Quantitative Analysis of Terrain Reflected Solar Radiation in Snow-Covered Mountains: A Case Study in Southeastern Tibetan Plateau
Abstract
Terrain reflected solar radiation in snow-covered mountains is nonnegligible in investigations of the energy budget. However, it has so far not been investigated thoroughly, especially with regard to the influence of snow cover. Several parameterization approaches have been raised but not yet evaluated in a more uniform and quantitative manner. Based on the three-dimensional (3-D) ray-tracing simulation, we explored the temporal and spatial characteristics of the terrain reflected radiation in 15 domains on the southeastern Tibetan Plateau, and comprehensively evaluated different parameterization approaches. The results indicate that the ratio of reflected radiation to total daily radiation ranges from 0.25% to 10.85% at the scale of 5 × 5 km2 in a winter clear day, and it is 57% higher at noon in spring and autumn due to the higher snow cover fraction. Snow cover not only enhances the magnitude of reflected radiation by increasing surface albedo but also changes the spatial distribution pattern of radiation in partial snow-covered mountains, causing more snow-reflected radiation to be received by the surrounding surfaces. Three forms of terrain configuration factors in common parameterization approaches were evaluated by the ray-tracing model. The complementary of sky view factor shows good consistency with ray-tracing model at domain-averaged scales, with a root mean square error (RMSE) of 2.55 (14%) W/m2, while the other two both underestimate the radiation. The parameterization approach involving the multi-reflection shows better performance with the normalized RMSE decreasing by 5%. However, the uncertainty of it increases with the surface spatial heterogeneity caused by the partial snow cover, especially at high resolution.
Key Points
-
A ray-tracing model was used to investigate terrain reflected solar radiation in snow-covered high mountains
-
Partial snow cover leads to more incident reflected radiation on snow-covered slopes in spatial distribution
-
Performance of different parameterization approaches was evaluated in detail
1 Introduction
An accurate estimation of incident radiation is important in the studies of the energy budget, biomass estimation, evapotranspiration, and climate change monitoring (Budyko, 1969; Dorno, 1920; Wild et al., 2013). For snow-covered mountains, the incident radiation is a major driving force of the snow-melting process. Garen and Marks (2005) pointed out that the solar radiation input was the most complex and difficult part in the mountainous snow-melting modeling. Due to the high albedo of snow or ice in the rugged terrain, high-altitude mountains usually get more multiple reflections between adjacent slopes, which increase the incident radiation at the ground surface (Dozier, 1980; Duguay, 1993; Kargel et al., 2014; Proy et al., 1989). Disregarding the terrain reflected solar radiation will lead to an underestimation of downward irradiance (Sirguey, 2009), thereby further influencing the estimation of surface albedo, biomass, snow cover, evapotranspiration, and so on (Corripio, 2004; Gratton et al., 1993).
Several studies on terrain reflected solar radiation in mountainous areas have been conducted. Dozier (1980) reported that the instantaneous terrain reflected irradiance contributed on average 17% of global radiation with a maximum of 66% for a partly snow-covered study site in the Sierra Nevada, with a grid resolution of 100 m. Lee et al. (2013) utilized a 3-D Monte Carlo model to investigate the topographic effect over the Tibetan Plateau (TP) at a 90-m resolution and found that large deviations in the reflected solar radiation can be 200 W/m2 over high-albedo snow-covered areas. Similarly, based on Monte Carlo simulations, Chen et al. (2006) found that the domain-averaged terrain reflected solar radiation can account for 3% of the total surface fluxes during the winter time with a high surface albedo, for a 109 × 106 km2 domain size at 1 km resolution. Apart from simulations conducted in a specific area at a given time, Helbig et al. (2009) developed a radiosity approach to precisely simulate the terrain reflected solar radiation with random model topographies and homogeneous surface albedo values, and further investigated the influence of slope, solar elevation, grid resolution, and domain size on the magnitude of terrain reflected solar radiation. There is already a consensus that the terrain reflected solar radiation is nonnegligible and important, especially in snow-covered high albedo areas. However, the systematic analysis of terrain reflected radiation in snow-covered mountains is still insufficient, how much reflected radiation can be obtained under different terrain features, albedo conditions and solar angle needs to be investigated in more detail, the influence of inhomogeneous surface albedos results from partial snow cover is also not yet discussed.
Achieving an accurate estimation of the reflected radiation is complicated and computationally expensive (Sirguey, 2009). There are basically three types of approaches to compute terrain reflected solar radiation in complex terrain: the 3-D radiative transfer (RT) model, accumulation method, and parameterization. The 3-D RT model is one important approach to simulate the terrain reflected solar radiation precisely, can be used to analyze the complex interactions of radiation with surface (Davies, 2005; Disney et al., 2000; Li et al., 1999). The ray-tracing model is one type of 3-D RT model, it is the most physically “exact” method for calculating radiative transfer involving mountains, and was used in some radiative studies over rugged terrain (Chen et al., 2006; Lee et al., 2013; Liou et al., 2007). However, 3-D RT model is usually computationally expensive, which makes it not efficient in large-scale applications considering the current limited computing power.
The accumulation method proposed by Proy et al. (1989) was also used in some researches (Wang et al., 2000; Wang et al., 2018; Yan et al., 2016, 2018). It searches all visible pixels for each target pixel and then accumulates the energy that the target pixel gets. The computational cost makes the accumulation method very limited (Mousivand et al., 2015). In the practical implementation, different sampling methods of searching directions and interpolation algorithms may also increase the uncertainty of the results.
In the applications in many fields, an efficient and operational approach is necessary to achieve the fast estimation of terrain reflected solar radiation. Hence, simplified parameterization approaches with terrain configuration factor (
) have been widely used in radiation modeling over rugged terrain (Dozier & Frew, 1990; Dubayah & Loechel, 1997; Kondratyev, 1977; Shepherd & Dymond, 2003; Wang et al., 2006). The product of averaged reflected radiation of surrounding area and 
is usually utilized to estimate the reflected radiation. The terrain configuration factor, also known as obstruction coefficient, represents the fraction of surrounding pixels visible to the target pixel, can be calculated from the sky view factor (
), which is the ratio of the diffuse sky irradiance at a point to that on an unobstructed horizontal surface (Dozier & Frew, 1990). However, the forms of 
are not unified. Due to the lack of effective observation data of terrain reflected radiation, the different parameterization approaches have not been well evaluated.
According to the accuracy of RT model and the practicality of parameterization approach, a 3-D ray-tracing based RT model was used in our study to simulate the surface downward solar radiation (DSR) for mountainous areas in the Southeastern TP, as a substitute for ground observations to evaluate the different parameterization approaches. We aim to explore the temporal and spatial characteristics of terrain reflected solar radiation over various terrains, quantitatively analyze the influence of surface albedo, topographic features, and solar angles on the magnitude and distribution of reflected radiation, and furthermore assess the accuracy and uncertainty of the parameterization approaches in different conditions.
This study is organized as follows: In Section 2 we describe the data and models we used in this study. In Section 3, we will present the simulation results in different scenarios and analyze the influencing factors and spatial characteristics of terrain reflected radiation. In Section 4, the parameterization approaches are evaluated by ray-tracing model, the error analysis of parameterization approaches is also presented. Section 5 discusses the impact of terrain reflected solar radiation on net solar radiation. Discussion and conclusions are given in Section 6.
2 Data and Methods
2.1 Study Area and Data
TP is called “The Roof of the World” or the “Third Pole,” with an average elevation exceeding 4,000 m. It has very complex terrain structure and is sensitive to the global climate change. The southeastern TP is one of the most rugged regions in TP, with high snow cover fraction and depth throughout the year, was selected as our study area. Fifteen 20 × 20 km2 mountain domains were chose to implement the simulation in this region, using the 30 m resolution digital elevation model (DEM) data from ASTER GDEM V2. ASTER GDEM is a product of Japan’s Ministry of Economy, Trade, and Industry (METI) and NASA. The domains were selected to consist of different terrain features, including steep and relatively flat mountains, snow-covered and non-snow-covered mountains. The elevation values in all domains range from 963 to 6,218 m, with an average of 4,494 m. The location of study area and all domains is shown in Figure 1.
Location of study area and distribution of 15 domains (red boxes). The base map of Tibetan Plateau is from Google Earth. The right subplot is the surface albedo derived from Landsat 8 on January 29, 2018.
The broadband albedo was used in our 3-D simulation instead of the spectral reflectance. As shown by Liou et al. (2007), an experiment of Monte Carlo simulation was conducted utilizing the MODIS albedo for the visible and near-IR band versus one spectrally invariant albedo, the result verified the negligible difference comparing to the radiation anomalies between mountains and a flat surface. The diurnal variation of albedo was also tested before our simulation. The MODIS BRDF parameter in MCD43A1 was used to calculate the diurnal shortwave albedo of ground pixel. Two sets of Monte Carlo simulations were conducted with the diurnal albedo and constant albedo. The result demonstrates the small difference in DSR within ±1 W/m2, even for the snow-covered area. Therefore, for the sake of simplicity, the shortwave albedo calculated from the Landsat-8 OLI surface reflectance data was assumed as valid for the entire day. The product of Landsat-8 OLI surface reflectance in 30 m resolution was downloaded from the website of the United States Geological Survey (USGS) EarthExplorer (http://earthexplorer.usgs.gov/), and was topographic corrected with semi-empirical C correction method afterward (Teillet et al., 1982). The narrowband to broadband conversion coefficients for snow-covered ground and snow-free ground by Wang et al. (2016) was used to predict the surface shortwave albedo from the Landsat-8 OLI spectral reflectance. The statistical information of surface albedos (January 29, 2018) and altitudes in 15 domains is shown in Figure 2.
Boxplot of surface albedos on January 29, 2018 and the altitudes of 15 domains.
2.2 The Ray-Tracing Simulation of DSR
A LargE-Scale remote sensing data and image Simulation framework (LESS) is a ray-tracing based 3-D RT model, which has the advantage of high computational efficiency in 3-D RT modeling (Qi et al., 2017, 2019). It employs forward photon tracing mode to simulate downwelling and upwelling radiation. The sum of incident photons on each pixel is counted as the downwelling radiation on it. The LESS was compared with other models from the radiation transfer model intercomparison (RAMI) experiment and validated with field measurements in previous study (Yan et al., 2020), guaranteeing the accuracy of radiative simulation. More information about the LESS model can be found on its website (http://lessrt.org). In this study, it was used to simulate DSR over rugged terrain to explore the topographic effect on terrain reflected solar radiation.
The number of photons in ray-tracing simulation is important. Generally, more accurate results are derived with more photons, but too many photons can make the computation extremely time-consuming. Thus, a test with different illumination photon densities, defined as the number of photons per square meter, was performed before the simulation. The scene with non-reflecting surface was constructed with given entering direct and diffuse radiation (photons) on the bottom of atmosphere (BOA). The atmospheric extinction and cloud radiative effects were not included in simulation. The simulation result shows that the maximum downward irradiance and the averaged downward irradiance become stable after increasing the photon density to 25 (m−2) (Figure 3). The theoretical maximal irradiance is the sum of the input direct normal irradiance and the diffuse irradiance. Thus, the photon density of 25/m2 was employed throughout the study. In this case, 3.6 × 109 photons were assigned to a 20 × 20 km2 domain with a 30 × 30 m2 resolution.
The maximum (black solid line) and average irradiance (red dash-dotted line) calculated by using different illumination photon densities in LargE-Scale remote sensing data and image Simulation framework. The theoretical maximum irradiance of simulation is given with the black dashed line, which equals to the sum of direct normal irradiance and diffuse irradiance.
In the radiative transfer modeling of LESS, the ground surface in the model was constructed by triangle mesh based on the DEM data, rather than the stair-like surfaces which could produce significant errors locally (Lee et al., 2013). In addition, the boundary of study area is designed to be cyclic. Photons exiting the eastern boundary will reenter the western boundary of domain. Therefore, only the central 15 × 15 km2 area of each 20 × 20 km2 domain was used in the following analysis to avoid the edge effect.
This study focuses on the topographic effect on surface solar radiation, therefore, the impacts of atmosphere and clouds were precluded, only the clear condition was considered. Considering the atmospheric module in LESS is not yet well-validated currently, the MODTRAN™5 radiative transport model (Berk et al., 2005) was first used to simulate the direct and diffuse irradiance on a virtual horizontal surface at BOA, at the altitude of regional average elevation. The direct solar beam and isotropic diffuse radiation at BOA was then used as the illumination source in LESS to simulate the surface interaction. Because the atmosphere in TP is clean and rarely affected by humans, the standard mid-latitude winter atmosphere was used in MODTRAN, along with rural aerosol and the meteorological range of 23 km.
By setting the surface reflectance to zero and non-zero in LESS, we achieved the DSR over rugged terrain without terrain reflected solar radiation and the DSR including terrain reflected radiation, respectively. Thus, three radiation datasets were collected from the radiative transfer simulations of MODTRAN and LESS: (a) the direct and diffuse irradiance on horizontal surface without topographic effects; (b) the sum of direct and diffuse irradiance on mountain surfaces with the topographic effects (hereafter referred to simply as “direct&diffuse radiation” with the abbreviation of “d&f radiation”); (c) the terrain reflected irradiance on mountain surfaces.
2.3 Parameterization Approaches
) is estimated using the product of average surrounding upward radiation and terrain configuration factor 
in parameterization approaches. The expression of the 
can be written as
(1)
and 
are the reflectance and downward solar radiation of surrounding terrain pixels, respectively. They are computed on a sliding window with specific size, which we will discuss in Section 4.2.1.
expressions were compared and evaluated in this study. The first one was deduced by Kondratyev (1977). According to Kondratyev, the 
of an infinitely long slope adjoining a horizontal surface was deduced from the portion of sky dome visible to the slope 
:
(2)
raised by Dozier & Frew (1990) is more commonly used in the estimation of topographic radiation (Chen et al., 2006; Dubayah et al., 1990; Gratton et al., 1993; Han et al., 2016; Lee et al., 2011; Wang et al., 2005). This 
is obtained from Equation 3:
(3)
and 
are zenith angle and azimuth angle, A is the aspect of slope surface, 
is the angle from zenith downward to the local horizon, 
means the angle of the ray parallel to the slope defined by
(4)
is the sky view factor, represents the magnitude of openness from the ground surface to the sky (Blennow, 1995). Here we used the typical method of Dozier & Frew (1990) to calculate it:
(5)
was also used by many researchers in the estimation of terrain reflected solar radiation (Corripio, 2004; Helbig & Löwe, 2012; Helbig et al., 2009, 2010; Ma et al., 2016; Müller & Scherer, 2005; Richter, 1998). In this case, the portion of visible terrain is simply considered as the complementary to the 
:
(6)
(7)
(8)3 Simulation Results
3.1 Diurnal Solar Radiation Simulated by LESS
To have a clearer understanding about how much terrain reflected solar radiation can be received by mountainous areas, the simulation for 15 domains in southeastern TP on a clear day in winter (January 29, 2018) was carried out with LESS. The ground albedo information from Landsat was used. Figure 4 shows an example of the simulation results for Domain-3. The simulated direct&diffuse irradiance and terrain reflected irradiance from 7:00 to 17:00 (GMT +6) is presented. The solar zenith angles (SZA) ranges from 50° to 90° in the day, the solar azimuth angles (SAA) ranges from 112° to 248° from sunrise to sunset. The results of central 15 × 15 km2 area were extracted from the simulation of 20 × 20 km2 domain.
The ray-tracing simulation result of direct&diffuse radiation (upper) and the terrain reflected radiation (lower) by LargE-Scale remote sensing data and image Simulation framework in Domain-3 on January 29, 2018 (unit: W/m2). The area is 15 × 15 km with resolution of 30 m.
(9)
(10)
(11)
, 
, and 
refer to the deviation of domain-averaged d&f radiation, terrain reflected radiation and total incident radiation for rugged terrain to its corresponding flat domain, respectively; A is the total flat area of domain, which equals 15 × 15 km2 here;
and 
are the d&f irradiance of rugged surface pixel 
and flat surface pixel 
, respectively, with unit of W/m2; 
is the area of each rugged pixel 
; n is the number of pixels inside the domain.The diurnal variation of 
, 
, and 
were shown in Figure 5. The DSR for flat surface in this area is around 782 W/m2 at noon. The anomalies of the d&f irradiance with reference to flat surface is related to the slope and aspect distribution, and varies in a day. The 
is always positive on account of the absence of mutual reflection over flat surface. The 
is almost symmetrical about noon, reaches high value at noon and lower at dawn and dusk, due to the highest direct and diffuse radiation around noon in clear day. The domain-averaged reflected radiation is obviously nonnegligible for most of areas, especially for snow-covered mountains. For Domain-4, the flattest domain without any snow cover, the 
is 4.5 W/m2 at noon, with a maximum 
of 49.6 W/m2 in local scale (30 m). For high mountains mostly covered by snow, for example, Domain-5, the 
can reach as high as 84.8 W/m2 at noon, which is quite a large amount of radiation for domain-average scale.
The diurnal variation of domain-averaged radiation (DDR) in all domains on January 29, 2018. The value indicates the deviation of the domain-averaged (15 × 15 km) radiation with reference to a horizontal surface, influenced by the terrain feature of each domain. The red, blue, black lines represent the 
, 
, and 
, respectively.
The simulation of diurnal 
was also implemented in different seasons in Domain-3. Domain-3 was chosen as an example for the illustration because of the representative distribution of surface albedo and terrain features. Figure 6 presents the surface features of Domain-3. Nearly half of the surface was covered by snow or ice. The histograms show a wide range of slope values close to the Gaussian distribution and the almost equally distributed aspect values. In the investigation of different seasons, another three clear days in 2018 were chosen as 3 April, 6 June, and 28 October, concurrent with the overpass of Landsat 8, which provides the observation data of ground reflectance. Figure 7 gives the diurnal variation of 
for Domain-3 on 4 days. As illustrated in the figure, the 
is similar near morning and evening on these 4 days, while big gap occurs near noon. In terms of the daily reflected radiation, the domain receives more on 3 April and 28 October than that on 29 January and 6 June. The 
of Domain-3 reaches 61 W/m2 at noon on 3 April and 28 October, which is 57% higher than that on 29 January.
The surface features of Domain-3. (a) Spatial distribution of digital elevation model, (b) spatial distribution of surface albedo calculated from Landsat-8 data on January 29, 2018, (c) histogram of slope values, (d) histogram of aspect values.
The simulated 
of Domain-3 on 29 January, 3 April, 6 June, 28 October in 2018. The low reflected solar radiation on January results from the low d&f radiation and comparatively little snow cover, the latter is related with the reduction of snowfall in winter, the blowing snow events caused by strong westerly winds and sublimation effect at high elevation region in Tibetan Plateau.
The information of surface albedo gives a clue to the high reflected solar radiation in spring and autumn (Figure 8a and Table 1). The snow cover fraction and albedo values on 3 April and 28 October are higher than that on 29 January and 6 June. This phenomenon is consistent with the researches of snow cover variation at different elevation ranges in TP (Chu et al., 2017; Li et al., 2018; Pu & Xu, 2009), which found the maximum of snow cover appeared in spring and autumn at high mountains with elevations above 5,000 m, and a relative minimum during winter along with the typical minimum in the summer. For areas with elevation above 6,000 m, the snow cover fraction in winter can be even less than summer. According to those researches, the increase of snow at high altitudes in November results in the lower air temperature and increased static stability of atmosphere, leading to a reduction in the probability of later snowfall. Additionally, the blowing snow events caused by strong westerly winds at upper levels in winter and the sublimation effect also contribute to the decreases in snow cover fraction during the winter at high elevation regions in TP (Moore, 2004; Qin et al. 2006). Figure 8 shows the distribution of d&f radiation and terrain reflected solar radiation on 4 days at noon. Under the combination of moderate downward d&f radiation and high surface albedo, the terrain reflected solar radiation reaches highest in April and October.
The distribution of albedo and downward radiation at noon in Domain-3 on 29 January, 3 April, 6 June, 28 October. (a) Surface albedo, (b) d&f irradiance, (c) terrain reflected irradiance.
| Albedo value | 29 January | 3 April | 6 June | 28 October |
|---|---|---|---|---|
| Minimum | 0.00 | 0.00 | 0.00 | 0.00 |
| Mean | 0.39 | 0.46 | 0.36 | 0.52 |
| Maximum | 1.00 | 0.89 | 0.78 | 1.00 |
| Standard Deviation | 0.23 | 0.22 | 0.21 | 0.25 |
3.2 Influencing Factors on Terrain Reflected Solar Radiation
We investigated the influence of surface roughness, surface albedo distribution, and solar angle on the ratio of 
by means of the simulation results in real scenes in Section 3.1 and an ensemble of simulations in virtual scenes based on Domain-3. Based on the simulated radiation data of all domains on January 29, 2018, the relationship between the ratio of daily 
and the albedo and surface roughness are displayed in Figure 9. The daily 
was integrated from instantaneous values in a day from sunrise to sunset. Each 15 × 15 km2 domain was separated into nine 5 × 5 km2 subdomains to increase the samples. The standard deviation of slope (SD(S)) is used as the expression of surface roughness in this study (Grohmann et al., 2010). The mean albedo and SD(S) were counted for every subdomain. As shown in Figure 9, the ratio of daily 
is within 0.25%–10.85% of total DSR for subdomains. Except some extremely rugged terrains, most of mountains without snow cover (low albedo) have a low ratio of 
, which is generally less than 3%. The effect of snow cover on the reflected radiation is significant.
The ratio of daily 
changes with the mean albedo and SD(S) of subdomains, based on the simulation on January 29, 2018 in Section 3.1.
A set of stretched terrains based on Domain-3 was also used to look deeply into the effects of topographic parameters. The elevation values in Domain-3 were multiplied by exaggeration factor of 0.05, 0.1, 0.2, 0.4, 0.6, 0.8, 1, 1.2 to generate terrains from flat to steep. The exaggerated DEMs were shifted vertically to keep the mean elevation value of each DEM the same. The topographic information of stretched DEMs are shown in Table. 2, including the mean, minimum, maximum, range of elevation, the mean slope, the surface roughness SD(S), and the mean 
. Based on the simulation with stretched terrains with SZA = 50.27°, SAA = 180° at noon and a set of homogeneous albedos ranging from 0.1 to 0.9, the relationship between the ratio of 
and topographic parameters is displayed in Figures 10a and 10b. The SD(S) and the mean 
at domain scale were used as x-coordinate, respectively. The ratio of domain-averaged 
shows the exponential relationship with SD(S), while it is negative linear relationship with the mean 
. The 
is proportional to the ground albedo value, which is in line with another test with single variable of albedo. For DEM-7, the ratio of 
ranges from 1.23% to 12.87% with albedo of 0.1 and 0.9, respectively.
| DEM series | Exaggerate factor | Mean (h) | Min (h) | Max (h) | Range (h) |  |
SD (S) |  |
|---|---|---|---|---|---|---|---|---|
| m | m | m | m | degree | degree | - | ||
| DEM-1 | 0.05 | 5159.4 | 5115.2 | 5198.3 | 83.0 | 1.5 | 0.78 | 0.9996 |
| DEM-2 | 0.1 | 5071.0 | 5237.1 | 166.1 | 2.9 | 1.56 | 0.9983 | |
| DEM-3 | 0.2 | 4982.7 | 5314.9 | 332.2 | 5.8 | 3.09 | 0.9934 | |
| DEM-4 | 0.4 | 4806.0 | 5470.4 | 664.4 | 11.4 | 5.90 | 0.9750 | |
| DEM-5 | 0.6 | 4629.4 | 5626.0 | 996.6 | 16.6 | 8.29 | 0.9455 | |
| DEM-6 | 0.8 | 4452.7 | 5781.5 | 1328.8 | 21.3 | 10.23 | 0.9129 | |
| DEM-7 | 1 | 4276.0 | 5937.0 | 1661.0 | 25.7 | 11.76 | 0.8791 | |
| DEM-8 | 1.2 | 4099.3 | 6092.5 | 1993.2 | 29.6 | 12.95 | 0.8411 |
- Abbreviation: DEM, digital elevation model.
The ratio of 
varies with (a) surface roughness standard deviation of slope (SD(S)), (b) mean 
of domain, and (c) solar angle. The simulation with various SD(S) and 
was set with homogeneous albedos from 0.1 to 0.9, and with solar zenith angles = 50.27° and solar azimuth angles = 180° at noon. Each marker corresponds to the result of one stretched digital elevation model. The simulation of solar angle was set with a homogeneous albedo of 0.39 for Domain-3.
The influence of solar angle was further investigated with fixed ground albedo of 0.39 for Domain-3. Virtual solar angles were used in this analysis. The SZA are set as 10°, 20°, 30°, 40°, 50°, 60°, 70°, 80°, 85°, while the SAA are set as 90°, 180°, 270°. Figure 10c shows the variation of the ratio with different solar angles. The ratio is approximately constant when the SZA is less than 60° and reaches a top around SZA of 70° or 80° and decreases with SZA higher than 80°. The domain gets more terrain reflected solar radiation when the sun is at west direction (SAA = 270°) than east or south, which is on account of more steep slopes facing west in Domain-3. Noted that the relationship between the ratio of 
and SZA changes while the surface albedo is not homogeneous.
3.3 Spatial Distribution and Influence Range of Terrain Reflected Radiation
For the spatial distribution of 
the comparison between inhomogeneous surface albedo and homogeneous albedo is worth noticing. When the surface albedo is a fixed value of 0.4 (the mean albedo of Domain-3 is 0.39), the domain gets high 
in low-altitude valleys due to more reflection in valleys (Figure 11b). The distribution of 
in this case shows strong negative correlation with the surface elevation. However, the radiation distribution is largely changed by the inhomogeneous surface albedo under partial snow-cover (Figure 11a). The 
is more evenly distributed, the value around valleys is not conspicuous any more, whereas more reflected radiation is received by the snow-covered high-altitude slopes. The probability density function (PDF) curves (Figure 11c) show the pretty similar altitude distribution between the snow-covered area and the high reflected radiation area (defined as the reflected radiation larger than 60 W/m2 in this study) in the partial snow-covered scene. Hence, for the mountains with partial snow covered, the high-altitude snow enhances the reflected radiation received by high-altitude slopes, reinforcing the incident radiation of these areas.
The influence of inhomogeneous surface albedos on the distribution of 
. (a) The simulated 
under the inhomogeneous Landsat albedo in Figure 6b; (b) the simulated 
under homogeneous albedo of 0.4; (c) the probability density function of altitude distribution for: the snow-covered area in the scene of (a), the high reflected radiation (>60 W/m2) area in (a), and the high reflected radiation (>60 W/m2) area in (b).
The influence range of 
was also investigated, reflecting the effective interaction distance in mountains. The result is presented for five random positions in Domain-3, include a deep valley (p1), small peak (p2), and three slopes (p3, p4, p5). The ray-tracing simulations were conducted for scenes centered on each position with different sizes with SZA of 50° and inhomogeneous albedo data from Landsat. Then the results of central 5 × 5 grids in each scene were extracted and the mean value of it was calculated. From Figure 12 we see that the mean 
of central grids rises and toward constant with the increase of scene size. For most of positions, the reflected energy mainly comes from the surrounding areas within 2–3 km radius. Though the mean 
at p3 keep rising within 4 km, indicating that the reflected solar radiation from terrain may has an impact from distances more than 4 km away for some locations in snow-covered area.
Mean 
received by the target areas around five positions (p1, p2, p3, p4, p5) in Domain-3 with increasing scene sizes. The inhomogeneous albedo on January 29, 2018 and the solar zenith angle of 50° were used in the simulation.
4 Evaluation of Parameterization Approaches
4.1 Evaluation of Terrain Configuration Factors
According to the introduction in Section 2.3, there are three different forms of 
used for the estimation of 
. From the perspective of formula structure, the trigonometric equation of 
from Kondratyev (1977) in Equation 2 holds only for an infinitely long slope adjoining a horizontal surface, without considering the reflected solar radiation from adjacent hills. The integral formula of Dozier & Frew (1990) carefully considers the contribution from adjacent slopes below local horizon in Equation 3. However, the simplified form after the approximately equal sign represents the difference between the 
of an infinitely long slope (
) and the 
of the pixel itself. Both 
calculates the visible ratio of upper hemisphere, missing the radiation coming from the terrain below the horizon (Ma et al., 2016). From Figure 13 the 
depict well the surrounding terrain visible to the target pixel.
The schematic diagram of sky view factor (
) and terrain configuration factor (
) for mountain pixel.
In order to quantitatively evaluate three 
expressions, we calculated the instantaneous 
for all domains with Equation 1 and compared it with LESS simulation results. The simulated results of LESS are considered as the relative truth due to the high accuracy of ray-tracing model. The 
was calculated as the sum of direct and diffuse radiation in the parameterization approach. Comparisons between the simulated domain-averaged 
from the parameterization approach and LESS model are displayed in Figure 14.
The comparison of domain-averaged 
simulated by LargE-Scale remote sensing data and image Simulation framework (LESS) and parameterization approaches with three different terrain configuration factors. The calculation was conducted for all domains on January 29, 2018. (a) Shows the results of parameterization approach with Equation 1, each marker represents the instantaneous domain-averaged 
; (b) shows the results of improved parameterization approach with consideration of the multiple reflection.
Figure 14a shows the evaluation result of normal parameterizations with Equation 1. The plot illustrates that all of the parametrization approaches underestimate the adjacent reflection, but the estimation using 
is much better than the others. The root mean square error (RMSE) for the parameterization method with 
is 2.55 W/m2 (14%), far less than the errors of the other two. However, all markers show underestimation of parameterization approach where the 
is large even using 
. For example, the top right markers in Figure 14a, corresponding to the result of Domain-5, which has the large elevation range and is mostly covered by snow (Figure 2), where is most likely to have strong multiple reflection. The improved parameterization approach with consideration of the multiple reflection was further evaluated by LESS (Figure 14b). The Equation 8 was used in parameterization estimation. The total RMSE decreases after considering the multiple reflection term. The results confirm that the multiple reflection is nonnegligible in parameterization approach, especially in extreme high reflective area. Hence, the improved parameterization approach was used for the error analysis in the following.
4.2 Error Analysis of Parameterization Approach
4.2.1 Influence of Sliding Window Size
The sliding window size in the process of computing averaged upward solar radiation 
was investigated based on the above simulation results of all domains. The mean radiation within the square-shaped sliding window was assigned to the central pixel. The estimated daily 
at 30 m resolution was aggregated into different pixel scales, for example, daily 
at resolution of 500, 2,000, 5,000 m. According to Figure 15, the error of parameterization approach is not sensitive to the pixel scale larger than 2,000 m. From our simulation results, the best sliding window size for parameterization approach is around 2,700 m. For some scales, the error becomes bigger when the window size is larger than 2,700 m, because the 
is mainly affected by the radiation condition in adjacent areas.
The error change of parameterization approach with different sliding window sizes. The daily 
at 30 m resolution was aggregated into different target scales.
4.2.2 Influence of Snow Cover
Suppose the ground albedo is homogeneously distributed, the 
is proportional to the magnitude of albedo. In parameterization approach, the situation of partial snow cover leads to the inhomogeneous albedo, enhancing the spatial heterogeneity of upward solar radiation.
The errors of parameterization approach were calculated in scenarios with inhomogeneous albedos and homogeneous albedos, respectively. The diurnal simulation results in all domains with Landsat albedo were used to calculate normalized RMSE (nRMSE) in inhomogeneous scenario. The mean albedo of each domain was used in the simulation for homogeneous scenario. The discrepancy of two nRMSE is shown in Table 3, along with the coefficient of variation for domain albedo (Cv(ρ)). For most domains, the nRMSE is bigger in inhomogeneous-albedo scene than homogeneous-albedo scene. The averaged increment of nRMSE is 2.55%. Figure 16 indicates that the error of parameterization approach is positive correlated with the spatial heterogeneity of domain albedo, which is easily influenced by the partial snow cover.
| Domain number | Cv ( ) |
nRMSE1 | nRMSE2 | nRMSE1-nRMSE2 |
|---|---|---|---|---|
| Unit | % | % | % | |
| 1 | 0.45 | 3.04 | 1.54 | 1.5 |
| 2 | 0.83 | 4.97 | 1.78 | 3.19 |
| 3 | 0.58 | 8.15 | 2.17 | 5.98 |
| 4 | 0.26 | 17.25 | 17.76 | −0.51 |
| 5 | 0.34 | 7.04 | 6.31 | 0.73 |
| 6 | 0.56 | 5.46 | 1.93 | 3.53 |
| 7 | 0.67 | 4.8 | 4.13 | 0.67 |
| 8 | 0.86 | 9.35 | 6.47 | 2.88 |
| 9 | 0.46 | 1.9 | 2.82 | −0.92 |
| 10 | 0.76 | 7.02 | 2.89 | 4.13 |
| 11 | 0.6 | 4.67 | 4.83 | −0.16 |
| 12 | 0.85 | 10.17 | 2.43 | 7.74 |
| 13 | 0.97 | 8.96 | 2.96 | 6.0 |
| 14 | 0.59 | 11.9 | 8.08 | 3.82 |
| 15 | 0.58 | 10.7 | 11.03 | −0.33 |
| Average | - | 7.69 | 5.14 | 2.55 |
-
Notes. Both for the LESS simulation and parameterization approach estimation, the real albedo from Landsat was used as the inhomogeneous albedo, the domain mean albedo was used as the homogeneous albedo. The instantaneous domain-averaged
was simulated to calculate the nRMSE.
- Abbreviations: LESS, LargE-Scale remote sensing data and image Simulation framework; nRMSE, normalized root mean square error.
Relationship between the coefficient of variation of domain albedo (Cv(
)) and the increment of normalized root mean square error of parameterization approach.
4.2.3 Spatial Scale Effect of Parameterization Approach
From Figure 15 we also see that the error of parameterization approach tends to be bigger with higher spatial resolutions. Figure 17 gives an example of estimated 
at 30 m resolution in Domain-8 simulated by LESS and parameterization, it shows large discrepancy in their spatial distribution. The result indicates that the result from parameterization approach maybe not reliable at very high resolutions, due to the strong heterogeneity of terrain reflection at small scale.
Instantaneous 
in the resolution of 30 m simulated from (a) LargE-Scale remote sensing data and image Simulation framework and (b) parameterization approach, and (c) histogram of the difference between former two. The simulation is in domain-8 at noon on January 29, 2018, with surface reflectance data from Landsat.
Though the accuracy of parameterization approach is not good at 30 m resolution, the aggregated 
at large scales is closer to the true value, for example, the domain-averaged 
in above analysis. To explore the applicable scale of parameterization approach, the simulated 
in 30 m resolution was averaged into different scales: 30, 90, 300, 500, 750, 1,000, 2,000, 3,000, 5,000, 7,500, 15,000 m, both for LESS simulation and parameterization approach. The RMSE of parameterization approach for noon and daily 
at different scales were calculated and shown in Figure 18. The error of parameterization approach decreases with the increasing target scales in Figure 18. For daily 
, the RMSE is lower than 3 W/m2 when the target pixel size is larger than 500 m. The error becomes stable and is less than 1 W/m2 (10%) for pixel larger than 5 km. The result for instantaneous estimation shows larger error, with RMSE of 8.3 and 3.21 W/m2 at target scale of 1 and 5 km, respectively.
The root mean square error of parameterization approach in daily and noon 
at different pixel scales. The result is based on the calculated 30 m radiation and then aggregated into target scales.
Parameterization approach is an alternative to the 3-D RT models in virtue of its simpler form and high computational efficiency. However, the results along with the impact analysis of snow cover in Section 4.2.2 indicate that the sophisticated 3-D RT simulation is necessary for the accurate estimation of 
, especially in the snow-covered mountains at high resolution.
5 Impact on Net Solar Radiation
can be derived by:
(12)
and 
is the direct radiation and diffuse radiation, respectively. According to the parameterization approach, the 
can be estimated by the sum of 
and 
, 
and Ct:
(13)The downward d&f radiation (
) can be nearly regarded as constant with the change of ground albedo. The changing in the ratio of 
to the d&f radiation was used to illustrate the variation of net radiation with albedo, as shown in Figure 19. Regardless of the terrain reflected radiation, the ratio of 
increases in the rate exhibited by the black solid line, with the decreased albedo. Under actual circumstances, both 
and 
decrease with the decline of snow cover, the growth rate of 
changes in this context. For the area with high albedo, the increase of net radiation is faster than that without considering of reflected radiation, as present by the gradient of line. While in the area with low albedo the rate is inversely lower. In the case of a given albedo value, the bigger the 
, which means that the mountain is steeper and more rugged, the ratio of net radiation to downward direct&diffuse radiation is more significant. This result implies that the neglect of terrain reflected radiation may result in the underestimation of net radiation increase in high albedo mountains.
The ratio of net radiation to d&f radiation changes with the ground albedo. The ratio without considering of 
is shown with the black solid line. Different dash-dotted lines correspond with different 
.
6 Discussion and Conclusion
The accurate estimation of terrain reflected solar radiation in high-altitude mountains is important for investigations of global change and the energy budget, especially for the deep understanding of snow-melting process in those areas. The 3-D RT model LESS based on the ray-tracing mechanism was used in this study to investigate the temporal and spatial distribution characteristics of terrain reflected solar radiation in mountainous areas in the southeastern TP, due to its high computation efficiency compared to other 3-D RT models.
The Southeastern TP was chosen as our study area because of its rugged terrain and the high sensitivity to climate change. Especially, there are a lot of high mountains covered by snow throughout the year. Fifteen 20 × 20 km2 domains were selected in this area with the DEM data at 30 m resolution. To avoid the edge effect, only the result of the central 15 × 15 km2 area in each domain was utilized. Based on the ray-tracing simulations over the 15 domains quantitative analyses of the influencing factors on terrain reflected solar radiation were conducted and different parameterization approaches were evaluated in detail.
The diurnal simulation on a winter clear day shows the nearly symmetrical variation of terrain reflected solar radiation about noon. The instantaneous domain-averaged reflected radiation reaches a remarkable value of 84.8 W/m2, while the maximum local reflected radiation can be much higher. The ratio of daily reflected radiation to total DSR ranges from 0.25% to 10.85% for 135 5 × 5 km2 subdomains in these conditions. Influenced by the special TP meteorology, for example, the strong westerly winds and the sublimation effect, there is more snow in spring and autumn than summer and winter over the very high mountains in the TP, resulting in a higher terrain reflected solar radiation in spring and autumn. For instance, the 
at noon on 3 April and 28 October in Domain-3 is 57% higher than the one on 29 January.
The simulation results indicate that the mountains without snow cover (with low albedo) generally get quite low reflected radiation compared to the sum of direct and diffuse radiation. However, snow and ice largely enhance the level of terrain reflected solar radiation, making it a nonnegligible term in the energy balance, which can heavily influence the energy budget in cryosphere and estimation of surface albedo. Limited by the available reflectance data in clear days, only 4 days were used in our study to figure out how the terrain reflected solar radiation performs in different seasons. Note that this result cannot represent the general situation in every year because the meteorological conditions change annually. However, it gives us the idea that under the effects of ground reflectivity and downward radiation, the high mountains in TP get more terrain reflected solar radiation in spring and autumn.
The influences of different factors on the magnitude of terrain reflected solar radiation were analyzed. The ratio of domain-averaged 
to DSR has an increase in large SZA. The standard deviation of the slope SD(S), used as the surface roughness of the domains, shows an exponential relationship with the ratio of 
, while the ratio is almost linear with the mean 
. As expected, 
is proportional to the mean albedo of the domain. However, the snow cover in mountains affects the distribution of reflected radiation a lot. In the case of homogeneous surface albedos, the valleys usually receive more reflected radiation than peak and slopes. While for mountains with partial snow cover, the snow and ice mostly distribute on the high-altitude slopes and peaks, resulting in more reflection between the upper mountains.
The analysis of the spatial distribution in partial snow-covered scene implies a limited influence range of reflection, hence followed by the investigation of effective distance from adjacent terrain at five positions in Domain-3. The results indicate that most positions gain reflected solar radiation from surrounding terrain within 2 km, whereas some slopes are still affected from distances more than 4 km away. In this case, the accumulation method for the calculation of reflected radiation (Proy et al., 1989) requires the operation in a large enough area, otherwise the underestimation of reflected radiation can easily occur for some areas. A wide searching range can largely increase the calculation time and cost in the accumulation method, making it not effective and practical in applications. However, the atmospheric attenuation in the path from the surrounding terrain to each location was neglected in the LESS simulation in this study, considering that the extinction of the atmosphere on clear days in the TP is relatively low.
The above simulation results reveal the importance of terrain reflected solar radiation on the energy budget of snow-covered mountains. The accurate calculation of terrain reflected solar radiation is necessary, especially for the research of snow-melting process in high mountains. Different forms of parameterization approaches were comprehensively compared and evaluated by the delicate 3-D RT model for the first time, including the evaluation of the three terrain configuration factors and the improved parameterization scheme, and the error analysis of parameterization approach in different conditions. The parameterization result with the third terrain configuration factor (
) agreed well with that simulated by LESS on a domain-averaged scale, with the RMSE of 2.55 W/m2 (14%) for instantaneous reflected radiation, the other two factors both underestimate the radiation, with the RMSE of 12.41 W/m2 (68%) and 13.28 W/m2 (73%), respectively. This finding confirms the ignoring of the radiation coming from the terrain below the horizon in Dozier’s factor. The improved scheme proposed by Sirguey adds the consideration of multi-reflection in the formula based on the common parameterization approach, and was also assessed by LESS. The nRMSE of the parameterization approach decreases from 14% to 9% for instantaneous reflected radiation, implying the necessity to take into account the multi-reflection term in parameterization.
Finally, we quantified the error of the parameterization approach with different sliding window sizes, surface heterogeneity and spatial scales. The sliding window size is hyposensitive in the application of the parameterization approach, 2,700 m (window radius of 1,350 m) is suggested according to our analysis. It should be noted that there is a difference between the sliding window size and the influence range of reflection. The former determines the window which the averaged reflection comes in, while the latter represents the effective surrounding area which contributes to the target pixel. According to the averaged strategy in parameterization, the method is more suitable for relatively homogeneous surfaces. The nRMSE of the parameterization approach for instantaneous radiation is around 5.14% on the domain scale with homogeneous surface albedos and raises by 2.55% with partial snow cover in the domains. The snow on mountains increases the heterogeneity of surface albedo and further enhances the uncertainty of the parameterization approach. According to the analysis of spatial scale effects, the parameterization approach has lower error on the large scale than on higher resolution. The RMSE for daily 
is generally less than 5 W/m2 and is lower than 1 W/m2 (10%) with a target pixel scale larger than 5 km. The analysis indicates that the uncertainty of parameterization approach is larger than 3-D RT model in calculating radiative processes in snow-covered areas at high resolution. More careful consideration is needed in the cryosphere-climate models using parameterization in those conditions, since the error could be relatively large.
Since this study focuses on the terrain reflection pattern between mountains, the atmospheric processes in MODTRAN and the simulation of topographic effect in LESS are independent, which may lead to some uncertainties in the results. However, the atmospheric extinction in the TP in clear sky conditions is usually low, hence the interaction between atmosphere and terrain can be ignored to a certain extent. In the future, we would like to integrate the atmospheric module into the ray-tracing model to achieve a more accurate simulation. Furthermore, limited by the data and model, this study only considers clear sky conditions. The clouds may decrease the total downward radiation, but may increases the terrain-cloud reflected solar radiation.
Acknowledgments
This work was supported by the key program of the National Natural Science Foundation of China (NSFC) (Grant No. 42090013) and the China’s National Key R&D Programmes (NKPs) (Grant No. 2020YFA0608702). We gratefully acknowledge the reviewers for their valuable comments and constructive suggestions to improve the manuscript substantially.
Open Research
Data Availability Statement
Landsat-8 surface reflectance data courtesy of the U.S. Geological Survey (http://earthexplorer.usgs.gov/). The DEM data were obtained from the Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) Global Digital Elevation Model (GDEM) of version 2.0 (https://lpdaac.usgs.gov/products/astgtmv002/). The LESS model can be downloaded from its website (http://lessrt.org). No new data were used, nor created for this research.
