Surface albedo is an essential variable to determine the Earth's surface energy budget, in particular for snow-covered areas where it is involved in one of the most powerful positive feedback loops of the climate system. In situ measurements of broadband and spectral albedo are therefore common. However they are subject to several artefacts. Here we investigate the sensitivity of spectral albedo measurements to surface slope, and we propose simple correction algorithms to retrieve the intrinsic albedo of a slope from measurements, as if it were flat. For this, we first derive the analytical equations relating albedo measured on a slope to intrinsic direct and diffuse albedo, the apportionment between diffuse and direct incoming radiation, and slope inclination and aspect. The theory accounts for two main slope effects. First, the slope affects the proportion of solar radiation intercepted by the surface relative to that intercepted by the upward-looking, horizontal, sensor. Second, the upward- and downward-looking sensors receive reduced radiation from the sky and the surface respectively and increased radiation from neighbouring terrain. Using this theory, we show that (i) slope has a significant effect on albedo (over 0.01) from as little as a

The solar irradiance absorbed by snow-covered surfaces (or net shortwave flux

A potential solution to overcome the effect of the slope and obtain intrinsic albedo is to set the sensors parallel to the terrain

Spectral albedo measurements are less frequent than broadband albedo but provide richer information, enabling us to not only establish the shortwave radiative budget, but also investigate if and how surface albedo is driven by snow microstructural properties

To develop slope correction methods for spectral albedo, the first step is to establish the equations linking measured albedo to intrinsic albedo accounting for the slope and illumination conditions. Mathematically, this problem is closely related to the widely addressed problems of the distribution of the solar radiation at the surface and remote sensing data correction in mountainous areas

The objective of the present paper is to (1) provide the theoretical framework to relate spectral apparent albedo to intrinsic albedo on a slope and (2) present four correction methods to be applied depending on the available information and the assumptions that can be reasonably made, for different measurement conditions.

We first describe the theory of apparent albedo over a snow-covered slope (Sect.

The theoretical equations are implemented in an open-source computer code, and a web application is made available to interactively explore the slope effect as a complement to the present paper (see “Code availability”).

The objective is to relate the albedo measured with perfectly horizontal sensors over a slope to given intrinsic direct and diffuse albedo of the snow surface and other variables (diffuse direct proportion of the incident radiation, solar angles, slope angles, etc.). To this end, we establish the equations for the downwelling and upwelling radiative fluxes on the slope and the neighbourhood (Sect.

The geometry and angles of the problem are depicted in Fig.

The incoming light irradiance

A slope is modelled here as a plane defined by its normal

In addition, the slope receives diffuse irradiance from the atmosphere

This reduced contribution from the atmosphere (

The four studied cases depending on the sensor position (mid-slope, top hill) and surface type of the horizontal surface: snow (blue), dark (black).
The neighbourhood (i.e. horizontal surface) is seen by the downward-looking sensor in the lower solid angle (green shading, as in Fig.

To continue, the term

Multiplying Eq. (

The upward- and downward-looking sensors are considered to be horizontal and to have a perfect cosine response with a 180

For the irradiance received by the upward-looking sensor, two cases shall be distinguished (Fig.

The second case (called T) is when the sensor is above or close to the top of the slope, but still low enough for the downward-looking sensor to mostly view the slope. This is a common case on small slopes.
Rigorously, the notion of “top of the slope” is incompatible with the assumption of infinite slope used before, but it is acceptable here as a trade-off between conducting analytical calculations and representing concrete practical situations. In case T, the irradiance received by the upward-looking sensor is given by

From this point, we have all the fluxes required to compute the albedo for four various cases, depending on whether the sensor is located at the top (T) or middle (M) of the slope and on whether the surrounding terrain is dark (D) or covered with snow (S).

Apparent albedo formulation

The measured apparent albedo can now be expressed as a function of the intrinsic direct and diffuse albedo, the diffuse-to-total ratio, and the geometrical parameters. We present here the analytical derivation for the case of “small slopes” and leave the general case of “large slopes” to the Appendix. A summary of all the results is provided at the end of the section.

The small-slope approximation mathematically corresponds to neglecting second-order variations in

For large slopes, the mathematical derivation of the apparent albedo is more complex because

We propose four methods to retrieve the surface albedo based on measured albedo on moderate slopes, when the small-slope approximation applies (Sect.

Given the slope parameters (

The slope parameters are often unavailable or the precision on these parameters is insufficient. If we further assume that surface snow contains negligible amounts of light-absorbing impurities, the intrinsic albedo in the visible wavelengths is nearly constant and close to a value of 1 over a wide range, typically between 400 and 500

Another practical case is when albedo is measured at different hours during a single day. If we can assume that snow properties have not evolved during that day (e.g. no precipitation, no melt),

Another method is derived for situations where the snow surface is known to be free of impurities. In such a case, it may be interesting to constrain the albedo value in the blue–green range,

Albedo data were acquired at the Col du Lautaret site (45

Solalb is composed of a single light collector fixed at the tip of a 3 m long arm and connected to a spectrometer (400–1050

Autosolexs has two fixed light collectors pointing upward and downward, which are successively connected to a spectrometer by an optical switch

The impact of the slope is studied considering both 100 % diffuse radiation (overcast conditions) and 100 % direct radiation for various zenith solar angles

Apparent albedo as a function of slope computed for various formulation and different illumination conditions. The flat albedo is here fixed to 0.8 and the no diffuse radiation is considered.

The differences between the scenarios appear on the graph starting from about a 10

The slope effect is largest under direct illumination, which occurs
most in the near-infrared domain under clear-sky conditions. Under diffuse radiation (overcast conditions or in the blue and UV ranges) the slope effect is null for the small-slope formulation (bottom right panel in Fig.

The dependence of the albedo variations with slope on the type of illumination is of practical interest to understand spectral albedo measured in natural conditions, which is addressed in the next section.

Under clear-sky or partially cloudy conditions, the proportion of direct and diffuse incident radiation varies as a function of the wavelength, and given the contrasted response to the slope between direct and diffuse albedo the shape of measured albedo spectra is distorted over slope. Figure

Spectra of apparent albedo for various slopes under blue-sky conditions (the diffuse irradiance decreases as a power of 4 of the wavelength). The grey curves are calculated with the small-slope approximation.

The flat surface case (second panel in Fig.

When the terrain is not flat, the shape of the albedo spectrum is affected and not only depends on the ice absorption variations but also on the proportion of direct and diffuse illumination. In the red and infrared range (

The distortion is very different for north- and south-facing slopes. For a north-facing slope, where the direct albedo is reduced by the slope and is thus systematically lower than the diffuse albedo, the apparent albedo shows a marked decrease from the blue to the red ranges (left column in Fig.

Measured, calculated, and corrected albedo for seven acquisitions (rows) taken in different terrain configurations. The first column compares apparent albedo calculated from the theory (small slope and SM case) to measured albedo. The second column compares corrected albedo using measured slope parameters to intrinsic diffuse albedo calculated for a flat surface using measured SSA (grey). The third column is similar to the second column, except that the measured albedo is corrected without using measured slope parameters but by assuming clean snow. The second and third columns also show measured albedo to highlight the change due to the correction.

A set of Solalb measurements acquired in winter 2018 on clean and dry snow has been selected to cover a wide range of slope inclinations and orientations. The comparison with the theoretical apparent albedo using measured slopes, SSA, and diffuse-to-total ratio is shown in Fig.

Figure

Measured (plain) and simulated (dashed) albedo acquired on 23 March 2018.

With the slope being south-south-east, most albedo spectra are affected by sun-facing distortions, i.e. a maximum higher than 1 in the visible range and a concave spectral shape, which is also predicted by the calculation. A greater distortion is observed in the morning when the sun is facing the slope, while the spectrum taken just before the local sunset is almost unaffected. The agreement between the measurements and the theory is overall good even though the adjustment of the slope inclination has a great influence on such a result. The largest discrepancy between measurements and calculations is found at the end of the day, in the near-infrared range, suggesting that snow metamorphism during the day might have decreased the specific surface area. Additional simulations with a SSA of 22

The correction methods described in Sect.

Figure

Overall, we also note that the quality of the correction does not depend on the slope inclination. The amplitude of the correction (i.e. the difference between the blue and the violet or red curves) can be large (e.g. case 7); still the correction is satisfactory. These results show that the correction of apparent albedo is possible for a relatively large range of slopes, even without knowing the slope parameters. However, this high performance relies on the assumption of clean snow and on the quality of the cross-calibration between the upwelling and downwelling flux acquisitions, which is usually not an issue for manual measurements

Figure

Albedo spectra measured with Autosolexs (grey) and corrected (orange, blue) on 23 March 2018. Measured albedo during overcast conditions on 2 March 2018 are also shown (green).

The albedo spectrum corrected without constraint (blue curve) shows an improvement compared to the measured albedo spectra (grey). The albedo value is more constant in the visible range and closer to 1 in the blue–green range. Nevertheless, it is still higher than 1 and slightly concave, which is a clear sign of insufficient correction. Furthermore, the method produces mixed results depending on the conditions (e.g. taking a subset of the inputs, results not shown). This unstable behaviour comes from the fact that the only constraint to infer the slope parameters is provided by the temporal variations induced by the course of the sun. In turn, any small uncertainty that affects the angular behaviour of the sensor (e.g. imperfect levelling or imperfect cosine response), of the snow surface albedo (e.g. roughness), or of the underlying theory (mainly Eq.

Assuming an impurity-free surface brings a strong constraint on the albedo in the visible, and, if it is true, precisely provides what was missing in the unconstrained method. The constrained method performs well in our case (orange versus green curves, in Fig.

Along with the corrected albedo, both methods provide estimates of the two slope angles, which was not possible with the methods using a single acquisition. We obtain a slope of 3

The theoretical developments and the measured spectra presented throughout this paper make it clear that the “apparent albedo” of a slope is not an albedo sensu stricto and must not be used to compute the energy absorbed by the surface with the usual equation

It is worth recalling that measured broadband albedo is similarly subject to the slope effect and that the usual equation

Ideally, the term “albedo” should not be used to refer to the ratio of upwelling and downwelling light fluxes over a slope, but in practice, we recommend at least considering this issue, systematically using the terms “apparent” or “measured”.

All the results presented in Sect.

Improvement of instruments (equipped with an inclinometer) and protocols (systematic acquisition of diffuse-to-total ratio and slope inclination) is a response to these requirements. Another response is the ad hoc correction of the measured albedo. In a first attempt, it was common in our community to normalize the measured albedo to 1 (or 0.98) in the visible range

Example of scaled spectrum of pristine snow affected by a small slope (2

We thereby recommend not applying rough scaling but instead using a proper correction method as described in this paper. Nevertheless, the risk of over or undercorrections also exists and may result in confusion with dirty snow. For instance overcorrecting measurements taken over pristine snow on a slope facing away from the sun result in a spectrum with a convex shape characteristic of dirty snow. Our conclusion is that small slopes, impurity content and calibration errors, sensor angular response, and operator shadows are interrelated, and only a global assessment of the main error sources and a consistent treatment of these sources can lead to properly corrected albedo spectra.

The results show that the correction of the slope effect is possible in the domain of validity of small slopes (

Here we used diffuse-to-total ratio measurements acquired at almost the same time as the albedo acquisitions, which is a very favourable case.

When a single albedo acquisition is available, the most favourable case is when the slope angles (inclination and aspect) or the slope parameter (

When multiple albedo acquisitions at the same location are available for a wide range of sun positions (e.g. time series of albedo), a correction method exists assuming constant snow properties throughout the acquisitions but without the requirement of assuming clean snow or a known SSA (Eq.

The correction method proposed by

Another important assumption used in the present paper is the perfect levelling of the sensors. It applies well to our measurements because the inclination angles are systematically recorded and proved to be stable within 0.2

The wide range of possible assumptions shows that many methods can be valuable depending on the conditions. Further work should perform a more systematic comparison and exploration of the sensitivity to input uncertainties of each method.

Spectral albedo measured with horizontal sensors is very sensitive to the slope of the underlying surface in clear-sky conditions, first because the illumination received by the slope from the sun is changed compared to that on a flat surface, and second because the upward- and downward-looking sensors are affected by additional illumination coming from the slope itself and the neighbouring slopes.

The first cause dominates up to about 15

The four spectral albedo correction methods proposed here for small slopes complement other methods presented in the literature for both spectral and broadband albedo. The diversity of methods is explained by the different possible assumptions that apply or not depending on the type of available measurements. More methods can be devised in the future, and the rigorous equation set provided in this paper should be helpful for this. Nevertheless, even though our results show that a satisfactory correction can be achieved in many situations (residual error better than 0.03), we emphasize that ancillary information is required to perform such a correction, implying higher complexity and cost of instruments and protocols. In this context, our main recommendation is that slope inclination and aspect and the diffuse-to-total irradiance ratio should be systematically recorded in future albedo measurement campaigns.

Combining Eqs. (

The measured ratio between the diffuse flux obtained by obstructing the sun and the total received is equal to

This equation is similar to that for a small slope, except that the direct term is scaled by

At mid-slope, the albedo has a different expression,

If the diffuse-to-total ratio is measured mid-slope by obstructing the sun, it is of the form

Another useful derivation is when the diffuse-to-total ratio is modelled using an atmospheric model, i.e. similar to when measured at the top of the hill. The ratio is then defined as in Eq. (

The energy received by the downward-looking sensor in the case of snow is obtained by plugging Eqs. (

Dividing by the incoming irradiance at the top of the slope

The energy received by the downward- and upward-looking sensors is given by Eqs. (

The expression of the albedo when the diffuse-to-total ratio is available from atmospheric modelling far above the topography is derived following Eq. (

When the sun is below the slope, we shall distinguish two cases, first when the upward-looking sensor is shadowed but the neighbouring surface is still illuminated and second when the neighbouring surface is also in the shadows. The first case is not tractable because

The theory presented in Sect.

GP and MD designed the study and performed the theoretical calculations. ML, FT, FL, and LA contributed to the measurements. All authors contributed to the reflection on the slope effect and to the manuscript.

The authors declare that they have no conflict of interest.

Alexander Kokhanovky provided very helpful comments on the manuscript.

This research has been supported by the Agence Nationale de la Recherche (grant no. 1-JS56-005-01), the Agence Nationale de la Recherche (grant no. ANR10 LABX56), the Centre National d'Etudes Spatiales (grant no. Miosotis), the Agence Nationale de la Recherche (grant no. ANR-16-CEO01-0006), and the Institut National des Sciences de l'Univers, Centre National de la Recherche Scientifique (ASSURANCE grant) and the academy of Finland (grant no. 304345) in the Financial support.

This paper was edited by Michiel van den Broeke and reviewed by two anonymous referees.