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HYDROLOGICAL PROCESSESHydrol. Process. 23, 2584–2599 (2009)Published online 29 June 2009 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/hyp.7346

Modelling snow melt and snowcover depletion in a smallalpine cirque, Canadian Rocky Mountains

Christopher M. DeBeer* and John W. PomeroyCentre for Hydrology, University of Saskatchewan, 117 Science Place, Saskatoon, Saskatchewan, Canada

Abstract:

Spatial and temporal patterns of areal snowcover depletion (SCD) were studied over a small (<0Ð6 km2) alpine cirque withinthe Canadian Rocky Mountains using a combined approach of daily acquisition of remotely sensed imagery, together withmeteorological observations and snowmelt modelling. Digital terrestrial photographs were georeferenced using a novel softwaretool together with a high-resolution digital elevation model and used to derive measurements of fractional snowcovered area(SCA) over the cirque. Manual snow surveys carried out in the pre-melt period were used to describe the initial frequencydistribution of snow water equivalent (SWE) values over the cirque, and indicated a lognormal distribution of SWE whensurveys were stratified by terrain features. Rates of snowmelt were simulated using a physically based snowmelt energybalance model, Snobal, driven by observed meteorological conditions at a nearby station, which were adjusted for slopeorientation and exposure by making corrections to observed incoming shortwave and longwave radiation components in thecold regions hydrological model platform. Simulated melt rates were then applied to the approximated SWE distributions tomodel the decline in SCA over the spring. The model was found to perform well for the simulation of snowmelt based on pointobservations of SWE at the meteorological station, and produced a close correspondence between simulated and observedSCD curves representing two opposing slopes within the cirque. The results show that both the pre-melt distributions of SWEand the spring melt rates exhibit considerable spatial variability between distinct slope units within the cirque, and that thisvariability has a significant impact on simulated SCD. Assuming a unimodal pre-melt frequency distribution and conditions ofspatially uniform snowmelt over complex terrain such as this can lead to large errors in the simulation results. It is suggestedthat modelling applications intended to represent snowmelt dynamics and areal SCD in similar alpine environments considerthe effects of spatial variation in SWE distribution and melt energetics between slopes. Copyright 2009 John Wiley & Sons,Ltd.

KEY WORDS snowcover depletion; snowmelt modelling; scaling; mountain hydrology; spatial variability

Received 30 June 2008; Accepted 25 March 2009

INTRODUCTION

In alpine environments, variations in elevation, slope,aspect, topographic shelter and vegetation structure giverise to complex patterns of both snowcover developmentthroughout the winter and surface energetics during melt(Elder et al., 1991, 1998; Pomeroy, 1991; Bloschl andKirnbauer, 1992; Allen and Walsh, 1993; Marks et al.,1998; Marks and Winstral, 2001; Pomeroy et al., 2003;Anderton et al., 2004). These patterns largely control thedynamics of areal snowcover depletion (SCD) over thelandscape during the spring and result in a multitude ofsnow patches of varying size with intervening snow-freeareas as the snowcover ablates. This disintegration of thesnowpack into patches over time is important hydrolog-ically as it controls the contributing area for snowmeltrunoff generation and thereby affects the rate and mag-nitude of meltwater production. From a climatologicalperspective, the fraction of snowcovered area (SCA) issignificant as it exerts a major influence on the sur-face energy fluxes through its effects on the albedo and

* Correspondence to: Christopher M. DeBeer, Centre for Hydrology,University of Saskatchewan, 117 Science Place, Saskatoon, SaskatchewanS7N 5C8, Canada. E-mail: [email protected]

temperature of the surface. Accurate representation ofSCA and SCD is therefore important for hydrologicaland atmospheric modelling applications in such environ-ments.

One approach is to explicitly model the spatial andtemporal variability in snow processes at very fine scalesusing fully distributed models (e.g. Davis et al., 1995;Hartman et al., 1999; Marks et al., 1999). These modelscan be run at spatial scales as small as several meters;however, this level of detail necessitates extensive spatialinformation on model parameters and forcing variablesand requires considerable computing resources. For thesereasons, the approach is limited to a relatively smallnumber of applications for which these requirementsare met.

More commonly, hydrological models and land sur-face schemes are applied at larger spatial scales andaccount for the variability at small scales by employ-ing a sub-grid or sub-model element parameterization ofthe areal snowcover state. This allows for the determi-nation of effective surface parameters or for weightingseparate flux calculations for snowcovered and snow-freeareas. These parameterizations are often based on meansnow water equivalent (SWE) or accumulated depth ofsnowmelt over time, and rely on some understanding

Copyright 2009 John Wiley & Sons, Ltd.

MODELLING SNOW MELT AND SNOWCOVER DEPLETION IN A SMALL ALPINE CIRQUE 2585

of the frequency distribution of SWE over the land-scape. The approach generally relies on several limitingassumptions, however, including spatially uniform meltrates over the model domain or computational grid cellarea (Liston, 1999; Luce et al., 1999; Luce and Tar-boton, 2004), some theoretical average depth of snowbeyond which fractional SCA is equal to unity (Donaldet al., 1995) or a single unimodal frequency distributionof SWE over the grid cell (Liston, 2004). Even at rela-tively small spatial scales in sharp alpine terrain, theseassumptions are often violated, and thus the applicabilityof models relying on such parameterizations is potentiallylimited. The problem is related, in part, to the choice ofgrid cell resolution or the method of disaggregation ofthe terrain for modelling. The use of an arbitrary gridsystem that does not conform to natural scales and loca-tion of variability (both stochastic and deterministic) inthe processes being modelled (e.g. Seyfried and Wilcox,1995) introduces artificial scaling problems in the mod-els where none exists in nature, and thus sophisticatedtechniques must be employed to handle the variabilityat sub-grid scales. The processes involved in areal SCDgenerally do not aggregate linearly and it is not possi-ble to assume that E�f�x�� D f�E�x��, where E is themathematical expectation (i.e. mean), x is location andf a function or variable (Bloschl, 1999). This must beconsidered when changing model scales and/or lumpingtogether model inputs and conditions over larger com-putational areas. It is likely that hydrological modellingapplications dealing with snowmelt and areal SCD canbe applied at intermediate spatial scales and objectivelychosen terrain units for calculation, which are consistentwith the variability in the relevant processes, and therebyavoid resorting to the use of finely distributed approachesor arbitrarily selected model grids. Several recent studies(e.g. Pomeroy et al., 2004; Davison et al., 2006; Dorneset al., 2008) have focused on the appropriate scale and/orlandscape segmentation for dealing with snow ablation,SCD and meltwater runoff generation. These studies haveshown that these processes are terrain-class-specific, andthat stratification of the landscape according to featuressuch as slope/aspect and elevation yields improved sim-ulation results relative to those from an inappropriatelysegmented or spatially aggregated treatment of the pro-cesses.

This study aims to build on these findings and explorethe effects of spatial variability in the relevant processesfor areal SCD in a region of sharp alpine topographyin the Canadian Rocky Mountains. We demonstrate theuse of a relatively new technique—oblique terrestrialphotography—for deriving daily measurements of SCAin this environment, where conventional remote sensingmethods are unfeasible due to persistent cloud cover andinfrequent return intervals of most sensors. The specificobjective of this study is to upscale a physically basedpoint-scale energy balance simulation of snowmelt tocalculate the basin scale SCA and SCD as well as thatfor two distinct and opposing slope units within the smallbasin. We use these simulations to assess the modelling

implications of spatially variable SWE and snowmeltrates within this environment.

THEORETICAL CONSIDERATIONS

The lognormal probability density functionAs a result of differences in snow deposition and

redistribution of snow over a landscape, snowcoverstend to exhibit substantial spatial variability in theirwater equivalent depth. A number of studies within avariety of environments (e.g. Donald et al., 1995; Shook,1995; Pomeroy et al., 1998, 2001; Faria et al., 2000)have found that distributions of SWE prior to melt canbe approximated by the lognormal distribution whenstratified by terrain classes. The lognormal distributionhas the advantage that it is a relatively simple two-parameter distribution with known statistical properties.

The lognormal distribution is expressed in terms of theprobability density function of the transformed variable,i.e. y D ln(SWE) as in the following Equation (1):

f�y� D 1

s2y

p2�

exp

(� �y � y�2

2s2y

), �1�

where y is the mean of the logarithmic values of SWE andsy is the standard deviation of the transformed variable.For practical applications, the lognormal distribution isoften expressed in the following linear form:

SWE D SWE�1 C KCV�, or �2a�

SWE D SWE C Ks, �2b�

where SWE is the value of the snow water equivalenthaving an exceedence probability equal to that of thefrequency factor, K and SWE, s and CV are the arithmeticmean, standard deviation and coefficient of variation (i.e.standard deviation divided by the mean) of the naturalvalues, respectively. When the values of the transformedvariable are normally distributed, K in Equation (2) isthe well-known z-statistic of the normal distribution andis given by the following Equation (3) (Chow, 1954):

K D 1

CV

[exp

(syKy � s2

y

2

)� 1

], �3�

where Ky D �y � y�/sy , the theoretical frequency factorof the transformed data.

Equation (2) describes the theoretical two-parameterlognormal distribution. Thus, observed values of SWEplotted against K should approximate a straight line witha slope equal to the standard deviation of SWE andan intercept at K D 0 equal to SWE, provided that theunderlying distribution is lognormal. Note that naturalvalues of K and SWE should be used. Values of Kfor observed data (excluding observations of zero SWEdepth) can be calculated as (Chow, 1954):

K Dexp

(syKy � s2

y

2

)� 1

√exp�s2

y� � 1. �4�

Copyright 2009 John Wiley & Sons, Ltd. Hydrol. Process. 23, 2584–2599 (2009)DOI: 10.1002/hyp

2586 C. M. DEBEER AND J. W. POMEROY

A value for observed Ky is required for Equation (4);this may be determined by noting that the exceedenceprobability, P, of Ky and SWE are equal for a givensample of n observations. The probability of Ky beingexceeded, P�Ky�, is provided in the Equation (5):

P�Ky� D 1p2�

∫ 1

Ky

exp

(�K2

y

2

)dKy. �5�

The exceedence probability of each measurement for aset of observations of SWE can be determined followingGumbel (1954) as in the following Equation (6):

P(SWE) D r

n C 1, �6�

where r is the rank of the SWE observation whenarranged in decreasing order of magnitude and n is thenumber of observations. In Equation (6), observations ofzero SWE depths should be included. Given that P(SWE)D P�Ky�, corresponding values for observed Ky can bedetermined by taking the inverse of the standard normalcumulative distribution for the values of 1 � P(SWE) foreach SWE observation (e.g. P(SWE) is the exceedenceprobability and therefore 1 � P(SWE) is the cumulativeprobability).

Areal SCD

Approximations of the snowcover distribution basedon the lognormal probability density function are wellsuited for the calculation of SCA and SCD over a giventerrain unit. For example, as expressed by Equation (2),the value of K is a function of the exceedence probabilityof the corresponding value of SWE. Therefore, thefrequency factor corresponding to specific SWE valuescan be used as an index of the probability of thatparticular value of SWE being exceeded, or alternatively,the areal fraction of the snowcover with a value ofSWE exceeding that particular value. The value of Kcorresponding to SWE D 0 (i.e. Kmin) is then an indexof the SCA over the landscape.

Estimates of SCA and SCD can be derived as afunction of the applied melt (i.e. the one-dimensional meltrate applied uniformly to the SWE distribution) and theinitial frequency distribution of SWE. The steps involvedin this procedure are as follows:

1. Establish the values of SWE and CV, and fromEquation (2) the value of Kmin is determined from thecorresponding x-intercept (i.e. SWE D 0).

2. Calculate Ky from Equation (4). This requires thestandard deviation of the log-transformed data, whichcan be estimated from (Chow, 1954):

sy D√

ln�CV2 C 1�. �7�

3. Based on this estimate of Ky , the SCA fraction(equivalent to P(SWE)) is quantified as P�Ky�, whichis determined as one minus the value of the standard

normal cumulative distribution function evaluated forKy .

4. For the next time step, the value of SWE is reducedby the amount of applied melt, and the procedure issubsequently repeated.

This procedure is equivalent to uniformly melting theinitial frequency distribution of SWE.

METHODS

Study site

This work focused on a small (¾0Ð6 km2) alpine cirqueon the eastern side of Mt. Allan (hereafter referred toas Mt. Allan cirque) within the Marmot Creek ResearchBasin (50Ð96 °N; 115Ð21 °W). The cirque ranges in ele-vation from ¾2300 m at local treeline up to 2831 mat the summit of Mt. Allan, and is characterized byseveral major slopes of different orientation with dis-tinct microtopographic variations superimposed over theterrain (Figure 1). Marmot Creek is situated within theFront Ranges of the Canadian Rockies, where climaticconditions are dominated by continental air masses. Win-ters are long and cold, with an average temperature of�15 °C for the months of January through March (i.e.for Mt. Allan cirque based on extrapolated readings fromKananaskis, AES Sta. 3053600; 1391 m). The springseason is generally cool and wet, often producing latesnowfall events at high elevations. Average temperaturesat the Mt. Allan cirque for May and June (during theprimary snowmelt period) are 2 and 8 °C, respectively.Historically, annual precipitation at Marmot Creek hasbeen observed to average about 900 mm, increasing toover 1140 mm at treeline near Mt. Allan cirque (Storr,1967), and recent observations more-or-less correspondto these values. Roughly 60–75% of the total precipita-tion falls as snow at Marmot Creek, while at the higherelevations here this fraction is likely greater.

Meteorological observations during the course of thisstudy were made at a permanent meteorological stationlocated on Fisera Ridge directly adjacent to the cirque(2318 m; Figure 1). Incoming shortwave radiation wasmeasured here using a Kipp & Zonen CM21 pyranometer(directional error < š 10 W/m2), and incoming longwaveradiation was measured using a Kipp & Zonen CG1 pyr-geometer (manufacturer estimated error < š 20 W/m2).Air temperature and relative humidity were measuredwith a Vaisala HMP45C212 hygrothermometer housedinside a Gill radiation shield. Wind speed and directionwere measured using an R.M. Young anemometer (Model05103-10). The hygrothermometer and anemometer weremounted at respective heights of 2Ð25 and 2Ð55 m abovethe ground surface. Rainfall was measured at FiseraRidge using a Campbell Scientific TB4-L tipping bucketrain gauge, while snowfall depths were measured with aCampbell Scientific SR50 sonic ranging sensor. Measure-ments of total precipitation were obtained using a GeonorT-200B strain gauge located at the mid-elevation Upper



Figure 1. Shaded relief map of the Mt. Allan cirque within the Marmot Creek Research Basin, Kananaskis, Alberta. Snow survey transects aremarked on the map (white dashed lines) along with the location of the meteorological station on Fisera Ridge (hatched circle). Map insets includean oblique aerial photograph of the cirque and a Landsat 7 image showing the location of Marmot Creek within the Rocky Mountain Front Ranges

Clearing site (1843 m) within the Marmot Creek basinroughly 2 km away.

Terrestrial photography acquisition and image analysis

In early May of 2007, a digital single lens reflexcamera (Pentax model K110D) with a high-precision lensto minimize radial image distortion (Pentax DA 21 mmF3Ð2AL Limited) was mounted to the Fisera Ridgestation inside a weatherproof housing. The camera’sshutter release was controlled remotely using a CampbellScientific CR23X datalogger, which was programmed totake photos several times daily. Photos taken at 12 : 00p.m. local time each day were selected for analysis,except in situations when low cloud cover or snowfallobscured the terrain.

To derive SCA measurements from the digital pho-tography, the images were projected orthogonally onto a1-m resolution digital elevation model (DEM) that wasgenerated from airborne LiDAR (Light Detection AndRanging) data collected in August 2007 over the entireMarmot Creek Research Basin. The re-projection of theseimages was performed using an IDL software tool, Geo-referencing Terrestrial Photography, described in detailby Corripio (2004). Initially, the DEM is projected vir-tually from the perspective of the camera position andorientation so that it forms a two-dimensional represen-tation of the relief information contained in the DEM.Orientation data consisting of three rotation parametersfor spatial rotations of the camera, location of the camera(x, y, z) and the central pixel of the photographic imagein the DEM coordinate system and principal distance (i.e.camera focal length) are required to achieve this represen-tation. This virtual projection of the DEM is then scaled

by the resolution of the image, and the correspondencebetween image pixels, the projected coordinates of theDEM cells and their geographic location is established.Finally, based on this information, the image pixels arere-projected over the DEM to the geographic location thatthey correspond to.

Orientation parameters were derived manually by trialand error for each image to achieve optimal correspon-dence between the image pixels and their geographiccoordinates. This was necessary since very subtle move-ments of the camera due to wind and periodic operatoraccess into the camera housing caused significant move-ments of the geographic locations of individual pixelsbetween subsequent images. Comparison of a series ofground control points within the cirque obtained using adifferential global positioning system and clearly identifi-able objects within the georeferenced images (e.g. bushes,small rock outcrops, etc.) revealed a root mean square(RMS) error of ¾3 m, but this error tended to be direc-tionally consistent, and thus the area of features waspreserved. Visual inspection of the georeferenced imageseries and a 1-m resolution shaded relief image of theDEM also indicated a very close correspondence betweendata sets over nearly all visible parts of the cirque.

Daily SCA measurements were derived from thegeoreferenced imagery for the entire cirque as well asfor two opposing slopes within the cirque (describedin more detail below). These measurements were madeusing ESRI ArcMap 9Ð1, in which a threshold was appliedto classify snow and non-snow areas. This classificationcould be easily performed because of the large differencein brightness values of the pixels representing snow andthose representing bare ground or vegetation. SCA was



then determined as the ratio of the number of ‘snow’pixels to the total number of pixels over the relevant area(excluding pixels containing ‘no-data’ values that werehidden from view of the camera).

Pre-melt snow surveys

Multiple snow surveys were carried out along lineartransects within and adjacent to the cirque on March29–30 before the main snowmelt period to characterizethe distribution of pre-melt SWE (Figure 1). Each ofthese surveys included between 25 and 100 measurementsof snow depth obtained using an aluminium rod graduatedin 1 cm increments and density measurements takenevery fifth-depth measurement using a Mount Rosesnow tube. In addition, several measurements of thedensity of shallow snow were made by weighing samplesobtained with a fixed volume triangular cutting device(Perla ‘Swedish Sampler’). The surveys were chosenwithin a variety of terrain types across the cirque toobtain representative distributions of SWE for the majortopographic slope, aspect and exposure classes withinthe cirque. Subsequent analysis of these data was carriedout to determine the relevant statistical parameters (i.e.SWE and CV) and test the applicability of the lognormaldistribution for describing pre-melt SWE within thecirque.

Snowmelt modelling and SCD simulation

Snowmelt rates were simulated within the cirque usingthe cold regions hydrological model (CRHM) platform(Pomeroy et al., 2007). CRHM is a flexible object-oriented modelling system that can be used to develop,support and apply dynamic hydrological process algo-rithms. These algorithms are applied over hydrologicalresponse units (i.e. homogeneous terrain units character-ized by their geometric and surface vegetation proper-ties), within which conditions and processes are repre-sented by single sets of parameters, state variables andenergy and mass fluxes. Various component modules rep-resenting basin characteristics, observations and hydro-logical processes are combined within CRHM to forman operational model of the system that has a level ofcomplexity specified by the needs of the user.

Melt rates were computed using the Snobal (snowmeltenergy balance model) (Marks et al., 1998, 1999) modulewithin CRHM. This module approximates the snowcoveras being composed of two layers: a surface activelayer of fixed thickness and a lower layer representingthe remaining snowpack. The module solves for thetemperature (°C) and the specific mass (kg/m2) or waterequivalent depth per unit area (mm) of each layer foreach timestep. The energy balance of the snowpack at apoint is expressed as in the following Equation (8):

Qm D QŁ C QH C QE C QG C QP � dU/dt, �8�

where Qm is the energy available for snowmelt, QŁ is thenet radiation composed of both shortwave and longwavecomponents, QH, QE and QG are the sensible, latent

and ground heat fluxes, respectively, QP is the energyadded to the snowpack by precipitation, and U is theinternal energy of the snowpack. The melt energy canbe expressed as a depth of melt, m, by the followingEquation (9):

m D Qm

�hfˇ, �9�

where � is the density of the snow, hf is the latent heat offusion (0Ð334 MJ/kg) and ˇ is the fraction of ice in snow(taken as 0Ð97). Melt is computed in either layer whenthe accumulated energy exceeds that required to bringthe snowpack to 0 °C, at which point positive values ofQm result in snowmelt. The reader is referred to Markset al. (1998, 1999) for a comprehensive description ofthe Snobal model.

The model was run in CRHM using observed meteoro-logical conditions from Fisera Ridge at 15-min time inter-vals as external forcing data. Melt rates were computedat a point from these data, and water equivalent meltdepths were summed over 24-h periods to produce val-ues of the simulated daily melt rate. These computationswere also made for two opposing major slopes within thecirque. The first of these is a predominantly north-facingslope ranging in elevation from 2325 to 2440 m situatedon the southern side of the cirque. The other slope facessouth to south-east and ranges in elevation from 2375to 2500 m. These slopes were distinguished on the basisof dominant terrain features (i.e. slope, aspect, elevation,exposure) that exhibit a relatively large degree of homo-geneity over the defined areas. Table I lists several ofthe relevant terrain parameters used to define each of theslope units.

To simulate melt rates over the north- and south-facing slopes, the direct and diffuse beam components ofsolar radiation were adjusted using the modules Globaland Slope Qsi within CRHM. Global calculates thetheoretical direct beam component of solar radiation toslopes, Qdir, using an expression proposed by Garnier andOhmura (1970) as in the Equation (10):

Qdir D I Ð pm[�sin � cos H�� cos A sin Z�

� sin H�sin A cos Z�

C �cos � cos H� cos Z] cos υ C [cos ��cos A sin Z�

C �sin � cos Z�] sin υ �10�

Table I. Terrain parameters for the two slopes within the Mt.Allan cirque and for the point location of the Fisera Ridge

meteorological station

Parameter North-facingslope

South-facingslope

Metstation

Average slope (°) 30 23 0Average aspect (°a) 20 160 NAMedian elevation (m) 2382 2438 2318Sky view factor 0Ð62 0Ð66 0Ð74

a Units are degrees clockwise from North.



where I is the intensity of extraterrestrial radiation, p isthe mean zenith path transmissivity of the atmosphere, mis the optical air mass, υ is the declination of the sun, �is the latitude, H is the hour angle measured from solarnoon positively towards west, A is the slope azimuth (i.e.aspect) measured from the north through east and Z isthe angle of the slope. Global uses a simple means ofcalculating the diffuse clear-sky radiation, Qdif (W/m2),given by List (1968) as in the following Equation (11):

Qdif D 0Ð5��1 � aw � ac�Qext � Qdir�, �11�

where aw is the radiation absorbed by water vapour(7%), ac is the radiation absorbed by ozone (2%), Qext

(W/m2) is the extraterrestrial radiation on a horizontalsurface at the outer limit of the earth’s atmosphere andQdir (W/m2) is the direct clear-sky radiation reaching theearth’s surface on a horizontal surface. The Slope Qsimodule estimates shortwave radiation for a slope from themeasured incoming shortwave radiation on the level. Theratio of measured shortwave radiation and the calculatedtheoretical clear-sky direct and diffuse radiation on ahorizontal plane is used to adjust the calculated clear-skyshortwave radiation value on the slope.

The snow albedo, ˛, was parameterized as an expo-nential decay during the melt period to an asymptoticminimum of 0Ð3 following Essery and Etchevers (2004).For each timestep with snowmelt, the albedo is updatedaccording to the Equation (12):

˛ ��! �˛ � 0Ð3� exp(�t

�

)C 0Ð3, �12�

where t is the timestep length and � is a time constantapplied to melting snow. We used a value of 106 s for �in our model. For time steps with snowfall, the albedo isincreased by the following Equation (13):

˛ ��! ˛ C �˛f � ˛�Sft

10, �13�

where Sf is the snowfall rate (mm/t), so that a 10 mmsnowfall refreshes the albedo to ˛f (set equal to 0Ð85in our model). This parameterization allows the albedoof snow to decline to artificially low values for puresnow, but in doing so, it effectively represents an arealalbedo that is characteristic of the mixed snow, vegetationand bare ground surface supplying energy to the melt-ing snow. Further refinements to the modelling shouldinclude explicit representation of small-scale advection.Our observations within the cirque also indicate that suchlow values are not unrealistic as the snowcover here iscovered in wind-blown debris and snow algae during thelate spring, which significantly reduced the albedo of thesnow.

Incoming longwave radiation was adjusted for theslopes using a modified version of the parameterizationsuggested by Sicart et al. (2006), in which the effectof varying sky view is accounted for. The sky viewfactor is the fraction of sky visible from a specificpoint, and is defined as the ratio of the projected area

of the visible hemisphere to the projected area of thewhole hemisphere. In our model, longwave irradianceon the slopes, L, was calculated as in the followingEquation (14):

L D L0 C VeffεsT4s , �14�

where L0 is the observed incoming longwave radiation,Veff is the effective terrain view factor (i.e. the differencebetween sky view factor at the observation site andthat over the slope), εs is the emssivity of the surface(taken as 0Ð98), is the Stefan–Boltzmann constant(5Ð67 ð 10�8 W/m2/K4), and T is the surface temperature(K). We used the daily average air temperature as anapproximation of the surface temperature. The parameterVeff accounts for the fact that a component of theobserved longwave radiation is contributed from adjacentterrain. This parameter effectively represents the relativeincrease (or decrease) in exposure to surrounding terrain,and thus provides a useful means for extrapolatinglongwave radiation measurements to nearby locationswith different sky view factors. To obtain representativevalues of sky view factor within the cirque, severaldigital hemispheric photographs were taken over the twoslopes and the floor of the cirque as well as at FiseraRidge. Sky view factor was then calculated from theseimages, following Corripio (2003), as r2/R2, where r isthe average radius of the visible horizon and R is theradius of the image.

The computed melt rates and accumulated melt overtime were then used to simulate the SCD curves for theMt. Allan cirque and for the north- and south-facingslopes within the cirque. These curves were generatedby applying the accumulated melt over time to approx-imations of the observed SWE based on the lognor-mal distribution according to the theoretical frameworkdescribed above. Observations from all surveys were usedto characterize the overall distribution of SWE withinthe cirque. Observations from the northern part of survey#5 (Figure 1) were used to characterize the south-facingslope, whereas observations from surveys #2 and #3 andthe southern part of survey #5 were used to characterizethe north-facing slope (access to other parts of the slopewas limited due to terrain hazards at that time). Thesesurveys and the subsets of survey #5 seemed to representthe major features of the snowcover that were visiblyapparent on both slopes; therefore, the sample observa-tions are considered to be representative of the snowcoveron both slopes at that time.

Because the month of April was characterized byfurther snow accumulation with cold temperatures andlittle or no snowmelt, the fitted SWE distributions weremodified to account for an increase in SWE. This wasdone by assuming that the values of CV from each ofthe surveys were conserved throughout the accumulationevents, which is supported from observations in thesub-arctic (Pomeroy et al., 2004) and from recent snowsurveys on Fisera Ridge and within Mt. Allan cirqueduring the late winter and spring of 2008. Based on



this assumption, values of pre-melt standard deviationwere recalculated using values of SWE that had beenadjusted to account for the additional accumulation thattook place during the month of April. By Equation (2),this is equivalent to increasing the slope of the line in aplot of SWE versus K.

The effect of accumulation events in the melt periodafter the end of April was accounted for by using arescaled depletion curve following Moore et al. (1999).After a snow accumulation event, the areal snowcoverfraction was defined so as to revert to the initial maximumuntil a certain fraction of the new snow had melted (takento be 0Ð5 in our model). A linear reversion to the originalpoint on the SCD curve (e.g. when plotted as a functionof Kmin) occurred for the melt of the remainder of thenew snow. This approach conceptually represents the factthat snowfall events during the spring are often wet andso less subject to wind redistribution via blowing snowas the threshold wind speed for transport is higher forwet snow (Li and Pomeroy, 1997). Therefore, springsnowfalls tend to produce a rather uniform distributionof snow over the landscape. For this reason, the Mooreapproach is considered to be superior to other reportedtechniques for handling snow accumulation (e.g. Luceet al., 1999; Liston, 2004).

RESULTS

Pre-melt SWE distributions

Table II gives the statistical parameters of observedSWE values from all surveys conducted within andadjacent to the Mt. Allan cirque. The maximum SWEvalue occurs for the mean of all surveys because theseincluded several surveys with deeper snow (i.e. surveys#1 and #4 and parts of survey #5) that were not usedto characterize the north- and south-facing slopes. Thedata from all surveys are shown in Figure 2, which plotsvalues of observed SWE against the corresponding valuesof the frequency factor K (calculated by Equation (4)).A straight line representing the theoretical lognormaldistribution is shown together with the observations ofSWE within the cirque, and it is clear that the observeddata deviate from this distribution considerably. It doesappear, however, that SWE values plotted against Kapproximate a straight line for specific ranges of SWE,such as the more shallow depths of SWE up to ¾700

Table II. Statistical parameters of observed SWE within theMt. Allan cirque

Survey location SWE(mm)

s(mm)

CV

North-facing slope 329Ð3 318Ð3 0Ð97South-facing slope 142Ð8 188Ð2 1Ð32Sout-facing (shallow SWE) 24Ð6 13Ð4 0Ð54South-facing (deep SWE) 339Ð8 178Ð7 0Ð53All surveys 427Ð2 438Ð7 1Ð03

Figure 2. Pre-melt snow water equivalent (SWE) distribution for the Mt.Allan cirque based on surveys conducted on 29–30 March, 2007. Dashed

line represents a theoretical lognormal distribution of SWE

mm, and some of the deeper snow from ¾900 to ¾1500mm SWE.

SWE distributions over both the north- and the south-facing slopes are shown in Figure 3 and the statis-tical parameters of the observed SWE are given inTable II. The data show that stratification of observedSWE according to major terrain features produces amuch better fit to the theoretical lognormal distribution.SWE values over the north-facing slope fit a straightline very well, particularly for shallow depths of SWE(Figure 3(a)). Over the south-facing slope, however,observed values of SWE still show deviations around thetheoretical lognormal distribution, despite the fact thatthe linear fit to the data has been considerably improvedfrom the combined SWE values for all surveys.

To attempt to obtain a better linear fit to the observedvalues of SWE over the south-facing slope, the data weredivided into two categories: ‘shallow’ snow (i.e. SWE<80 mm) and ‘deep’ snow (i.e. SWE >80 mm). Themeasurements of SWE in these two categories more-or-less correspond to locations within the south-facingslope that are exposed and windswept, and areas that aresheltered and form drifts (i.e. surface depressions andsmall gullies). Observed SWE values are plotted againstK for both the shallow and deep snow on the south-facingslope and shown in Figure 3(c) and (d). By distinguishingobserved SWE in this manner, a strong linear fit isobtained for both the shallow and the deep snow overthis slope, and the values of CV are significantly reduced(Table II).

Observed areal SCD

Figure 4 shows several of the digital terrestrial pho-tographs of the Mt. Allan cirque acquired during thesnowmelt period, together with the georeferenced ver-sions of these images. Previous studies have used obliqueaerial photography for analysis of snowcover patternsin alpine terrain (e.g. Bloschl and Kirnbauer, 1992),but terrestrial-based photography allows for more fre-quent and consistent acquisition of imagery, and the pro-cedure of Corripio (2004) is easily implemented with



Figure 3. Pre-melt snow water equivalent (SWE) distributions for (a) north-facing and (b) south-facing slopes within the cirque as well as pre-meltSWE distributions for (c) ‘shallow’ and (d) ‘deep’ SWE over the south-facing slope, shown together with fitted theoretical lognormal distributions

Figure 4. Examples of (a) original and (b) georeferenced terrestrial photographs of the Mt. Allan cirque at different times throughout the snowmeltperiod in 2007



Figure 5. (a) Observed snowcover depletion curves for the entire Mt. Allan cirque and for the north- and south-facing slopes within the cirque,(b) total daily precipitation and (c) daily average air temperature at Fisera Ridge during the snowmelt period

accurate results. This imagery is very insightful as itshows how the snowcover disintegrated into a complexmosaic of snow patches and bare ground over time.The south-facing slope became predominantly snow-freemuch sooner than other parts of the cirque, includingthe north-facing slope. By early July, the only locationsretaining snowcover were the depressions, gullies and leeslopes that formed drifts or accumulated deep snow dur-ing the winter.

The areal SCD curves that were obtained fromthe georeferenced terrestrial photography are shown inFigure 5(a) and indicate considerable differences in thetiming and rate of SCD between the different slopeswithin the cirque. The primary period of snowmelt andSCD did not occur until mid to late May over muchof the cirque, but this period began slightly earlier onthe south-facing slope and progressed much faster here.Periods of cool weather and intermittent snowfall events(Figure 5(b)) kept melt rates low and frequently renewedthe snowcover over the entire cirque until a period ofwarm and sunny weather in late May and early June,when high melt rates led to rapid rates of SCD. Anothermajor snowfall event occurred in mid-June, which blan-keted the entire cirque with nearly 80 mm water equiv-alent of fresh snow. As this snow melted during thefollowing days, SCD rates were at first low and becameincreasingly rapid with warming temperatures and greaterfractions of exposed ground. The SCD curves followingthis event then gradually approached the original curvesas the more recent snow disappeared and the remainingareas of deeper snow became progressively depleted untilmid or late July.

Figure 6. Comparison of modelled snow water equivalent (SWE) usingcold regions hydrological model together with Snobal and observed SWE

values at the Fisera Ridge meteorological station

Simulated snowmelt and SCD

Figure 6 shows both the modelled and observed SWEat the Fisera Ridge meteorological station. SWE wasmodelled for this test case using an initial value of170 mm SWE depth beginning on 1 April. ObservedSWE values were derived from the measurements ofsnow depth obtained from the SR50, together with mea-sured snow density at several times throughout the meltperiod and estimated density (i.e. interpolated betweenmeasurements and adjusted for new snow by assuminga fresh snow density of 150 kg/m3) for the interveningperiods. The model clearly performs well for the simula-tion of snowmelt, as it accurately represents the timing of



both melt onset and snow disappearance, and conformswell to the observed melt rates at all times during thesnowmelt period. The model also performs reasonablywell in simulating snow accumulation, but does have thetendency to misrepresent short-term variability, which islikely due to snow redistribution and drifting. Table IIIgives the RMS error between modelled and observedSWE (excluding times when SWE D 0). The evaluationof the model at a point confirms that it is suitable forsimulations of snowmelt at Fisera Ridge, and representsan important step towards applying it over other remoteareas within the cirque.

Simulated daily melt rates for the two opposing slopesare shown in Figure 7(a), whereas Figure 7(b) and (c)shows the modelled shortwave and longwave radiationinputs that were partially driving simulated melt ratesover the two slopes. Melt rates simulated by the modelwere relatively low or zero throughout April and muchof early May, and became greater in magnitude in lateMay/early June, and again in late June and throughoutJuly. During most of the spring, the south-facing slopewas characterized by higher melt rates as a result of

Table III. RMS errors between measurements and simulations ofSWE and fractional SCA

Simulation RMS error

Fisera Ridge SWE 18.4 mmMt. Allan cirque SCA 0.20North-facing slope SCA 0.08South-facing slope SCA 0.15South-facing slope SCA (after 20 May) 0.10

greater incident solar radiation. Towards the summer sol-stice, melt rates became similar in magnitude betweenthe north- and south-facing slopes because of high solarangles, longer daylight hours and a longer solar pathwith more northerly solar azimuths in the morning andlate afternoon. Due to the greater exposure to surround-ing terrain, modelled incident longwave radiation to thenorth-facing slope was greater throughout the spring.Thus, at times when conditions were cloudy and long-wave radiation contributed a relatively greater proportionof the total melt energy, simulated melt rates becamecomparable between the slopes, or even greater on thenorth-facing slope. Over the duration of the simulationperiod (i.e. 1 April to 20 July, 2007), total incomingshortwave radiation to the south-facing slope was ¾751MJ/m2, or 36% greater than that to the north-facing slope,whereas total longwave radiation was ¾134 MJ/m2, or4% greater to the north-facing than to the south-facingslope. Total potential accumulated melt over the south-facing slope was 316 mm, or 19% greater than that overthe north-facing slope as a result of the net incomingradiation difference.

Simulated melt rates based on the observed meteoro-logical conditions at Fisera Ridge were used together withthe theoretical lognormal distribution of SWE in Figure 2to generate the SCD curve shown in Figure 8(a). Thiscurve clearly fails to capture the timing and rate of arealSCD over the cirque. The modelled SCD curve does notbegin to decline until nearly 2 weeks after observed SCDbegan in late May, and once this does begin to occur, therate of depletion is far too rapid during the first severaldays. The initial phase of minor SCD during the early

Figure 7. (a) Simulated daily melt rates for the north- and south-facing slopes within the Mt. Allan cirque; (b) simulated values of total daily incidentshortwave radiation for both slopes; and (c) simulated values of total daily incident longwave radiation for both slopes



Figure 8. Observed and modelled snowcover depletion curves for (a) the spatially lumped representation of the Mt. Allan cirque, (b) the north-facingslope and (c) the south-facing slope

melt period is missed by the model, and the SCD curvetakes too long to decline during the late melt period.

The modelled SCD curves for both the north- andthe south-facing slopes were similarly generated usingthe corresponding distributions of SWE and the simu-lated melt rates for both slopes. The SCD curve for thesouth-facing slope was derived by making separate cal-culations of fractional SCA over time for each of thetwo fitted distributions (i.e. Figure 3(c) and (d)), andsubsequently weighting the calculations of SCA by therelative proportion of the sample observations in eachcategory (i.e. 62Ð5% and 37Ð5% for ‘shallow’ and ‘deep’SWE, respectively). The results in Figure 8(b) and (c)and Table III indicate that by treating initial SWE distri-butions and melt rates separately over the two slopes, themodel performs far better at predicting fractional SCA,and the timing and rate of simulated SCD are substan-tially improved. The approach used for rescaled SCDfollowing snow accumulation events has also produceda good correspondence with the observed SCD at thesetimes. Despite the improvements, however, the results ofthe model still suffer some errors, particularly during theearly snowmelt period over the south-facing slope.

DISCUSSION

Spatial variability of snowmelt and SWEThe results show that the SWE values from surveys

from several distinct slope units, when grouped together,

are not well represented by the lognormal distribution.This is in concurrence with Pomeroy et al. (2004) whofound the lognormal distribution failed when snow sur-veys were aggregated over several terrain classes in aYukon mountain environment, but worked well whensnow surveys were stratified by terrain following Step-puhn and Dyck (1974). When a lognormal approxima-tion is fitted to the aggregated snow survey data andused to predict the decline in areal snowcover over thespring, the results are subject to considerable error anddo not conform well to observations. This is due eitherto deviation of the SWE values from the lognormal dis-tribution, spatial variability of snowmelt rates within thecirque or some combination of both. Figure 2 shows thatthe lognormal approximation of the observed values ofSWE overpredicts these values at low values of K (i.e.<�0Ð2), which correspond to observed values of SWE<350 mm and P(SWE) >0Ð43. A significant portion ofthe total SWE distribution is therefore misrepresented,which would lead to an overestimation of areal snow-cover as this snow melted and disappeared. The observedKmin for the distribution is �0Ð36, where the fitted curvepredicts a SWE value of 215 mm (Figure 2). This indi-cates that 215 mm of accumulated melt is required toproduce any decline in SCA. Based on our assumptionthat the CV is conserved, however, the actual requiredmelt depth is even greater as the slope of the line wasincreased to account for the increase in SWE. Accordingto the model, a net accumulated melt depth of 262 mm



had occurred before the predicted areal snowcover beganto decline. This is an obvious error that is directlyattributable to the poor fit of observed SWE values tothe lognormal distribution, and explains why modelledSCD does not begin until late in the snowmelt period.

Due to similar misrepresentation of observed SWE bythe lognormal distribution over all ranges of K, it isdifficult to infer the magnitude of errors that may be dueto any spatial variability in snowmelt over the cirque. Itis noted that late in the melt period, the rate of modelledSCD began to more closely approximate the observed rateof depletion, despite a delay of several days (Figure 8(a)).This is possibly due to a closer approximation of SWEover the range of K from �0Ð1 to 2Ð0, together withincreasingly uniform melt rates over the cirque with time.It is more likely, however, that this is a result of thecancellation of multiple errors in the modelling procedureproducing realistic appearing results, but for the wrongreasons.

A significant improvement in the results was achievedby separately treating initial SWE distributions and meltrates for the two slopes within the cirque. Based on theclose fit of observed SWE values to the theoretical log-normal distribution (when stratified by terrain features)and the specific modelling approach used in this study,which assumed uniform melt energy receipt by slopeunit, the results suggest that the variability in both pre-melt SWE and melt energetics over differing slopes hasa major effect on areal SCD in this environment. Thedifferences between the initial distributions of SWE overthe two slopes are attributed to factors such as the direc-tion of prevailing winds on blowing snow transport andredistribution over local topography and the effect ofdifferences in energy receipt over the winter on the ener-getics of ablation on the north- and south-facing slopes.An interesting result is that ‘shallow’ and ‘deep’ snowover the south-facing slope can be treated separately toreduce the CV of SWE and further improve the lognor-mal fit to the observations. This implies that small-scale(i.e. 1–100 m) snow wind redistribution processes mayhave a large impact on the overall distribution of SWEwithin individual slope units. Similar results were notfound for the north-facing slope, possibly due to differ-ences in wind-loading. The fact that the modelled SCDcurve closely approximated the observed curve atteststhat the single approximated frequency distribution cansufficiently represent the snowcover on this slope whensnowcover is stratified by terrain features.

Melt rate differences at this scale are primarily relatedto differences in radiation receipt as influenced by topog-raphy. These differences are primarily a result of differen-tial receipt of shortwave radiation between slopes, whichin the Rocky Mountain Front Ranges and similar envi-ronments is relatively pronounced in contrast with morehumid and cloudy environments. The effect of differencesin the exposure to surrounding terrain on longwave radi-ation also has an important effect on melt energetics. Theapproach used here approximated the temperature of thesurrounding terrain using mean daily air temperature, but

differences in the effective surface temperature of the ter-rain that slope units are exposed to could lead to furthervariation in longwave irradiance over the cirque. Exposedsoil, rocks and vegetation can be significantly warmerthan melting snow (� 0 °C), especially during clear days.For example, in the spring of 2008 when the snowcoverwas near maximum spatial extent, we measured temper-atures of exposed rocks in the cirque of more than 22 °Cusing a thermal infrared camera. Thus, as different partsof the cirque become free of snow at different rates andtimes, the effect could be to increase, or alternativelydampen, the spatial variation in longwave irradiance overdifferent parts of the cirque.

Despite the improvements, the modelled SCD curvesstill exhibit some obvious errors. On the north-facingslope, the minor reductions in SCA in mid-May werenot well simulated, whereas on the south-facing slope,the magnitude of SCA was greatly underestimated duringthe early melt period. These features are likely due tomisrepresentations of the actual distributions of SWEon the slopes at these specific times. For example, ourapproach was to adjust the pre-melt SWE distributionsby assuming that CV values are conserved while SWEincreases in the early spring. This assumption may notentirely hold, and may be responsible for some of theerrors in the early stages of areal SCD. The fitting methodused to estimate the parameters for this distribution isbiased towards estimating the mean of the distributionrather than the tails. This would have the largest impactover the south-facing slope, where a large percentage ofthe terrain is exposed and windswept, and representedby a distribution with a very low value of SWE.Furthermore, the shallow snow on much of this slopeis highly transient and likely more difficult to representbased on the approach used here. This may partly explainthe large deviations between modelled and observed SCDcurves on this slope in the early melt period, and it ispossible that an improved physical basis for representingsnow redistribution following accumulation events wouldresult in an improvement of the simulated curve.

Another possible cause of the poor model performancein the early simulation period may include spatial vari-ability in snowmelt at scales smaller than the individualslopes. This could result from small-scale advection ofsensible heat from adjacent areas of bare ground, exposedrocks and vegetation as well as local variations in inci-dent solar radiation due to small variations in microto-pography. Both sources of variation would be relativelymore important over the south-facing slope due to thegreater amount of incident solar radiation. Local advec-tion of energy could result in significant enhancementof snowmelt in certain locations as SCA declines (Marshet al., 1997; Shook and Gray, 1997; Granger et al., 2002).It is unclear over what spatial scales this process operatesin this environment, thus implying the need for furtherinvestigation on the relative importance of advection onsnowmelt. The effects of variable SWE at small scalesmay also be important in terms of variable melt rates due



to variable internal energy storage. This point is discussedfurther below.

Modelling implications

The results of this study indicate that at the scale ofindividual slope units in this particular environment (i.e.¾100–500 m), there are spatial variations in both the dis-tributions of pre-melt SWE and the melt rates applied tothese distributions. This variability strongly influences thetiming and rate of areal SCD, and the SCD curve is notadequately represented by considering only the overall(basin scale) distribution of SWE and spatially uniformmelt rates, in accordance with the findings of Dorneset al. (2008) for sub-arctic complex terrain. Related workon the effects of variability in pre-melt SWE and meltrates in mountainous terrain has shown that both sourcesof variation are important for accurate representation ofareal SCD and meltwater runoff generation (e.g. Tarbotonet al., 2000 and references therein). These previous stud-ies as well as the present study have demonstrated thatsimply applying effective, or basin average parametersresults in large errors; this is due to the nonlinear natureof snowmelt processes (Bloschl, 1999). However, in theseother studies, the optimal results were only achieved bythe use of a fully distributed model. In this study, we haveshown that such an approach was not necessary to achievea good correspondence between observations and modelresults. Rather, we have used objective means to stratifythe terrain, and in so doing, were able to account explic-itly for the major sources of variability in the radiativecomponents of snowmelt energetics over the landscape.Dornes et al. (2008) also showed that this works in sub-arctic mountainous landscapes. At smaller spatial scales(i.e. below that of the individual slope units), we haveaccounted for the variability in SWE using a stochasticapproach based on the theoretical lognormal distribution,and therefore simulated SCA and SCD with accuracy.This is ideal because the parameters for this distributioncan likely be determined from a limited number of snowsurveys based on representative landforms (e.g. Pomeroyet al., 1998). Such simulations have not been done suc-cessfully before in complex terrain without resorting tohighly parameterized, fully distributed models, and showsthat simpler modelling strategies that employ both top-down understanding of system behaviour and bottom-uprepresentation of physical processes can have consider-able success (Dornes et al., 2008; Savenije, 2009).

These results suggest that the appropriate scale fordisaggregation of the terrain depends on the associatedscale dependence of the variability of both SWE andmelt rates. In similar environments, large variations insnowmelt energetics due to variations in radiation receipt,as well as air temperature variation with elevation, arelikely to occur at the scale of the individual slope units(e.g. ¾100–500 m length scale in this case). Distinct dis-tributions of SWE are likely to become obscured anddistorted beyond such scales as homogeneous slope unitsare combined, but this may occur at much smaller scales

in some instances. Therefore, this scale seems to be anupper limit for modelling applications, beyond whichexplicit spatial representation is necessary, and withinwhich small-scale variability may be represented stochas-tically. However, the appropriate landscape stratificationis not only scale-dependent (e.g. Cline et al., 1998) butalso location dependent, so as to conform to the underly-ing variations in terrain (i.e. distinct units of slope/aspect,exposure and elevation) that are responsible for the vari-ation in snow processes.

In this environment, snowmelt is frequently inter-rupted by snowfall events that refresh the snowcoverand delay melt to some extent. These events must beproperly accounted for to produce reasonable simulationsof snowmelt and SCD. Although the method of Mooreet al. (1999) is not physically based, it performs well insimulating the areal depletion of snow over the periodsimmediately following snowfall events. Other publishedapproaches would have either resulted in too rapid anareal depletion of snowcover (e.g. by failure to accountfor reduced wind redistribution of snow and predictionof an immediate decline in SCA; Luce et al., 1999), orwould have failed to restore the snowcover to its max-imum value following snowfall events (e.g. by assum-ing SCA to follow the same depletion curve regardlessof such accumulation events; Liston, 2004). The albedorefresh and decay parameterization given by Essery andEtchevers (2004) has also been found to work well fol-lowing snowfall events.

The finding that an improved linear fit to the SWEobservations is achieved when snow on the south-facingslope is stratified by depth of SWE into exposed andsheltered terrain has influenced the modelling procedureemployed and has had a large influence on the results.It is likely that this may be an important feature ofthe snowcover on other, predominantly windswept slopesin this environment and other windy alpine regions.Therefore, an objective means of classifying the exposedversus the sheltered terrain may need to be employed incircumstances where detailed snow measurements (e.g.surveys or remotely sensed imagery) are lacking. Thiscould include indices such as terrain slope or curvature,which are useful surrogates for zones of drift formation(Tabler, 1975; Bloschl et al., 1991; Liston and Sturm,1998), or more detailed indices taking into accountupslope length and terrain curvature in the direction ofprevailing winds (e.g. Winstral and Marks, 2002).

An issue remains in that it may be problematic to sim-ulate uniform melt rates over a nonuniform snowcoverthat is characterized by large differences in depth anddensity over short distances. The internal energetics (i.e.dU/dt) of the snowpack are dependent on snow depth,density and temperature, which may all exhibit differingdegrees of spatial variance and covariance. Simulating anareal snowcover by applying point-scale simulations thatrely only on areal mean values may therefore introducedeviations from reality due to the effects of this variation(Horne and Kavvas, 1997). This issue could perhaps beresolved by considering separate classes of SWE depth



for melt rate computation within individual slope units.The approach is similar to disaggregating the slope intoexposed and sheltered areas for improving the linear fit tothe observed SWE values, and weighting relative propor-tions of the slope by the areal extent of these zones. Anideal solution is to compute melt rates separately over thesame subsets of the total SWE distribution used to defineexposed and sheltered terrain. Alternatively, a stochasticapproach may be necessary to simulate melt rate vari-ability and its effects on SCD as a result of covariancebetween the joint SWE and melt rate distributions (Esseryand Pomeroy, 2004; Pomeroy et al., 2001).

Improvements in simulations might also result from amore accurate representation of the temporal variation inmelt energetics, so that towards the later stages of thesnowmelt period, sufficient melt rates are applied to sim-ulate the depletion of the remaining snow in shelteredareas such as snowdrifts and snow-filled gulleys. This islinked to improved spatial representation of melt as it iscontrolled by the spatial distribution of pre-melt SWE.During the pre-melt period, energy inputs to the deeperand denser snow in these areas are used to warm thesnowpack up to 0 °C, while energy inputs to adjacentareas of shallow isothermal snowcover are more likelyto be expended in snowmelt. In the later stages of themelt period, the remaining snow in sheltered areas is sub-jected to greater energy inputs than those applied earlierin melt due to advection of sensible heat from local bareground, warmer air masses, longer daylight periods andsignificantly reduced snow albedo as a result of wind-blown debris, snow algae, etc., that have accumulated onthe snow surface. The fact that we allowed the surfacealbedo in our model to decline towards a value of 0Ð3 inthe late melt period has likely accounted for differencesin energetics in later melt and the increase in melt ratesby advection to late-lying snow patches. Further work isrequired to elucidate the effects of such spatial and tem-poral variations in snowmelt on large scale SCD curves.

CONCLUSIONS

This study examined the spatial variability in arealdepletion of the snowcover over a small alpine cirquein the Front Ranges of the Canadian Rocky Mountainsduring the spring of 2007. The approach consisted ofdirect observation using oblique terrestrial photography,together with SCD simulation based on measured pre-melt distributions of SWE and point-scale snowmeltmodelling using observed meteorological conditions atthe cirque. The results of the model were found to bein reasonable agreement with both the observations ofSWE at the meteorological station and the timing andrate of SCD over two opposing slopes within the cirque.The key findings and implications of this research are thefollowing:

1. Distributions of SWE before the onset of melt fitthe theoretical lognormal distribution function well

when stratified by major terrain features such as slopeorientation and topographic exposure.

2. Spatial variation in surface energetics resulting fromdifferences in radiation receipt due to differential slopeorientation and sky/terrain view has a significant effecton snowmelt rates and areal SCD at the scale ofindividual slope units.

3. Prediction of the timing and rate of areal SCD issignificantly improved when the simulation procedureaccounts for the difference in both pre-melt SWEdistributions and the radiative components of snowmeltenergetics that occur between slopes.

4. The scale and spatial extent of individual slope units(i.e. ¾100–500 m length scale for this particular basin)in complex alpine environments likely represent anupper limit for model applications dealing with arealSCD, with major variations in radiative componentsof snowmelt energetics represented explicitly, andvariability in SWE within these units representedstochastically.

5. Terrestrial photography provides a reliable and usefulmeans of deriving high-resolution and high-frequencyimagery for environmental monitoring in alpine areaswhere conventional remote sensing techniques areunfeasible.

Further work needs to address several remainingissues. One issue involves the effects of small-scale spa-tial variability in pre-melt SWE and melt energetics onSCD within individual slope units. The implications ofapplying uniform melt rates should be more rigorouslyassessed through a combined modelling and observationapproach. Furthermore, the representativeness of limitedsnow surveys for estimates of SWE and CV over morebroad parts of the landscape should be demonstrated.Repeat measurements using airborne LiDAR (e.g. withand without snowcover) provide an ideal tool that hasnot yet been fully used in this context. This will also pro-vide insight into the extent to which the landscape mustbe disaggregated to achieve distinct SWE distributions.Future model applications can then use these findings tocouple over-winter snow accumulation and redistributionprocesses with snowmelt processes, and thereby improvesimulations of the overall basin scale SCA and SCD.

ACKNOWLEDGEMENTS

Funding for this work was provided by the Cana-dian Foundation for Climate and Atmospheric Sciences(CFCAS) through their support of the IP3 Research Net-work, the Natural Sciences and Engineering ResearchCouncil of Canada (NSERC), the University of CalgaryBiogeoscience Institute and the Canada Research Chairsprogram. Nakiska Ski Resort kindly provided logisticalsupport. The authors gratefully acknowledge the fieldassistance of Richard Essery (Univ. Edinburgh), GregLangston (Univ. Calgary), Chad Ellis, Matt MacDonaldand Jim MacDonald (Univ. Sask). We also thank Michael



Solohub (Univ. Sask) for technical support with the cam-era and meteorological stations and for help in the field,Javier Corripio (Univ. Innsbruck) for helpful instructionon the setup of the camera, Kevin Shook (Univ. Sask) forinsightful discussion on the lognormal frequency distribu-tion and SCD calculation and Tom Brown (Univ. Sask)for CRHM development and support. Two anonymousreviewers provided comments that greatly improved themanuscript.

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