in the physical features of the land had a significant impact on the correction of the land’s topography. This correction was most effectively achieved using the MCMSko method, which allows the final LST value to closely match the LST obtained using Landsat L2.

However, Nugraha et al. (2024) reported that validating the LST over a large area is difficult. Therefore, using a pixel condition is more appropriate. As a result, pixel validation results will be adopted as a standard for all other larger areas. When using MCMSko for topographic correction to achieve the LST, it is important to consider the variation of land physiography, land coverage, and geological situation. These three factors have influenced the LST value in various areas. Furthermore, it is necessary to explore the non-Lambertian topographic correction as an alternative option for further investigation. This method involves the use of a C value to establish a correlation between the illumination and radiance/reflectance of the remote sensing image (Gupta and Shukla, 2020; Ma et al., 2021; Vanonckelen et al., 2013).

Physiographic variations associated with changes in land elevation and LST value proved an inverse relationship, where higher elevations correspond to lower LST values. However, the LST range at a similar altitude does not necessarily have identical values for every area. The variations depend on the humidity, temperature, and vegetation coverage as the external factors. The internal factors, such as topographic correction, emissivity, and LST methods, also contribute to this issue. Furthermore, it is worth noting that the majority of the lands in the research areas, i.e., Situbondo and Tuban, with an elevation of 100 meters or less, were exclusively used for cultivating rice. In contrast, the areas with an elevation of 800 m or higher in Situbondo, Bondowoso, and Probolinggo consist mostly of woods with moderate to dense vegetation, apart from the Wurung and Bromo craters. Thus, it can maintain the object’s temperature in line with that of the surrounding environment, which is relatively low. Despite applying topographic correction, the LST trend still decreased, as seen in Fig. 6.

Meanwhile, the validation results of applying the Lambertian model indicate that the SCS and Cosine Correction methods tend to provide a more balanced L2 product compared to results without applying the Lambertian model in hilly and mountainous regions. Conversely, in flat plains, LST results without the Lambertian model tend to be more optimal compared to those with SCS and Cosine Correction. Understanding this, the Lambertian model directly influences outcomes in surface temperature estimation in areas characterized by hilly to mountainous physiography. In contrast, the validation results comparing the Lambertian model LST and non-Lambertian model LST with field measurements indicate that physiography affects the magnitude of the error (K). However, LST in hilly and mountainous areas using the Lambertian model shows the smallest discrepancy compared to the LST obtained without the Lambertian model in flatland regions.

Based on the analysis, it is evident that LST is capable of providing information on surface temperature dynamics and potential phenomena in specific physiographic contexts. Furthermore, the application and utility of LST can influence analytical outcomes, as demonstrated by Nugraha and Kurniawan (2024), who considered temperature as a parameter in vegetation analysis(Forest Canopy Density). Their study revealed significant differences in areas of low vegetation and grassland. Additionally, LST is useful for providing information on drought potential, Surface Urban Heat Island (SUHI) effects, and vegetation health, which can be correlated with land cover or other meteorological phenomena (Nugraha et al., 2022; Sobrino et al., 2013; Kogan, 2002; Sultana and Satyanarayana, 2020). Therefore, LST is essential in various research fields as a key parameter, impacting the selection of appropriate and optimal LST methods for study areas, depending on the necessity of applying the Lambertian model.

6. Conclusions

The implementation of topographic correction using the empirical Lambertian model, specifically the SCS and Cosine Correction, indicated a notable change in the thermal bands across several estimating aspects to produce emissivity and LST value. The Lambertian model had no substantial impact on the estimation of emissivity; it merely enhanced the spatial information for identifying vegetation coverage. Consequently, the measurement of emissivity value using NDVITHM had a negligible impact, regardless of whether it was corrected or uncorrected. In addition, the measurement must consider the heterogeneity and homogeneity of the studied area. Contrary to the calculation of LST value, the Lambertian model and without Lambertian model values exhibited substantial differences in range, which had an impact on the spatial distribution. Hence, the Lambertian model should consider the physiography of a particular area (especially the flat plains), as this method may lead to an overestimation of the LST value. In areas characterized by abundant hills and mountainous terrains, the LST distribution generally follows a regular pattern, even though there may be some variations in certain locations due to specific circumstances or events. Besides, the optimal application of the Lambertian model occurs when it encompasses a broad spectrum of places and all types of LST. Thus, it is highly recommended to conduct further research exploring the non-Lambertian model for measuring the LST and emissivity.

Acknowledgments

This research was supported by the 2024 Lecturer Independent Research Program funded by the Faculty of Geography at Universitas Gadjah Mada, Indonesia, assigned to Muhammad Kamal(Grant No. 130/UN1/GE/KPT/2024). The authors express their gratitude to the Doctoral Program in Geographical Sciences at the Faculty of Geography, Universitas Gadjah Mada for their provision of research facilities.

Conflict of Interest

No potential conflict of interest relevant to this article was reported.

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