Improve the Simulation of Radiation Interception and Distribution of the Strip-Intercropping System by Considering the Geometric Light Transmission

Abstract

Intercropping radiation interception model is a promising tool for quantifying solar energy utilization in the intercropping system. However, few models have been proposed that can simulate intercropping radiation interception accurately and with simplicity. This study proposed a new statistical model (DRT model), which enables the simulation of daily radiation distribution by considering the geometric light transmission in the intercropping system. To evaluate model performance, the radiation interception and distribution in two wheat/maize strip intercropping experiments (A and B) were simulated with the DRT model and other two statistical models, including the horizontal homogeneous canopy model (HHC model) and the Gou Fang model (GF model). Experiment A was conducted in different intercropping configurations, while Experiment B was conducted in soils with different salinity levels. In both experiments, the HHC model exhibited the poorest performance (0.120 < RMSE < 0.172), while the DRT model obtained a higher simulation accuracy in the fraction of photosynthetically active radiation (PAR) interception, with RMSE lower by 0.008–0.022 and 0.022–0.125 than the GF and the HHC models, respectively. Especially, the DRT model showed stronger stability than the other two models under soil salinity stress, with R² higher by 0.129–0.354 and RMSE lower by 0.011–0.094. Moreover, the DRT model demonstrated a relatively ideal simulation of the daily radiation distribution in Experiment A (0.840 < R² < 0.893, 0.105 < RMSE < 0.140) and Experiment B (0.683 < R² < 0.772, 0.111 < RMSE < 0.143), especially when the continuous canopy formed during the later crop growth stages. These results indicate the superiority of the DRT model and could improve our understanding of radiation utilization in the intercropping system.

1. Introduction

Intercropping, a productive practice of growing two or more crops simultaneously in the same field, has received more interest and is widely used across the world [1,2]. Due to the critical demand for mechanized agriculture, relay strip intercropping gets more attention and development [3]. For relay strip intercropping, adequate radiation interception of the crop aboveground is a primary advantage of intercropping, it affects the crop photosynthetic process and increases yields [2,4]. To enhance agronomic management and optimize yields in intercropping systems, it is essential to quantify the radiation transmission of the intercropping system accurately.

The radiation interception model is an effective tool for simulating radiation interception and partitioning of field crops in strip intercropping systems. Two types of models can be classified according to their modeling principles: the geometrical model and the statistical model [5]. The geometrical model considered instantaneous light transmission in the canopy and more details such as heterogeneous canopies [6], border row effects [7], and strip orientation [8]. Gijzen and Goudriaan [9] may first propose a geometrical model considering the direction of incident light and the canopy architecture. Subsequent studies applied the model in practice and improved its accuracy [10,11,12]. Munz, Graeff-Honninger, Lizaso, Chen, and Claupein [11] used the geometrical model to calculate the available radiation at the top of the subordinate crop canopy to describe the incoming light distribution of the subordinate crop. The geometrical model improves our understanding of the radiation transfer in the intercropping system and generally can obtain high simulation accuracy; however, the geometrical model is restricted in use due to its complexity [10,13]. Numerous input variables need to be measured in the geometrical model, such as row orientation, leaf area density, canopy extinction coefficient (g_ψ and k_ψ), plant height (H), and the exact corresponding time.

Different from the geometrical model, the statistical model simplifies plant canopies as homogeneous and turbid cuboids using the strip-crop approach [14]. Based on the hypothesis, the statistical model can estimate the daily radiation interception of crops with fewer variables, such as the horizontal homogeneous canopy model (HHC model), employed in the popular crop model APSIM, which has been widely used in radiation interception simulation. Pronk et al. [15] introduced the “view factor” and “compressed canopy” theories and proposed an empirical model based on Beer’s law to calculate the radiation interception by only four easily obtained input variables/parameters, including H, leaf area index (LAI), strip and path width (R and P), and extinction coefficient (k). Zhang, et al. [16] used this method to calculate light interception in wheat-cotton relay intercrops. Wang et al. [17] attempted to allocate the residual indirect light of the “view factor” theory to each individual crop of wheat/maize strip intercropping system. To further improve model accuracy, Gou et al. [18] separated the taller intercrop canopy into the upper layer (strip-planted sole crop) and the under layer (two crops with the same height), considering the penetrated light from the upper layer as the incident light of the under layer. Numerous studies have proven the effectiveness of Gou’s model (GF model) and applied it in strip intercropping simulations [2,19,20,21,22]. For example, Pinto, van Dam, van Lier, and Reichardt [22] chose the latest GF model to calculate the radiation interception of each crop in the newly proposed intercropping model. And Pierre et al. [23] also chose the GF model as the method of calculating radiation interception in DSSAT.

However, there are still several limitations regarding the statistical model. The accuracy of the simplified statistical models could be restricted [10]. The statistical model generally fails to capture spatial differences in instantaneous light transmission due to the simplification of the light angle in simulation [11]. Meanwhile, though the radiation interception of the whole canopy can be verified using existing statistical models, the radiation interception partitioning for each crop is difficult to quantify [13]. Explaining the light transmission in the canopy may be a persuasive method for the radiation interception partitioning of statistical models. To address these issues, we proposed a novel radiation interception model for the relay strip intercropping system, the DRT model, which considers the angles of light in a two-dimensional plane and calculates the temporal and spatial integration for both direct and indirect light. Other two statistical models, a horizontally homogenous model (HHC model) and a GF model, were selected to compare. To comprehensively assess the accuracy and stability of these models, this study utilized two sets of independent experimental data. The first experiment focused on the accuracy of the model under different intercropping configurations, while the second experiment targeted the model’s accuracy and stability under different soil salinity stresses. The objectives of this study were to (1) develop a new radiation interception model for relay strip intercropping, (2) test the model performances for simulating radiation interception in different intercropping experiments, and (3) propose a simplified approach to simulate daily light distribution and test its performances.

2. Materials and Methods

2.1. Model Theory

2.1.1. Horizontally Homogeneous Canopy Model (HHC Model)

Taking a two-crop intercropping system as an example, the whole canopy is divided into two horizontally homogeneous layers in the HHC model. The layer boundaries are at the top of the two crops’ canopy. The fraction of light transmitted out from the bottom of the upper layer is the fraction entering the lower layer. The upper layer contains only the taller crop, encompassing a fraction of its LAI. While the lower layer is a two-crop mixed canopy, comprising the remaining LAI of the taller crop and the LAI of the short crop, LAI is assumed to follow the power of five functions with the relative height as follows:

$$LAI(h)=h^5$$

(1)

where h is the relative height (0 at the bottom and 1 at the top of the canopy). In the lower mixed canopy, the fraction of radiation interception of each crop with the same height is calculated as Equations (2)–(4):

$$f_{\mathrm{intercropps}}=1-e^{-k_{1}LA{I}_1-k_{2}LA{I}_2}$$

(2)

$$f_{\mathrm{intercropps}\_1}=\frac{k_{1}LA{I}_1}{k_{1}LA{I}_1+k_{2}LA{I}_2}{f}_{\mathrm{intercropps}}$$

(3)

$$f_{\mathrm{intercropps}\_2}=\frac{k_{2}LA{I}_2}{k_{1}LA{I}_1+k_{2}LA{I}_2}{f}_{\mathrm{intercropps}}$$

(4)

where f_{intercrop_1} and f_{intercrop_2} are the fractions of radiation interception of crop 1 and crop 2 in the mixed canopy, respectively; k₁ and k₂ are the light extinction coefficients of crop 1 and crop 2; LAI₁ and LAI₂ are the leaf area indexes of crop 1 and crop 2. The HHC model has been applied in the Canopy module of the APSIM (Version 7.9) [2]; more details can be found on the website (http://www.apsim.info, accessed on 3 May 2017) [24].

2.1.2. Gou Fang Model (GF Model)

Similar to the HHC model, the GF model divided the canopy into two layers according to the height difference between the two intercropping crops. The upper layer can be assumed to be a strip-planted canopy with a single crop with some rows omitted, which creates a compressed canopy and empty paths (Figure 1B). The fraction of light interception with the same LAI of a strip-planted canopy is numerically between the homogeneous canopy (as the maximum, Figure 1A) and the compressed canopy (as the minimum, Figure 1C). A weight (w) is used in the empirical function to calculate the fraction of light interception by a strip-plant canopy (f_strip) as follows:

$$f_{\mathrm{strip}}=f_{\mathrm{homo}}\times \left(1-w\right)+f_{\mathrm{compr}}\times w$$

(5)

The fraction of light interception in the homogeneous canopy (f_homo) and compressed canopy (f_compr) are both calculated by Beer’s Law. For the compressed canopy, the LAI in the green part of the field can be converted to a compressed LAI by strip width (R) and path width (P):

$$f_{\mathrm{homo}}=1-e^{-k\times LAI}$$

(6)

$$f_{\mathrm{compr}}=\left(1-e^{-k\times LA{I}_{\mathrm{compr}}}\right)\times \frac{R}{R+P}$$

(7)

$$LA{I}_{\mathrm{compr}}=LAI\times \frac{R+P}{R}$$

(8)

where k is the light extinction coefficient.

The value of w is between 0 and 1, and can be calculated as follows:

$$w=\frac{SP-SR}{1-e^{-k\times LA{I}_{\mathrm{compr}}}}$$

(9)

where SP and SR are the fractions of light transmitted to the soil surface in the path and under the strip, respectively. The larger w means more light is transmitted to the soil surface in the path than that in the strip, indicating that the strip-plant canopy is closer to a compressed canopy. On the contrary, a smaller w indicates the strip-plant canopy is closer to a homogeneous canopy. SP and SR can be calculated as follows:

$$SP=I{P}_{\mathrm{black}}+\left(1-I{P}_{\mathrm{black}}\right)\times e^{-k\times LAI}$$

(10)

$$I{P}_{\mathrm{black}}=\frac{\sqrt{H^2+P^2}-H}{P}$$

(11)

$$SR=I{R}_{\mathrm{black}}\times e^{-k\times LA{I}_{\mathrm{compr}}}+\left(1-I{R}_{\mathrm{black}}\right)\times e^{-k\times LAI}$$

(12)

$$I{R}_{\mathrm{black}}=\frac{\sqrt{H^2+R^2}-H}{R}$$

(13)

Equations (11) and (13) are obtained by spatial (from 0 to P/R, respectively) and angular (from θ to π/2) integration. The detailed integration process was shown in Pronk, Goudriaan, Stilma, and Challa [15]. From Equations (10) and (12), we can know SP and SR are coming from two origins. For radiation transmitted to the soil surface of the path (SP), one is direct radiation, which depends on the view factor over the path (IP_black), and the other is transmitted from neighboring one or several crop rows (1 − IP_black). The precise calculation of transmitted radiation (1 − IP_black) is supposed to be a spatial (from 0 to P/R) and angular (from 0 to θ) integration integrated like direct radiation. The integrand is the radiation that falls on the soil surface after being intercepted by a certain thickness of compressed canopy, which is determined by the geometric relationship. The remaining radiation from the upper canopy interception along the path serves as the incoming light for the compressed canopy of the shorter crop, while the remaining radiation from the strip acts as the incoming light for the compressed canopy of the taller crop with a part of its LAI.

2.1.3. Daily Radiation Transmission Model (DRT Model)

We developed a statistical daily radiation transmission model (DRT model), referring to the quantifications of instantaneous radiation transmission in the geometrical model. The DRT model comprises three modules: crop skip-row growth, co-growth of two crops, and spatial distribution of daily radiation transmission.

Module 1: Crop skip-row growth

For the crop skip-row growth (Figure 2A), the radiation interception of the single crop (f), for example, an intercropping unit (strip and path, d + p), is the total radiation minus the radiation reaching the soil surface as follows:

$$f_{}=1-\left(I_{p,d}+I_{p,i}\right)-\left(I_{d,d}+I_{d,i}\right)$$

(14)

where I_p,d and I_d,d are the direct radiation on the path and the strip; I_p,i and I_d,i are the indirect radiation (through the canopy) on the path and strip, respectively. These variables can be calculated based on view factor theory and radiation geometry principles. The detailed calculation equations for these variables are in Appendix A.

Module 2: Co-growth of two crops

In the co-growth of two crop situations (Figure 2B), the radiation interception of crop 1 and crop 2 (F1 and F2) corresponds to the summation of their respective direct and indirect radiation components along strip d and strip p:

$$F1=F{1}_{d,d}+F{1}_{p,i}+F{1}_{d,i}$$

(15)

$$F2=F{2}_{p,d}+F{2}_{p,i}+F{2}_{d,i}$$

(16)

where F1_d,d and F1_d,I are the radiation interception of direct and indirect light along strip d of crop 1, respectively; F1_p,i is the radiation interception of indirect light along strip p of crop 1; F2_p,d and F2_p,i are the radiation interception of direct and indirect light along strip p of crop 2, respectively; F2_d,i is the radiation interception of indirect light along strip d of crop 2. The calculation of these variables is shown in Appendix B.

Module 3: Spatial distribution of daily radiation transmission

The spatial daily light interception can be calculated in the DRT model, it is a function of the geometrical models but not available in other statistical models. It has two steps. First, due to the position of simulated points (x) being controlled by the position of Gaussian points (a_n), it is necessary to ensure the alignment between the positions of the Gaussian points and the measured PAR points. The method for selecting Gaussian points is in Appendix C. Secondly, unlike calculating the total direct radiation interception on the strip, it is necessary to calculate the light interception of the direct radiation at every Gaussian point. The calculation of the indirect radiation (rp(x) and rd(x)) in Module 3 is identical to Module 1 and Module 2. The angle of integration of direct radiation at the Gaussian point is calculated as follows:

The crop skip-row growth:

$$I_{p,x}=rp(x)+\frac{{\displaystyle {\int }_{{\theta }_{1,x}}^{\frac{\pi }{2}}\sin \theta }\mathrm{d}\theta }{{\displaystyle {\int }_0^{\frac{\pi }{2}}\sin \theta }\mathrm{d}\theta }=rp(x)+\cos {\theta }_{1,x}=rp(x)+\frac{x}{\sqrt{x^2+h^2}}$$

(17)

$$I_{d,x}=rd(x)+\frac{x}{\sqrt{x^2+h^2}}·e^{-k·LA{I}_{\mathrm{c}}}$$

(18)

Co-growth of two crops:

$$I_{p,x}=rp(x)+\frac{x}{\sqrt{x^2+h^2}}·e^{-k_2·LA{I}_{2,c}}$$

(19)

$$I_{d,x}=rd(x)+\frac{x}{\sqrt{x^2+h^2}}·e^{-k_1·LA{I}_{1,c}}$$

(20)

where x represents the position of that Gaussian point; x = a_n × d when calculating I_d,x; x = a_n × p when calculating I_p,x. I_p,x is the fraction of radiation that falls on the x point along the strip of the shorter crop (p); I_d,x is the fraction of radiation that falls on the x point along the strip of the taller crop (d).

The angle integration in Module 3 is from 0 to π/2, while the daily radiation is from 0 to π. Therefore, the results need to be converted into daily radiation interception. Due to the geometric symmetry in the DRT model, the radiation interception at the points exhibits horizontal symmetry as well. The symmetric points need to be averaged to represent the radiation interception at the two points.

2.2. Experimental Data Collection

In this study, two wheat-maize intercropping experiments A and B were selected to evaluate the model performances. Experiment A was conducted in the Wageningen field experiment (51°59′20″ N, 5°39′16″ E). Five wheat-maize intercropping configurations were designed in Experiment A, including 6:0 WM (wheat skip growth), 0:2 WM (maize skip-row growth), 6:2 WM, 6:3 WM, and 8:2 WM (three different treatments of co-growth of wheat and maize). The details of Experiment A can be found in Gou et al. [18].

Experiment B was conducted in an experimental station at Wuhan University, Hubei Province, China (30°32′36″ N, 114°22′09″ E). Two sole crop systems of wheat (Treatment 1) and maize (Treatment 2) and a wheat-maize intercropping system (Treatment 3, the intercropping ratio is 6:2 WM) were designed (Figure 3). Moreover, the crops in each treatment were planted in fields with three soil salinity levels (S1, S2, and S3). Specifically, the average electrical conductivity of the saturation extract (EC_e) values in the three fields were 0.89–1.18, 1.98–2.96, and 2.63–6.42 dS m⁻¹ in S1, S2, and S3, respectively. The row spaces of each treatment are shown in Figure 3. The wheat variety is Zhengmai9032, and the maize variety is Hunong101. The row orientation was north-south direction. Winter wheat was sown on 20 November 2022, and harvested on 31 May 2023. Maize was sown on 15 March 2023, and harvested on 1 August 2023. Before wheat sowing, a compound fertilizer of N-P₂O₅-K₂O is applied, and the application rates of nitrogen, phosphorus, and potassium were 180 kg ha⁻¹, 75 kg ha⁻¹, and 100 kg ha⁻¹, respectively.

In Experiment B, the leaf area and height of two crops in each field were determined by the average status of three independent measurements. The leaf areas of the maize and wheat were calculated based on LA = 0.75 × L × W and LA = 0.85 × L × W (L and W are the length and width of the measured leaf), respectively. The plant height is defined as the distance from the soil surface to the top of the canopy. The crop height measurement was conducted on the same day, with LAI measurements about once every two weeks from crop emergence to the final harvest. During the crop growth period, we measured light interception 12 times. Each measurement involved three separate recordings within one day, conducted at 10:00 a.m., 12:00 p.m., and 2:00 p.m. The daily light interception at each measurement point was determined by the average of three recordings. Photosynthetically active radiation (PAR) above and below the canopy (about 5 cm above the soil surface) was measured using a Beam Fraction Sensor (BFS, a quantum sensor). Five values were collected during each measurement. The SunScan probe was parallel to the rows at the measured points. Four different points were measured in Treatments 1 and 2, and nine different points were measured in Treatment 3 and its different growth stages (Figure 3). In Treatments 1 and 2, measurements were used to calibrate the k of wheat and maize. In Treatment 3, measurements were made at approximately 19 cm intervals across a width of 170 cm. The overall fraction of light interception in the heterogeneous canopy was calculated as the arithmetic average of measurements at the nine different measured points.

2.3. Statistical Indicators

In this study, R-square (R²) and root mean square error (RMSE) were used to estimate the accuracy of simulations as compared to measured values.

$$R^2=\frac{{\left[{\displaystyle {\sum }_i^m\left(o_i-\overline{o}\right)\left(p_i-\overline{p}\right)}\right]}^2}{{\displaystyle {\sum }_i^m{\left(o_i-\overline{o}\right)}^2{\displaystyle {\sum }_i^m{\left(p_i-\overline{p}\right)}^2}}}$$

(21)

$$RMSE=\sqrt{\left(1/m\right){\displaystyle {\sum }_i^m{\left(o_i-p_i\right)}^2}}$$

(22)

where o_i and p_i are the measured value from the field experiment and predicted value from the model simulation, respectively; $\overline{O}$ and $\overline{p}$ are the average of measured and predicted values, respectively; and n is the number of measurements.

3. Results

3.1. Model Evaluation in Experiment A

3.1.1. Fraction of Photosynthetically Active Radiation Interception

The simulated fraction of PAR interception in skip-row crop growth by the three models (HHC model, GF model, and DRT model) is shown in

Table 1. The measured and simulated fraction of PAR interception of the three models at crop skip-row growth.

Treatment	Days	Measured	HHC	GF	DRT
0:2 WM	150	0.074	0.044	0.043	0.041
	154	0.045	0.058	0.057	0.055
	164	0.173	0.236	0.226	0.219
	171	0.261	0.319	0.304	0.294
	178	0.353	0.432	0.407	0.393
	184	0.437	0.608	0.566	0.544
	192	0.678	0.712	0.668	0.642
	199	0.714	0.781	0.741	0.714
	205	0.791	0.812	0.773	0.746
	212	0.748	0.813	0.775	0.749
	223	0.769	0.814	0.776	0.750
	223 (6:2 WM)	0.561	0.684	0.657	0.635
	223 (6:3 WM)	0.709	0.860	0.818	0.790
	223 (8:2 WM)	0.597	0.631	0.609	0.587
6:0 WM	122	0.185	0.233	0.194	0.187
	136	0.241	0.563	0.374	0.351
	143	0.434	0.698	0.448	0.414
	150	0.447	0.792	0.487	0.450
	154	0.474	0.832	0.520	0.479
	164	0.467	0.785	0.539	0.495
	171	0.473	0.728	0.515	0.472
	178	0.472	0.663	0.485	0.445
	184	0.349	0.614	0.460	0.422
	192	0.447	0.536	0.418	0.384
	199	0.358	0.451	0.367	0.338
	205	0.396	0.393	0.328	0.304
	212	0.410	0.361	0.306	0.283
	122 (8:2 WM)	0.265	0.318	0.247	0.238
	136 (8:2 WM)	0.366	0.584	0.382	0.358
R2		-	0.745	0.931	0.930
RMSE		-	0.172	0.061	0.053

and Figure 4. The calibrated parameters (

Table 2. The strip width of different treatments as model input parameter.

Treatment	Crop Height	d (cm)	p (cm)
0:2 WM	Maize skip-row	150	75
6:0 WM	Wheat skip-row	75	150
6:2 WM	W higher M	75	150
	M higher W	150	75
6:3 WM	W higher M	75	150
	M higher W	150	75
8:2 WM	W higher M	100	125
	M higher W	125	100

) were determined according to Gou et al. [18], including the light extinction coefficient of two crops (k_maize = 0.69, k_wheat = 0.63) and the strip width (d and p) in different intercropping ratios.

In the skip-row crop growth situation, the worst performances were obtained in the HHC model; the DRT model exhibited slight superiority over the GF model, with RMSE lower by 0.008. The simulated fraction of radiation interception by the HHC model was consistently greater than those of the GF model and DRT model. In the 0:2 WM treatment (maize skip-row), the simulated values of the three models exhibit little difference, ranging between 0 and 0.07. Additionally, the DRT model obtained higher accuracy than the GF model in the 0:2 WM treatment, with the RMSE lower by 0.01. In 6:0 WM (wheat skip-row), the HHC model obviously overestimates the fraction of radiation interception, while similar better performances were obtained in the GF and the DRT models.

In the two-crop co-growth situation (

Table 3. The measured and simulated fraction of PAR interception of the three models at the co-growth of two crops.

Crop Height	Treatment	Days	Measured	HHC	GF	DRT
W higher M	6:2 WM	143	0.434	0.681	0.464	0.438
		150	0.501	0.773	0.521	0.502
		154	0.478	0.813	0.562	0.544
		164	0.543	0.776	0.582	0.588
		171	0.599	0.742	0.584	0.610
		178	0.606	0.732	0.638	0.667
	6:3 WM	143	0.434	0.698	0.488	0.461
		150	0.543	0.797	0.566	0.545
		154	0.516	0.838	0.615	0.595
		164	0.579	0.832	0.659	0.665
		171	0.660	0.829	0.688	0.712
		178	0.759	0.843	0.766	0.788
	8:2 WM	143	0.560	0.737	0.540	0.522
		150	0.585	0.835	0.598	0.591
		154	0.616	0.874	0.637	0.630
		164	0.661	0.848	0.640	0.669
		171	0.661	0.821	0.643	0.684
		178	0.737	0.798	0.679	0.719
M higher W	6:2 WM	184	0.622	0.732	0.684	0.712
		192	0.767	0.737	0.720	0.742
		199	0.737	0.746	0.730	0.763
		205	0.795	0.759	0.741	0.781
		212	0.777	0.755	0.736	0.781
	6:3 WM	184	0.749	0.849	0.819	0.838
		192	0.859	0.866	0.849	0.869
		199	0.849	0.883	0.857	0.886
		205	0.882	0.898	0.866	0.898
		212	0.890	0.896	0.864	0.897
	8:2 WM	184	0.730	0.771	0.654	0.706
		192	0.813	0.747	0.738	0.753
		199	0.811	0.737	0.723	0.753
		205	0.843	0.748	0.728	0.766
		212	0.828	0.735	0.714	0.758
R2			-	0.101	0.852	0.894
RMSE			-	0.170	0.055	0.045

), the HHC model overestimates the PAR interception and exhibits the lowest simulation accuracy (R² = 0.101, RMSE = 0.170). Moreover, the DRT model was superior to the GF model, with R² higher by 0.042 and RMSE lower by 0.01. As shown in Figure 5, the simulated values of the DRT model were closer to the measured values than the GF model. Specifically, the simulated values of the GF model are greater than the DRT model during 143–154 days, while smaller than the GF model during 164–212 days. Additionally, the R² of the DRT model in the 6:3 WM treatment (R² = 0.962) was higher than that in the 6:2 WM (R² = 0.925) and 8:2 WM (R² = 0.908) treatments.

3.1.2. Simulated Spatiotemporal Photosynthetically Active Radiation Interception in the DRT Model

Figure 6 shows that the DRT model demonstrated satisfactory performance in simulating the fraction of PAR interception at measured points in different intercropping situations, with R² ranging from 0.840 to 0.893 and RMSE ranging from 0.105 to 0.140. Specifically, Figure 7 shows the DRT model exhibited higher precision in simulating PAR interception at measured points during the period when wheat was taller than maize, such as on the 154th and 171st days, with R² ranging from 0.822 to 0.973 and RMSE ranging from 0.076 to 0.127. On the 205th day, when maize was taller than wheat, the DRT model demonstrated lower R² values but comparable RMSE values. Additionally, the DRT model shows the highest simulation accuracy in the 6:3 WM treatment (0.065 < RMSE < 0.105).

As shown in Figure 8, the DRT model showed high accuracy in simulating spatial PAR interception of the 6:0WM treatment (wheat skip-row) during the whole growth period, with R² ranging from 0.932 to 0.955 and RMSE ranging from 0.057 to 0.111. The DRT model also performed excellently (0.861 < R² < 0.934, RMSE < 0.057) in the later growth stages (such as days 192, 205, and 223) of the 0:2 WM treatment (maize skip-row). However, in the early growth stage of maize (days 154), the R² was low. It is worth noting that, between days 154 and 171, the measured fraction of PAR interception showed two distinct peaks at the 8th and 13–14th measured points. Additionally, the DRT model aligns well with the growth and development of wheat and maize skip-row treatments.

3.2. Model Evaluation in Experiment B

3.2.1. Fraction of Photosynthetically Active Radiation Interception

The measured and simulated fractions of radiation interception of the three models for skip-row crop growth are shown in

Table 4. The measured and simulated fractions of PAR interception of the three models at different growth stages.

Field	Days (Stage)	Measured	HHC	GF	DRT
S1	68 (6:0 WM)	0.436	0.812	0.641	0.606
	82 (6:0 WM)	0.606	0.851	0.688	0.647
	90 (6:0 WM)	0.600	0.84	0.703	0.662
	105 (6:2 WM)	0.675	0.832	0.667	0.674
	116 (6:2 WM)	0.758	0.853	0.635	0.717
	128 (6:2 WM)	0.735	0.823	0.655	0.732
	139 (6:2 WM)	0.782	0.732	0.710	0.708
	152 (0:2 WM)	0.598	0.547	0.516	0.491
	166 (0:2 WM)	0.534	0.566	0.535	0.510
	180 (0:2 WM)	0.527	0.56	0.530	0.505
	196 (0:2 WM)	0.469	0.503	0.478	0.454
	213 (0:2 WM)	0.381	0.459	0.438	0.416
R2		-	0.380	0.480	0.697
RMSE		-	0.162	0.090	0.068
S2	68 (6:0 WM)	0.492	0.741	0.599	0.568
	82 (6:0 WM)	0.657	0.862	0.691	0.65
	90 (6:0 WM)	0.635	0.787	0.665	0.626
	105 (6:2 WM)	0.683	0.814	0.658	0.658
	116 (6:2 WM)	0.741	0.809	0.619	0.682
	128 (6:2 WM)	0.744	0.682	0.578	0.625
	139 (6:2 WM)	0.789	0.659	0.635	0.638
	152 (0:2 WM)	0.564	0.532	0.500	0.475
	166 (0:2 WM)	0.607	0.595	0.560	0.534
	180 (0:2 WM)	0.566	0.561	0.530	0.505
	196 (0:2 WM)	0.529	0.508	0.481	0.457
	213 (0:2 WM)	0.401	0.425	0.406	0.384
R2		-	0.375	0.508	0.729
RMSE		-	0.120	0.087	0.076
S3	68 (6:0 WM)	0.458	0.742	0.598	0.567
	82 (6:0 WM)	0.591	0.721	0.601	0.569
	90 (6:0 WM)	0.579	0.749	0.638	0.602
	105 (6:2 WM)	0.594	0.807	0.657	0.655
	116 (6:2 WM)	0.673	0.796	0.611	0.669
	128 (6:2 WM)	0.681	0.683	0.572	0.62
	139 (6:2 WM)	0.727	0.619	0.585	0.595
	152 (0:2 WM)	0.560	0.469	0.440	0.417
	166 (0:2 WM)	0.540	0.556	0.522	0.497
	180 (0:2 WM)	0.473	0.503	0.476	0.453
	196 (0:2 WM)	0.458	0.459	0.436	0.414
	213 (0:2 WM)	0.252	0.412	0.393	0.372
R2		-	0.359	0.455	0.584
RMSE		-	0.140	0.091	0.080

and Figure 9. The light extinction coefficients of wheat (k_maize) and maize (k_wheat) were calibrated at 0.57 and 0.53, respectively. The strip width and path width (d and p) in Experiment B are in

Table 5. The strip width of different growth stages as model input parameter.

Growth Stage	Crop Height	d (cm)	p (cm)
6:0 WM	Wheat skip-row	90	80
6:2 WM	Wheat higher Maize	90	80
	Maize higher Wheat	80	90
0:2 WM	Maize skip-row	80	90

.

Compared with Experimental A, the performance of the three models exhibited a more significant disparity in Experimental B. In all fields with different salinity levels (S1, S2, S3), the DRT model outperformed the GF model with higher R² (0.129–0.221) and lower RMSE (0.011–0.022), while the HHC model (0.359 < R² < 0.380, 0.120 < RMSE < 0.162) exhibited the poorest performance (Figure 9). As shown in Table 4, although the GF model and the DRT model overestimated values on the 68th and 90th days (6:0 WM), the DRT model’s simulated values were relatively lower about 0.04 than those of the GF model. On the 105th and 139th days (6:2 WM), both of the two models were underestimated, while the DRT model’s simulated values were higher by approximately 0.06. Additionally, the simulation accuracy of both the GF and DRT models was slightly lower in the high-salinity field S3 (0.455 < R² < 0.584) compared to the low-salinity field S1(0.480 < R² < 0.697) and the medium-salinity field S2 (0.508 < R² < 0.729).

3.2.2. Simulated Spatiotemporal Photosynthetically Active Radiation Interception in the DRT Model

Compared with Experimental A (0.840 < R² < 0.893), the DRT model exhibited a lower R² in Experimental B (0.683–0.772) in the simulation of the fraction of PAR interception at measured points (Figure 10). Moreover, the accuracy of the DRT model is higher in low-salinity field S1 (R² = 0.762, RMSE = 0.111) compared to its performance in high-salinity field S3 (R² = 0.683, RMSE = 0.143). Additionally, the simulation results of DRT exhibit an overestimation within the 0.1–0.3 range while tending to underestimate within the 0.7–0.9 range. Figure 11 shows During the different growth stages, the accuracy of the DRT model is higher in crop skip-row growth (6:0 WM and 0:2 WM) than in the co-growth of wheat and maize (6:2 WM). The R² of the 6:0 WM stage and the 0:2 WM stage is mainly larger than 0.9, and the RMSE is mainly lower than 0.1 in all fields. However, in the 6:2 WM stage (105th day), the simulated values were larger than the measured values at the 6–8 measured points. On the 128th day (6:2 WM), the simulated values were obviously larger than the measured values, with RMSE ranging from 0.122 to 0.207.

4. Discussion

4.1. The Factors Contributing to the Performances of the Three Models

This study evaluated the simulation performances of three statistical models in two distinct wheat-maize intercropping experiments. Experiment A was conducted in different intercropping configurations (strip width, sowing density, and sowing time). We intended to test the model’s accuracy under conditions without abiotic stress. Differently, Experiment B was conducted in different soil conditions (soil salinity) and focused on the model stability of varying growth conditions caused by salinity stress. Both experiments show the GF model obviously outperforms the HHC model (Figure 4, Figure 5, and Figure 9). The reason behind this is that the HHC model considers the strip intercropping canopy as a homogeneous canopy, ignoring light directly falling on the path, thereby overestimating simulated results [2,21]. Differently, the GF model applied a weighted average of the compressed canopy (w) into the strip intercropping to mitigate the overestimation. The GF model further quantifies the value of w based on the view factor theory by calculating the radiation interception along the strip and path.

Although the GF and DRT models both obtained acceptable simulation results, the DRT model exhibits superior accuracy and stability (higher R² and lower RMSE) in Experiments A and B (Figure 4, Figure 5, and Figure 9). The lower simulation accuracy of the GF model could be attributed to it assuming that the indirect radiation completely passes through a homogenous canopy and ignoring the angular characteristics of the indirect radiation. But in practice, some of the indirect radiation traverses a small compressed canopy before reaching the soil surface, while other portions of indirect radiation need to pass through one or more compressed canopy before reaching the soil surface. Moreover, the radiation falling onto the strip and path may potentially be absorbed by both crops before reaching the soil surface. Different from the GF model, the DRT model employs both spatial and temporal integration using the Gaussian-Legendre methods for indirect radiation, enhancing model stability compared to the GF model. Such as during the period from the 68th to 128th days in the S1 field in Experiment B (Table 4), the simulated results of the DRT model were generally closer to the observed values. Furthermore, owing to its geometric structure, the DRT model provides more accurate radiation interception simulations for shorter crops compared to the GF model. In Experiment B, soil salinity influenced the model accuracy of both the DRT model and GF model, with lower accuracy observed in S3 compared with S1 and S2. This may be due to salinity stress impeding the growth of maize, restricting both plant height and LAI expansion, while the accuracy of radiation interception models relies on the assumption that the canopy is a rectangular hedge. Due to similar reasons, the R² value of the DRT model in the 6:3 WM treatment, which increased the plant density of maize, is better than that of the 6:2 WM and 8:2 WM treatments. However, compared with the GF model, relatively satisfactory simulation performance in the high salinity field was obtained in the DRT model (R² = 0.584, RMSE = 0.08), which indicates the adaptability of the DRT model in soil stress conditions. The DRT model is therefore well-suited for semi-arid regions characterized by elevated soil salinity levels, offering a viable alternative to the HHC model in both maize-bean and cereal-legume intercropping systems [10,25]. Additionally, the DRT model may be more suitable for soybean-maize intercropping systems compared to the HHC model and the GF model because it accommodates a longer co-growth period [2,26,27].

4.2. The Spatial Daily Radiation Interception of the DRT Model

Based on geometric principles, the DRT model simulates spatial variations of radiation interception in the intercropping system with few parameters (light extinction coefficient, LAI, width of strip, and path), which significantly reduces the computational complexity compared to the traditional geometric model [8]. Meanwhile, the simulations of the fraction of daily radiation interception at measured points by the DRT model generally exhibit relatively high accuracy and strong stability, with R² values typically exceeding 0.85 and RMSE usually being less than 0.12.

However, it should be noted that the simulated spatial daily radiation interception in the DRT model is strictly symmetrical due to its simplifications of parameters and assumptions. This may result in a simulated range that is narrower than the observed range. Furthermore, both geometrical models and statistical models rely on a critical assumption; the crop canopy is assumed to be a chaotic rectangular hedge [9,15]. However, during the early growth stages of maize, the height and leaf area are small, the canopy does not completely cover the ground, and there are huge horizontal interspaces within the canopy structure, making it challenging for maize to meet the assumption of a whole rectangular canopy [28]. Therefore, the DRT model is challenging to simulate the two distinct peaks observed in the measured fraction of PAR interception at measured points, as shown on the 154th and 171st days of the 0:2 WM treatment (Figure 8), and restricted the model’s accuracy as demonstrated on the 128th day (Figure 11). But as maize grows and both plant height and LAI increase, the simulation accuracy of the DRT model is significantly improved.

5. Conclusions

This study developed a novel radiation interception model, the DRT model, for strip intercropping systems. The direct and indirect radiations were detailed and quantified using temporal and spatial integrations combined with the view factor theory. To access the model performances, the radiation interception of the wheat-maize intercropping system was simulated with the DRT model and the other two statistical models of HHC and GF in two distinct experiments. In both experiments, the DRT model outperformed the GF model and HHC model (higher R² by 0.042–0.221 and lower RMSE by 0.008–0.022). In experiment A, the DRT model was slightly superior to the GF model in different intercropping configurations, with a lower RMSE of 0.008–0.01. In experiment B, the soil salinity may negatively affect the stability of the three models, causing a decrease in model accuracy. The DRT model showed stronger stability than the GF model and the HHC model (R² higher by 0.129–0.221 and 0.225–0.354, respectively, and RMSE lower by 0.011–0.022 and 0.044–0.094, respectively). The DRT model offers a simple way to accurately describe the spatiotemporal radiation distribution of the strip intercropping system compared with the geometrical model. In the different treatments, the DRT model showed high accuracy and a strong ability to simulate the daily radiation distribution of experiment A (0.840 < R² < 0.893, 0.105 < RMSE < 0.140) and experiment B (0.683 < R² < 0.772, 0.111 < RMSE < 0.143). It is promising to couple the DRT model with the crop or soil physics models to explore the interactions among radiation utilization, water-heat-carbon cycle, crop growth, and yield formation of intercropping systems in diverse environments.