SGOT: A Simpliﬁed Geometric-Optical Model for Crown Scene Components Modeling over Rugged Terrain

: Topography affects the fraction of scene components of the canopy and background, resulting in the observed reﬂectance distortion. Modeling the canopy reﬂectance over rugged terrain needs to account for topographic effects. For this purpose, the existing models greatly increased the mathematical complexity while improving description of terrain and crown structure, which dramatically decreased the computational efﬁciency so as to limit their universal application. In this study, we developed a simpliﬁed geometric-optical model (SGOT) for simulating the scene components over rugged terrain. The geotropism of tree growth was considered to make SGOT physically sound. The internal structure of crown was simpliﬁed to make SGOT mathematically simpler. Scene component observations derived from Persistence of Vision Ray-tracer (POV-Ray) on surfaces with different normal directions and simulations were made using Geometric-Optical and Mutual Shadowing Coupled with Topography Model (GOMST) and Geometric-Optical for Sloping Terrains Model GOST; models were combined to test the SGOT model. In addition, topographic factors and crown density effect on the scene components modeling were analyzed. The results indicated that SGOT has good accuracy (R 2 for the areal proportions of sunlit crown ( Kc ), sunlit background ( Kg ), shaded crown ( Kt ), and shaded background ( Kz ) are 0.853, 0.857, 0.914, and 0.838, respectively) compared with POV-Ray simulation, and performs better than GOMST, especially in scenes with high crown density. Moreover, SGOT outperformed the compared models in computational efﬁciency (4% faster than GOMST and 29.5% faster than GOST). Finally, the simulations of the scene components distribution in different topographic factors and crown density were further discussed. SGOT and GOST can both capture scene component variations caused by terrain better than GOMST, but comparatively, SGOT provides a more efﬁcient tool to simulate the crown scene components because of its physical soundness and mathematical simplicity, and consequently, it will facilitate the modeling of canopy reﬂectance over mountainous regions. were used for comparison. The results showed that the SGOT model can accurately simulate the scene components over different sloping surfaces (R 2 for the areal proportions of sunlit crown ( Kc ), sunlit background ( Kg ), shaded crown ( Kt ), and shaded background ( Kz ) are 0.853, 0.857, 0.914, and 0.838, with RMSE of 0.0347, 0.0342, 0.0267, and 0.0374, respectively) and reproduce the topographic effects on the simulations of scene components well. Compared with the GOMST model, the SGOT model has higher simulation accuracy in the forests, especially with dense crown density. In addition, the computational time of the SGOT model is 29.5% faster than the GOST model; the efﬁciency of the SGOT model has been signiﬁcantly improved. This contribution demonstrated that the SGOT model can improve simulation accuracy and computational efﬁciency in scene components modeling over rugged terrain. It can help us to understand the topographic effects on scattered components and improve the performance of the radiation contributions calculation, and also provide an efﬁcient and reliable tool for upscaling the reﬂectance of canopy components to the canopy. The projection method and the simpliﬁed scheme of tree crown proposed in SGOT will be expected to be a paradigm to extend and improve the performance of the existing canopy reﬂectance models. Further improvements to the SGOT model would include a more applicable projection algorithm, accounting for tree trunk effects, and exploration of the potential of the SGOT model to improve the performance of canopy reﬂectance models.


Introduction
Canopy reflectance models provide causal relationships between the vegetation structural parameters and remote sensing observations, and they have served as effective tools to explore biophysical variables from remote sensing observations [1], including radiative transfer (RT) [2,3], geometric-optical (GO) [4,5], hybrid GO-RT [6,7], and computer simulation models [8,9]. For geometric-optical (GO) models, the observed reflected signals are composed of scattering contributions from canopy and background in sunlight and shadow, which are called crown scene components that are the key parameters to control the area fraction of different radiation-scattering components [4,10]. However, with the variations of the surface normal direction, the fraction of the crown scene components of sloping simulation accuracy and computational efficiency is required in the development of the crown scene components simulation model.
In this paper, we presented a simplified geometric-optical model for crown scene components over rugged terrain based on extension of the GOMST model (hereafter referred to as the SGOT model), in which geotropism of tree growth and gap fraction within crowns were taken into account, and we ensured that the mechanism would be complete as well as easy to implement. To verify the feasibility of the model, we compared the simulation scene components of the SGOT model and POV-Ray over the surfaces of different normal directions. The variations of the scene components over rugged terrains are related to the canopy structure parameters and topographic factors; in particular, slope, aspect, and crown density were recognized as the main factors affecting the scene components simulation. This paper also compared the simulation accuracy and computational efficiency of the SGOT model with the GOMST and GOST models. The development of the SGOT model is described in Section 2. The design of the experiment is given in Section 3. The validation results and analysis are given in Section 4, followed with the discussion in Section 5 and a conclusion in Section 6.

SGOT Model Development
According to the GO-like model [4], the forest canopy reflected signals received by sensors are modeled as the sum of the reflectance of individual components weighted by their areas within the pixel: where Kc, Kg, Kt, and Kz represent the areal proportions of four crown scene components, i.e., sunlit crown, sunlit background, shaded crown, and shaded background, respectively; C, G, T, and Z are the reflectance factors of four scene components. The derivation of areal proportions of scene components is the key step for canopy reflectance simulation. In the proposed model, a new projection algorithm of tree crowns on the sloping surface is established, which can retain the geotropism of tree growth. In particular, the tree crown on a sloping surface is represented as a volume consisting of numerous discrete leaves; therefore, the gap fraction is a crucial variable to determine the sunlit components. Unlike the GOMST model, in which the crown is treated as an opaque rigid body, the new model needs to consider the gap fraction within crowns. Here, we referred to the gap fraction formula in the GOST model, which is computed from input parameters such as leaf area index and extinction coefficient.

Crown Shape Transformations
According to the GO model derivation, the shape of a tree was generally assumed to be an ideal 3-D geometric shape according to the geometric characteristics of tree species [5,10,17,23,30]. For example, the "cone + cylinder" crown shape was assumed in the GOST model [17], and the ellipsoidal crown shape was assumed in the GOMST model [22]. The GOST model can be adapted to different crown shapes by adjusting the calculation formula of the crown projected area. However, the GOMST model can only be applicable to simulate the scene components of ellipsoid-shaped tree crowns. Considering that the proposed model needs to be compared with GOMST and GOST models, to facilitate the comparison among different models, the ellipsoidal crown shape was assumed in SGOT and GOST models in this study.
The computation of the canopy projected area over a sloping surface contains two steps, including the crown shape transformation and the projected area topographic transformation. The first step is scene stretching (Figure 1); the purpose of this step is to change the crown shape into a sphere, to facilitate the calculation of the crown projected area. The scene stretching can alter all the geometric parameters in the same ratio of major axis to minor axis (b/r); therefore, it has no effect on the simulation of scene components [5,26]. The second step is to establish the mathematical relationship between the vertically pro-Remote Sens. 2022, 14, 1821 4 of 24 jected area on a horizontal surface and the projected area on a sloping surface. After scene stretching, the ratio of major axis to minor axis is converted from b: r to r: r. Then, the ellipsoid crown is replaced with spheres that cast the same shadow area. The solar zenith angle θ s and the slope angle α will be changed in the above steps, and the equations are as following [22]: where θ s and θ s are the solar zenith angles before and after scene stretching; b and r are the major and minor axes of the ellipsoid crown; after scene stretching to spherical crown shape, b equals r ( Figure 1b); α and α are the slope angles before and after scene stretching. Similarly, using the view zenith angle θ v instead of θ s , Equation (2) can also be used to calculate the view zenith angle θ v after scene stretching. h and h are the mean height of the crown center before and after scene stretching.
nents [5,26]. The second step is to establish the mathematical relationship between vertically projected area on a horizontal surface and the projected area on a sloping face. After scene stretching, the ratio of major axis to minor axis is converted from b: r r. Then, the ellipsoid crown is replaced with spheres that cast the same shadow area solar zenith angle θs and the slope angle α will be changed in the above steps, and equations are as following [22]: where θs and θ's are the solar zenith angles before and after scene stretching; b and the major and minor axes of the ellipsoid crown; after scene stretching to spherical cr shape, b equals r ( Figure 1b); α and α' are the slope angles before and after s stretching. Similarly, using the view zenith angle θv instead of θs, Equation (2) can al used to calculate the view zenith angle θ'v after scene stretching. h and h' are the m height of the crown center before and after scene stretching. After scene stretching, the crown shape becomes spherical. The meani each parameter is the same as that in Equations (2)-(4).

Projection of Tree Crowns over Sloping Surfaces
The projection of tree crowns is the foundation of scene components simulatio the SGOT model, we designed a new projection relationship between the vertically jected area on horizontal surface and the projected area on sloping surface (as show Figure 2); the projected area in the sunlight direction can be defined as: where A is the vertically projected area on the horizontal surface; Aα is the projected on the sloping surface; φs and φα are solar azimuth angle and aspect; σ is the local s angle, which refers to the included angle formed by the intersection line of the obs After scene stretching, the crown shape becomes spherical. The meaning of each parameter is the same as that in Equations (2)-(4).

Projection of Tree Crowns over Sloping Surfaces
The projection of tree crowns is the foundation of scene components simulation. In the SGOT model, we designed a new projection relationship between the vertically projected area on horizontal surface and the projected area on sloping surface (as shown in Figure 2); the projected area in the sunlight direction can be defined as: where A is the vertically projected area on the horizontal surface; A α is the projected area on the sloping surface; ϕ s and ϕ α are solar azimuth angle and aspect; σ is the local slope angle, which refers to the included angle formed by the intersection line of the observation plane with the horizontal ground and the sloping surface [31]. The local slope angle contains effective information about slope, aspect, and solar/view azimuth angle; it is newly introduced into the crown projection method in SGOT, suitable for calculating the directional projection area of crowns. One must replace subscript s with  (5) and (6) can also give the projected area and the local slope angle in the observation direction, respectively.
tion plane with the horizontal ground and the sloping surface [31]. The local slo contains effective information about slope, aspect, and solar/view azimuth an newly introduced into the crown projection method in SGOT, suitable for calcul directional projection area of crowns. One must replace subscript s with v; Equa and (6) can also give the projected area and the local slope angle in the observ rection, respectively. Compared to the coordinate transformation method adopted by GOMST, projection transformation method in SGOT has two advantages. First, it cons geotropic nature of tree growth; second, the steps of establishing slope coordi tems and recalculating the incident and observation angles of slope coordinat are omitted. According to Equation (5), the topography transformations con steps: firstly converting vertical projection to incident/observation projection 2a,b), and secondly, converting the horizontal projection of the incident/observ rection to the sloping projection of the incident/observation direction (Figure 2b, It is generally known that crown shape simply assumed to be the ellipsoid shape will significantly limit its practical application because the natural crow various shapes. To facilitate the comparison between SGOT and GOMST, in th iment, we only assumed that the crown is ellipsoid. By adjusting the horizontal p area in Equation (5) and introducing the shape-adjusting factor (for the details, p Equations (A1)-(A3) in Appendix A), other geometric shapes can also be used t it in SGOT if needed.

Crown Gap Fraction over Sloping Surfaces
The variation of gap fraction over sloping surfaces affects the upper bound dition of the radiative transfer process within a canopy through redistributing dent irradiance [32]. Introducing a simplified gap fraction as a critical intermed iable into the SGOT model is considered to be a clear improvement on modelin scene components. In the view direction, the gap fraction between crowns (P sloping surfaces can be defined as [17]: where S is the projection area of the sloping surface in the view direction; D is the of trees in a pixel; r is the radius of the crown. In this paper, we defined the distribution of trees as being randomly in sp assumed the canopies to be the "turbid medium" filled with randomly distribute Compared to the coordinate transformation method adopted by GOMST, the new projection transformation method in SGOT has two advantages. First, it considers the geotropic nature of tree growth; second, the steps of establishing slope coordinate systems and recalculating the incident and observation angles of slope coordinate system are omitted. According to Equation (5), the topography transformations contain two steps: firstly converting vertical projection to incident/observation projection (Figure 2a,b), and secondly, converting the horizontal projection of the incident/observation direction to the sloping projection of the incident/observation direction (Figure 2b,c).
It is generally known that crown shape simply assumed to be the ellipsoidal crown shape will significantly limit its practical application because the natural crowns have various shapes. To facilitate the comparison between SGOT and GOMST, in this experiment, we only assumed that the crown is ellipsoid. By adjusting the horizontal projected area in Equation (5) and introducing the shape-adjusting factor (for the details, please see Equations (A1)-(A3) in Appendix A), other geometric shapes can also be used to replace it in SGOT if needed.

Crown Gap Fraction over Sloping Surfaces
The variation of gap fraction over sloping surfaces affects the upper boundary condition of the radiative transfer process within a canopy through redistributing the incident irradiance [32]. Introducing a simplified gap fraction as a critical intermediate variable into the SGOT model is considered to be a clear improvement on modeling crown scene components. In the view direction, the gap fraction between crowns (P vg_b ) over sloping surfaces can be defined as [17]: where S is the projection area of the sloping surface in the view direction; D is the number of trees in a pixel; r is the radius of the crown.
In this paper, we defined the distribution of trees as being randomly in space and assumed the canopies to be the "turbid medium" filled with randomly distributed leaves. Therefore, according to Lambert-Beer's law, the gap fraction within a tree crown (P vg_i ) in the view direction can be represented as [33]: where L 0 is the leaf area index in the view direction, K is the extinction coefficient in the view direction, and can be written as: where G is the leaf projection function [3] defined as the area of a unit LAI projected along the view direction, and l is the mean of path length through a tree crown. In the view direction, it is calculated by the ratio of the volume of crown to the projected area in the view direction: Therefore, the total gap fraction over a sloping surface in the view direction (P vg ) can be represented as the sum of gap fraction between crowns and within crowns: With solar zenith angle θ s instead of view zenith angle θ v in the above equations, Equation (10) can be used to calculate the total gap fraction over a sloping surface in the sunlight direction (P sg ).

Areal Proportions of Scene Components over Sloping Surfaces
According to the GOMST model derivation [22], the scene components of the background consist of the projections of sunlit and shaded background area in the view direction. In particular, the shaded background may be obscured by the crowns in the view direction; the overlap between the sunlight shadow and the observing shadow is given as: where t is given as [5]: where ϕ sv is the relative azimuth angle between the sun and the sensor, h is mean height of crown center after scene stretching, r is the radius of the crown after scene stretching. Therefore, using Boolean theory [34] and considering the gap fraction within crowns, the areal proportions of sunlit background and shaded background can be defined as: where λ is the crown count density, defined as the ratio of the number of tree crowns in a pixel to the area of the pixel, P sg_i is the gap fraction within crowns in the solar direction.
With Kg defined, the mutual shadowing factor f can be used to establish a relationship with Kc and Kg. The mutual shadowing factor f is given as: The derivation of f can be found in previous studies [5,22]. Finally, with Kc defined, Kt is given as:

Strategy for Evaluating the Performance of the SGOT Model
The evaluation of the results of crown scene components modeling over sloping surfaces is still a great challenge, because the measurement in real forests is very difficult and expensive [17]. The computer graphics technique provides us with a feasible verification method [23]. It is used to construct the 3-D virtual crowns, and then the information about the crowns and background can be retrieved from the multi-angle observation images of forest scenes [35]. In this study, the computer 3-D virtual model Persistence of Vision Ray-tracer (POV-Ray) was used to construct the 3-D virtual crowns and then to compare the crown scene components with those of the simulation by the proposed model over sloping surfaces. At first, taking the simulated 3-D virtual scenes by POV-Ray as the reference, we evaluated the performance of the SGOT model in crown scene components simulation. The simulation accuracy of the SGOT model can be quantified by the determination coefficients (R 2 ) and root-mean-square error (RMSE). Then, the mechanism correctness of the SGOT model was analyzed according to the basic GO physical principle; for example, the characteristics of scene components should change with topographic factors and view directions, e.g., the hotspot phenomenon [36]. In addition, to reveal the improvement of the SGOT model, two classical GO-like modes (GOMST and GOST) were selected to compare with SGOT on crown scene components simulation; the advantages and disadvantages of the compared models under different conditions were discussed. Considering the simplified scheme adopted in SGOT, the improvement of the calculation efficiency of the SGOT model was also evaluated subsequently.

Simulation of Scene Components with Computer 3-D Virtual Model
In this study, the scene components of forest scenes with different canopy density (sparse, medium, and dense) over sloping surfaces were generated by POV-Ray. POV-Ray is a ray-tracing software and is often devoted to image rendering and synthesis, which can simulate the interactions between ray and objects [37,38]. As shown in Figure 3, the forest canopy scene was generated by POV-Ray; we can clearly see the canopy scene components from different perspectives. As efficient and easy-to-operate software, POV-Ray has been widely used in building various 3-D scenes [39].  The forest virtual scene was set in an area of 100 m × 100 m with 55, 138, and 220 randomly positioned spherical crowns. The crowns' vertical axis and horizontal axis were set as 4.5 m and 3.4 m, the average crown center height was set as 5 m; correspondingly, the vegetation coverage of three forest scenes was 20% (sparse), 50% (medium), and 80% (dense), respectively. The LAIs of three virtual forest scenes were set as 1, 2.5, and 4, to ensure the identical leaf area density of a crown. To accomplish the multi-angle observation of the forest scenes, the solar zenith and azimuth angles were 20 • and 0 • , respectively, the view zenith angle ranged from 0 • to 80 • , and the view azimuth angle ranged from 0 • to 360 • , which basically covers the whole observation field. The slope increased from 0 • to 60 • with a step of 10 • , including horizontal surface, medium slope surface, and steep slope surface, and the aspect was set as 0 • , 90 • , 180 • , and 270 • , which represents the slope surface facing north, east, south, and west, respectively.
The specifications of the POV-Ray input parameters for forest scenes generation are summarized in Table 1. A total of 84 forest scenes with 65 view directions were constructed by POV-Ray simulation, covering various crown densities, terrain conditions, and view geometry. The scene can be separated into sunlit and shaded parts under the virtual parallel white light in the orthographic projection camera. The multi-angle images of the virtual forest scene were rendered by changing the positions and the view directions of camera. The output of the POV-Ray is an image of the virtual forest scene, in which different scene components have obvious differences in DN values in each layer, and the scene components can be easily classified from the image.

Input Parameter Settings of Each Compared GO-like Model
The SGOT was compared with GOMST and GOST in this study. Three GO-like models were set as the same input parameters, and the same as the settings of POV-Ray simulation, shown in Table 1. To give prominence to the difference of simulation results over different slopes, the relative azimuth angles between the sun and the sloping surfaces were set to 0 • . The simulation of view angle should be as wide as possible to reveal the trend of scene components along the view principal plane; therefore, the view zenith angle ranged from 0 • to 80 • with the step of 10 • , and the view azimuth angle was set at 0 • and 180 • .

SGOT Model Validation through 3-D Virtual Canopy Model Simulations
In this section, the POV-Ray was used to generate reference scene components for validating the SGOT model. Density scatterplots between the scene components simulated by the SGOT model and generated by POV-Ray are shown in Figure 4. The scene components over different sloping surfaces and different view directions (see Table 1 upper part of the crowns to exceed the edge of the observable background. In general, the scene components simulated by the SGOT model basically coincide with the POV-Ray simulations over different surfaces and view directions, which indicates that the SGOT model has the ability to accurately simulate the scene components over sloping surfaces. that each scene component simulated by POV-Ray and SGOT model has high con-sistency. The SGOT simulated scene components are consistent with POV-Ray simulations, with the RMSE (R 2 ) of Kc, Kg, Kt, and Kz being 0.0347 (0.853), 0.0342 (0.857), 0.0267 (0.914), and 0.0374 (0.838), respectively. However, Figure 4d also shows that Kz is slightly overestimated in the high-value region, where the view azimuth is near the nadir. This may cause the upper part of the crowns to exceed the edge of the observable background. In general, the scene components simulated by the SGOT model basically coincide with the POV-Ray simulations over different surfaces and view directions, which indicates that the SGOT model has the ability to accurately simulate the scene components over sloping surfaces. As illustrated in Figure 5, the similar distribution patterns can be figured out between the SGOT model simulated scene components (Figure 5a,c,e,g) and the POV-Ray As illustrated in Figure 5, the similar distribution patterns can be figured out between the SGOT model simulated scene components (Figure 5a,c,e,g) and the POV-Ray simulated scene components (Figure 5b,d,f,h) in the principal plane. The results show that the SGOT and POV-Ray results have good consistency over different sloping surfaces. It indicates that the SGOT model has the ability to separate the scene components from sloping forest scenes. In addition, by comparing the scene components of different slopes in the same view direction, when the view direction is at the nadir in the down-slope direction, the relative differences of Kc between horizontal and sloping surfaces can reach up to 51.7% (Figure 5a,b); when the view zenith angle is 0 • , the differences of Kg between horizontal and sloping surfaces can reach up to 29.3% (Figure 5c,d). Therefore, neglecting the effects of topography in GO-like models can lead to significant errors in the scene components simulation, and errors can consequently pass to canopy reflectance modeling.
down-slope direction, the relative differences of Kc between horizontal and sloping surfaces can reach up to 51.7% (Figure 5a,b); when the view zenith angle is 0°, the differences of Kg between horizontal and sloping surfaces can reach up to 29.3% (Figure 5c,d). Therefore, neglecting the effects of topography in GO-like models can lead to significant errors in the scene components simulation, and errors can consequently pass to canopy reflectance modeling.

Analysis of Topographic Effects on Scene Components by SGOT Modeling
To investigate the topographic effects on scene components simulation, the scene components over surfaces with different slopes and aspects were simulated using the SGOT model. The hotspot is an important phenomenon that can be used for retrieving canopy structural parameters [40]. In this experiment, the solar zenith angle is set to 20 • ; when the view zenith angle is equal to the solar zenith angle, the hotspot occurs (see Figure 5). The SGOT model can successfully capture the significant increases in the simulations of scene components at the hotspot direction. The sunlit crown and background reach the peak and the shaded crown and background are 0% at the hotspot because the shaded scene components cannot be observed. The sum of the gap fraction between crowns and within crowns increases with the increase of slope; therefore, the sunlit background increases and the sunlit crown decreases in the hotspot direction (Figure 5a,c).
The lines in Figure 5a,c,e,g are smoother than those in Figure 5b,d,f,h; these slight differences may be caused by the image information extraction of POV-Ray. The setting of observation conditions (such as image resolution and camera height) in POV-Ray can affect the accuracy of image segmentation, and then affect the scene component results. Despite the lines of POV-Ray results being unsmooth, the scene components simulated by the SGOT model are very close to the POV-Ray results, and this indicates that SGOT is reliable for simulating the scene components over sloping terrain.
The gap fraction is a critical variable that can affect the scene components. Figure 6 shows the angular distributions of gap fraction in view directions over different sloping surfaces; the input parameters are the same as those in Figure 5. With the slope increasing, the gap fraction increases in the down-slope direction and decreases in the up-slope direction. Therefore, the probability of observing the background increases with the increases of slope in the down-slope direction, corresponding to the variations of Kg and Kz when the view zenith angle ranges from 20 • to 80 • in Figure 5c

Analysis of Topographic Effects on Scene Components by SGOT Modeling
To investigate the topographic effects on scene components simulation, the scene components over surfaces with different slopes and aspects were simulated using the SGOT model. The hotspot is an important phenomenon that can be used for retrieving canopy structural parameters [40]. In this experiment, the solar zenith angle is set to 20°; when the view zenith angle is equal to the solar zenith angle, the hotspot occurs (see Figure 5). The SGOT model can successfully capture the significant increases in the simulations of scene components at the hotspot direction. The sunlit crown and background reach the peak and the shaded crown and background are 0% at the hotspot because the shaded scene components cannot be observed. The sum of the gap fraction between crowns and within crowns increases with the increase of slope; therefore, the sunlit background increases and the sunlit crown decreases in the hotspot direction ( Figure  5a,c).
The lines in Figure 5a,c,e,g are smoother than those in Figure 5b,d,f,h; these slight differences may be caused by the image information extraction of POV-Ray. The setting of observation conditions (such as image resolution and camera height) in POV-Ray can affect the accuracy of image segmentation, and then affect the scene component results. Despite the lines of POV-Ray results being unsmooth, the scene components simulated by the SGOT model are very close to the POV-Ray results, and this indicates that SGOT is reliable for simulating the scene components over sloping terrain.
The gap fraction is a critical variable that can affect the scene components. Figure 6 shows the angular distributions of gap fraction in view directions over different sloping surfaces; the input parameters are the same as those in Figure 5. With the slope increasing, the gap fraction increases in the down-slope direction and decreases in the up-slope direction. Therefore, the probability of observing the background increases with the increases of slope in the down-slope direction, corresponding to the variations of Kg and Kz when the view zenith angle ranges from 20° to 80° in Figure 5c   To explore the scene component angular distributions over different terrain conditions, we simulated the scene components over different slopes and aspects based on the SGOT model, as shown in Figures 7 and 8 To explore the scene component angular distributions over different terrain conditions, we simulated the scene components over different slopes and aspects based on the SGOT model, as shown in Figures 7 and 8, respectively.  As illustrated in Figure 7, the first to the fourth columns represent the Kc, Kg, Kt, and Kz, respectively. The first to fourth lines represent that the slope is 0 • , 20 • , 40 • , and 60 • , respectively. The polar path represents view zenith angle, and the polar angle represents view azimuth angle. Especially when the view zenith angle is greater than (π/2)-arctan(tanαcos(ϕ v -ϕ α )) along the up-slope direction, and the sensor cannot observe the target sloping surface, the topographic mask appears. Considering that the relative azimuth angle between the sun and the sloping surface is 0 • , the results show that the scene components exhibit bilateral symmetry along the view principle plane over both the horizontal and sloping surfaces. Comparing the scene component distributions over the different sloping surfaces, with the slope becoming steeper, the distortion of scene components images becomes serious along the vertical plane. , and Kz (j-l), respectively. The first to third rows represent that the aspect is 90°, 180°, and 270°, respectively.
As illustrated in Figure 7, the first to the fourth columns represent the Kc, Kg, Kt, and Kz, respectively. The first to fourth lines represent that the slope is 0°, 20°, 40°, and 60°, respectively. The polar path represents view zenith angle, and the polar angle represents view azimuth angle. Especially when the view zenith angle is greater than (π/2)-arctan(-tanαcos(φv-φα)) along the up-slope direction, and the sensor cannot observe the target sloping surface, the topographic mask appears. Considering that the relative azimuth angle between the sun and the sloping surface is 0°, the results show that the scene components exhibit bilateral symmetry along the view principle plane over both the horizontal and sloping surfaces. Comparing the scene component distributions over the different sloping surfaces, with the slope becoming steeper, the distortion of scene components images becomes serious along the vertical plane.
As shown in Figure 7b-d, in the up-slope direction, the magnitudes of Kc distributed on both sides of the vertical plane are larger than those distributed in the middle. According to the variations of Kt in Figure 5e, because the azimuth of the sun and the viewer is opposite, the probability of observing the shaded crown increases at the nadir view direction along the view principal plane. Therefore, as the relative azimuth angle between the viewer and the sloping surface decreases from 180°, the probability of observing the sunlit middle and lower parts of the crown increases gradually. Figure 8 shows how the scene component changes with the aspect ranging from 90° As shown in Figure 7b-d, in the up-slope direction, the magnitudes of Kc distributed on both sides of the vertical plane are larger than those distributed in the middle. According to the variations of Kt in Figure 5e, because the azimuth of the sun and the viewer is opposite, the probability of observing the shaded crown increases at the nadir view direction along the view principal plane. Therefore, as the relative azimuth angle between the viewer and the sloping surface decreases from 180 • , the probability of observing the sunlit middle and lower parts of the crown increases gradually. Figure 8 shows how the scene component changes with the aspect ranging from 90 • to 270 • in 90 • intervals. Comparing the scene components over the sloping surfaces with aspects of 90 • (Figure 8a,d,g,j) and 270 • (Figure 8c,f,i,l), their distributions are completely symmetrical. However, there are some obvious differences between the distributions of scene components with aspects of 0 • (Figure 7a,e,i,m) and 180 • (Figure 8b,e,h,k). Different aspects alter the gap fraction in the solar direction, resulting in the increase of the sunlit area when the relative azimuth angle between the sun and the sloping surface is less than 90 • , or the decrease of the sunlit area when the relative azimuth angle between the sun and the sloping surface is more than 90 • . Therefore, when the sloping surface faces the sun, Kt and Kz decrease in the direction close to the sun (see Figure 7j,n) because the shadow area observed in the direction near the sun is less than that of 180 • in Figure 8h,k; meanwhile, the distributions of Kc (Figures 7b and 8b) and Kg (Figures 7f and 8e) show the opposite characteristics to Kt and Kz. To reveal the effects of slopes and aspects, the experiments show that the different slope and relative azimuth angles among the sun, the viewer, and the slope can cause significant changes in scene components. The SGOT model shows good mechanism performance in the estimations of scene components and can accurately capture the variations of scene components under different illumination, observation, and terrain conditions.
In addition, as shown in Figure 9, the scene components of Kt over sloping surfaces with different crown density were also compared. The results show that crown density is another important factor affecting scene components, except for terrain factors. Different from the sparse forest (the first row in Figure 9), with the increase of crown density (see from row 2 to row 3 in Figure 9), the contours are more densely distributed in the up-slope direction and more dispersed in the down-slope direction. The higher crown density means the gap size between crowns becomes smaller, the probability of observing the crowns increased and that of the background decreased, respectively. Especially when the view direction is far from the solar direction, the magnitude of Kt increases significantly with the increase of crown density. With the increase of slope, Kt shows similar trends in the three vegetation coverage areas.
90°, or the decrease of the sunlit area when the relative azimuth angle between the su and the sloping surface is more than 90°. Therefore, when the sloping surface faces th sun, Kt and Kz decrease in the direction close to the sun (see Figure 7j,n) because th shadow area observed in the direction near the sun is less than that of 180° in Figure 8h, meanwhile, the distributions of Kc (Figures 7b and 8b) and Kg (Figures 7f and 8e) sho the opposite characteristics to Kt and Kz. To reveal the effects of slopes and aspects, th experiments show that the different slope and relative azimuth angles among the su the viewer, and the slope can cause significant changes in scene components. The SGO model shows good mechanism performance in the estimations of scene components an can accurately capture the variations of scene components under different illuminatio observation, and terrain conditions.
In addition, as shown in Figure 9, the scene components of Kt over sloping surface with different crown density were also compared. The results show that crown density another important factor affecting scene components, except for terrain factors. Differen from the sparse forest (the first row in Figure 9), with the increase of crown density (se from row 2 to row 3 in Figure 9), the contours are more densely distributed in th up-slope direction and more dispersed in the down-slope direction. The higher crow density means the gap size between crowns becomes smaller, the probability of observ ing the crowns increased and that of the background decreased, respectively. Especiall when the view direction is far from the solar direction, the magnitude of Kt increase significantly with the increase of crown density. With the increase of slope, Kt show similar trends in the three vegetation coverage areas.

Slope increasing
Crown density increasing To summarize, the above analysis of GOST simulations shows that the main factors affecting the scene components over sloping surfaces include slope, aspect, and crown density. Among them, the slope and the crown density can alter the magnitude of scene components by adjusting the gap fraction, while the aspect can affect the distribution pattern of scene components by altering the relative azimuth angle among the sun, the viewer, and the sloping surface.

Comparison with Typical GO-Like Models
As shown in Figure 10, the scene component simulations of SGOT model and GOMST model have good consistency, just with a slight difference on different surfaces. Considering the formulation of the gap fraction within crowns, the area of background observed in SGOT scenes is larger than that of GOMST scenes. Therefore, Kg and Kz simulated by GOMST are slightly higher than SGOT simulations, and the difference between GOMST and SGOT reaches the maximum at the hotspot. Similarly, Kc and Kt simulated by GOMST are slightly lower than SGOT simulations.
However, there are systematic differences between the simulations of the SGOT model and the GOST model. Figure 10 also shows that the scene components from crowns (Kc and Kt) simulated by SGOT are higher than the GOST simulations. In addition, the scene components from background (Kg and Kz) simulated by SGOT are lower than the GOST simulations as shown in Figure 10b,d,f,h,j,l. These comparison results indicate that more sunlight can reach the background through the crowns in GOST simulation than in SGOT and GOMST. By comparing the tree distribution patterns in two models, the Neyman type-A distribution [41] and the Poisson distribution were assumed in the GOST model and the random distribution was assumed in the SGOT model and the GOMST model. The different tree distribution patterns can affect the projected area of the background in the solar and view directions, and the difference between the SGOT and GOST simulations increases with the increase of view zenith angle, because of the larger number of overlapping crowns. Therefore, there is a certain gap between the magnitude of scene component simulations of the SGOT model and the GOST model.
In addition, compared with the GOMST model, the SGOT model takes the gap fraction within canopies into account. The gap fraction within canopies has different effects on the results in the areas with different crown density. Therefore, the scene component simulations between the SGOT model and the GOMST model in the higher crown density area may be quite different. Figure 11 shows the scatterplots between simulated scene components from the different models with spare, medium, and dense crown densities, which were set as 0.2, 0.5, and 0.8, respectively. For the spare crown density forest scenes (Figure 11a-d), the gap fraction within crowns accounts for a small proportion in the whole scene; the simulation accuracy of the SGOT model is almost the same as that of the GOMST model. However, with the increase of crown density, the difference between the simulation results of the GOMST model and the SGOT model increases gradually. For the medium and dense crown density forest scenes, Figure 11e Figure 11a,c,e,g,i,k, the scene components of crown simulated by the GOST model are underestimated, and as shown in Figure 11b,d,f,h,j,l, the scene components of background simulated by the GOST model are overestimated, which may also be caused by the different tree distribution patterns assumed in SGOT and GOST. Therefore, different endogenous mechanisms of the compared models lead to different results, as mentioned in Figure 11; with the increase of view zenith angle, the probability of observing the background in the GOST scene is higher than that in the SGOT and GOMST scenes. It can be concluded that the simulation accuracy of the SGOT model is better than that of the GOST model when the distribution of forest stand is random. Therefore, the SGOT model appears to have better performance in scene component simulation, especially in areas with high crown density.

Comparison with Typical GO-Like Models
As shown in Figure 10, the scene component simulations of SGOT model and GOMST model have good consistency, just with a slight difference on different surfaces. Considering the formulation of the gap fraction within crowns, the area of background observed in SGOT scenes is larger than that of GOMST scenes. Therefore, Kg and Kz simulated by GOMST are slightly higher than SGOT simulations, and the difference between GOMST and SGOT reaches the maximum at the hotspot. Similarly, Kc and Kt simulated by GOMST are slightly lower than SGOT simulations. Figure 10. Comparisons between scene components models along the view principal plane over the horizontal surface. The first to fourth rows represent Kc, Kg, Kt, and Kz, respectively. The first to third columns represent the slope of 0°, 20°, and 40°, respectively.  Considering the difference in internal mechanism of each model, different view directions, slopes, and crown densities may have different effects on the results. To figure out the influence of each factor, the simulation accuracy of each model in different view azimuths (the interval of the azimuth was set to 45°) and various crown densities (the same as the settings in Figure 11) over the slight (<10°), moderate (10°-20°), and steep (>20°) sloping surfaces (the aspect was set to 0°) was compared in this section. Figure 12 shows the correlation between the simulated sunlit crown of different models and POV-Ray results; the correlation coefficients of the SGOT, GOMST, and GOST simulations were represented by the green, black, and red lines, respectively; the closer the line is to the outermost circle, the better the simulation result of the model. With the slope increasing, the gap between the correlation coefficients of SGOT and GOMST results is Considering the difference in internal mechanism of each model, different view directions, slopes, and crown densities may have different effects on the results. To figure out the influence of each factor, the simulation accuracy of each model in different view azimuths (the interval of the azimuth was set to 45 • ) and various crown densities (the same as the settings in Figure 11) over the slight (<10 • ), moderate (10-20 • ), and steep (>20 • ) sloping surfaces (the aspect was set to 0 • ) was compared in this section. Figure 12 shows the correlation between the simulated sunlit crown of different models and POV-Ray results; the correlation coefficients of the SGOT, GOMST, and GOST simulations were represented by the green, black, and red lines, respectively; the closer the line is to the outermost circle, the better the simulation result of the model. With the slope increasing, the gap between the correlation coefficients of SGOT and GOMST results is gradually increasing, especially when the view azimuth is close to 0 • , because there are fewer crowns overlapped in the direction of sight in the down-slope direction, and the gap fraction within crowns has a significant effect on the results. As shown in Figure 12c,f,i, with the increase of crown density, the improvement of SGOT becomes more obvious. This comparison indicates that SGOT can adapt to a larger range of view azimuth, slope, and crown density. gradually increasing, especially when the view azimuth is close to 0°, because there are fewer crowns overlapped in the direction of sight in the down-slope direction, and the gap fraction within crowns has a significant effect on the results. As shown in Figure  12c,f,i, with the increase of crown density, the improvement of SGOT becomes more obvious. This comparison indicates that SGOT can adapt to a larger range of view azimuth, slope, and crown density. With respect to the computational efficiency of each model, as shown in Figure 13, in this experiment, the computational efficiency of the SGOT model is slightly better than the GOMST model and is obviously better than that of the GOST model. The timer recording shows that it requires 29.5% less time to run the SGOT model than the GOST model and 4% less time than the GOMST model for scene component simulations. With respect to the computational efficiency of each model, as shown in Figure 13, in this experiment, the computational efficiency of the SGOT model is slightly better than the GOMST model and is obviously better than that of the GOST model. The timer recording shows that it requires 29.5% less time to run the SGOT model than the GOST model and 4% less time than the GOMST model for scene component simulations. Compared with the GOST model, the SGOT model ignored the 3-D complex structure of the crowns; the distribution of leaves in the crown was not cons SGOT. Therefore, the calculation of gap fraction in SGOT was simpler than th GOST model, which greatly reduced the computational time. It can be specul considerable time can also be saved in the subsequent canopy reflectance calcula

Discussion
Crown scene components play an important role in canopy reflectance m but topography affects the fraction of scene components of the canopy and bac resulting in the observed reflectance distortion. For correcting this distortion, the models greatly increased the mathematical complexity while improving descr terrain and crown structure, which dramatically decreased the computational e so as to limit their universal applications. In this paper, we developed a simpli metric-optical model (SGOT) to simulate crown scene components over slopin and investigated the effects of topographic factors on scene components. To achi goals, we presented an extension of the GOMST model, mainly focusing on tw in scene components modeling. First, a new projection relationship between the projected area on a horizontal surface and a sloping surface projected area is pro make the projection conversion process more convenient and this can retain th pism of tree growth. Second, the study gives a detailed account of a simplified sc the internal structure of the tree crown to make the SGOT model more physical and computationally efficient.
However, the crown shape was assumed to be ellipsoid in this study; this hy refers to the GOMST model to facilitate comparison between models. Previou have shown that the crown shape has significant effects on the simulation components [42]; the crown projection area in different planes and the volum crown need to be adjusted according to other specific shapes [17]. As mentione the projected area of the tree crown can be derived from the vertically projected the shape-adjusting factor; the SGOT model has the ability to simulate the sc ponents of tree crowns with different shapes (for details, please see Appendix A tionally, the crown was assumed to be uniformly filled with leaves in the p model; this assumption referred to extending ROSS [3] to sloping surfaces by C Compared with the GOST model, the SGOT model ignored the 3-D complex internal structure of the crowns; the distribution of leaves in the crown was not considered in SGOT. Therefore, the calculation of gap fraction in SGOT was simpler than that in the GOST model, which greatly reduced the computational time. It can be speculated that considerable time can also be saved in the subsequent canopy reflectance calculation.

Discussion
Crown scene components play an important role in canopy reflectance modeling, but topography affects the fraction of scene components of the canopy and background, resulting in the observed reflectance distortion. For correcting this distortion, the existing models greatly increased the mathematical complexity while improving description of terrain and crown structure, which dramatically decreased the computational efficiency so as to limit their universal applications. In this paper, we developed a simplified geometricoptical model (SGOT) to simulate crown scene components over sloping terrain and investigated the effects of topographic factors on scene components. To achieve these goals, we presented an extension of the GOMST model, mainly focusing on two aspects in scene components modeling. First, a new projection relationship between the vertically projected area on a horizontal surface and a sloping surface projected area is proposed to make the projection conversion process more convenient and this can retain the geotropism of tree growth. Second, the study gives a detailed account of a simplified scheme for the internal structure of the tree crown to make the SGOT model more physically sound and computationally efficient.
However, the crown shape was assumed to be ellipsoid in this study; this hypothesis refers to the GOMST model to facilitate comparison between models. Previous studies have shown that the crown shape has significant effects on the simulation of scene components [42]; the crown projection area in different planes and the volume of the crown need to be adjusted according to other specific shapes [17]. As mentioned above, the projected area of the tree crown can be derived from the vertically projected area and the shape-adjusting factor; the SGOT model has the ability to simulate the scene components of tree crowns with different shapes (for details, please see Appendix A). Additionally, the crown was assumed to be uniformly filled with leaves in the proposed model; this assumption referred to extending ROSS [3] to sloping surfaces by Combal et al. [27]. In fact, the leaves are not uniformly and randomly distributed within crowns [10,17,23], e.g., in conifer stands, needles are grouped into shoots, branches, and whorls; all these substructures within crowns have a significant effect on the bidirectional reflectance properties [10]. However,