**1. Introduction**

In recent canopy modeling studies, row modeling has been extensively studied. The canopy modeling of crops based on the row model can more accurately estimate the seasonal changes of biophysical canopy parameters in remote sensing [1]. In the inversion of remote sensing, physical modeling is key, and its approach can be separated into three categories based on di fferent physical mechanisms, name the computer simulation (CS) approach, radiative transfer (RT) approach, and geometric optical (GO) approach [2]. The GO approach can describe the geometric characteristics of an individual canopy. It is most suitable for heterogeneous canopy modeling [3]. The CS approach has the highest accuracy and has been applied primarily for understanding radiation regimes. It has been used as a "truth value" ("surrogate truth" or benchmark in Radiation Transfer Model Intercomparison (RAMI)) [4–6] to validate GO and RT approaches. However, compared to the GO approach, the computational time of the CS approach is too slow; hence, it is rarely used in remote sensing inversion. The RT approach is based on volume scattering in the radiative transfer equation. It is very suitable for describing the scattering issue in canopies [7]. However, compared to the GO approach, the RT approach lacks a description of the geometric characteristics of an individual canopy and is mostly used for high-density homogeneous (or continuous) canopy modeling [2]. The row crop

is a heterogeneous canopy. It has the typical geometric characteristics of an individual canopy, such as row structure (e.g., height, row width, between-row distance, etc.) [8–11]. Therefore, considering the advantages of accurately describing the row structure, the GO approach is widely used in the canopy modeling of row crops [9,10,12].

As canopy reflectance models, GO models have to deal with the interactions of light occurring within and between individual plant canopies and calculate the reflectance at the top of the canopy [13]. These interactions are divided into two basic physical processes. The first physical process is surface reflectance in the GO approach, also known as the single-scattering contribution. The single-scattering contribution represents the one-order interaction between light and medium, which is calculated by the four-component area fraction (sunlit and shaded canopy and sunlit and shaded soil [14]) and corresponding representative reflectance for each component in the GO approach [3,15]. For calculating the four-component area fraction, gap probabilities that reflect the transmission of light in the canopy are key [16,17]. An initial approach [18,19] for calculating gap probabilities only considered the overlapping relationship between leaves and ignored the overlapping relationship between plant canopies, which caused computational deviations in reflectance near the hotspot (a reflectance peak around a viewing direction that is exactly opposite the solar illumination direction [2]) [20]. The reflectance peak (sum of the reflectance) near the hotspot is caused by a single-scattering contribution [2]. To improve computational accuracy for the single-scattering contribution near hotspots, Li et al. [21] and Chen et al. [22] attempted to establish a new approach considering the overlapping relationship between leaves and discrete crowns (individual tree crowns) to calculate gap probabilities. With the improvement of the mathematical description of the gap probabilities, the accuracy of the single scattering near the hotspot is further improved.

The second physical process is a multiple-scattering contribution. When a single-scattering contribution interacts with the medium again, two-order and high-order interactions between light and medium are formed. The cumulative sum of these interactions is called the multiple-scattering contribution [23]. From the models presented in the GO approach, previous studies largely ignored multiple-scattering contributions [3,15,24,25]. GO models are generally accurate in the visible part of the solar spectrum but less accurate in the near-infrared (NIR) part where multiple scattering in plant canopies is the strongest [26]. In general, this issue can be tackled by two different methods to establish a multiple-scattering equation in the GO approach. In the first method, similar to the single-scattering principle, the multiple-scattering equation continues to be established using the GO approach. The second method uses the radiative transfer (RT) approach to establish the multiple-scattering equation, combines it with the first scattering equation constructed by the GO approach, and calculates the reflectance at the top of the canopy. In the first method, Chen et al. [26] attempted to introduce view factors between the various sunlit and shaded components within a canopy to perfect the multiple scattering framework in the GO approach. However, Chen et al. [23,26] suggested that some treatments remain to be improved in the multiple-scattering equations they proposed: (1) because calculating the view factors is quite complicated, the study used an a priori numerical experiment to simplify the complex integral involved in the view factor, which caused some situations to be inapplicable; (2) because the geometric relationship for two-order and high-order scattering is unclear, the two-order and high-order scattering angles are difficult to accurately determine. Regarding the second method, Li et al. [27] developed a GORT (an Analytical Hybrid geometric optical and radiative transfer approaches) model. They used the GO approach to describe the surface reflectance (single-scattering contribution) and adopted the RT approach (numerical method of successive order) to estimate the multiple-scattering contribution. Subsequently, Ni et al. [13] reported that the GORT model was overestimated in the multiple scattering of a sunlit canopy. Therefore, they established a simplified analysis model considering the path scattering effect on a sunlit canopy, thereby reducing the overestimation of the multiple scattering of the sunlit canopy. However, the results shown by Ni et al. have a computational deviation at a small solar zenith angle, and this phenomenon is especially obvious in the principal plane of the NIR part. Although the established multiple-scattering equation of the RT approach was coupled with the GO approach to address some issues of improvement in reflectance accuracy, it still has a limitation. The major di fficulty is that the geometric characteristics of independent canopies are not considered in the multiple-scattering contribution [26].

The above studies for the single- and multiple-scattering contributions all aimed at heterogeneous forest modeling but rarely row modeling. Therefore, it is necessary to consider these issues of the single- and multiple-scattering contributions in a row model based on the GO approach. Similar to the heterogeneous forest, row crops are also heterogeneous canopies in agriculture [15], and the tree crown mentioned in the forest is similar to the canopy closure (vegetation material area, also called a row). Therefore, row crops also have an overlapping relationship between individual plant canopies, which implies that the calculation of gap probabilities in the row crops should also consider the overlapping relationship between leaves and canopy closures. Similar to the heterogeneous forest canopy, gap probabilities in the row crops are accurately calculated. The single-scattering contribution near the hotspot can be accurately calculated further.

In addition to the issue of the single-scattering contribution in row modeling, the multiple-scattering contribution is more important and has often been ignored in previous studies [14,15,28,29]. The multiple scattering of row crops includes the multiple scattering of the between-row area (the between-row area is the area of bare soil between the canopy closures) in addition to the multiple scattering of the canopy closure [30,31]. The calculation area involved in the multiple scattering of the canopy closure is an area with vegetation material, and its multiple scattering characteristics are similar to those of continuous crops. Therefore, its calculation is uncomplicated. However, in the between-row area, the multiple scattering of the between-row area involves multiple-scattering contributions between the soil and the adjacent canopy closures (two mediums: soil particles and vegetation leaves) and needs to consider the row structure in the calculation [30–32]. Therefore, the calculation for multiple scattering in the between-row area is more complicated. Although an integral form of the radiative transfer equation was proposed to address the multiple scattering of the between-row area, its study remains in the RT field [30]. The multiple scattering of the between-row area used in the GO approach in row crops is an issue requiring further study.

This study focuses on the limitations of the GO approach currently used in row modeling, especially in describing the multiple scattering framework. To achieve this goal, there are two subproblems to be dealt with for the single- and multiple-scattering contributions: (1) How can we address the issue of ignoring the geometric characteristics of independent canopies in the establishment of multiple-scattering equations using the RT approach? (2) How can we establish a gap probability approach that considers the overlapping relationship between leaves and canopy closures and addresses computational deviations in the reflectance near the hotspot in a single-scattering contribution? Specifically, we introduced the adding method and the mathematical solution of integral radiative transfer equation into row modeling. On the basis of improving the overlapping relationship of the gap probabilities involved in the single-scattering contribution, we derived multiple-scattering equations suitable for the GO approach. Finally, we established a row model to accurately calculate the canopy reflectance of row crops.
