Comparison between Physical and Empirical Methods for Simulating Surface Brightness Temperature Time Series

Abstract

Land surface temperature (LST) is a vital parameter in the surface energy budget and water cycle. One of the most important foundations for LST studies is a theory to understand how to model LST with various influencing factors, such as canopy structure, solar radiation, and atmospheric conditions. Both physical-based and empirical methods have been widely applied. However, few studies have compared these two categories of methods. In this paper, a physical-based method, soil canopy observation of photochemistry and energy fluxes (SCOPE), and two empirical methods, random forest (RF) and long short-term memory (LSTM), were selected as representatives for comparison. Based on a series of measurements from meteorological stations in the Heihe River Basin, these methods were evaluated in different dimensions, i.e., the difference within the same surface type, between different years, and between different climate types. The comparison results indicate a relatively stable performance of SCOPE with a root mean square error (RMSE) of approximately 2.0 K regardless of surface types and years but requires many inputs and a high computational cost. The empirical methods performed relatively well in dealing with cases either within the same surface type or changes in temporal scales individually, with an RMSE of approximately 1.50 K, yet became less compatible in regard to different climate types. Although the overall accuracy is not as stable as that of the physical method, it has the advantages of fast calculation speed and little consideration of the internal structure of the model.

1. Introduction

Land surface temperature (LST) has been considered a key variable in many fields, including agriculture, hydrology, and meteorology [1], and many studies have reported its applicability to vegetation monitoring and drought prediction at varying scales based on remotely sensed techniques [2,3,4,5]. Although LST products have been obtained from remote sensing at various spatial and temporal resolutions, it is still difficult to meet the requirements of applications such as high spatial–temporal resolution, eliminating the terrain and angular effects, and all-weather conditions. In recent years, there has been increasing interest in improving LST products [6,7,8]. One of the most important foundations for the studies above is a theory to understand how to model LST with various influencing factors, such as canopy structure, solar radiation, and atmospheric conditions. In addition, a study on the simulation of LST can help scale-up point LST measurements to match the remote sensing LST resolution in validation [9].

Many physical-based soil–vegetation–atmosphere transfer (SVAT) models have been proposed for simulating LSTs by combining radiative transfer and energy balance theories, such as one-dimensional CUPID [10] and soil canopy observations of photochemistry and energy flux (SCOPE) [11] models, three-dimensional discrete anisotropic radiative transfer-energy balance (DART-EB), and thermal radiosity-graphics combined model-energy balance (TRGM-EB) [12,13,14]. This method has received considerable critical attention associated with remote sensing LST products [9,15,16,17]. Because of the straightforward modeling framework and low computational cost, several researchers have focused on empirical methods such as machine learning and even deep learning [18,19]. Based on the regression relationships between LST and auxiliary data, such as the reflectance of visible and near-infrared (VNIR) bands and soil water content, the spatial and temporal adaptive reflectance fusion model (STARFM) and spatiotemporal adaptive data fusion algorithm for temperature mapping (SADFAT) methods were proposed [20,21,22]. The random forest (RF) regression method was applied for applications such as the normalization of the temporal and terrain effects of LST [6,18] and LST regional downscaling [23]. Relative to natural surfaces, the downscaling, unmixing and time evolution of surface temperature in urban surfaces are more complex. Stewart. et al. [24,25] reviewed many physical and empirical methods for urban surfaces.

Currently, both physical-based and empirical methods have been widely adopted in the improvement of LST products from remote sensing data. There are significant differences between these two categories of methods because of their physical and empirical assumptions. However, few studies focus on the comparison of these methods. Although some evaluation studies for individual methods have been performed when proposed or applied [18,26], it is still a challenging task to select a suitable method for a specific case because most of these evaluated results were obtained based on different conditions and different datasets. Moreover, different methods have advantages and disadvantages in theory. The physical-based model is always considered to be an accurate method to show the exchange of radiative and heat fluxes between the surface and atmosphere, whereas its application can be limited due to modeling complexity and slow computing speed [13]. In contrast, stability is one of the most frequently stated problems with applications of empirical methods. The direct application of an empirical method for surface quantitative studies remains challenging. No one method can be applied to all situations yet. Therefore, a comparison study is necessary for people to select a suitable method for a specific application. Moreover, a comparison study can also aid in developing a better understanding of these methods for the improvement or exploration of a combined strategy to make full use of these methods by addressing their shortcomings.

Therefore, the main aim of this study is to investigate the differences between different methods in simulating surface brightness temperatures. The physical-based method SCOPE, the machine learning method RF regression, and the recurrent neural network (RNN) method long short-term memory (LSTM) were selected for comparison. The data sources used for evaluation were obtained from measurements in the Heihe River Basin, in which a comprehensive remote sensing experiment, Heihe watershed allied telemetry experimental research (HiWATER), was performed [27,28]. These three methods were synchronously evaluated by TOC thermal emissions, and the comparison was performed from different perspectives, i.e., within the same surface type, between different years, and between different climate types. The outline of this paper is as follows. Section 2 begins with the descriptions of study area and method inputs. Section 3 describes different methods; then, Section 4 provides the evaluation results of these three methods using field measurements. The influencing factors of these three methods are discussed in Section 5. Finally, Section 6 provides a summary and stresses some perspectives.

2. Dataset

2.1. Study Area and Experiment

In this study, the Heihe River Basin, located in northwest China (Gansu Province), was selected as the study area, which has cold and arid climate conditions, as shown in Figure 1a. The data source used in this study was the Heihe watershed allied telemetry experimental research (HiWATER) experiment, in which three key experimental areas were selected to conduct intensive and long-term observations, including the cold region in the upstream mountainous area, the artificial oasis in the midstream area, and the natural oasis in the downstream area; the annual air temperatures of these key experimental areas were approximately 0.5 °C, 7.0 °C and 9.9 °C, respectively. Because of the difference in climate, the surface type in the Heihe River Basin displays a significant variation. The valley bush, alpine meadow, and swamp are common in the cold landscape upstream, in which the elevation varies from approximately 2640 m to 5000 m. In the middle reaches of the Heihe River Basin, oasis agriculture with typical vegetation types of corn, wheat, and vegetables has been developed. Because of the arid and semiarid conditions downstream, desert and barren soil occupy a large area, and Populus euphratica and Tamarix are mainly distributed in small areas along the Heihe River.

2.2. Meteorological Station Data

In this study, the data from meteorological stations during the HiWATER experiment were used to drive and evaluate the SCOPE and RF regression methods. The HiWATER intensive observation period fieldwork started in May 2012 in the middle stream. In 2012, many stations were set up in 5 km × 5 km key experimental areas in the artificial oasis to obtain intensive water and energy fluxes on a heterogeneous surface. After 2012, most of these stations were withdrawn in addition to the DM superstation. Some of the stations were moved to key experimental areas upstream and downstream to obtain long-term measurements. Three superstations were set up in the study area, one for each key experimental area, namely, AR, DM, and SDQ for upstream, midstream, and downstream, respectively. At the ordinary automatic meteorological station (AMS), the meteorological conditions, i.e., air temperature, air humidity, soil temperature and moisture, wind speed, and four-component radiation (downward/upward and shortwave/longwave radiations), were automatically obtained at a height of 5 m/10 m above ground level. The superstations were outfitted with an EC system, a Bowen ratio energy balance system, and a lysimeter (optional) to measure fluxes at multiple scales, and some variables, such as air temperature, wind speed, and wind direction, were also measured at other heights, such as 3 m, 15 m, and 20 m [29]. An intelligent monitoring system was developed for the ground-based observation networks, and for details on this dataset, refer to [30]. As mentioned above, the comparison between the SCOPE and empirical methods was performed in three dimensions by using different data. The data measured from ten meteorological stations in artificial oases during the 2012 intensive experiment were selected for comparison within the same surface type. These ten meteorological stations all corresponded to corn canopies similar to Figure 1c. The comparison between the three methods for different climate types/surface types was based on three superstations corresponding to Tamarix, corn, and alpine meadow canopies, as shown in Figure 1b–d, respectively. The performances of the three methods during different years were evaluated using 5-year measurements from 2013 to 2017 from three superstations. The information of these stations can be found in

Table 1. Stations of the evaluated dataset.

Stations	Latitude (°)/ Longitude (°)	Altitude (m)	Year	DOY	Land Cover
M1	38.8869/ 100.3541	1559	2012	150–250	Corn
M2	38.8905/ 100.3763	1543	2012	150–250	Corn
M3	38.8757/ 100.3507	1567	2012	150–250	Corn
M4	38.8767/ 100.3652	1556	2012	150–250	Corn
M5	38.8725/ 100.3765	1550	2012	150–250	Corn
M6	38.8757/ 100.3957	1534	2012	150–250	Corn
M7	38.8652/ 100.3663	1559	2012	150–250	Corn
M8	38.8607/ 100.3785	1550	2012	150–250	Corn
M9	38.8587/ 100.331	1570	2012	150–250	Corn
M10	38.8555/ 100.3722	1556	2012	150–250	Corn
DM	38.8555/ 100.3722	1556	2013–2017	150–250	Corn
AR	38.0473/ 100.4643	3033	2013–2017	150–250	Grass/ alpine meadow
SDQ	42.0012/ 101.1374	873	2013–2017	150–250	Tamarix

. Note that M10 and DM in Table 1 correspond to the same station. According to the data policy of the HiWATER, the experimental data can be timely and adequately utilized. We downloaded the dataset from the National Tibetan Plateau Data Center with the websites of http://data.tpdc.ac.cn and http://data.tpdc.ac.cn/en/ (accessed on 1 October 2019), corresponding to Chinese and English versions, respectively.

3. Methods

3.1. Physical-Based SCOPE Method

The SCOPE model is an SVAT model based on radiative transfer, micrometeorology, and plant physiology theories, which aims to provide combined simulations of directional TOC or top-of-atmosphere (TOA) VNIR reflectance, emitted brightness temperatures, sun-induced fluorescence signals, and energy, water, and carbon dioxide (CO₂) fluxes. The SCOPE model has undergone a series of upgrades since it was proposed in [11], such as replacing the PROSPECT model with the FLUSPECT model, providing several leaf-level fluorescence and photosynthesis models for alternatives, and improving calculation efficiency [31,32]. To date, the SCOPE model has been widely used in surface forward simulations and inversions using remote sensing data, particularly for the applications of vegetation monitoring [17,32,33]. SCOPE version 1.70 was used in this study.

This study of the SCOPE model aims at a one-dimensional soil–vegetation system without considering tree trunks or branches. The basic units of soil are the sunlit and shaded areas. In addition to the solar illumination effect, the basic unit of leaves is also related to the height and inclination angle. The leaves are divided into 60 layers, and the angular distributions of shaded and sunlit leaves are quantified by using 13 zenithal angles and 13 zenithal $\times$ 36 azimuthal angles, respectively, for each layer. The modules related to surface thermal emissions mainly include radiative transfer and energy balance, in which the temperature of each unit is first calculated and then the TOC/TOA emission at the canopy scale is simulated. The radiative transfer process of SCOPE is based on the unified four-stream scattering by the arbitrarily inclined leaves (4SAIL) model, which is separately performed in the VNIR and TIR domains corresponding to spectral ranges varying from 400 nm to 2400 nm and from 2500 nm to 50 μm, respectively. Net radiation and unit temperatures are the bridge between the radiative transfer and energy balance modules. Based on preset temperatures, the net radiation of each unit can be calculated after a series of radiative transfer simulations as follows:

$$R_{n,l}\left(x,{\theta }_l,{\phi }_l\right)=\left[\left|f_s\right|·E_{sun}+E^{-}\left(x\right)+E^{+}\left(x\right)-2H_l\left(x,{\theta }_l,{\phi }_l\right)\right]·\left(1-{\rho }_l-{\tau }_l\right)$$

(1)

$$R_{n,s}\left(x,{\theta }_l,{\phi }_l\right)=\left[\left|f_s\right|·E_{sun}+E^{-}\left(x\right)+E^{+}\left(x\right)-2H_l\left(x,{\theta }_l,{\phi }_l\right)\right]·\left(1-{\rho }_l-{\tau }_l\right)$$

(2)

where $R_{n,l}$ and $R_{n,s}$ represent the net radiation of leaf and soil units, respectively; $x$ represents the relative depth of a leaf in the canopy; ${\theta }_l$ and ${\phi }_l$ represent leaf zenithal and azimuthal angles, respectively; $f_s$ represents the leaf area projection factor in the direction of the sun; $E_{sun}$ represents the direct solar irradiance in the canopy; $E^{-}$ and $E^{+}$ represent the downward and upward irradiances, respectively; $H_l$ and $H_s$ represent the blackbody emissions by leaves and soil, respectively; and $\rho$ and $\tau$ represent the reflectance and transmittance, respectively. Subsequently, the sensible and latent heat fluxes and surface heat flux can be calculated based on the energy balance method as follows:

$$R_n=H+\lambda E+G$$

(3)

$$H={\rho }_a{c}_p\frac{T_u-T_a}{r_a}$$

(4)

$$\lambda E=\lambda \frac{q_s\left(T_u\right)-q_a}{r_a+r_s}$$

(5)

$$T_s\left(t+\mathsf{\Delta }t\right)-T_s\left(t\right)=\frac{\sqrt{2\omega }}{\mathsf{\Gamma }}·\mathsf{\Delta }t·G\left(t\right)-\omega ·\mathsf{\Delta }t\left[T_s\left(t\right)-\overline{T_s}\right]$$

(6)

where $H$ and $\lambda E$ represent sensible and latent heat fluxes, respectively; ${\rho }_a$ represents the air density; $c_p$ represents the air heat capacity; $\lambda$ represents the evaporation heat of water; $T_u$ represents the surface temperature of a unit; $T_a$ represents the air temperature; $q_s$ represents the humidity of a unit; $q_a$ represents the humidity above the canopy; $r_a$ and $r_s$ represent the aerodynamic and surface/stomal resistance resistances, respectively;$G$ represents the surface heat flux, which is only considered for soil; $T_s$ represents soil temperature; $t$ represents the time node; $\mathsf{\Delta }t$ represents the time interval between two successive simulations; $\overline{T_s}$ represents the average annual temperature of soil; $\omega$ represents the frequency of the diurnal cycle, and $\mathsf{\Gamma }$ represents the thermal inertia of the soil.

The calculated heat fluxes from preset or previous unit temperatures may not be close to the energy balance Equation (3); therefore, the temperatures of each unit need to be optimized based on the energy balance constraint. In this process, the aerodynamic and surface/stomal resistances play important roles in determining the distribution of the radiation to fluxes according to equations analogous to Ohm’s Law. In SCOPE, the aerodynamic resistance is calculated based on [34]. In addition to the Ball–Berry model, leaf stomal resistance can also be calculated using the leaf-level photosynthesis and fluorescence model Von Caemmerer-MD12 [32]. Based on an iteration between radiative transfer and energy balance modules, the unit temperatures are calculated step by step until the local thermodynamic equilibrium state of all units is realized. Notably, the VNIR radiative transfer process must be executed once outside the iteration loops because of its independence of surface thermal information. The first guess for the input of unit temperature is based on air temperature. In this study, the SCOPE model was run in MATLAB code with MATLAB software R2016b. The SCOPE model can be obtained on GitHub with a URL of https://github.com/Christiaanvandertol/SCOPE (accessed on 1 January 2019).

3.2. Random Forest Regression Method

The RF method has become popular in the remote sensing community because of its applications in the fields of classification and regression [35]. An RF can produce multiple decision trees to use randomly selected subsets of training samples and variables with features of high variation and low bias. The error estimate of the RF is known as an out-of-bag error. Each tree in RF can independently grow to its maximum size based on approximately two-thirds of the samples, which refers to in-bag samples, and the remaining one-third, which are referred to as out-of-bag samples, are used to cross-validate the fitting performance. Then, final classification or regression results can be obtained by averaging all possible results from each decision tree. Therefore, RF regression can be considered a nonlinear statistical ensemble bagging method. In the RF regression, several parameters should be set in advance, including but not limited to the number of decision trees ($n_{tree}$), the number of variables ($f_{max}$) to be selected and tested for the best split when growing trees, the maximum depth of a decision tree growing ($d_{max}$), the minimum number of samples required to split an internal node ($s_{min}$), and the minimum number of samples required to be at a leaf node ($l_{min}$). In this study, a grid search algorithm was used to determine the fittest parameters. In particular, $n_{tree}$ was set to 200 because of low sensitivity and no overfitting problem in the results, $f_{max}$ was set to the square root of the number of input variables, and $d_{max}$, $s_{min}$, and $l_{min}$ were set to 80, 2, and 1, respectively. The RF regression method was used by the Sklearn package version 1.0.2 in Python, which a Python package for automated algorithm configuration of standard machine learning algorithms provided by Scikit-Learn [36].

3.3. Long Short-Term Memory Method

Considerable progress and wide application have been witnessed for deep learning techniques in recent years. According to the difference in cell structure and input data characteristics, deep learning can be classified into several types, including recurrent neural networks (RNNs), convolutional neural networks (CNNs) and deep neural networks (DNNs), etc., each of which has been widely adopted to solve regression and classification problems. In regard to dealing with sequential data such as text, audio, and video, RNNs are dominant and extremely powerful [37]. A typical RNN architecture is a cyclic connection, which is very suitable for processing time-series datasets and can maintain the dependency relationship of inputs. However, RNNs suffer from both exploding and vanishing gradients caused by RNN’s iterative nature [38]. The long short-term memory (LSTM) emerges as the times require [39], which successfully addresses the vanishing gradients problem by reparameterizing. The core concept of LSTM consists of memory cells whose state can be updated by the previous cells’ states along with the current input with feedback connections and nonlinear gating units that adjust the information between cells. A typical LSTM with a forgetting gate can be mathematically expressed as follows:

$$\begin{array}{l}{f}_t=\sigma (W_{fh}{h}_{t-1}+W_{fx}{x}_t+b_f),\\ i_t=\sigma (W_{ih}{h}_{t-1}+W_{ix}{x}_t+b_i),\\ {\tilde{c}}_t=\tanh (W_{\tilde{c}h}{h}_{t-1}+W_{\tilde{c}x}{x}_t+b_{\tilde{c}}),\\ c_t=f_t\cdot c_{t-1}+i_t\cdot {\tilde{c}}_t,\\ o_t=\sigma (W_{oh}{h}_{t-1}+W_{ox}{x}_t+b_o),\\ h_t=o_t\cdot \tanh (c_t).\end{array}$$

(7)

where$x_t,h_t,$and $c_t$ represent the input, the recurrent information, and the cell state at time t, respectively;$f_t$,$i_t$, and $o_t$ represent the forget gate, the input gate, and the output gate, respectively.$W_{fh}$, $W_{fx}$, $W_{ih}$, $W_{ix}$, $W_{\tilde{c}h}$, $W_{\tilde{c}x}$, $W_{oh}$, and $W_{ox}$ are the weights, and $b_f$, $b_i$, $b_{\tilde{c}},$ and $b_o$ are the biases. Figure 2 shows the LSTM architecture with a forget gate.

Ever since LSTM’s proposal, most achievements based on RNNs have been achieved by LSTM, which include handwriting recognition [40,41], audio detection [41,42], video description [43,44], and many other subjects. Numerous variants of LSTM architecture have emerged since its inception, yet none of the variants can improve significantly upon the standard LSTM architecture [45]. Thus, in this study, a stacked LSTM network was realized using the PyTorch package version 1.9.1 in Python. In addition, for the purpose of better adapting to the problem, the LSTM architecture is set to two layers, with 128 cells at each layer. The processed meteorological data have a time resolution of half an hour, so there are 48 pieces of data in a day, and the sequence length is set to 48. For the model to be fully trained but not overfitting, the learning rate is set to 0.001, and the batch size is set to 64. The RF and LSTM methods used have been uploaded to GitHub with a URL of https://github.com/airtteam/lst_simulate (accessed on 9 June 2022).

3.4. Input and Output

During the simulation of the SCOPE model, the scene inputs associated with the physical properties of the inside components and canopy structure are given first, and then the physical processes, including radiative transfer, energy budget, and leaf photosynthesis and fluorescence, are driven by inputs. In SCOPE, the canopy structure mainly corresponded to the leaf area index (LAI), canopy height, and leaf angle distribution function (LADF). In 2012, the LAI values of corn canopies were regularly measured using LAI 2200 instruments around each station during the vegetation growing season. At the same time, canopy height was also measured. For evaluations from 2013 to 2017, because of the lack of field measurements, LAI values at three superstations were obtained from the LAI product MCD15A3H retrieved by Moderate Resolution Imaging Spectroradiometer (MODIS) data [46]. The time interval of this LAI product was four days, and an average value of 3 × 3 pixels corresponding to a 1.5 km × 1.5 km area with the station as the center was adopted for LAI input. The LADF of all vegetation canopies was assumed to follow a spherical pattern with parameters LADFa and LADFb of −0.35 and −0.15, respectively. The physical properties of the components were mainly used to determine the spectral information and photosynthesis capacity. The VNIR reflectance and transmittance of all vegetation leaves were calculated using the FLUSPECT model [47], in which input parameters, such as chlorophyll, dry material, water and senescent material concentrations, and the thickness parameter were set, as shown in

Table 2. Inputs of SCOPE model.

	Corn	Grass	Tamarix
Canopy structure information
LAI	LAI2200/ MCD15A3H	MCD15A3H	MCD15A3H
Canopy Height	0–2.2	0.15	1.5
LADFa	−0.35	−0.35	−0.35
LADFb	−0.15	−0.15	−0.15
Component spectral information
Leaf VNIR spectral	FLUSPECT model
Leaf structure parameter (Ns)	1.518	1.70	1.80
Chlorophyll a and b content (Cab)	58	45	100
Dry matter content (Cm)	0.0036	0.0030	0.0070
Equivalent water thickness (Cw)	0.0131	0.0150	0.0038
Soil VNIR spectral	ASTER spectral library
Leaf broad emissivity	0.975	0.975	0.975
Soil broad emissivity	0.955	0.955	0.955
Component biochemical information
Vegetation Type	C4	C3	C3
Maximum carboxylation capacity (Vcmo)	35	80	80
Ball–Berry slope (m)	4	9	9
Respiration as fraction of Vcmo (Rdparam)	0.025	0.015	0.015
Slope of cold temperature decline (slti)	0.2	0.2	0.2
Slope of high temperature decline (shti)	0.3	0.3	0.3
Temperature below which photosynthesis is lower than half that predicted by Q10 (Thl)	288	281	278
Temperature above which photosynthesis is lower than half that predicted by Q10 (Thh)	313	308	313
Temperature at which respiration is lower than half that predicted by Q10 (Trdm)	328	328	328
Soil surface resistance (rss)	using soil water content [26]
Surface meteorological information
Air temperature (Ta)	Meteorological station
Air humidity (RH)
Wind speed (WS)
Downward shortwave radiation (DS)
Downward longwave radiation (DL)

according to [48,49]. The VNIR reflectance of the soil was from the ASTER spectral library [50]. In the infrared domain (>2.5 $\mathsf{\mu }\mathrm{m}$), the emissivity of leaves and soil were set to 0.975 and 0.955, respectively, according to surface measurements by ABB BOMEM MR304 Fourier transform infrared spectroscopy [51], and their transmittance was assumed to be 0. The default option of leaf-level photosynthesis and the fluorescence function Ball–Berry were used to calculate the leaf stomal resistances of both C3 and C4 vegetation types, in which the recommended biochemical parameters in SCOPE were adopted in this study, as shown in Table 2 [32]. The surface resistance of the soil was calculated using soil moisture according to [26]. The meteorological input in SCOPE included air temperature, wind speed, air humidity, and downward shortwave and longwave radiation, which were directly obtained from meteorological stations during HiWATER.

According to previous studies, different auxiliary data were examined for the normalization and simulation of LSTs, such as vegetation index, surface albedo, surface elevation, and LAI [23,52]. Despite the fact that both RF and LSTM are empirical methods, the link between LST and input influencing factors should also follow physical laws associated with radiative transfer and energy balance processes. Therefore, the main inputs of SCOPE were also adopted for the empirical methods in this study, including LAI, soil moisture, air temperature, wind speed, air humidity, downward shortwave radiation, and longwave radiation, as follows:

$$T_{RF/LSTM}=f\left(LAI, SM,T_a,RH,DS,DL\right)$$

(8)

The performance of the SCOPE RF and LSTM methods was evaluated using measured TOC brightness temperatures. The 101-day measurements from day of the year (DOY) 150 to 250 were used with a time interval of 30 min. The measured TOC brightness temperatures ($T_b$) were calculated using measured downward and upward longwave radiation as follows [53,54]:

$$T_b={\left[\frac{F^{\uparrow }-\left(1-{\epsilon }_b\right)·F^{\downarrow }}{\sigma }\right]}^{1/4}$$

(9)

where $F^{\uparrow }$ and $F^{\downarrow }$ represent upward and downward longwave radiation at the surface, respectively, ${\epsilon }_b$ represents surface broadband emissivity, and $\sigma$ represents Stefan-Bolzman’s constant (5.67 $\times {10}^{-8}{ \mathrm{Wm}}^{-2}{\mathrm{K}}^{-4}$). Since the output of the empirical methods was consistent with its training data, the empirical methods were trained using the measured TOC brightness temperatures in advance. To evaluate the applicability of empirical methods for different cases, the training and test data may be from different stations or during different years. To compare methods more intuitively and clearly, both empirical and physical methods were evaluated by the same index, including RMSE, $R^2$, and bias.

4. Results

In this paper, the comparison was performed from different perspectives. All three methods are evaluated by the same index, including RMSE, $r^2,$ and bias. For the physical method, LST can be simulated as long as the input data are available. The simulated results were tested by measured brightness temperatures under three circumstances, including within different types, between different years, and between different climate types. The latter two case simulation results for SCOPE are merged into Section 4.2.1 for the use of the same data. For empirical methods, because of the lack of training data from measurements or suitable well-trained models, the RF and LSTM regression methods were trained using data from a case and applied to different cases in practical applications, which led to more results. The evaluation of the empirical method was the same as that of the physical method, but to clearly demonstrate its migration ability, it was expressed in the way of a heatmap. The process can be described as shown in Figure 3, in which the training data are set to the color yellow. The computer we used to simulate in this study has an 11th Gen Intel(R) Core(TM) i7-11700 @ 2.50 GHz CPU and an NVIDIA GeForce RTX 3070 GPU.

4.1. Evaluation within the Same Vegetation Type

4.1.1. Evaluation of SCOPE Model

Figure 4a–j displays the scattering plots between the measured and SCOPE-simulated TOC brightness temperatures at stations M1–M10. The results with differences between the simulation and measurement larger than 10 K were identified as bad data and removed. The root mean square errors (RMSE) of these ten stations were all found to be lower than 2.50 K, which indicated a stable and acceptable performance of SCOPE. The good agreement of the SCOPE-simulated results with the measurements is also demonstrated by the coefficients of determination ($R^2$) with values all larger than 0.85. The large discrepancies of some data can be explained by the fact that a systematic overestimation appeared in the SCOPE-simulated results with positive biases at all stations, particularly for the results at station M4 with biases and RMSEs of 1.60 K and 2.50 K, respectively. The overall RMSE, $R^2$, and bias for all SCOPE-simulated results are 2.01 K, 0.91 and 0.79 K, respectively. Figure 4k displays the boxplots of differences between simulated and measured brightness temperatures for each station. The medium and quartiles of these differences also depict the overestimation of the SCOPE-simulated results despite many outliers lower than the minimum range.

4.1.2. Evaluation of the RF Method

Figure 5 displays the RMSEs of the results predicted by the RF regression method at stations M1–M10. The corresponding $R^2$ and bias for these cases are also provided in the Appendix A. Because of the lack of training data from measurements, the RF regression method may be trained using data from a case and applied to different cases in practical applications. Therefore, in addition to an evaluation under good calibration, the RF regression was also trained and tested using data from cross stations. The RF regression method displayed excellent performance with RMSEs lower than 0.46 K and biases lower than 0.01 K when the training and test data were from the same station. When the training and test data were from different stations, the RMSEs increased to approximately 1.68 K for most cases, indicating that the performance of the RF regression method remains remarkable. In addition, the RMSEs reached a maximum when the model trained by other stations was used to verify M4 all above 2.40 K, yet the corresponding biases were still under 1.19 K and $R^2$ were still over 0.83, indicating that the model trained by a single station is practically usable even under the worst circumstances. In addition, the performance of the RF regression method was quite impressive when the model was trained by data from all stations and tested by data from each station with RMSEs lower than 0.52 K, which indicates that adequate training data are an important way to improve the predicted results. In this case, the overall RMSE for all predicted results is 0.30 K with a bias lower than 0.01 K.

4.1.3. Evaluation of the LSTM Method

Figure 6 displays the RMSEs of the results predicted by the LSTM regression method at stations M1–M10. The corresponding $R^2$ and bias for these cases can be found in the Appendix A. Similar to RF regression, LSTM was also trained and tested using data from cross stations. For cases when the training and test data were from the same station, LSTM regression showed a promising result, with RMSEs fluctuating from 0.71 K to 1.99 K and biases between −0.33 K and 0.18 K. Unlike RF regression, the LSTM’s performance remained moderately stable when the training and test data were from distinguishing stations with RMSEs of approximately 1.78 K and $R^2$ of approximately 0.92. Additionally, similar to RF regression, the LSTM showed its worst result only in the cases related to M4, whether being used as a training or test dataset. Furthermore, LSTM presented a gradual increasing trend in terms of performance when the model was trained by data from all stations and tested by data from each station with RMSEs of approximately 1.22 K, which supports the statement that larger data have less influence on LSTM performance than RF regression does. In general, for datasets within the same vegetation type, RF performance slightly overcomes LSTM, while LSTM is more stable and constant; both empirical methods exceed the physical SCOPE model according to RMSE results.

4.2. Evaluation of Different Surface Types during Different Years

4.2.1. Evaluation of SCOPE Model

Figure 7 displays the scatter plot between the SCOPE-simulated and measured TOC brightness temperatures at three superstations during 2013–2017. Figure 7a–e displays the simulation results at the DM station from 2013 to 2017. Low RMSEs (<2.0 K) and large $R^2$ (>0.90) indicate a satisfactory performance of SCOPE, which is similar to the evaluation results in Figure 4. The overall RMSE, $R^2,$ and bias for all simulations at the DM station were 1.61 K, 0.95 and −0.09 K, respectively. No obvious variations in these evaluation results appeared, indicating that the performance of SCOPE was stable regardless of the year. Figure 7f–j displays the simulation results at the AR station from 2013 to 2017. The overall RMSE, $R^2,$ and bias of all simulations at station AR were 2.11 K, 0.95, and −0.18 K, respectively. According to the evaluation results, SCOPE performed slightly worse at the AR station than at the DM station. The worse evaluation result at the AR station can be mainly attributed to some points having large discrepancies (>5.0K). The worst performance of SCOPE appeared at the SDQ station, as shown in Figure 7k–o, with an overall RMSE, $R^2,$ and bias of all simulations of 2.32 K, 0.95, and 0.07 K, respectively. The stable performance of SCOPE during different years was also shown at the AR and SDQ stations because there were no obvious interannual variations in the evaluation results. Figure 7p displays the boxplots of differences between the SCOPE-simulated and measured brightness temperatures. Compared with simulations at the DM and SDQ stations, more outliers appeared at the AR station. The interquartile ranges in the boxplots for the SDQ station were larger than those for other stations, although fewer outliers appeared. The large RMSEs of the predicted results at the SDQ station can be explained by the overestimation of some data with values lower than 290 K and the underestimation of some data with values larger than 310 K. The differences in RMSEs among the three vegetation types were lower than 1.0 K despite the discrepancies that appeared in different ways.

4.2.2. Evaluation of the RF Method

In Figure 5, the RF regression was evaluated by training and testing data from stations within the same surface type. In practical applications, an RF regression method may be trained using data from a given year but applied to another year or trained using data for a given surface type but applied to another surface type. Therefore, the applicability of the RF regression method in both spatial and temporal dimensions is an essential factor. In this study, the RF regression method was evaluated using data from stations in different surface types during different years.

Figure 8 shows the RMSEs of the results predicted by the RF regression method using training and test data from the same stations from 2013 to 2017, and the corresponding $R^2$ and bias results can be found in the Appendix A. Similar to the evaluation results in Figure 5, when the training and test data were from the same station during the same year, low RMSEs with values lower than 0.30 K and biases between −0.01 K and 0.01 K indicate an excellent performance of the RF regression method. The performance of the RF regression method became worse when the test data were from another year at the same station, and the average RMSEs for the predicted results at the DM, AR, and SDQ stations were 1.28 K, 1.00 K, and 1.48 K, respectively, and the corresponding $R^2$ values were 0.96, 0.98, and 0.97. These evaluation results indicate that the applicability of the RF regression method to a temporary scale was acceptable.

Figure 9 shows the RMSEs, $R^2,$ and bias of the predicted results by the RF regression method of five years of full data training at one site to verify the data of another site for all five years from 2013 to 2017. When the trained RF regression was tested using data from stations in another surface type, the RMSEs for the predicted results were all found to increase dramatically at other stations. The RF regression model trained by the SDQ station shows great performance in predicting the DM station from 2013 to 2017, with an overall RMSE of 2.13 K and biases of 0.22 K. In contrast, the model trained by the AR station performs the worst among all six cases when testing the SDQ station with RMSE at approximately 6.17 K with $R^2$ at merely 0.52 and biases as low as −4.60 K, indicating that the prediction results are generally smaller than the actual value. In addition, the DM to SDQ result (trained by the DM station and tested by the SDQ station from 2013 to 2017, similarly hereinafter) does not perform as well as the SDQ to DM result, and the AR to SDQ result does not perform as bad as the SDQ to AR result, indicating that there is no strong similarity or dependence on upstream and downstream meteorological data. Overall, these results indicate that the RF regression method has relatively general applicability on the spatial and time scales of different climate types.

4.2.3. Evaluation of the LSTM Method

Figure 10 shows the RMSEs of the results predicted by the LSTM regression method using training and test data from the same stations from 2013 to 2017, and the corresponding $R^2$ and bias results can be found in the Appendix A. LSTM performs similarly remarkably when the training and test data were from the same station in the same year. In the meantime, nothing much of LSTM’s property changed when dealing with cases in which the test data were from another year at the same station, with RMSEs of approximately 1.13 K, 0.96 K, and 1.82 K and $R^2$ of approximately 0.97, 0.98, and 0.95 for the DM, AR, and SDQ stations, respectively, showing a more stable performance than RF does. In particular, the SDQ station only possesses 56 days of available meteorological data in 2013 (101 days of data for the other stations each year), yet the 2013 and 2015 SDQ meteorological data lack soil moisture data and were filled from the adjacent station, the HHL station. These two factors resulted in a rather poor performance for the station SDQ in 2013 to predict other years, with RMSEs plateauing at 1.82 K and bias plateauing at 1.08 K in the year of 2017. Furthermore, the RMSEs reached a maximum when using the model trained by 2014 or 2015 and test 2017 of the SDQ station at 5.08 K and 3.42 K, showing that under certain circumstances, the year gap between training and test data being large could contribute to apparent low capability. Although there are some outliers, the transferability and stability of LSTM are acceptable on a time scale.

Finally, Figure 11 shows the RMSEs of the predicted results by the LSTM regression method using a station’s five-year dataset as the training dataset and testing other stations from 2013 to 2017, and the corresponding $R^2$ and bias results can be found in the Appendix A. As mentioned above, the SDQ station lacks an adequate amount of meteorological data in 2013 and misses the moisture soil variable in 2013 and 2015, so the model of the SDQ station was only trained by 2014, 2016, and 2017 (the same was carried out for RF). Unlike RF, the LSTM regression that used training and test data from cross stations generally performed more stably, with $R^2$ all above 0.81. Among all cases, the AR to DM result, SDQ to DM result, DM to AR result, and SDQ to AR result performed relatively well, with RMSEs of approximately 2.16 K, 2.17 K, 2.27 K, and 2.98 K, respectively, and biases of approximately −0.86 K, 0.17 K, −0.50 K, and −0.20 K. In the meantime, LSTM regression shows the same tendency yet better than RF regression when dealing with cases testing the SDQ station, indicating that LSTM regression is more sensitive and can extract the features of input more powerfully. In general, despite the limitations of the data, the LSTM regression method shows a strong ability to migrate and predict with moderate accuracy at spatial and temporal scales for different climate types.

5. Discussion

5.1. The Effect of $Vcmo$ in SCOPE

Based on the evaluation results above, the accuracy of SCOPE can be identified as acceptable, and its performance for different surface types and years is stable. However, compared with the RF regression method, a number of variables need to be set in advance. This is not conducive to an application of a model in a case with little prior information. In [26], the evaluation results for SCOPE-simulated brightness temperatures were found to be better with RMSEs lower than 1.5 K. The large RMSEs in this study are because default values of 35 ${\mathsf{\mu }\mathrm{molm}}^{-2}{\mathrm{s}}^{-1}$ and 80 ${\mathsf{\mu }\mathrm{molm}}^{-2}{\mathrm{s}}^{-1}$ were used for the maximum carboxylation capacity ($Vcmo$) of the C3 and C4 plants, respectively. However, the $Vcmo$ values were calibrated day by day with measured $\mathrm{C}{\mathrm{O}}_2$ and evapotranspiration fluxes in [26], and the calibrated $Vcmo$ displayed a large variation even over two consecutive days. To date, the estimation or calibration of $Vcmo$ on each day is still a difficult task in practical applications, which is why the recommended fixed $Vcmo$ values were directly used in this study. To analyze the effect of $Vcmo$ on the SCOPE-simulated results, an additional evaluation was performed using different $Vcmo$ values.

Table 3. Statistical information of SCOPE-simulated results using different Vcmo values.

	Unit	DM			AR			SDQ
Vcmo	$\mathsf{\mu }\mathrm{mol}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$	20	35	50	50	80	110	50	80	110
RMSE	K	2.03	1.83	1.85	2.16	2.25	2.06	2.07	2.12	2.24
Bias	K	0.22	−0.29	−0.35	0.52	−0.32	0.06	1.05	0.75	0.53
$R^2$	-	0.91	0.92	0.92	0.94	0.92	0.93	0.97	0.97	0.97

shows the evaluation results at three superstations based on measurements in 2017. In the trend, the biases decreased as the $Vcmo$ values increased. This may be explained by the fact that as leaf photosynthesis capacity increased, the leaf stomal resistance decreased, and then, the additional heat was easier to remove. In addition, the simulated results using calibrated $Vcmo$ values were also evaluated following [26]. The calibration process was performed day by day, and the $Vcmo$ value that can make a minimal RMSE between 1-day measured and simulated brightness temperatures was considered to be the result of calibration. The RMSEs of all 101-day simulations would therefore be the minimal, with values of 1.86 K, 1.59 K, and 2.02 K corresponding to the AR, DM, and SDQ stations, respectively. Nevertheless, the evaluation results display slight changes in RMSE and $R^2$ due to different $Vcmo$ values.

5.2. The Contribution of Inputs

Figure 12 displays the contribution importance of inputs in RF regression for simulating TOC brightness temperatures. The air temperature is the first driving force for the results predicted by RF regression at ten stations within corn canopies, as shown in Figure 12a, and the downward shortwave radiation occupies the second position of contribution importance. The LAI, downward longwave radiation, and air humidity play weaker roles when predicting the results, and the wind speed and soil water content contribute less. The high contribution importance of air temperature and downward shortwave radiation can be explained by the fact that their frequency of change was consistent with LST. The wind speed, downward longwave radiation, and soil water content are mainly used to account for the fluctuations in the temporal variations in LST rather than the heating/cooling effect for the diurnal temperature circle. The frequency of rapid changes in wind speed can make LST change within one minute, whereas the time step of predicted results by the RF regression method is ten minutes. The large variations in downward longwave radiation and soil water content mainly correspond to thick clouds and precipitation/irrigation, respectively. These factors do not have a lasting effect during the whole period, although if they occur, the surface temperature displays a significant change. The change in LAI was obviously slower than that in LST. Figure 12b displays the contribution importance of inputs to the predicted results at three superstations for 2013–2017. Relative to Figure 12a, similar input contribution importance can be found at the DM and SDQ stations, whereas a large difference appears at the AR station. The contribution importance of downward shortwave radiation significantly increases and becomes the first driving variable at the AR station. A possible explanation for this may be that the air temperature remains in a lower range at higher altitudes. The sums of the contribution importance of inputs, except for the two variables of air temperature and downward shortwave radiation at the DM station, were slightly larger than those at the AR and SDQ stations. A possible explanation is that there are more human interventions at the DM station, such as farming, irrigation, and harvesting. The larger contribution importance of downward longwave radiation at the AR station can be attributed to the cloudy weather in the mountains. These differences may be the reason for the worse performance of the RF and LSTM methods when the training and test data were from stations in different surface types. Figure 12c represents the contribution importance of inputs with different time intervals. As the time interval of simulations increased, the sum of the contribution importance of the air temperature, downward short radiation, and air relative humidity decreased. The LAI, wind speed, and soil water content were more important. This can be explained by the fact that the number of these samples, which display a large variation in temperature but with a similar LAI, wind speed, and soil water content, decreased.

5.3. Intellectual Merits and Limitations

Although land surface temperature products can be obtained from remote sensing data at various temporal and spatial scales, it is still difficult to meet the requirements in applications of high spatial–temporal resolution of land surface temperature. Currently, many physical and empirical methods that simulate land surface temperatures have been used when studying or improving land surface temperature from remote sensing data. Different methods have their advantages and disadvantages, and no one method can be applied to all situations yet. This study selects three methods, including one physical and two empirical methods that have been used for remote sensing data, and highlights the difference among them in simulating surface temperature with the same input. We hope our results can provide useful information for readers to choose a suitable method. If they have selected a method, we also hope such comparison results can improve the understanding of the selected method from the perspective of other methods. Considering the characteristics of physical and empirical methods, it is possible to combine them for robust and fast performance. In this case, this comparison study can provide some references for future combinations, such as the RF/LSTM method trained using data simulated by SCOPE before practical application. Nevertheless, limitations of this study also appeared:

• The transferability of an empirical model after being trained is an important factor in evaluation. Therefore, the comparison was performed from different perspectives, i.e., within the same surface type, between different years, and between different climate types. To meet this need, only limited measurements were selected in this study; in particular, the comparison with different surface types is insufficient. Additional work is therefore required based on measurements for more surface types.
• Moreover, because the differences among methods were focused on simulating land surface temperature, the comparison was performed by a forward simulation type at a point scale. When they are used with remote sensing data in practical applications, there are some differences due to the uncertainty of remote sensing products and the introduction of other data, which also deserves further analysis in the future.
• In addition, the SCOPE and RF/LSTM models were selected for the physical and empirical methods, respectively. Even similar models have some differences, especially for machine learning models. The structure of LSTM itself is complex, so a series of simplifications have been made, leading to the proposal of gated recurrent units (GRUs) and minimal gate units (MGUs). Although some research shows that none of the variants can improve significantly upon the standard LSTM architecture, different structures have different performances according to different specific problems [45], which has not been discussed in this study.

6. Conclusions

Land surface temperature is a vital parameter in many fields. Methods based on physics and experience are the two most common methods for land surface temperature simulation, each of which has its advantages and disadvantages. In this study, a physical-based method, soil canopy observation of photochemistry and energy fluxes (SCOPE), and two empirical methods, random forest (RF) and long short-term memory (LSTM), were realized based on HiWATER experimental datasets for comparison under three different circumstances: within the same surface type, between different years, and between different surface types. Regardless of surface types and years, the physical-based SCOPE method performed relatively stably with a root mean square error (RMSE) of approximately 2.0 K but required many inputs and a high computational cost. When the training and test data were from the same surface type, the empirical methods presented promising RMSE results of approximately 1.50 K and 1.70 K for RF and LSTM, respectively. Meanwhile, the empirical methods also performed well, with RMSEs of approximately 1.25 K and 1.30 K for RF and LSTM, respectively, dealing with cases whose training and test data were from the same station but different years. These two experiments indicated that the empirical methods have high precision and surpass the physical method when migrated at spatial or temporal scales individually. In regard to different surface types and different years, the empirical methods clearly worsened. Even with adequate training data, the RMSE of RF regression rose to approximately 3.50 K with outliers larger than 6.17. However, LSTM regression remained steady, with an RMSE of approximately 2.80 K and few outliers. The empirical methods are practical with moderate accuracy at spatial and temporal scales for different surface types and different years. Although the overall accuracy is not as good as that of the physical method, it has the advantages of fast calculation speed, few input parameters, and little consideration of the internal structure of the model. This experiment provides a reference for subsequent applications. The results show that if supportive data of the same surface type, or at the same station but different years, are available for the target station, empirical methods can give full play to their advantages and have high accuracy and stability at the same time. However, when there is a lack of supportive data or a great difference between supportive data and target data, physical methods are recommended. In the future, a strategy combined with physical and empirical methods can be explored to inherit the advantages of the two methods, and this kind of combined method can be a promising option in practical applications.