Accurately determining the position of the sun is essential in order to properly calculate the amount of incoming solar radiation. This is also necessary when considering topographic effects. Assuming a flat plane, the solar position is described by a combination of the zenith (Z) and azimuth angles, which are calculated using standard methods (Iqbal, 1983).

We model potential clear-sky direct solar radiation (Ia), as it passes through the atmosphere, at 15 min time intervals throughout the melt season (1 April–31 September): Ia=I0RmR2ψaPP0cos⁡, where I0 is the solar constant (∼1368 W m-2), R is the sun–earth distance (subscript m refers to mean), ψa is atmosphere clear-sky transmissivity (a constant of 0.75 is used; Hock, 1999), P is atmospheric pressure calculated using a simple lapse rate for dry air, P0 is mean atmospheric pressure at sea level, and Z is local zenith angle (which accounts for the size of air mass that radiation must travel through before arriving at the surface). Equation (1) has been modified from Hock (1999) to exclude the component responsible for attenuation at the surface. This term will be added later in the topographic model.

In alpine terrain, the combination of slope and aspect is generally presumed to be the most influential topographic factor regulating absorbed solar radiation at the surface (Dozier and Frew, 1990). However, some areas surrounded by steep terrain can also be highly influenced by topographic shading (Arnold et al., 2006). Due to the high spatial and temporal variability in the incident angle and topographic shading throughout the day, comparing the impact of these topographic components side by side to assess relative importance can be challenging. Here we present Eqs. (3)–(6) as a means by which to address this issue. We use these equations to calculate the change in irradiance averaged over the course of a day for a given topographic component.

We incorporate two additional terms, incident angle and topographic shading, in conjunction with Eq. (1) to determine the distribution of solar radiation at the surface due to topographic effects. Potential clear-sky solar radiation arriving on an inclined surface is Ic=Iacos⁡θiS, where Ia is potential atmospheric clear-sky direct solar radiation from Eq. (1); θi is the incident angle; and S is topographic shading, a binary value indicating whether a given cell is in “shade” (0) or “sun” (1). Topographic shading is calculated with a modified ray-tracing algorithm that uses a solar illumination plane perpendicular to the sun's zenith angle in order to determine if a cell is blocked by surrounding cells at a certain zenith and azimuth angle (Corripio, 2003). This method incorporates both self-shading from relief and cast shadows. The incident angle is the zenith angle (Z) modified for a surface with a specific slope and aspect (Iqbal, 1983).

We use variations of Eq. (2) in order to determine the daily mean change in solar radiation from specific topographic components shown in Fig. 1. Equations (3)–(6) show the mean change in solar irradiance due to slope and aspect (incidence angle), shaded relief, cast shadows, and the combined effect of these topographic components: ΔISA‾=1t∫0tIaScos⁡θi-cos⁡Zdt,ΔISR‾=1t∫0t-Iacos⁡Zdt,θi≥90∘0,θi<90∘,ΔICS‾=1t∫0tIacos⁡θiS-1dt,ΔICom‾=1t∫0tIaScos⁡θi-cos⁡Zdt. Figure 3 shows the derivation of these equations with respect to Eq. (2) (model 1) and illustrates the topographic components of interest. In order to determine the influence of each component, a second model is created excluding the component of interest. Equations (3)–(6) calculate the difference between Eq. (2) (model 1) and a model that excludes the topographic component of interest, which is then integrated over the course of a day. The result of these equations is a change in irradiance due to a specific topographic component. Equation (3) shows the daily mean change in solar radiation due to the slope and aspect relative to a flat plane. By incorporating S in this equation, values considered to be in shade are excluded. The mean change in irradiance due to shaded relief is also relative to a flat plane (Eq. 4), as shaded relief also relies on the presence of slope and aspect values. Shaded relief only occurs when the incident angle is greater than or equal to 90∘. Equation (5) shows the mean change in irradiance due to cast shadowing relative to an inclined plane void of cast shadows. This accounts for any change in irradiance due to the slope and aspect, as well as shaded relief. By default, shaded relief is incorporated in both the calculation of the incident angle (θi) and the shading algorithm used to determine topographic shading (S). This allows us to parse out the individual impact from each of these topographic components, for comparison. Finally, Eq. (6) is the combined effect from these three components relative to a flat plane. For each of the equations described above, the mean change in irradiance is equivalent to the integrated difference over the course of a day.

Our results present a daily mean change in irradiance due to each topographic component, averaged across the entire melt season. For simplicity, we will refer to the daily mean change in irradiance averaged over the melt season as the mean change in irradiance.

This study utilizes the 30 m resolution Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) global digital elevation model (GDEM) to simulate topographic terrain in the models. Although a 30 m resolution DEM does not fully capture the actual topographic complexity in a glacial valley, it serves as an adequate representation in order to measure the overall effect of surrounding topography on solar radiation within our theoretical framework. DEM resolution is further considered in the discussion in Sect. 4.1 section of this paper.

Glacier boundaries are determined with the latest shapefiles available from the Randolph Glacier Inventory 5.0 and ICIMOD (Arendt et al., 2015; Bajracharya and Shrestha, 2011). Both inventories delineate glacier ice using a variety of techniques to determine glacier boundaries. A buffer of 5 km is generated around the glacier shapefiles in order to include all surrounding topography. Because valley glaciers are constrained by the immediate surrounding topography, only nearby features are able to affect incoming solar radiation. However, at higher glacier elevations, the visible horizon can become much larger, in which case the extent must incorporate topographic features within visibility. For this study, a buffer of 5 km proved to be sufficient. For example, topographic shading was altered by less than 0.01 % when changing DEM extent from 5 to 3 km beyond the boundary of Satluj Glacier.

We implement two idealized scenarios in order to test the sensitivity of the shaded relief and cast shadows to changes in overall glacier aspect and latitude. The components of topographic shading are calculated for two different glaciers as they are rotated in each of the main cardinal direction (north, east, south, and west). This allows us to observe changes in direct irradiance due to different valley aspects for the two glaciers. We then calculate the relative change in irradiance due to topographic shading across varying degrees of latitude (20–50∘) for the same two glaciers. Glacier size, slope, and surrounding topography are constant in these idealized scenarios, while aspect and latitude are systematically varied.

We also apply our cast shadow model to four glaciated regions across the greater Himalaya, shown in Fig. 2. These regions span multiple latitudes and degrees of topographic relief, and include a larger sampling of glacier geometries throughout HMA. Only glaciers larger than 3 km2 were used in the analysis in order to exclude small cirque glaciers and focus on valley glaciers with a developed tongue. These glaciers are then grouped based on general valley aspect of the ablation zone (ice below mid-elevation), as we are most interested in the cast shadowing effect from the valley walls along lower elevations where melt is dominant. The valley aspect for each glacier is determined based on the mode of pixels of a reclassified aspect value. They are then grouped into two categories: north/south- and east/west-oriented glaciers. Glaciers with less than 10 % majority aspect are manually corrected and labeled based on visual inspection. This introduces some complication as glacier aspect and morphology can be extremely irregular throughout the lower elevation for these HMA glaciers. For this reason, we include a large sample of valley glaciers for each region, ranging from 80 to 117 glaciers. A single mean value of change in solar radiation due to topographic shading is calculated across the defined ablation zone of each glacier. A kernel density estimation is fit to the mean values in each region for both north/south and east/west glaciers in order to see how the distribution of each group varies from one another.

We selected a glacier in the Panjnad basin of the Indus watershed in the Himachal Pradesh region of the western Himalaya and one located in the eastern Himalaya on the China–Bhutan border for our detailed analysis and idealized scenarios. We refer to the glacier in the western Himalaya as Satluj Glacier, due to the subbasin in which it resides. Similarly, we refer to the glacier in the eastern Himalaya as Nianchu Glacier, after its subbasin name. We chose these two valley glaciers because they both have clear north-facing aspects and well-developed glacier tongues. However, Satluj Glacier is in an area of steep topography and high relief, while Nianchu Glacier spans a slightly higher range of elevation and is surrounded by less topographic relief.

We perform idealized scenarios in order to quantify how topographic shading changes over different valley aspects and latitudes. In particular, we rotate Satluj and Nianchu glaciers in each of the main cardinal directions and move each glacier across varying degrees of latitude.

Only slight changes in the shaded relief occur when Satluj Glacier is rotated in each cardinal direction. On the other hand, cast shadowing shows a significant mean decrease in irradiance at lower elevations for both north and south valley aspects, compared to east and west (Fig. 6). This is expected, as shading is largest when the sun is low in the sky and nearby topography is oriented such that it blocks direct solar rays. This occurs during the early and late hours of the day when large terrain features are situated to the east and west. East and west valley aspects lose the orientation of steep adjacent topography able to cast significant shadows; thus the impact is greatly reduced. However, we see an increase in cast shadowing in the upper elevations for east and west aspects, as the upper tributaries in Fig. 4 are now predominantly facing north and south.

Nianchu Glacier shows a less clear pattern in aspect sensitivity; however, some differences are notable. Cast shadowing appears to be most significant for lower and mid-elevations with north and south valley aspects (Fig. 6). Additionally, the upper elevations for the south direction show a large increase in cast shadows. Shaded relief also shows a larger mean change in upper elevations for north and south directions. One interesting point when comparing both glaciers side by side is that the impact of shaded relief only surpasses cast shadowing in the north-facing direction. Additionally, in these scenarios, cast shadows and combined shading are largest when the glaciers are in a south-facing valley.

In general, we see that topographic shading is more significant on glaciers with a north or south valley aspect, particularly in the lower elevations of valley glaciers. This is of particular interest for regions like the Himalayas where valley glaciers are dominantly north- or south-facing. Although we continue to see variability in shading specific to each glacier, this general result can be a useful proxy to determine how impactful shading might be on a given glacier.

When moving these same glaciers across varying degrees of latitude, we see a relative decrease in mean irradiance with increasing degree of latitude for both Satluj and Nianchu glaciers (Fig. 7). The trend of relative change in mean irradiance for each glacier closely follows a simple mathematical trend of negative tangent of the change in degree latitude (β), -tan⁡(β)dβ. This mathematical trend arises from a simple trigonometric relationship between the zenith angle (Z) and length of cast shadows (S) for topography of a given height (H). As the zenith angle increases, the length of shadows cast also increases; tan⁡(Z)=S/H. Apart from variation over the course of a day, zenith angles increase proportionally to an increase in latitude (Iqbal, 1983). As such, we assume that the change in irradiance from cast shadows across changing latitude, for a glacier with fixed topography, is largely a function of -tan⁡(β)dβ, with variation based on changes in the intersection of azimuth angles and surrounding topography. While this simple mathematical trend captures the overarching changes in irradiance at differing latitudes, it overpredicts for Satluj Glacier and underpredicts for Nianchu Glacier.

Outside of latitude and aspect, the impact of topographic shading is determined by combined characteristics of the immediate surrounding topography. As such, unique basin morphology gives rise to significant variability in the impacts of shading on direct solar radiation across glaciated regions. We apply the topographic models to the full Satluj basin including the surrounding glacial valleys, seen in Fig. 8. This expanded view offers a variety of valley aspects and different morphologies, and a means to explore the spatial variability in the shading components. A hillshade is used to visually enhance topography.

Shaded relief (Fig. 8a) appears to be significant only on extremely steep terrain that is not south-facing. Although this may be a significant topographic component in the upper cirques of some glaciers, the impact is minimal in glacier valleys and does not play a large role in the ablation zones where differences in solar irradiance can impact the mass balance. Cast shadows (Fig. 8b), on the other hand, are most pronounced in low-elevation north- and south-facing valleys, as predicted in our aspect sensitivity tests. Interestingly, although the mean change in irradiance increases due to slope and aspect for some south-facing valleys (Fig. 8c), this effect is offset, and in some cases overwhelmed, by cast shadows (Fig. 8d). By calculating the influence of these topographic components on a larger scale, we are able to confirm the results from our aspect sensitivity analysis in Fig. 6. Spatial patterns throughout the basin show that cast shadows significantly impact mean daily irradiance along deep north and south valley aspects.

We also apply these models to four glaciated regions spanning large portions of the greater Himalaya (Fig. 2), to further test our idealized scenarios and assess the degree of spatial variability in shading on valley glaciers. Again, we see that north/south-facing glaciers are more impacted by cast shadowing than east/west-facing glaciers (Fig. 9). Additionally, we see that the peak values in our north/south distributions (mode) of solar irradiance changes show a greater change in regions of higher latitude and relief (regions 1 and 2), fluctuating from a mean glacial change of -28.7 W m-2 at the highest latitude to -20.7 W m-2 at the lowest. Although we see significant variability between regions, there is generally less spread for east/west- than north/south-facing glaciers. Our regional model validates the findings in our idealized scenarios, confirming that the impact of topographic shading is more prominent on north- and south-facing valleys with a general increase in the mean change in irradiance between regions located at higher latitudes.

ASTER GDEM, used in this study, builds surface elevation using orthorectification of two ASTER images, producing a product near 30 m resolution. Although ASTER provides greater detail and generally good elevation accuracy, orthorectification tends to fail in completely snow-covered regions as orthoimages become difficult to align. These issues, along with the resolution, introduce some inaccuracy as flat surface features become exaggerated and ridgelines are smoothed, which can artificially increase or lower the effects of shading from topographic features (Arnold et al., 2006). It should also be noted that, although the sampling interval of the GDEM is 30 m resolution, the spatial resolution may be up to 3 times coarser in some regions (Hengl and Reuter, 2011). Despite the shortcomings of this DEM product, the findings in this study still demonstrate clearly the impacts of DEM resolution on incoming shortwave radiation.

Hopkins et al. (2010) found a linear increase in modeled glacier melt as DEM resolution decreases from 1 to 1000 m. This is due to decreased textural relief as DEM resolution is reduced. In certain scenarios, Hopkins et al. found that total melt increased with DEM resolution by 4 %. In order to illustrate the potential impact of DEM resolution on topographic shading, we calculate a mean glacial change in irradiance as we systematically decrease the ASTER GDEM resolution. Figure 10 shows a significant decrease in accurately calculating topographic shading as resolution declines for Satluj and Nianchu glaciers. As resolution coarsens, the modeled mean glacial change in irradiance abruptly decreases (a nearly 17 % loss for Nianchu Glacier as resolution increases from 30 m to ∼100 m) and continues to steadily decrease for both glaciers until the modeled change in irradiance is nearly 50 % below the initial level. Topographic shading, particularly from cast shadowing, relies on the ability to simulate a change in energy based on detailed features of the surrounding topography. As detail degrades, so does the true effect of topographic shading. Higher-resolution and higher-accuracy DEMs, therefore, will significantly improve the accuracy of the shading models. This begs the question as to whether the impact of modeled shading would continue to increase at the same rate when using DEM resolutions higher than 30 m.

We focus specifically on direct solar radiation in this study, which is the major component of the global radiation. However, other global radiation terms are also impacted by topography. While surrounding topography decreases the amount of direct clear-sky irradiance received at the surface, diffuse sky radiation (Ds) and reflected radiation from surrounding terrain (Dt) can also alter the net global radiation. For example, Arnold et al. (2006) calculates the total diffuse radiation as Qdif=FsDs+Dt, where reflected radiation from surrounding topography (Dt) is calculated as Dt=αt1-FsQglobal, where αt is the mean albedo of the surrounding terrain, Fs is the sky-view factor, and Qglobal is measured global radiation. From Eqs. (7) and (8), it is apparent that diffuse sky and terrain-reflected radiation have an inverse relationship with the sky-view factor, which is directly related to the height of the surrounding topography. As such, we would expect the effects of these two components of global radiation to somewhat offset one another in the presence of topography. In Eq. (2), daily mean direct solar radiation decreases in the presence of higher topography; diffuse sky will also decrease as the view factor decreases, and terrain-reflected radiation will increase the radiation received at the surface.

Depending on the orientation of topography, local meteorological conditions, and the mean albedo of the surrounding terrain, terrain-reflected radiation could significantly offset the effects of topographic shading on global radiation received on a glacier surface. The relationship between direct, diffuse sky, and terrain-reflected radiation should be further investigated, especially for valley glaciers in steep terrain.

Most glaciers, under current trends of increasing global temperature, are expected to thin and retreat in response. As they do so, they will thin and retreat into varying amounts of topographic shading. For example, Satluj Glacier will likely thin and retreat up-valley. However, this response may be dampened somewhat by the decrease in incoming solar radiation as a portion of the ablation zone resides in the area of maximum shading (Fig. 4). Ultimately, once the glacier tongue has receded beyond the most shaded region in Fig. 4, the ablation zone will receive a higher amount of direct solar radiation and the mass loss will increase. The link between climate, glacier dynamics, and shading is also relevant for Nianchu Glacier. As the ablation zone moves up-valley into regions of increased shading, the surface will be less affected by direct solar radiation throughout the day, reducing the melt rate and dampening the response to changes in climate. These scenarios suggest that shading may not only be a useful tool for improving our understanding of the current mass balance of valley glaciers throughout HMA but could also likely improve our understanding of the magnitude of glacier response to climatic change under future-climate scenarios. Similarly, the variability in shading on a glacier surface over time is likely to play a role in explaining glacier sensitivity to historical climate changes. Future work should focus on quantifying the role of topographic shading in glacier response to climate changes.