Quasi‐analytical treatment of spatially averaged radiation transfer in complex terrain

Löwe, Henning; Helbig, Nora

doi:10.1029/2012jd018181

Cited by 15 publications

(30 citation statements)

References 29 publications

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“…Second, the L / ξ ratio corrects for finite coarse‐scale grid cells. Helbig and Löwe [] generalized the analytical approximation for the domain‐averaged sky view factor of Löwe and Helbig [], which solely depended on mean square slope μ , to derive equation . To do so, they introduced the terrain parameter L / ξ to systematically correct for finite L / ξ ratios in coarse‐scale grid sizes.…”

Section: Resultsmentioning

confidence: 99%

“…Partial derivatives ∂ x z and ∂ y z in orthogonal directions were used to derive slope angles ζ from

\overset{2}{\tan} ζ = {(\partial_{x} z)}^{2} + {(\partial_{y} z)}^{2}

. Based on the assumption of isotropy, a joint (slope) probability density for ∂ x z and ∂ y z factorizes into two Gaussians with the standard deviation

μ = \sqrt{2} σ / ξ

[see Löwe and Helbig , ]. The terrain parameter μ can also be associated with the mean square slope by

μ = {({}, \overset{([], {(\partial_{x} z)}^{2} + {(\partial_{y} z)}^{2})}{false} / 2)}^{1 / 2},

via

2 μ^{2} = \overset{{(\partial_{x} z)}^{2} + {(\partial_{y} z)}^{2}}{false} = \overset{\tan^{2} ζ}{false} = 4 {((), σ / ξ)}^{2}

, following the definition of the Gaussian covariance in equation [ Adler , ].…”

Section: Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Parameterizing surface wind speed over complex topography

Helbig¹,

Mott²,

Herwijnen³

et al. 2017

JGR Atmospheres

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Subgrid parameterizations are used in coarse‐scale meteorological and land surface models to account for the impact of unresolved topography on wind speed. While various parameterizations have been suggested, these were generally validated on a limited number of measurements in specific geographical areas. We used high‐resolution wind fields to investigate which terrain parameters most affect near‐surface wind speed over complex topography under neutral conditions. Wind fields were simulated using the Advanced Regional Prediction System (ARPS) on Gaussian random fields as model topographies to cover a wide range of terrain characteristics. We computed coarse‐scale wind speed, i.e., a spatial average over the large grid cell accounting for influence of unresolved topography, using a previously suggested subgrid parameterization for the sky view factor. We only require correlation length of subgrid topographic features and mean square slope in the coarse grid cell. Computed coarse‐scale wind speed compared well with domain‐averaged ARPS wind speed. To further statistically downscale coarse‐scale wind speed, we use local, fine‐scale topographic parameters, namely, the Laplacian of terrain elevations and mean square slope. Both parameters showed large correlations with fine‐scale ARPS wind speed. Comparing downscaled numerical weather prediction wind speed with measurements from a large number of stations throughout Switzerland resulted in overall improved correlations and distribution statistics. Since we used a large number of model topographies to derive the subgrid parameterization and the downscaling framework, both are not scale dependent nor bound to a specific geographic region. Both can readily be implemented since they are based on easy to derive terrain parameters.

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Section: Resultsmentioning

confidence: 99%

“…Partial derivatives ∂ x z and ∂ y z in orthogonal directions were used to derive slope angles ζ from

\overset{2}{\tan} ζ = {(\partial_{x} z)}^{2} + {(\partial_{y} z)}^{2}

. Based on the assumption of isotropy, a joint (slope) probability density for ∂ x z and ∂ y z factorizes into two Gaussians with the standard deviation

μ = \sqrt{2} σ / ξ

[see Löwe and Helbig , ]. The terrain parameter μ can also be associated with the mean square slope by

μ = {({}, \overset{([], {(\partial_{x} z)}^{2} + {(\partial_{y} z)}^{2})}{false} / 2)}^{1 / 2},

via

2 μ^{2} = \overset{{(\partial_{x} z)}^{2} + {(\partial_{y} z)}^{2}}{false} = \overset{\tan^{2} ζ}{false} = 4 {((), σ / ξ)}^{2}

, following the definition of the Gaussian covariance in equation [ Adler , ].…”

Section: Methodsmentioning

confidence: 99%

Parameterizing surface wind speed over complex topography

Helbig¹,

Mott²,

Herwijnen³

et al. 2017

JGR Atmospheres

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“…Thus, its sign is pixel-dependent and is determined by the interaction between the sun, the topography and the surface reflectance. Nevertheless, Löwe and Helbig (2012) showed in the simplified case of a Lambertian BRDF, atmospheric effects being neglected and with single terrain reflections, that the higher the subgrid orography, the lower the effective albedo. This yields that α/ TE has probably a negative sign in a first approximation, and this confirms that including topographic effects in reflectance calculations results in a lower apparent reflectance, which is converted to a smaller SSA by the retrieval algorithms.…”

Section: Influence Of Topographic Correctionmentioning

confidence: 99%

Intercomparison of retrieval algorithms for the specific surface area of snow from near-infrared satellite data in mountainous terrain, and comparison with the output of a semi-distributed snowpack model

Mary¹,

Dumont

Dedieu

et al. 2013

The Cryosphere

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Abstract. This study compares different methods to retrieve the specific surface area (SSA) of snow from satellite radiance measurements in mountainous terrain. It aims at addressing the effect on the retrieval of topographic corrections of reflectance, namely slope and aspect of terrain, multiple reflections on neighbouring slopes and accounting (or not) for the anisotropy of snow reflectance. Using MODerate resolution Imaging Spectrometer (MODIS) data for six different clear sky scenes spanning a wide range of snow conditions during the winter season 2008-2009 over a domain of 46 × 50 km in the French Alps, we compared SSA retrievals with and without topographic correction, with a spherical or non-spherical snow reflectance model and, in spherical case, with or without anisotropy corrections. The retrieved SSA values were compared to field measurements and to the results of the detailed snowpack model Crocus, fed by driving data from the SAFRAN meteorological analysis. It was found that the difference in terms of surface SSA between retrieved values and SAFRAN-Crocus output was minimal when the topographic correction was taken into account, when using a retrieval method assuming disconnected spherical snow grains. In this case, the root mean square deviation was 9.4 m 2 kg −1 and the mean difference was 0.1 m 2 kg −1 , based on 3170 pairs of observation and simulated values. The added-value of the anisotropy correction was not significant in our case, which may be explained by the presence of mixed pixels and surface roughness. MODIS retrieved data show SSA variations with elevation and aspect which are physically consistent and in good agreement with SAFRAN-Crocus outputs. The variability of the MODIS retrieved SSA within the topographic classes of the model was found to be relatively small (3.9 m 2 kg −1 ). This indicates that semi-distributed snowpack simulations in mountainous terrain with a sufficiently large number of classes provides a representation of the snowpack variability consistent with the scale of MODIS 500 m pixels.

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“…using 2µ 2 = (∂ x z) 2 + (∂ y z) 2 = tan 2 ζ = 4(σ/ξ ) 2 as outlined by Löwe and Helbig (2012). We also use the L/ξ ratio where a large ratio indicates that more topographic features are included in a domain size L. Note that the typical width of topographic features ξ in a domain size L can be obtained via ξ = √ 2σ z /µ, with the standard deviation of the summer DSM σ z .…”

Section: Terrain Characteristicsmentioning

confidence: 99%

“…For selecting terrain parameters, we exploited the fact that real topographic slope characteristics are reasonably well described by Gaussian statistics . Gaussian random fields with a Gaussian covariance such that topography is reduced to only two underlying large length scales in a model domain of size L, were previously used to systematically investigate radiative transfer in complex terrain via the radiosity approach (Helbig et al, 2009;Helbig and Löwe, 2012;Löwe and Helbig, 2012) as well as to develop a parameterization for domain-averaged sky view factors in complex terrain (Helbig and Löwe, 2014).…”

Section: Terrain Characteristicsmentioning

confidence: 99%

Fractional snow-covered area parameterization over complex topography

Helbig¹,

Herwijnen²,

Magnusson³

et al. 2015

Hydrol. Earth Syst. Sci.

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118

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Abstract. Fractional snow-covered area (SCA) is a key parameter in large-scale hydrological, meteorological and regional climate models. Since SCA affects albedos and surface energy balance fluxes, it is especially of interest over mountainous terrain where generally a reduced SCA is observed in large grid cells. Temporal and spatial snow distributions are, however, difficult to measure over complex topography. We therefore present a parameterization of SCA based on a new subgrid parameterization for the standard deviation of snow depth over complex topography. Highly resolved snow depth data at the peak of winter were used from two distinct climatic regions, in eastern Switzerland and in the Spanish Pyrenees. Topographic scaling parameters are derived assuming Gaussian slope characteristics. We use computationally cheap terrain parameters, namely, the correlation length of subgrid topographic features and the mean squared slope. A scale dependent analysis was performed by randomly aggregating the alpine catchments in domain sizes ranging from 50 m to 3 km. For the larger domain sizes, snow depth was predominantly normally distributed. Trends between terrain parameters and standard deviation of snow depth were similar for both climatic regions, allowing one to parameterize the standard deviation of snow depth based on terrain parameters. To make the parameterization widely applicable, we introduced the mean snow depth as a climate indicator. Assuming a normal snow distribution and spatially homogeneous melt, snow-cover depletion (SCD) curves were derived for a broad range of coefficients of variations. The most accurate closed form fit resembled an existing fractional SCA parameterization. By including the subgrid parameterization for the standard deviation of snow depth, we extended the fractional SCA parameterization for topographic influences.For all domain sizes we obtained errors lower than 10 % between measured and parameterized SCA.

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Quasi‐analytical treatment of spatially averaged radiation transfer in complex terrain

Cited by 15 publications

References 29 publications

Parameterizing surface wind speed over complex topography

Parameterizing surface wind speed over complex topography

Intercomparison of retrieval algorithms for the specific surface area of snow from near-infrared satellite data in mountainous terrain, and comparison with the output of a semi-distributed snowpack model

Fractional snow-covered area parameterization over complex topography

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