Snow Parameter Caused Uncertainty of Predicted Snow Metamorphism Processes
Report on the Research Performed during the REU Program at the University of Alaska Fairbanks,
Geophysical Institute, 903 Koyukuk Drive, Fairbanks, AK 99775, USA
By
Anastasia Gennadyevna Yanchilina (REU summer intern)
Nicole Mölders (advisor)
2006
Abstract
Simulated snow metamorphism processes, snow fluxes and snow state variables depend on one or more
empirical parameters that have a standard deviation. Thus, simulated quantities will have a stochastic
error or standard deviation because of the parameters’ uncertainty. The prediction limitations of the
hydro-thermodynamic soil-vegetation scheme’s snow model that are due to stochastic errors,
parameterization weaknesses, and critical parameters are examined by Gaussian Error Propagation (GEP)
principles. The standard deviations in snow quantities determined by the GEP procedure are used to
prioritize which parameters to measure with higher accuracy.
In the predicted value of the solar radiation absorbed and scattered within the snowpack, snow
porosity contributes only up to 3% uncertainty. In snow density, snow water equivalent, effective
porosity, and the rate of change of snow by sublimation, and snow heat capacity snow porosity
contributes from about to 5 to no more than 30% uncertainty, i.e. in an acceptable range. Measuring snow
porosity with higher accuracy provides some potential for more precise predictions.
In the prediction of percolation through the snowpack, snow hydraulic conductivity, snow heat
flux, change of snow temperature, change of volumetric water content in snow, change of snow density
by compaction and destructive metamorphism, snow porosity dominate the uncertainty and yields
unacceptably high relative errors. Therefore, the parameterizations of these processes should be replaced
with formulations that are less sensitive to snow porosity uncertainty.
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1. Introduction
Any empirical parameters are accompanied with errors from their respective standard deviations.
These standard deviations can be as high as the parameters themselves. Thus, the simulated quantities
predicted by use of these parameters will have a stochastic error or standard deviation resulting from the
parameters. Knowing the degree of uncertainty in the simulated quantity caused by the parameters, one
can understand better assess (1) the accuracy of snow model simulations, (2) how sensitive snow models
are to their respective empirical parameters, (3) which parameters could provide potential for model
improvement if they were know with a better accuracy than today, and (4) which parameterizations
should be replaced in future model development.
Gaussian Error Propagation (GEP) techniques (e.g. Kreyszig, 1970; Meyer, 1975) are the
particular statistical methods used in this study to identify the uncertainties cause in the simulations of
upward directed longwave radiation Rls↑, snow heat flux Gsnow, infiltration SF, ponding time after onset of
melting tp, snow water equivalent hw, snow density ρsnow, effective porosity π, maximum volumetric water
content that the matured snowpack can hold against gravity θret, rate of change of snow depth by
sublimation/deposition dhsnow/dt, snow density at the surface ρsnow,surf, rate of change of snow density by
compaction and destructive metamorphism ∂ρsnow/∂t, percolation J, hydraulic conductivity kw, rate of
snow temperature change ∂Tsnow/∂t, shortwave energy absorbed and scattered within the snowpack R↓ss,z,
volumetric heat capacity of snow Csnow, and rate of change of the volumetric water content in the
snowpack∂θ/∂t.
The statistical uncertainties here are analyzed with typical as well as extreme data for polar and
mid-latitude regions to assess the uncertainty in snow model simulations for a wide range of conditions.
2. Method.
The aforementioned quantities are functions of various empirical parameters like emissivity of the snow
εsnow, snow porosity φ, saturated hydraulic conductivity Kws, saturated soil volumetric water content (soil
porosity) ηs, soil pore-size distribution index b, saturated soil water potential ψs, snow albedo αsnow,
sbsorption coefficient kext,z, and various empiricallz determined fitting coefficients. Thus, the
aforementioned quantities are functions of the empirical parameters as follows:
Rls↑=f(εsnow), Gsnow=f(φ), SF=f(Kws, ηs, b, ψs), tp=f(Kws, ηs, b, ψs), hw=f(φ), ρsnow=f(φ), π=f(φ), θret=f(φ),
∂hsnow/∂t=f(φ), ρsnow,surf=f(A, B, C), ∂ρsnow,compaction/∂t =f(C1, C2, 0.08, φ), ∂ρsnow,destruction/∂t=f(C3, C4, ρc, φ,
0.046), J=f(φ), kw=f(φ), ∂Tsnow/∂t=f(φ), R↓ss,z=f(αsnow, kext,z), Csnow=f(φ), ∂θ/∂t=f(φ). Consequently, these
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quantities are burdened by an error or standard deviation (uncertainty) σφ resulting from the random
variability of standard deviations (Mölders et. al. 2005).
To determine model uncertainty caused by the snow parameters at different atmospheric and snow
conditions for these quantities, we use the Gaussian Error Propagation principles. The uncertainty of any
predicted quantity can be calculated from these individual derivations (∂φ/∂χi) and the standard deviations
σχi of the empirical parameters χi by (e.g. Kreyszig 1970; Meyer 1975)
2
⎛ ∂φ ⎞
⎟⎟ σ xi 2 =
σ φ = ∑ ⎜⎜
i =1 ⎝ ∂χ i ⎠
n
∑ {φ, σ } ,
n
i =1
2
xi
(1)
where n is the number of parameters, σ2χi are the variances, and (∂φ/∂χi)σχi:= {φ, σχi) is the denoted
contribution while σφ is the uncertainty. Note that one standard deviation means that 68.27% of all values
fall within φ ± σφ (Mölders et. al., 2005).
To distinguish critical parameters we estimate the contributions of individual parameters to the
total uncertainty by analyzing each term {φ, σχi}. The ratio of the standard deviation to the mean value is
the fractional standard deviation or relative error ε = σ φ φ .
A good parameterization is one in which all of the terms have uncertainty that is of the same order
of magnitude. Parameterizations or parts of them will be considered as critical if a given parameter, whose
standard deviation contributes greatly to the standard deviation of a parameterization, does not cause any
trouble in a different parameterization of the snow model. A parameter is said to be critical if it is of
greater magnitude than all of the other terms contributing to the total uncertainty (Mölders et. al. 2005). If
an empirical parameter causes an uncertainty greater than 20% in several parameterizations, knowing
these parameters with higher accuracy may provide potential for improvement in snow modeling
(Mölders et. al. 2005).
Remote sensing, field and/or lab measurements have provided the standard deviation for many
snow and soil characteristics (e.g. albedo, emissivity, porosity) (e.g. Clapp and Hornberger 1978; Cosby
et al., 1984). Thus, we use standard deviations published in the literature where possible. For the other
empirical parameters we arbitrarily assume their standard deviation to be 10% of the parameter itself.
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3. Results
1. Upward Directed Longwave Radiation
The empirical parameter introducing uncertainty in upward directed longwave radiation (Mölders and
Walsh 2004)
R ls↑ = ε snow σT 4 snow ,surf + (1 − ε snow )R ls↓
(2)
Is the emissivity of snow εsnow. Here Tsnow,surf is snow surface temperature, and Rls↓ is the downward
directed longwave flux. The value of emissivity of snow is assumed to be 0.98, since the typical values
for all kinds of snow emissivity range from 0.96 to 1.00 (e.g. Pielke 1984). Its standard deviation is
assumed to be 10%. We vary Tsnow,surf from 233.15 to 273.15 K, and Rls↓ from 0 to 300 W/m2.
Fig. 1. Upward directed longwave radiation (color) and its uncertainty (dashed) and relative error (dashed dotted) caused by
uncertainty in snow surface emissivity.
Rls↑ decreases linearly with increasing snow temperature, and increases linearly with increasing
incoming longwave radiation (Fig. 1). This is because the more energy the snowpack receives, the more
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longwave radiation is available to be emitted back. The higher values are to be found for cloudier weather
conditions.
The uncertainty σRls↑, is smaller for snow temperatures close to the melting point, and higher for
colder (Artic) snow temperatures. The relative uncertainty ranges from about 0 to 10%, corresponding to
higher values for colder and cloudier conditions and lower values for snow surface temperatures that
exceed 250 K. It also shows an increasing nonlinear pattern for warmer snow surface temperatures and
less incoming longwave radiation.
In conclusion, since the uncertainty ranges from 0 to 10% for all weather conditions (e.g.
Antarctica – cloudy and cold, and midlatitude – warm), this uncertainty caused in longwave upward
radiation cause by uncertainty in snow emissivity is acceptable.
2. Snow Heat Flux
The snow heat flux is given by (Mölders and Walsh 2004)
G snow = − λ snow
∂Tsnow
∂q
− L v ρ w k v snow
∂z snow
∂z snow
,
(3)
Where λ snow = 0.02 + 2.5 ⋅ 10 −6 ρ 2 snow is the thermal conductivity of snow,λsnow, depending on snow
density ρsnow = (1 − φ)ρice + θρ w (Anderson,1976) where θ is the volumetric water content, and ρice is the
density of ice. Furthermore, pw = 1000 kg/m3, kv, Lv, are the density of water, molecular diffusion of
water vapor within air-filled pores of the snowpack, the latent heat of condensation,. The term
∂qsnow/∂zsnow represents the gradient of the mixing ratio for ice at saturation with depth, and ∂Tsnow/∂zsnow
represents the change of snow temperature with snow depth.
Snow porosity φ and its standard deviation are taken to be 0.7 m3m-3 and 0.07 m3m-3, respectively.
The snow temperature gradient is varied between -100 K/m and 100 K/m. The gradient of the saturated
ice mixing ratio with depth is varied between -0.1 and 0.1 kgkg-1m-1.
According to Figure 2, the snow heat flux increases linearly in magnitude for increasing absolute
values of snow temperature gradient, but decreases linearly with the increasing gradient of saturated
specific humidity over ice. As the gradient increases, more energy is consumed for phase transitions. As
a consequence the snow heat flux is reduced, and the snow liquid water content is increased, the
dependency on the gradient of specific humidity at saturation decreases. This is because as more pore
space is filled, there is less room for water vapor diffusion.
Uncertainty is nearly independent of the saturation gradient. Uncertainty in snow heat flux is
lower for low absolute value temperature gradients. The trend of higher relative uncertainties occurs for
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lower negative temperature gradients and higher specific humidity at saturation gradients is mirrored for
higher positive temperature gradients and lower specific humidity at saturation gradients.
As the
volumetric water content increases, the areas where relatively higher uncertainties occur decreases, which
leads to more accurate results, even though the nominal value of uncertainty increases. The relative
uncertainties range from 10 to 60% (Fig. 2).
Fig. 2. Snow heat flux, its uncertainty and relative error as obtained for various conditions at two different constant snow liquid
water content values.
At a constant specific saturated humidity gradient, snow heat flux decreases (increases)
nonlinearly with increasing snow liquid water content for higher positive (negative) values of the
temperature gradient (Fig. 3). This increase of the snow heat flux is stronger for higher values of the
temperature gradient because more energy is required for phase transitions. This nonlinear behavior
changes from slightly nonlinear to stronger behavior as the absolute values of temperature gradients
increases. In small temperature gradients with high water content in the snow pores corresponds to a
quasi-thermal snow-pack in the melting season. High temperature gradients with no liquid water content
occur in winter at temperatures below freezing in response to the diurnal cycle of air temperature.
Uncertainty increases nonlinearly with the increasing snow water content and the increasing
absolute value of the temperature gradients. The nonlinear behavior is stronger for higher values of the
uncertainty. Lower (higher) relative uncertainties occur for lower (higher) volumetric water contents and
absolute value of the temperature gradients, ranging from 20 to 60 %.
The recommendation is that this equation could be improved when developing a parameterization
of snow heat flux that is less sensitive to inaccuracies in porosity.
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Fig. 3. Snow heat flux, its uncertainty and relative error as obtained for various conditions at constant depth gradient of the
specific humidity at saturation with respect to ice.
3. Infiltration
As snow melts and reaches the soil surface it may infiltrate or pond on the soil. In the snow model
infiltration is given as (Schmidt 1990)
P / ρw
⎧
− 0.5
⎪
⎡
SF = ⎨
(P / ρ w − K ws ) 2 ⎤
⎪P / ρ w ⎢1 + 2t K ψ (η − η ) ⎥
⎥
ws
k
s
o ⎦
⎣⎢
⎩
ψs
⎛ψ
and η 0 = ηs ⎜⎜ s
With ψ k =
(1 + 3 / b)
⎝ψ
⎞
⎟⎟
⎠
P / ρ w < K ws
P / ρ w ≥ K ws
(4)
1/ b
being the initial soil volumetric water content at the time the
meltwater reaches the soil surface. Here P/ρw is the meltwater rate amount normalized by water density.
It is varied from 0 to 1·10-5 m/s, t is for the length of precipitation and was varied from 1 to 43200 s
(corresponds to half a day), and ψ is the soil water potential varied from -35.5 to -3.55 m. Here again Kws,
ηs, b, ψs are soil hydraulic conductivity, soil porosity, soil pore-size distribution index, and soil water
potential at saturation. This means the infiltration of snow meltwater depends upon are empirical soil
parameters. The value for Kws is taken to be 3.38·10-6, and its standard deviation was taken to be 50%.
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The values for ηs, b, ψs were taken to be 0.439 m3m-3, 5.25, and - 0.355 m, and their respective standard
deviations were 0.074 m3m-3, 1.66, and 0.0457 (Cosby et al., 1984).
Fig. 4. Snow meltwater infiltration into the soil, its uncertainty and relative error as obtained for various conditions at constant
meltwater rate.
If the meltwater rate reaching the soil surface is less than the hydraulic conductivity of the soil at
saturation, the infiltration rate equals the normalized meltwater rate. Then, uncertainty in infiltration is
only dependent upon that of the hydraulic conductivity at saturation. This is independent of the time after
the onset of melting and the initial soil surface moisture at onset of melting.
If the normalized meltwater rate remains constant with time and it exceeds the hydraulic
conductivity at saturation, infiltration is higher for lower initial soil moisture η0.
Infiltration rate
decreases non-linearly with increasing time after the onset of meltwater (Fig. 4).
The uncertainty slightly nonlinearly decreases with increasing duration of the melting event. It
will be lower if the initial soil moisture is low rather than high. The increase in uncertainty is higher with
increasing duration of the melting event and with increasing initial soil moisture. Relative uncertainty
follows the same trend as the total uncertainty, ranging from 500 to 3500% error, which is very high.
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Fig. 5. Snow meltwater infiltration into the soil, its uncertainty and relative error as obtained for various conditions at a given
time after meltwater reaches the soil surface.
Looking at a point in time, for increasing meltwater rates, infiltration decreases for initial soil
moisture values that exceed about 0.21 m3m-3 (Fig. 5). The infiltration rate is higher for lower initial soil
moisture values because such are farther away from saturation.
Uncertainty shows a weak nonlinear behavior for lower initial soil moisture content, but shows a
stronger nonlinear behavior as the initial soil moisture content is increased. Uncertainty also increases
nonlinearly with increasing meltwater rate. Relative uncertainties follow the same behavior as do the
uncertainties, ranging from 500 to 2700% in this case.
Infiltration values follow an increasing linear pattern until the precipitation rate exceeds the soil
hydraulic conductivity. Afterwards, infiltration rapidly increases until the duration of the melting event is
about 21,000s, a value dependent on soil type and initial soil water content. Then, the infiltration follows
a nonlinear decreasing pattern, which slows down with the increasing duration of the melting event (Fig.
6).
Uncertainties follow a highly nonlinear behavior in this case. They follow a slight increase with
the increase of the meltwater rate and its duration until a certain period in time. Although the larger
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uncertainties occur at lower meltwater rates and higher duration, higher relative uncertainties occur at
higher lower meltwater rates and higher duration (Fig. 6).
Fig. 6. Snow meltwater infiltration into the soil, its uncertainty and relative error as obtained for various conditions at constant
initial soil volumetric water content.
The uncertainty and relative uncertainty patterns of ∂SF/∂η0, ∂SF/∂b, and ∂SF/∂ψs closely follow
the respective patterns of the total uncertainty. Although ∂SF/∂Kws follows a similar pattern as does the
total uncertainty for a constant meltwater rate that exceeds the hydraulic conductivity, behaviors for at a
point in time and at a specific initial soil moisture differ. At a point in time, the uncertainty nonlinearly
increases with increasing meltwater rate and initial soil moisture content until a certain initial soil
moisture is reached, after which it nonlinearly decreases. Relative uncertainty follows a more extreme
nonlinear behavior, but follows a similar increase/ decrease pattern. At a specific initial soil moisture
content, uncertainty increases with meltwater rate and duration until a certain point in time is reached
(dependent on the soil properties), after which it decreases (nonlinearly). Relative uncertainty follows a
less extreme nonlinear behavior, and range from 2 to 65% (Fig. 6).
The highest relative uncertainty was found to be ∂SF/∂ψs contributing to 500 to 3500% of error of
the total uncertainty, and is identified as the most critical parameter in this case. Relative uncertainties for
∂SF/∂b ranged from 2 to 55%, and for ∂SF/∂ηs ranged from 3.6·10-6 to 6.5·10-5 m/s. Note that the highest
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standard deviations do not necessarily correspond with the highest uncertainties (e.g. standard deviation
for ψs is 12.9%, but its relative uncertainty came close to 3500%, while the standard deviation for Kws was
50% but its uncertainty came close to 65%). Although one parameter provides very small uncertainties,
η0, other parameters are critical, all exceeding 50% uncertainty, particularly the water potential at
saturation parameter. Thus, the parametrization for infiltration should be replaced.
4. Onset of Ponding Time
The ponding time
t p = K w ,s ψ k (η s − η 0 )[P / ρ w (P / ρ w )]
P / ρ w ≥ K w ,s
(5)
The soil and snow parameters and their standard deviations are the same as those used for
infiltration. Likewise, meltwater rate was ranged from 0 to 1·10-5 m/s, and ψ was varied from -35.5 to 3.55 m.
Fig. 7. Onset of snow meltwater ponding, its uncertainty and relative error as obtained for various conditions.
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The onset of ponding linearly decreases with increasing melting rate and decreases with the
volumetric water content in the snow. It occurs later for initially wet rather than dry soils. It is more
dependent on the melting rate than on the volumetric water content in the snow (Fig. 7).
The uncertainty in the onset time of ponding caused by the pore size distribution index and
saturated water potential is 2.6 and 2.9%, respectively.
The uncertainty caused by hydraulic conductivity decreases with increasing melting rate and
increasing initial soil volumetric water content (Fig. 8). Thus the uncertainty is more dependent on the
melting rate than on the initial soil volumetric water content, and the relative uncertainty is strongly
dependent upon the melting rate. For melting rates greater than 7·10-6 m/s, the uncertainty exceeds 100%.
Fig. 8. Onset of snow meltwater ponding, its uncertainty and relative error caused by uncertainty in saturated hydraulic
conductivity of the soil as obtained for various conditions.
Uncertainty caused by soil porosity decreases with increasing melting rate and is independent of
the initial soil volumetric water content. Thus, the relative error increases with increasing initial soil
volumetric water content independent of the melting rate. The relative error amounts to 47.48% if the
initial soil volumetric water content is at its greatest potential.
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The total uncertainty increases with increasing melt-water rate and decreasing initial soil
volumetric water content. For typical melt-water rates it exceeds 90%. Since uncertainty in hydraulic
conductivity is consistent in both the infiltration and ponding time equations, 2 to 65% and 80 to 100%, it
has potential for model improvement. Uncertainty in soil porosity, pore size distribution index, and water
potential at saturation, however, are not consistent. Uncertainty is low (high) in soil porosity and high
(low) in water potential at saturation and pore size distribution index parameters for infiltration (ponding
time). Thus, there may be potential for model improvement by replacing the simulation of infiltration by
a less sensitive one to water potential at saturation, and the simulation of ponding time by a less sensitive
one to soil porosity.
Herein the total uncertainty in the onset of ponding time is governed primarily by the uncertainty
in soil hydraulic conductivity followed by soil porosity. Although, highest uncertainty in ponding time is
caused by hydraulic conductivity, uncertainty due to the pore-size distribution index is relatively small,
and the highest uncertainty in infiltration is caused by the water potential at saturation parameter.
5.Snow Water Equivalent
The water equivalent (Dingman 1994)
h w = h snow
ρ snow
ρw
(6)
Is the depth of water that would result after complete melting of the snow-cover (e.g. Dingman). Snow
depth, hsnow, increases by deposition of new snow and decreases by sublimation, outflow of melt-water,
and the increase of snow density by windbreak, compaction, settling, melt-water, percolation, and
freezing (Mölders and Walsh 2004). This equation was differentiated with respect to snow porosity, φ,
and was varied for different values of snow volumetric water content, θ from 0 m3m-3 (pores completely
empty), to 0.329 m3m-3 (pores partially full).
The water equivalent of snow increases non-linearly with increasing snow depth and increasing
volumetric water content of the snow. The relative uncertainty in water equivalent of the snow increases
with increasing volumetric water content and strongly increases with increasing snow depth (Fig. 9).
The absolute uncertainty is independent of the volumetric water content and increases with snow
depth. This means that simulations have higher accuracy for mid-latitude relatively thin snow-packs rather
than for relatively thick snow-packs as in Greenland and Antarctica or at high elevations. One should also
note that the relative error increases with progression of the snowmelt. The relative errors remain below
10% for dry snowpacks less than 2.3 m thick and water saturated snowpacks less than 1.5 m thick. For
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typical snowpack thicknesses less than 1 m in the subarctic, arctic, and midlatitude regions during melting
season, the error is less than 2%.Thus, we conclude that uncertainty in snow porosity marginally affects
the simulated snow water equivalent.
Fig. 9. Snow water equivalent, its uncertainty and relative error caused by uncertainty in snow porosity for various conditions.
6. Snow Density
Snow density
ρ snow = (1 − φ)p ice + θρ w
(7)
depends on θ (varied from 0 to 0.329 m3m-3), the volumetric water content, and snow porosity φ, the
empirical parameter in this equation. Here, ρice=916 kg/m3 is the density of ice.
The density of snow linearly increases with increasing snow liquid water content. The uncertainty
in snow density caused by the uncertainty in snow porosity amounts to 64.12 kgm-3. Thus the relative
error slightly decreases with increasing snow volumetric water content from about 22.9% to 10.7%. This
means that the density of snow is simulated with higher accuracy for wet rather than dry snow-packs.
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7. Maximum Volumetric Water Content that the Matured Snowpack can hold against Gravity
The maximum volumetric water content that the matured snowpack can hold against gravity is given by
(e.g., Dingman 1994)
θ ret
⎧
⎪⎪
3.528 ⋅ 10 − 4
=⎨
⎪ (2.67 ⋅ 10 − 4 ρ snow − 0.0735) ρ snow
⎪⎩
ρw
ρ snow ≤ 280kg / m 3
ρ snow > 280kg / m 3
(8)
Porosity is the empirical parameter in this equation. The snow volumetric water content was varied
between 0 and 0.329 m3m-3.
The retention rate slightly increases with increasing snow volumetric water content. For snow
density less than or equal to 280 kgm-3, the relative error is zero because here the retention rate will be
independent of snow porosity if one ignores that the density of snow has the previously discussed
uncertainty.
For snow densities greater than 280 kgm-3, the relative error nonlinearly decreases with increasing
volumetric water content in snow, ranging from those highly exceeding 100% to 30.14% as the
volumetric water content reaches the amount that can be held against gravity. Uncertainty due to snow
porosity in the retention rate increases with increasing snow liquid water content at a lower rate than the
retention.
8. Effective Porosity
Snow has an effective porosity
π=
ρ snow − ρ ice
θ ret ρ w − ρ ice
(9)
since in contrast to saturated soils, the pore-space is not totally filled by water in saturated snowpacks
(Dunne et al. 1976). The water volumetric content was varied between 0 and 0.329 m3m-3. As snow
density depends on φ, the uncertainty of effective porosity can be affected by that of snow porosity.
The effective porosity linearly decreases with increasing snow liquid water content (Fig. 10). The
uncertainty in snow-pack porosity increases slightly with increasing snow liquid water content. This
behavior yields to a nonlinear increase in the relative error readings from 9.57% to 18.69% as the
volumetric water content reaches the retention rate. This means that the effective porosity can be
simulated with higher relative accuracy for dry snow-packs than for melting snow.
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Since the retention rate occurs in the denominator, the relatively high uncertainty in snow porosity
still leads to acceptable uncertainty in effective porosity and is mostly governed by the relative
uncertainty in snow density that is caused by snow porosity.
Fig. 10. Effective snow porosity, its uncertainty and relative error caused by uncertainty in snow porosity for various
conditions.
9. The rate of change of snow depth by sublimation
The rate of change of snow depth by sublimation is given by (Mölders and Walsh 2004)
∂h snow
E
= s
∂t
ρ snow
Snow porosity is the empirical parameter.
(10)
If the atmosphere is unsaturated with respect to ice,
sublimation occurs. If the water vapor pressure of the atmosphere exceeds the saturation water vapor
pressure with respect to ice, frost or ripe will occur. Both changes the snow depth with time, where
sublimation means a decrease and frost/ripe an increase in snow depth. The dependency of uncertainty in
snow density from snow porosity has been discussed already. Values for the snow volumetric water
17
content were varied between 0 and 0.329 m3m-3, and the values for Es were varied between -2.3·10-5 and
2.3·10-5 kgm-2s-1.
The snow depth decreases (increases) with increasing sublimation (deposition). The decrease is
higher for dry rather than for wet snowpacks (Fig. 11). More energy is needed for sublimation (only a
process for a completely dry snow-pack), than for evaporation that occurs concurrently with sublimation
whenever there is liquid water in the snowpack at the surface. The increase in snow depth decreases with
increasing snow liquid water content. If the atmosphere is saturated with respect to ice, but subsaturated
with respect to water, liquid snow water at the snow surface may evaporate and increase the water content
of air.
Fig. 11. Temporal changes in snow depth due to sublimation/deposition, their uncertainty and relative error caused by
uncertainty in snow porosity for various conditions.
The uncertainties in snow depth change are about one order of magnitude lower than the value,
thus ranging from 10.6 to 23.3%, lower (higher) values corresponding to snow-packs with lower (higher)
volumetric water contents in snow (Fig. 11).
The error increases with increasing sublimation and
deposition and decreasing snow liquid water content. Thus, one can conclude that changes in snow depth
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due to sublimation/deposition can be simulated with higher accuracy for mature wet snow-packs rather
than dry.
10. Snow Density at the Surface
Initial snow density at the surface is given as (Boone 2002, Mölders and Walsh 2004)
ρ snow ,surf = 84 kgm-3
psnow,surf < 84 kgm-3
ρ snow ,surf = A + B(TR − T0 ) + C v
psnow,surf ≥ 84 kgm-3
(11)
With A, B, and C being constants equivalent to 109 kg/m3, 6 kg/m3/K, and 26 kg s1/2 m7/2, respectively.
These are the empirical parameters in this equation. Since no standard deviation for these parameters
were available, standard deviations were assumed to be 10% of their empirical values. TR and T0 are the
air temperature at reference height, and the freezing point temperature in K. The air temperature at
reference height was varied between 223.15 and 278.15K, while the wind speed, │v│, was varied
between 0 and 20 m/s.
Fig. 12. Snow density at the surface, its uncertainty and relative error caused by uncertainty in snow porosity for various
conditions.
19
Initial snow surface density is equal to 84kg/m3 at minimum. It increases with increasing air
temperature and wind speed. This is because the surface air temperature above the freezing point will
warm the top layer of the snow, induce melting, increase the water content in snow, and thus an increase
in its density. Increase in wind speed will increase the density of the surface snow layer by compaction. If
the initial snow surface density is equal to 84kg/m3, the uncertainty of snow surface density becomes
independent of the uncertainty in the empirical parameters. Above that critical value the total uncertainty
in snow density decreases with increasing wind speed (Fig. 12). Herein it shows a strong non-linear
behavior with air temperature with the lower values around air temperatures at the freezing point than
below and above this point. All empirical parameters contribute to the total uncertainty in similar order of
magnitude. The relative total error decreases with increasing air temperature and decreases with
increasing wind speed. This decrease is stronger for low than high wind speed. Relative errors remain
below 10% for typical mid-latitude conditions and below 20% for Arctic conditions.
11. Rate of Change of Snow Density by Compaction
The rate of change of snow density by compaction is given by (e.g. Anderson, 1976)
∂ρ snow
= C1 exp(−0.08(T0 − Tsnow )) Wsnow ⋅ exp(−C 2 ρ snow ) ⋅ ρ snow
∂t
(12)
Where Wsnow, is the weight of the overlying snowpack, Tsnow is snow temperature, and C1 = 2.777·10-4m1 -1
s , and C2 = 2.1·10-2 m3kg-1. The empirical parameters include C1, C2, and 0.08, and φ. Since no
standard deviations for these empirical constants were available, we arbitrarily assumed the standard
deviations to be 10% of their values. The weight of the snow was varied between from 1 to 1000 kg,
snow temperature was varied between 233.15 and 273.15 K, and the snow volumetric water was varied
between 0 and 0.329 m3m-3. Snow density increases by settling compact and melting.
At constant volumetric water content, snow density change nonlinearly increases with increasing
snow temperature and increasing weight of the snow-pack above (Fig. 13). Thus the highest snow density
changes will occur during melting seasons in midlatitude and polar regions. Total uncertainty ranges
from about 130% to 145%, and is dominated by the error in snow porosity. Because the uncertainty
caused by C1, C2, snow porosity follow a similar pattern as does the snow density change, it is equivalent
to a constant value of 9.975%, 74.98%, 116.7 %, respectively. The uncertainty caused by the constant
0.08, increases with increasing amounts of snow weight above, but increases with snow temperature until
it reaches about 262 K, after which it decreases. The relative uncertainty caused by 0.08, increases
linearly with strong, independent of snow weight from 30 to 2%.
20
Fig. 13. Temporal change in snow density due to compaction, its uncertainty and relative error for various conditions at
constant snow liquid water content.
Fig. 14. Temporal change in snow density due to compaction, its uncertainty and relative error for various conditions at
constant weight of the snow above.
21
At constant snow weight above a certain layer, snow density change increases linearly with snow
temperature and decreases with the snow water volumetric water content (Fig. 14). This is because the
snow has more volumetric water content, and though not compress as much as snow with low water
content, since air is less dense than air. Thus the greatest changes will occur in midlatitude and polar
melting seasons. Total uncertainty ranges from 130% to 160% and is dominated by the uncertainty
caused by snow porosity, which ranges from about 114% to 121%. The uncertainties caused by C1, C2,
and snow porosity follow similar patterns as does the snow density change, with C1 equivalent to 9.975%,
C2 ranging from 60 to 100%. The uncertainty caused by 0.08 follows a different behavior, decreasing
linearly with increasing values of the volumetric water content, and increasing snow temperature until 262
K, after which it decreases with increasing snow temperature.
Relative uncertainty decreases with
increasing snow temperature, and is independent of the snow volumetric water content. Thus, better
accuracies are desired in snow porosity and C2, both reaching high relative uncertainties in snow with
high volumetric water content.
12. Rate of Change of Snow Density by Destructive Metamorphism
Destructive metamorphism is given by (Anderson 1976)
∂ρ snow
⎧exp(−0.046(ρ snow − ρ c ))
= C 3 exp(C 4 (T0 − Tsnow )) ⋅ ⎨
1
∂t
⎩
ρ snow ≥ ρ c
ρ snow < ρ c
(13)
The empirical constants in this equation of rate of change of snow density by destructive metamorphism
(Anderson, 1976) include C3 = 2.777·10-6s-1, C4 = 0.04 K-1, and ρc = 150 kg/m3. The empirical parameters
include C3, C4, ρc, φ, and 0.046. The standard deviations for these constants were assumed to be 10% of
their values. The snow temperature was varied between 233.15 and 273.15 K, and the snow volumetric
water content was varied between 0 and 0.329 m3m-3. Note that if Tsnow exceeds T0, all energy will be
used to produce meltwater. If Tsnow becomes colder than T0, meltwater present in the snowpack will
freeze until the concurrently released heat raises Tsnow to T0.
The empirical parameters introducing uncertainty in the change of snow density with time (caused
by destructive metamorphism) are four empirical coefficients and snow porosity. Uncertainty in snow
density increases with increasing snow liquid water content and slightly decreases with increasing snow
temperature (Fig. 15). For all thermal and hydrological conditions of the snowpack the uncertainty is
unacceptably high with values higher than the changes in snow density. The empirical parameters
unequally contribute to the uncertainty in snow density change. The uncertainty in the coefficients C3 and
C4 provide errors of similar magnitude that are about an order of magnitude lower than the change in
22
snow density. The uncertainty is critical in snow porosity for onset of a density impact on deconstructive
metamorphism contributes an order of magnitude more to the total uncertainty than the aforementioned
coefficients. The uncertainty in the coefficient C5 and snow porosity exceed 60% and 270%, respectively.
Thus, these parameters will provide high potential for improvement in simulating snow density changes,
if they are measured with higher accuracy. Moreover, since the contribution of uncertainty in snow
porosity even exceeds that of C5, the uncertainty in snow porosity determines the total error. Note that for
the changes of snow density by compaction porosity also dominated the uncertainty in snow density
changes. Therefore, priority should be given to measure snow porosity with higher accuracy. Based on
these findings one can conclude that unless snow porosity is known with higher accuracy than today
inclusion of snow density changes by destructive metamorphism and compaction should not be included
in determining the snow density changes with time.
Fig. 15. Temporal change in snow density due to destructive metamorphism, its uncertainty and relative error for various
conditions.
23
13. Percolation and Snow Hydraulic Conductivity
The percolation of meltwater through the matured snowpack once retention capacity is exceeded is given
by (Mölders and Walsh 2004)
J = ρ w θk w
Here
.
(14)
3
gρ w
gρ
⎛θ⎞
KS∗3 = w K⎜ ⎟
,
(15)
νw
νw ⎝ π ⎠
Is the snow hydraulic conductivity where νw (= 1.792⋅10-3 kg/(ms)) is the viscosity of water and g is the
kw =
acceleration of gravity. Here, K ( = 0.077d 2 exp[− 7.8ρ snow / ρ w ] ) is the permeability which depends on the
grain diameter, which is a function of snow density (e.g. Colbeck, 1978) and the diameter of the snow
(
)
crystals d ( = 2 ⋅10 −4 exp 5 ⋅10 −3 ρ snow ). Snow porosity is the empirical parameter in this equation. Snow
volumetric water content was varied between 0 and 0.329 m3m-3.
Fig. 16. Hydraulic conductivity of the snow (left) and meltwater percolation within the snow (right), their uncertainty and
relative error for various conditions.
The behaviors of hydraulic conductivity and percolation of snowmelt through the snow behave very
similarly, and differ by two orders of magnitude (Fig. 16). Once snowmelt water exceeds the retention
value meltwater starts to percolate downward in the snowpack. This process depends on snow density and
the hydraulic conductivity of the snow layer beneath. Uncertainty in snow porosity propagates into
uncertainty in snow density, hydraulic conductivity and the meltwater flux. The uncertainty behavior of
snow density has been already discussed above.
24
Hydraulic conductivity and percolation of meltwater through the snowpack is higher at high than
low snow liquid water content. Its uncertainty shows a similar behavior with slightly lower values than
hydraulic conductivity. Due to the dependency of percolation on hydraulic conductivity the meltwater
percolates quicker at high rather than low snow liquid water content. The uncertainty shows a similar
behavior but with lower values than does percolation. For both the relative error amounts about 74% at
maximum liquid water content that can be hold against gravity and increases with increasing snow liquid
water content from 42%. Again one has to conclude that the accuracy of snow porosity is critical for
modeling snow processes and that snow modeling can be improved if this quantity is known with a high
accuracy.
14. Rate of Change of Snow Temperature
Heat transport within the snowpack is given by (Mölders and Walsh 2004)
∂Tsnow
=
∂t
λ snow
↓
∂ 2 Tsnow
∂θ
∂Tw ∂R ss ,z
− Lf ρw
+ Csnow w
−
∂z 2
∂t
∂z
∂z
Csnow
(16)
The equation for rate of change of snow temperature within the snow includes terms that describe the
effects of 1) heat diffusion, 2) consumption and release of latent heat by phase transition processes, 3)
advection of temperature by percolating melt-water, 4) temperature change due to the solar energy
scattered and absorbed within the snowpack (e.g. Dunkle and Bevans, 1956). Snow porosity is the
empirical parameter in this equation. λsnow, thermal conductivity is dependent upon snow volumetric
water content, which was varied between 0 and 0.329 m3m-3. Snow temperature gradient, ∂2Tsnow/∂z2,
varied between -5.29·106 and 5.29·106 K2/m2; volumetric water content gradient, ∂θ/∂t, varied between 2.74·10-4 and 2.74·104 m3m-3s-1; water temperature gradient, ∂Tw/∂z, varied between 500 and 500 K/m;
speed of water through the snow, w, varied between 1·10-3 to 1·10-2; and the incoming shortwave radiation
gradient, ∂R↓ss,z /∂z, varied between 0 and 200000 Wm-2m-1.
The uncertainty in prediction of snow temperature is due to the uncertainty in snow porosity. Due
to the different processes that contribute to the changes in snow temperature, namely thermal diffusion,
melting/freezing, advection of heat by percolating meltwater, and absorption of shortwave radiation
penetrating into the snow the changes are a very complex system. For better insight, we vary two
variables at a time, while holding the other constant. Note that the snow liquid water content θ is the
(
)
value at the previous time step, while ∂θ ∂t ≈ θt −1 − θt ∆t denotes to the difference in snow liquid water
content between the previous and current time step. To get a good idea of how the uncertainties in snow
25
temperature change are affected by the change in several of the variables, we examine closely the
behavior of the uncertainties in some of the combinations. Then, we look at the change of which
variables cause the greatest uncertainty.
Fig. 17. Temporal change of snow temperature, its uncertainty and relative error for various conditions. Variables hold constant
are given at the top of the figure.
At constant snow liquid water content, absorption of radiation, diffusion and constant change of
snow liquid water content with time, the absolute snow temperature changes increase with increasing soil
temperature gradient (Fig. 17). These absolute changes are higher for high positive temperature gradients
with slowly flowing meltwater than at negative meltwater temperature gradients. There are combinations
of meltwater speed and soil temperature gradients at which the change in snow temperature is zero. At
high negative meltwater temperature gradients a zero change in snow temperature requires a higher
meltwater speed than at high positive temperature gradients. The uncertainty is constant for the
aforementioned constant condition, but the relative error follows the pattern of the snow temperature
changes, but increases as snow temperature changes decrease and vice versa.
At constant snow liquid water content, diffusion, meltwater speed and constant change of snow
liquid water content with time, the snow temperature changes decrease with increasing gradient of
shortwave radiation with depth (Fig. 18). The snow temperature changes hardly increase with increasing
meltwater temperature gradient. Thus, under the given conditions, the uncertainty decreases with
26
increasing temperature gradient and hardly depends on the radiation gradient. The relative error increases
to 70% with increasing radiation absorption to a critical value and then increases. For thin snowpacks the
uncertainty remains less than 10%.
Fig. 18. Temporal change of snow temperature, its uncertainty and relative error for various conditions. Variables hold constant
are given at the top of the figure.
At constant snow liquid water content, diffusion, meltwater speed and constant radiation
absorption, the snow temperature changes decrease with increasing changes in snow liquid water content
and slightly increase with increasing meltwater temperature gradient (Fig. 19). The uncertainty hardly
depends on the snow liquid water content changes and decreases with meltwater temperature gradient.
The relative error will remain below 20% at all meltwater temperature gradients if the snow water content
decreases. Such decreases are either associated with percolation or freezing. During snowmelt, when
meltwater or rain percolate from above, the liquid water content of snow increases and relative errors
exceed 20%. This means that uncertainty in predicted snow temperatures is higher during thawing than
during freezing.
At constant snow liquid water content, meltwater speed, constant radiation absorption, and change
in snow liquid water content, the snow temperature changes increase with increasing meltwater
temperature gradient and decreasing snow thermal diffusion. The uncertainty mirrors around the zero
temperature gradient. Relative errors range from 10 to 40%.
27
At constant snow liquid water content, radiation absorption, meltwater temperature gradient and
change in snow liquid water content, the snow temperature changes increase with increasing meltwater
speed and absorption. The uncertainty hardly depends on the meltwater speed and increases marginally
with increasing absorption. Relative errors are 39.5% for no movement of meltwater and decrease as the
speed of the meltwater increases.
Fig. 19. Temporal change of snow temperature, its uncertainty and relative error for various conditions. Variables hold constant
are given at the top of the figure.
At constant snow liquid water content, thermal diffusion, radiation absorption, and meltwater
temperature gradient, the snow temperature change decreases with increasing change in snow liquid water
content independent of the meltwater speed. The relative error remains below 15% for most combinations
of snow liquid water content change and meltwater speed.
At constant snow liquid water content, radiation absorption, and meltwater temperature gradient,
and change in snow liquid water content, the snow temperature change increases with increasing thermal
diffusion gradient and decreases with increasing speed of the meltwater. The uncertainty mirrors around
the zero diffusion. Herein uncertainty increases with increasing absolute value of thermal diffusion. The
relative errors range between 6 and 40%.
Snow temperature changes have an uncertainty less than 15% for all snow liquid water content
and absorption combinations and for all liquid water content values and their temporal changes when the
other variables are held constant. Snow temperature changes have an uncertainty less than 20% for (1) all
28
meltwater speed and meltwater temperature, (2) all temporal changes in snow liquid water content and
meltwater speed, (3) temporal changes in snow liquid water content and absorption, (4) thermal diffusion
and meltwater speeds, and (5) all snow liquid water content and meltwater speed when the respective
other variables are held constant. Uncertainty exceeds 40% in many cases for certain ranges of absorption
at all gradients in meltwater temperature. It also exceeds 40% for high liquid water content at all gradients
of meltwater temperature. These two mentioned cases, however, seldom occur in nature. For all other
combinations not explicitly mentioned the uncertainty remains below 40%. Note that the relative higher
percentage errors (>20%) occur for situations that seldom occur in nature. Thus, one may conclude that
for wide ranges of typical combinations the relative error remains within the 20% percentage.
15. Incoming Solar Energy Scattered and Absorbed within the Snowpack
Solar energy scattered and absorbed within the snowpack is given by (e.g., Dunkle and Bevans 1956)
R ss↓ ,z = R ss↓ (1 − α snow ) exp(− k ext ,z (h snow − z))
(17)
Where kext,z is the extinction coefficient for the snow-layer from the surface at hsnow to the level z in the
snowpack (Mölders and Walsh 2004). The empirical parameters include the exctinction coefficient, snow
albedo, and snow porosity. The extinction coefficient was assumed to be the average of the typical values
for dry and wet snow, 0.1207 m-1 and its standard deviation was taken to be 10%. The value for the snow
albedo was an average of snow albedos integrated over the visible and solar IR (Grenfell, 2004),
0.703813, and its standard deviation is assumed to be 0.021969. The standard value for snow porosity
and its standard deviation used was the same as the equations above. The incoming shortwave radiation
was varied between 0 and 200 W/m2, the snow depth was varied between 0.01 and 3m, and the level
within the snowpack was varied from 0.01 to 1m.
At constant solar radiation, the scattered and absorbed solar radiation decreases with the thickness
of the snowpack and increases with depth traveled through the snowpack (Fig. 20). Although the total
uncertainty slightly nonlinearly decreases with increasing snow thickness and increases with depth
traveled through the snowpack, the total relative uncertainty increases. The total relative uncertainty
follows a nonlinear for low values of snow thickness, but as the snow depth increases, this behavior
becomes more nonlinear, ranging from 7.4 to 8.5%.
29
Fig. 20. Absorbed solar radiation in the snowpack, its uncertainty and relative error for various conditions at constant solar
downward radiation.
Fig. 21. Absorbed solar radiation in the snowpack, its uncertainty and relative error for various conditions at constant snow
depth.
30
At constant snowpack thickness, the scattered and absorbed solar radiation increases linearly with
a strong dependence on increasing solar radiation and the depth traveled through the snowpack (Fig. 21).
This is because the more solar radiation is incident on the snow layer, the more it will be scattered and
absorbed within the layer. The total uncertainty follows a similar pattern. Total relative uncertainty,
however, increases with the depth traveled through the snowpack, independent of the solar radiation.
Fig. 22. Absorbed solar radiation in the snowpack, its uncertainty and relative error for various conditions at a given depth in
the snowpack.
At constant depth traveled through the snowpack, the scattered and absorbed radiation increases
with the solar radiation and decreases with the thickness of the snowpack (Fig. 22). The total uncertainty
increases with increasing solar radiation, and decreases with snow thickness until a certain critical value is
reached, after which it increases. The total relative uncertainty increases with decreasing solar radiation
and increasing depth traveled through the snowpack.
All of the parameters contribute an amount of equal magnitude to the total uncertainty. The
relative uncertainty caused by the extinction coefficient ranges from 0 to 3%. At constant solar radiation,
uncertainty and relative uncertainty are higher at higher values of snowpack thickness and lower values of
depth traveled through the snowpack. At constant snow thickness, uncertainty nonlinearly increases with
solar radiation, but decreases with depth traveled within the snowpack until a certain critical value of
about 0.7m is reached, after which it increases. Relative uncertainty shows similar behavior, but follows a
linear trend instead. At constant depth traveled through the snowpack, uncertainty nonlinearly increases
with snowpack thickness and solar radiation. The relative uncertainty caused by the snow albedo ranges
31
from about 4 to 7.5%. It is equivalent to a constant value throughout the different combinations of
holding one value constant, while varying another. The relative uncertainty caused by snow porosity
ranges from about 0 to 3%. At constant solar radiation, uncertainty increases linearly (nonlinearly) for
low (high) values of snowpack thickness with increasing depth traveled through the snowpack. At
constant snowpack thickness, relative uncertainty is equivalent to 2.28%. At constant depth traveled
through the snowpack, uncertainty increases nonlinearly with snowpack thickness and solar radiation.
16. Snow Volumetric Heat Capacity
The volumetric heat capacity of snow (Fröhlich and Mölders, 2002)
C snow = (1 − φ)c ice ρ ice + θc w ρ w + (φ − θ)c p ρ a
(18)
depends on the composition of snow, where cice = 2105 Jkg-1K-1 is the specific heat capacity of ice, φ-θ is
the air-filled pore-space, and cp = 1004 Jkg-1K-1 is the specific heat capacity of air at constant pressure.
Snow porosity served as the critical parameter, and the snow volumetric water content was varied
between 0 and 0.329 m3m-3.
The volumetric heat capacity of snow linearly increases with increasing liquid water content from
608,577 to 3.507·106 Jkg-1K-1m-3 when all snow pores are filled. The uncertainty in volumetric water
content amounts to 134,888 Jkg-1K-1m-3. The relative error nonlinearly decreases from less than 25% to
less than 5% as the snowpack melts. This means that simulated volumetric heat capacity of dry
snowpacks is higher than for mature melting snowpacks.
17. Rate of Change of the Volumetric Water Content
The change in the volumetric water content in a snow model-layer is given by (Mölders and Walsh 2004)
∂Tsnow
∂J c ρ
∂θ
= − + ice snow
∂z
Lf ρw
∂t
∂t
(19)
The first term describes the change of percolation through the matured snow-layer with depth, and the
second term describes freezing and thawing. The critical parameter in this equation is snow porosity.
The snow water volumetric content was varied between 0 and 0.329 m3m-3. The percolation
gradient, ∂J/∂z depends on the volumetric water content gradient, ∂4θ/∂z4, which was varied between 117.16 and 117.16 m12m-12m-4,and the rate of change of the snow temperature, ∂Tsnow/∂t which was varied
between -0.1917 and 0.1917 K/s.
The temporal change in volumetric snow water content is a function of the change of the water
flux with depth and freezing/thawing (Fig. 23). In a wet snowpack of given water content, it increases
32
with increasing value of the snow temperature change and slightly increases with increasing percolation
from the layer above. Herein a temperature decrease denotes to a melting snow layer and an increase
denotes to a freezing snow layer. In a completely dry snowpack, the volumetric liquid water change in
snow is lower than the snowpack that contains some water. The percolation from the layer above (into the
layer below) contributes with a positive increase (reduction). The empirical parameter causing uncertainty
in the prediction of the change in volumetric snow water content is snow porosity. For given volumetric
water content, the uncertainty in predicted liquid water content changes will be the lowest if a melting
layer gains appreciably water from above or if a freezing layer looses appreciable amounts of water to the
layer below. The relative uncertainty remains quite high in these cases, reaching up to 9000%. The reason
for this high error is because relatively low uncertainty exists in the effective porosity which is located in
the denominator of the percolation gradient of the equation. In a completely dry snowpack, the error is
smallest, which leads to the conclusion that later in the snowmelt, the predictions have lower accuracies.
Fig. 23. Temporal change in snow water content, its uncertainty and relative error for various conditions for a dry (left) and wet
(right) snowpack.
4. Conclusion
We used the Gaussian Propagation Techniques to study snow model uncertainty in the predicted
values of Rls↑, Gsnow, SF, tp, hw, ρsnow, π , θret , dhsnow/dt, ρsnow,surf, ∂psnow/∂t (change in density caused by
compaction), ∂psnow/∂t_destructive (change in density caused by destructive metamorphism), J, kw,
∂Tsnow/∂t, R↓ss,z , Csnow, ∂θ/∂t caused by statistical uncertainty of empirical snow and soil parameters. The
uncertainty in the upward directed longwave radiation flux caused by the emissivity of snow remains
below 10% for most climates (polar and midlatitude). The uncertainty in infiltration is dominated by the
water potential at saturation parameter, while the ponding time is dominated by soil porosity. The soil
33
hydraulic conductivity at saturation contributes high uncertainty to these two equations as well. Herein,
potential in model improvement exists for three parameters for the predicted values of infiltration and
ponding time.
The accuracy of snow porosity needs to be improved for better prediction of some, but not all
snow quantities. In the predicted value of the solar radiation absorbed and scattered within the snowpack,
snow porosity contributes only up to 3% uncertainty. In snow density, snow water equivalent, effective
porosity, and the rate of change of snow by sublimation, and snow heat capacity snow porosity
contributes from about to 5 to no more than 30% uncertainty, i.e. in an acceptable range. Measuring snow
porosity with higher accuracy would also give more precise predictions.
In the predicted values of percolation through the snowpack, and of hydraulic conductivity, snow
porosity parameter contributes from 40 to 70% uncertainty. In snow heat flux, snow porosity contributes
up to 60% uncertainty. In rate of change of snow temperature, it amount up from 0 to 70% uncertainty.
In the rate of change of volumetric water content in snow, it amount up to 9000% uncertainty, highest at
the 0 value of the rate of change of temperature. In the rate of change of snow density by compaction and
destructive metamorphism, snow porosity dominates the uncertainty, since it contributes about 100 and
200% uncertainty, respectively. The relative uncertainty due to snow porosity increases with volumetric
water content in the predicted values of snow water equivalent, effective porosity, rate of change of snow
density by compaction and destructive metamorphism, percolation, hydraulic conductivity, and the solar
radiation absorbed and scattered within the snowpack, but decreases in snow heat flux, rate of change of
volumetric water content, snow density, maximum volumetric water content, snow surface density, rate of
change of snow depth by sublimation, and snow heat capacity. Thus, we conclude that improvement in
snow modeling can be achieved by finding parameterizations for these processes that are less sensitive to
snow porosity uncertainty.
Acknowledgements
This study was funded by NSF within the GI REU program.
34
References
Anderson EA (1976) A point energy and mass balance model of snow cover. NOAA Technical Memorandom NWS Hydro-17.
US Department of Commerce, Silver Springs, MD, pp217
Boone AA (2000) Modelisation des processus hydrologiques dans le schema de surface ISBA: Inclusion d’un reservoir
hydrologique, du gel et modelisation de la neige. Ph.D. thesis (available from CNRM)
Colbeck SC (1978) The physical aspects of water flow through snow. Advances in Hydroscience 11: 165-604
Cosby BJ, Hornberger GM, Clapp RB, Ginn TR (1984) A Statistical exploration of the relationship of soil moisture
characteristics to the physical properties of soils. Water Resour Res 20: 682-690
Dingman SL (1994) Physical hydrology. Sydney: Macmillan Publishing Company, New York, Oxford, Singapore, 575 pp
Dunkle RU, Bevens JT (1956) An approximate analysis of the solar reflectance and transmittance of a snow cover. J Meteorol
13: 212-216
Dunne T, Price AG, Colbeck SC (1976) The generation of runoff from subarctic snowpacks. Water Resource Res 12: 677-685
Frölich K, Mölders N (2002) Investigations on the impact of explicitly predicted snow metamorphism on the microclimate
simulated by a meso-β/γ-scale non-hydrsostatic model. Atmos Res 62: 71-109
Grenfell, Thomas C. (2004) Seasonal and spatial evolution of albedo in a snow-ice-land-ocean environment. Journal of
Geophysical Research, Vol. 109
Kreyszig, E, (1970) Statistische Methoden und ihre Anwendung, Vanden Hoeck & Ruprecht, Göttingen, pp. 422
Meyer, SL (1975) Data analysis for scientists and engineers, J. Wiley & Sons Inc., New York, London, Sydney, Toronto,
pp.513.
Mölders, N, Walsh JE (2004) Atmospheric response to soil-frost and snow in Alaska in March. Theor Appl Climatol 77: 77105.
Mölders, N (2005) Plant and soil parameter caused uncertainty of predicted surface fluxes. Mon. Wea. Rev. 133: 3498-3516.
Mölders N, Jankov M, Kramm G (2005) Application of Gaussian Error Propagation Principles for Theoretical Assessement of
Model Uncertainty in Simulated Soil Processes Caused by Thermal and Hydraulic Parameters. Journal of
Hydrometeorology, 1045-1062
Pielke, RA (1984) Mesoscale Meteorological Modelling. Academic Press, Inc., London.
Schmidt B (1990) Derivation of an explicit equation for infiltration on the basis of the Mein_Larson Model. Hydrol Sci J 35:
197-208
35