Snow Parameter Caused Uncertainty of Predicted Snow Metamorphism Processes

Transcription

1 Sno Parameter Caused Uncertainty of Predicted Sno 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

2 Abstract Simulated metamorphism processes, fluxes and state variables depend on one or more empirical parameters that have a standard deviation. Thus, simulated quantities ill have a stochastic error or standard deviation because of the parameters uncertainty. The prediction limitations of the hydro-thermodynamic soil-vegetation scheme s model that are due to stochastic errors, parameterization eaknesses, and critical parameters are examined by Gaussian Error Propagation (GEP) principles. The standard deviations in quantities determined by the GEP procedure are used to prioritize hich parameters to measure ith higher accuracy. In the predicted value of the solar radiation absorbed and scattered ithin the pack, porosity contributes only up to 3% uncertainty. In density, ater equivalent, effective porosity, and the rate of change of by sublimation, and heat capacity porosity contributes from about to 5 to no more than 30% uncertainty, i.e. in an acceptable range. Measuring porosity ith higher accuracy provides some potential for more precise predictions. In the prediction of percolation through the pack, hydraulic conductivity, heat flux, change of temperature, change of volumetric ater content in, change of density by compaction and destructive metamorphism, porosity dominate the uncertainty and yields unacceptably high relative errors. Therefore, the parameterizations of these processes should be replaced ith formulations that are less sensitive to porosity uncertainty. 2

3 1. Introduction Any empirical parameters are accompanied ith 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 ill have a stochastic error or standard deviation resulting from the parameters. Knoing the degree of uncertainty in the simulated quantity caused by the parameters, one can understand better assess (1) the accuracy of model simulations, (2) ho sensitive models are to their respective empirical parameters, (3) hich parameters could provide potential for model improvement if they ere kno ith a better accuracy than today, and (4) hich 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 upard directed longave radiation R ls, heat flux G, infiltration S F, ponding time after onset of melting t p, ater equivalent h, density ρ, effective porosity π, maximum volumetric ater content that the matured pack can hold against gravity θ ret, rate of change of depth by sublimation/deposition dh /dt, density at the surface ρ,surf, rate of change of density by compaction and destructive metamorphism ρ / t, percolation J, hydraulic conductivity k, rate of temperature change T / t, shortave energy absorbed and scattered ithin the pack R ss,z, volumetric heat capacity of C, and rate of change of the volumetric ater content in the pack θ/ t. The statistical uncertainties here are analyzed ith typical as ell as extreme data for polar and mid-latitude regions to assess the uncertainty in model simulations for a ide range of conditions. 2. Method. The aforementioned quantities are functions of various empirical parameters like emissivity of the ε, porosity φ, saturated hydraulic conductivity K s, saturated soil volumetric ater content (soil porosity) η s, soil pore-size distribution index b, saturated soil ater potential ψ s, albedo α, sbsorption coefficient k ext,z, and various empiricallz determined fitting coefficients. Thus, the aforementioned quantities are functions of the empirical parameters as follos: R ls =f(ε ), G =f(φ), S F =f(k s, η s, b, ψ s ), t p =f(k s, η s, b, ψ s ), h =f(φ), ρ =f(φ), π=f(φ), θ ret=f (φ), h / t=f(φ), ρ,surf =f(a, B, C), ρ,compaction / t =f(c 1, C 2, 0.08, φ), ρ,destruction / t=f(c 3, C 4, ρ c, φ, 0.046), J=f(φ), k =f(φ), T / t=f(φ), R ss,z=f(α, k ext,z ), C =f(φ), θ/ t=f(φ). Consequently, these 3

4 quantities are burdened by an error or standard deviation (uncertainty) σ φ resulting from the random variability of standard deviations (Mölders et. al. 2005). To determine model uncertainty caused by the parameters at different atmospheric and conditions for these quantities, e use the Gaussian Error Propagation principles. The uncertainty of any predicted quantity can be calculated from these individual derivations ( φ/ χ i ) and the standard deviations σ χi of the empirical parameters χ i by (e.g. Kreyszig 1970; Meyer 1975) σ φ = n i= 1 φ χ i 2 n 2 2 σx = {, }, i φ σx i (1) i= 1 here n is the number of parameters, σ 2 χi are the variances, and ( φ/ χ i) σ χi := {φ, σ χi ) is the denoted contribution hile σ φ is the uncertainty. Note that one standard deviation means that 68.27% of all values fall ithin φ ± σ φ (Mölders et. al., 2005). To distinguish critical parameters e estimate the contributions of individual parameters to the total uncertainty by analyzing each term {φ, σ χi }. The ratio of the standard deviation to the mean value is the fractional standard deviation or relative error ε = σφ φ. A good parameterization is one in hich all of the terms have uncertainty that is of the same order of magnitude. Parameterizations or parts of them ill be considered as critical if a given parameter, hose standard deviation contributes greatly to the standard deviation of a parameterization, does not cause any trouble in a different parameterization of the 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, knoing these parameters ith higher accuracy may provide potential for improvement in modeling (Mölders et. al. 2005). Remote sensing, field and/or lab measurements have provided the standard deviation for many and soil characteristics (e.g. albedo, emissivity, porosity) (e.g. Clapp and Hornberger 1978; Cosby et al., 1984). Thus, e use standard deviations published in the literature here possible. For the other empirical parameters e arbitrarily assume their standard deviation to be 10% of the parameter itself. 4

5 3. Results 1. Upard Directed Longave Radiation The empirical parameter introducing uncertainty in upard directed longave radiation (Mölders and Walsh 2004) R (2) = ε σ 4 + ε T,surf (1 ls ) R ls Is the emissivity of ε. Here T,surf is surface temperature, and R ls is the donard directed longave flux. The value of emissivity of is assumed to be 0.98, since the typical values for all kinds of emissivity range from 0.96 to 1.00 (e.g. Pielke 1984). Its standard deviation is assumed to be 10%. We vary T,surf from to K, and R ls from 0 to 300 W/m 2. Fig. 1. Upard directed longave radiation (color) and its uncertainty (dashed) and relative error (dashed dotted) caused by uncertainty in surface emissivity. R ls decreases linearly ith increasing temperature, and increases linearly ith increasing incoming longave radiation (Fig. 1). This is because the more energy the pack receives, the more 5

6 longave radiation is available to be emitted back. The higher values are to be found for cloudier eather conditions. The uncertainty σ Rls, is smaller for temperatures close to the melting point, and higher for colder (Artic) temperatures. The relative uncertainty ranges from about 0 to 10%, corresponding to higher values for colder and cloudier conditions and loer values for surface temperatures that exceed 250 K. It also shos an increasing nonlinear pattern for armer surface temperatures and less incoming longave radiation. In conclusion, since the uncertainty ranges from 0 to 10% for all eather conditions (e.g. Antarctica cloudy and cold, and midlatitude arm), this uncertainty caused in longave upard radiation cause by uncertainty in emissivity is acceptable. 2. Sno Heat Flux The heat flux is given by (Mölders and Walsh 2004) G T q = λ L v ρ k v, (3) z z Where λ 6 2 = ρ is the thermal conductivity of,λ, depending on density ρ ( ρ + θρ (Anderson,1976) here θ is the volumetric ater content, and ρ ice is the = 1 φ) ice density of ice. Furthermore, p = 1000 kg/m 3, k v, L v, are the density of ater, molecular diffusion of ater vapor ithin air-filled pores of the pack, the latent heat of condensation,. The term q / z represents the gradient of the mixing ratio for ice at saturation ith depth, and T / z represents the change of temperature ith depth. Sno porosity φ and its standard deviation are taken to be 0.7 m 3 m -3 and 0.07 m 3 m -3, respectively. The temperature gradient is varied beteen -100 K/m and 100 K/m. The gradient of the saturated ice mixing ratio ith depth is varied beteen -0.1 and 0.1 kgkg -1 m -1. According to Figure 2, the heat flux increases linearly in magnitude for increasing absolute values of temperature gradient, but decreases linearly ith the increasing gradient of saturated specific humidity over ice. As the gradient increases, more energy is consumed for phase transitions. As a consequence the heat flux is reduced, and the liquid ater 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 ater vapor diffusion. Uncertainty is nearly independent of the saturation gradient. Uncertainty in heat flux is loer for lo absolute value temperature gradients. The trend of higher relative uncertainties occurs for 6

7 loer negative temperature gradients and higher specific humidity at saturation gradients is mirrored for higher positive temperature gradients and loer specific humidity at saturation gradients. As the volumetric ater content increases, the areas here relatively higher uncertainties occur decreases, hich 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. Sno heat flux, its uncertainty and relative error as obtained for various conditions at to different constant liquid ater content values. At a constant specific saturated humidity gradient, heat flux decreases (increases) nonlinearly ith increasing liquid ater content for higher positive (negative) values of the temperature gradient (Fig. 3). This increase of the 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 ith high ater content in the pores corresponds to a quasi-thermal -pack in the melting season. High temperature gradients ith no liquid ater content occur in inter at temperatures belo freezing in response to the diurnal cycle of air temperature. Uncertainty increases nonlinearly ith the increasing ater content and the increasing absolute value of the temperature gradients. The nonlinear behavior is stronger for higher values of the uncertainty. Loer (higher) relative uncertainties occur for loer (higher) volumetric ater contents and absolute value of the temperature gradients, ranging from 20 to 60 %. The recommendation is that this equation could be improved hen developing a parameterization of heat flux that is less sensitive to inaccuracies in porosity. 7

8 Fig. 3. Sno heat flux, its uncertainty and relative error as obtained for various conditions at constant depth gradient of the specific humidity at saturation ith respect to ice. 3. Infiltration As melts and reaches the soil surface it may infiltrate or pond on the soil. In the model infiltration is given as (Schmidt 1990) S F = P / ρ P / ρ (P / ρ 1 + 2t K s ψ k 2 K s ) ( ηs ηo ) / b ψ s 0 = ηs P / ρ P / ρ < K K ψ s With ψ k = and η being the initial soil volumetric ater content at the time the (1 + 3/ b) ψ meltater reaches the soil surface. Here P/ρ is the meltater rate amount normalized by ater density. It is varied from 0 to m/s, t is for the length of precipitation and as varied from 1 to s (corresponds to half a day), and ψ is the soil ater potential varied from to m. Here again K s, η s, b, ψ s are soil hydraulic conductivity, soil porosity, soil pore-size distribution index, and soil ater potential at saturation. This means the infiltration of meltater depends upon are empirical soil parameters. The value for K s is taken to be , and its standard deviation as taken to be 50%. s s (4) 8

9 The values for η s, b, ψ s ere taken to be m 3 m -3, 5.25, and m, and their respective standard deviations ere m 3 m -3, 1.66, and (Cosby et al., 1984). Fig. 4. Sno meltater infiltration into the soil, its uncertainty and relative error as obtained for various conditions at constant meltater rate. If the meltater rate reaching the soil surface is less than the hydraulic conductivity of the soil at saturation, the infiltration rate equals the normalized meltater 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 meltater rate remains constant ith time and it exceeds the hydraulic conductivity at saturation, infiltration is higher for loer initial soil moisture η 0. Infiltration rate decreases non-linearly ith increasing time after the onset of meltater (Fig. 4). The uncertainty slightly nonlinearly decreases ith increasing duration of the melting event. It ill be loer if the initial soil moisture is lo rather than high. The increase in uncertainty is higher ith increasing duration of the melting event and ith increasing initial soil moisture. Relative uncertainty follos the same trend as the total uncertainty, ranging from 500 to 3500% error, hich is very high. 9

10 Fig. 5. Sno meltater infiltration into the soil, its uncertainty and relative error as obtained for various conditions at a given time after meltater reaches the soil surface. Looking at a point in time, for increasing meltater rates, infiltration decreases for initial soil moisture values that exceed about 0.21 m 3 m -3 (Fig. 5). The infiltration rate is higher for loer initial soil moisture values because such are farther aay from saturation. Uncertainty shos a eak nonlinear behavior for loer initial soil moisture content, but shos a stronger nonlinear behavior as the initial soil moisture content is increased. Uncertainty also increases nonlinearly ith increasing meltater rate. Relative uncertainties follo the same behavior as do the uncertainties, ranging from 500 to 2700% in this case. Infiltration values follo an increasing linear pattern until the precipitation rate exceeds the soil hydraulic conductivity. Afterards, infiltration rapidly increases until the duration of the melting event is about 21,000s, a value dependent on soil type and initial soil ater content. Then, the infiltration follos a nonlinear decreasing pattern, hich slos don ith the increasing duration of the melting event (Fig. 6). Uncertainties follo a highly nonlinear behavior in this case. They follo a slight increase ith the increase of the meltater rate and its duration until a certain period in time. Although the larger 10

11 uncertainties occur at loer meltater rates and higher duration, higher relative uncertainties occur at higher loer meltater rates and higher duration (Fig. 6). Fig. 6. Sno meltater infiltration into the soil, its uncertainty and relative error as obtained for various conditions at constant initial soil volumetric ater content. The uncertainty and relative uncertainty patterns of S F / η 0, S F / b, and S F / ψ s closely follo the respective patterns of the total uncertainty. Although S F / K s follos a similar pattern as does the total uncertainty for a constant meltater 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 ith increasing meltater rate and initial soil moisture content until a certain initial soil moisture is reached, after hich it nonlinearly decreases. Relative uncertainty follos a more extreme nonlinear behavior, but follos a similar increase/ decrease pattern. At a specific initial soil moisture content, uncertainty increases ith meltater rate and duration until a certain point in time is reached (dependent on the soil properties), after hich it decreases (nonlinearly). Relative uncertainty follos a less extreme nonlinear behavior, and range from 2 to 65% (Fig. 6). The highest relative uncertainty as found to be S F / ψ s 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 S F / b ranged from 2 to 55%, and for S F / η s ranged from to m/s. Note that the highest 11

12 standard deviations do not necessarily correspond ith the highest uncertainties (e.g. standard deviation for ψ s is 12.9%, but its relative uncertainty came close to 3500%, hile the standard deviation for K s as 50% but its uncertainty came close to 65%). Although one parameter provides very small uncertainties, η 0, other parameters are critical, all exceeding 50% uncertainty, particularly the ater potential at saturation parameter. Thus, the parametrization for infiltration should be replaced. 4. Onset of Ponding Time The ponding time t p [ P / ρ (P / ρ )] = K,s ψ k ( ηs η0 ) P / ρ K, s (5) The soil and parameters and their standard deviations are the same as those used for infiltration. Likeise, meltater rate as ranged from 0 to m/s, and ψ as varied from to m. Fig. 7. Onset of meltater ponding, its uncertainty and relative error as obtained for various conditions. 12

13 The onset of ponding linearly decreases ith increasing melting rate and decreases ith the volumetric ater content in the. It occurs later for initially et rather than dry soils. It is more dependent on the melting rate than on the volumetric ater content in the (Fig. 7). The uncertainty in the onset time of ponding caused by the pore size distribution index and saturated ater potential is 2.6 and 2.9%, respectively. The uncertainty caused by hydraulic conductivity decreases ith increasing melting rate and increasing initial soil volumetric ater content (Fig. 8). Thus the uncertainty is more dependent on the melting rate than on the initial soil volumetric ater content, and the relative uncertainty is strongly dependent upon the melting rate. For melting rates greater than m/s, the uncertainty exceeds 100%. Fig. 8. Onset of meltater 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 ith increasing melting rate and is independent of the initial soil volumetric ater content. Thus, the relative error increases ith increasing initial soil volumetric ater content independent of the melting rate. The relative error amounts to 47.48% if the initial soil volumetric ater content is at its greatest potential. 13

14 The total uncertainty increases ith increasing melt-ater rate and decreasing initial soil volumetric ater content. For typical melt-ater 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 ater potential at saturation, hoever, are not consistent. Uncertainty is lo (high) in soil porosity and high (lo) in ater 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 ater 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 folloed 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 ater potential at saturation parameter. 5.Sno Water Equivalent The ater equivalent (Dingman 1994) ρ h h ρ = (6) Is the depth of ater that ould result after complete melting of the -cover (e.g. Dingman). Sno depth, h, increases by deposition of ne and decreases by sublimation, outflo of melt-ater, and the increase of density by indbreak, compaction, settling, melt-ater, percolation, and freezing (Mölders and Walsh 2004). This equation as differentiated ith respect to porosity, φ, and as varied for different values of volumetric ater content, θ from 0 m 3 m -3 (pores completely empty), to m 3 m -3 (pores partially full). The ater equivalent of increases non-linearly ith increasing depth and increasing volumetric ater content of the. The relative uncertainty in ater equivalent of the increases ith increasing volumetric ater content and strongly increases ith increasing depth (Fig. 9). The absolute uncertainty is independent of the volumetric ater content and increases ith depth. This means that simulations have higher accuracy for mid-latitude relatively thin -packs rather than for relatively thick -packs as in Greenland and Antarctica or at high elevations. One should also note that the relative error increases ith progression of the melt. The relative errors remain belo 10% for dry packs less than 2.3 m thick and ater saturated packs less than 1.5 m thick. For 14

15 typical pack thicknesses less than 1 m in the subarctic, arctic, and midlatitude regions during melting season, the error is less than 2%.Thus, e conclude that uncertainty in porosity marginally affects the simulated ater equivalent. Fig. 9. Sno ater equivalent, its uncertainty and relative error caused by uncertainty in porosity for various conditions. 6. Sno Density Sno density ρ = ( 1 φ) + θρ (7) p ice depends on θ (varied from 0 to m 3 m -3 ), the volumetric ater content, and porosity φ, the empirical parameter in this equation. Here, ρ ice =916 kg/m 3 is the density of ice. The density of linearly increases ith increasing liquid ater content. The uncertainty in density caused by the uncertainty in porosity amounts to kgm -3. Thus the relative error slightly decreases ith increasing volumetric ater content from about 22.9% to 10.7%. This means that the density of is simulated ith higher accuracy for et rather than dry -packs. 15

16 7. Maximum Volumetric Water Content that the Matured Snopack can hold against Gravity The maximum volumetric ater content that the matured pack can hold against gravity is given by (e.g., Dingman 1994) θ ret = ( ρ ρ ) ρ ρ ρ 280kg / m > 280kg / m Porosity is the empirical parameter in this equation. The volumetric ater content as varied beteen 0 and m 3 m -3. The retention rate slightly increases ith increasing volumetric ater content. For density less than or equal to 280 kgm -3, the relative error is zero because here the retention rate ill be independent of porosity if one ignores that the density of has the previously discussed uncertainty. For densities greater than 280 kgm -3, the relative error nonlinearly decreases ith increasing volumetric ater content in, ranging from those highly exceeding 100% to 30.14% as the volumetric ater content reaches the amount that can be held against gravity. Uncertainty due to porosity in the retention rate increases ith increasing liquid ater content at a loer rate than the retention. 3 3 (8) 8. Effective Porosity Sno has an effective porosity ρ θ ice π = (9) ret ρ ρ ρ ice since in contrast to saturated soils, the pore-space is not totally filled by ater in saturated packs (Dunne et al. 1976). The ater volumetric content as varied beteen 0 and m 3 m -3. As density depends on φ, the uncertainty of effective porosity can be affected by that of porosity. The effective porosity linearly decreases ith increasing liquid ater content (Fig. 10). The uncertainty in -pack porosity increases slightly ith increasing liquid ater content. This behavior yields to a nonlinear increase in the relative error readings from 9.57% to 18.69% as the volumetric ater content reaches the retention rate. This means that the effective porosity can be simulated ith higher relative accuracy for dry -packs than for melting. 16

17 Since the retention rate occurs in the denominator, the relatively high uncertainty in porosity still leads to acceptable uncertainty in effective porosity and is mostly governed by the relative uncertainty in density that is caused by porosity. Fig. 10. Effective porosity, its uncertainty and relative error caused by uncertainty in porosity for various conditions. 9. The rate of change of depth by sublimation The rate of change of depth by sublimation is given by (Mölders and Walsh 2004) h E s t = ρ Sno porosity is the empirical parameter. If the atmosphere is unsaturated ith respect to ice, sublimation occurs. If the ater vapor pressure of the atmosphere exceeds the saturation ater vapor pressure ith respect to ice, frost or ripe ill occur. Both changes the depth ith time, here sublimation means a decrease and frost/ripe an increase in depth. The dependency of uncertainty in density from porosity has been discussed already. Values for the volumetric ater (10) 17

18 content ere varied beteen 0 and m 3 m -3, and the values for E s ere varied beteen and kgm -2 s -1. The depth decreases (increases) ith increasing sublimation (deposition). The decrease is higher for dry rather than for et packs (Fig. 11). More energy is needed for sublimation (only a process for a completely dry -pack), than for evaporation that occurs concurrently ith sublimation henever there is liquid ater in the pack at the surface. The increase in depth decreases ith increasing liquid ater content. If the atmosphere is saturated ith respect to ice, but subsaturated ith respect to ater, liquid ater at the surface may evaporate and increase the ater content of air. Fig. 11. Temporal changes in depth due to sublimation/deposition, their uncertainty and relative error caused by uncertainty in porosity for various conditions. The uncertainties in depth change are about one order of magnitude loer than the value, thus ranging from 10.6 to 23.3%, loer (higher) values corresponding to -packs ith loer (higher) volumetric ater contents in (Fig. 11). The error increases ith increasing sublimation and deposition and decreasing liquid ater content. Thus, one can conclude that changes in depth 18

19 due to sublimation/deposition can be simulated ith higher accuracy for mature et -packs rather than dry. 10. Sno Density at the Surface Initial density at the surface is given as (Boone 2002, Mölders and Walsh 2004) ρ 84 kgm -3 p,surf < 84 kgm -3,surf = ρ = + B(T T ) C v p,surf 84 kgm -3 (11),surf A R 0 + With A, B, and C being constants equivalent to 109 kg/m 3, 6 kg/m 3 /K, and 26 kg s 1/2 m 7/2, respectively. These are the empirical parameters in this equation. Since no standard deviation for these parameters ere available, standard deviations ere assumed to be 10% of their empirical values. T R and T 0 are the air temperature at reference height, and the freezing point temperature in K. The air temperature at reference height as varied beteen and K, hile the ind speed, v, as varied beteen 0 and 20 m/s. Fig. 12. Sno density at the surface, its uncertainty and relative error caused by uncertainty in porosity for various conditions. 19

20 Initial surface density is equal to 84kg/m 3 at minimum. It increases ith increasing air temperature and ind speed. This is because the surface air temperature above the freezing point ill arm the top layer of the, induce melting, increase the ater content in, and thus an increase in its density. Increase in ind speed ill increase the density of the surface layer by compaction. If the initial surface density is equal to 84kg/m 3, the uncertainty of surface density becomes independent of the uncertainty in the empirical parameters. Above that critical value the total uncertainty in density decreases ith increasing ind speed (Fig. 12). Herein it shos a strong non-linear behavior ith air temperature ith the loer values around air temperatures at the freezing point than belo and above this point. All empirical parameters contribute to the total uncertainty in similar order of magnitude. The relative total error decreases ith increasing air temperature and decreases ith increasing ind speed. This decrease is stronger for lo than high ind speed. Relative errors remain belo 10% for typical mid-latitude conditions and belo 20% for Arctic conditions. 11. Rate of Change of Sno Density by Compaction The rate of change of density by compaction is given by (e.g. Anderson, 1976) ρ t = C exp( 0.08(T 1 0 T ))W exp( C ρ 2 Where W, is the eight of the overlying pack, T is temperature, and C 1 = m - 1 s -1, and C 2 = m 3 kg -1. The empirical parameters include C 1, C 2, and 0.08, and φ. Since no standard deviations for these empirical constants ere available, e arbitrarily assumed the standard deviations to be 10% of their values. The eight of the as varied beteen from 1 to 1000 kg, temperature as varied beteen and K, and the volumetric ater as varied beteen 0 and m 3 m -3. Sno density increases by settling compact and melting. At constant volumetric ater content, density change nonlinearly increases ith increasing temperature and increasing eight of the -pack above (Fig. 13). Thus the highest density changes ill occur during melting seasons in midlatitude and polar regions. Total uncertainty ranges from about 130% to 145%, and is dominated by the error in porosity. Because the uncertainty caused by C1, C2, porosity follo a similar pattern as does the density change, it is equivalent to a constant value of 9.975%, 74.98%, %, respectively. The uncertainty caused by the constant 0.08, increases ith increasing amounts of eight above, but increases ith temperature until it reaches about 262 K, after hich it decreases. The relative uncertainty caused by 0.08, increases linearly ith strong, independent of eight from 30 to 2%. ) ρ (12) 20

21 Fig. 13. Temporal change in density due to compaction, its uncertainty and relative error for various conditions at constant liquid ater content. Fig. 14. Temporal change in density due to compaction, its uncertainty and relative error for various conditions at constant eight of the above. 21

22 At constant eight above a certain layer, density change increases linearly ith temperature and decreases ith the ater volumetric ater content (Fig. 14). This is because the has more volumetric ater content, and though not compress as much as ith lo ater content, since air is less dense than air. Thus the greatest changes ill occur in midlatitude and polar melting seasons. Total uncertainty ranges from 130% to 160% and is dominated by the uncertainty caused by porosity, hich ranges from about 114% to 121%. The uncertainties caused by C 1, C 2, and porosity follo similar patterns as does the density change, ith C 1 equivalent to 9.975%, C 2 ranging from 60 to 100%. The uncertainty caused by 0.08 follos a different behavior, decreasing linearly ith increasing values of the volumetric ater content, and increasing temperature until 262 K, after hich it decreases ith increasing temperature. Relative uncertainty decreases ith increasing temperature, and is independent of the volumetric ater content. Thus, better accuracies are desired in porosity and C 2, both reaching high relative uncertainties in ith high volumetric ater content. 12. Rate of Change of Sno Density by Destructive Metamorphism Destructive metamorphism is given by (Anderson 1976) ρ exp( 0.046( ρ ρc )) ρ ρ c = C3 exp(c 4 (T0 T )) (13) t 1 ρ < ρ c The empirical constants in this equation of rate of change of density by destructive metamorphism (Anderson, 1976) include C 3 = s -1, C 4 = 0.04 K -1, and ρ c = 150 kg/m 3. The empirical parameters include C 3, C 4, ρ c, φ, and The standard deviations for these constants ere assumed to be 10% of their values. The temperature as varied beteen and K, and the volumetric ater content as varied beteen 0 and m 3 m -3. Note that if T exceeds T 0, all energy ill be used to produce meltater. If T becomes colder than T 0, meltater present in the pack ill freeze until the concurrently released heat raises T to T 0. The empirical parameters introducing uncertainty in the change of density ith time (caused by destructive metamorphism) are four empirical coefficients and porosity. Uncertainty in density increases ith increasing liquid ater content and slightly decreases ith increasing temperature (Fig. 15). For all thermal and hydrological conditions of the pack the uncertainty is unacceptably high ith values higher than the changes in density. The empirical parameters unequally contribute to the uncertainty in density change. The uncertainty in the coefficients C 3 and C 4 provide errors of similar magnitude that are about an order of magnitude loer than the change in 22

23 density. The uncertainty is critical in 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 C 5 and porosity exceed 60% and 270%, respectively. Thus, these parameters ill provide high potential for improvement in simulating density changes, if they are measured ith higher accuracy. Moreover, since the contribution of uncertainty in porosity even exceeds that of C 5, the uncertainty in porosity determines the total error. Note that for the changes of density by compaction porosity also dominated the uncertainty in density changes. Therefore, priority should be given to measure porosity ith higher accuracy. Based on these findings one can conclude that unless porosity is knon ith higher accuracy than today inclusion of density changes by destructive metamorphism and compaction should not be included in determining the density changes ith time. Fig. 15. Temporal change in density due to destructive metamorphism, its uncertainty and relative error for various conditions. 23

24 13. Percolation and Sno Hydraulic Conductivity The percolation of meltater through the matured pack once retention capacity is exceeded is given by (Mölders and Walsh 2004) J θk. (14) Here 3 gρ 3 gρ θ k = KS = K, (15) ν ν π Is the hydraulic conductivity here ν (= kg/(ms)) is the viscosity of ater and g is the 2 acceleration of gravity. Here, K ( 0.077d exp[ 7.8ρ / ρ ] = ) is the permeability hich depends on the grain diameter, hich is a function of density (e.g. Colbeck, 1978) and the diameter of the 4 3 crystals d ( = 2 10 exp( 5 10 ρ ) volumetric ater content as varied beteen 0 and m 3 m -3. ). Sno porosity is the empirical parameter in this equation. Sno Fig. 16. Hydraulic conductivity of the (left) and meltater percolation ithin the (right), their uncertainty and relative error for various conditions. The behaviors of hydraulic conductivity and percolation of melt through the behave very similarly, and differ by to orders of magnitude (Fig. 16). Once melt ater exceeds the retention value meltater starts to percolate donard in the pack. This process depends on density and the hydraulic conductivity of the layer beneath. Uncertainty in porosity propagates into uncertainty in density, hydraulic conductivity and the meltater flux. The uncertainty behavior of density has been already discussed above. 24

25 Hydraulic conductivity and percolation of meltater through the pack is higher at high than lo liquid ater content. Its uncertainty shos a similar behavior ith slightly loer values than hydraulic conductivity. Due to the dependency of percolation on hydraulic conductivity the meltater percolates quicker at high rather than lo liquid ater content. The uncertainty shos a similar behavior but ith loer values than does percolation. For both the relative error amounts about 74% at maximum liquid ater content that can be hold against gravity and increases ith increasing liquid ater content from 42%. Again one has to conclude that the accuracy of porosity is critical for modeling processes and that modeling can be improved if this quantity is knon ith a high accuracy. 14. Rate of Change of Sno Temperature Heat transport ithin the pack is given by (Mölders and Walsh 2004) T t λ = 2 T z 2 L ρ f θ + C t C T z R ss,z z (16) The equation for rate of change of temperature ithin the 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-ater, 4) temperature change due to the solar energy scattered and absorbed ithin the pack (e.g. Dunkle and Bevans, 1956). Sno porosity is the empirical parameter in this equation. λ, thermal conductivity is dependent upon volumetric ater content, hich as varied beteen 0 and m 3 m -3. Sno temperature gradient, 2 T / z 2, varied beteen and K 2 /m 2 ; volumetric ater content gradient, θ/ t, varied beteen and m 3 m -3 s -1 ; ater temperature gradient, T / z, varied beteen 500 and 500 K/m; speed of ater through the,, varied beteen to ; and the incoming shortave radiation gradient, R ss,z / z, varied beteen 0 and Wm -2 m -1. The uncertainty in prediction of temperature is due to the uncertainty in porosity. Due to the different processes that contribute to the changes in temperature, namely thermal diffusion, melting/freezing, advection of heat by percolating meltater, and absorption of shortave radiation penetrating into the the changes are a very complex system. For better insight, e vary to variables at a time, hile holding the other constant. Note that the liquid ater content θ is the t 1 t value at the previous time step, hile θ t ( θ θ ) t denotes to the difference in liquid ater content beteen the previous and current time step. To get a good idea of ho the uncertainties in 25

26 temperature change are affected by the change in several of the variables, e examine closely the behavior of the uncertainties in some of the combinations. Then, e look at the change of hich variables cause the greatest uncertainty. Fig. 17. Temporal change of temperature, its uncertainty and relative error for various conditions. Variables hold constant are given at the top of the figure. At constant liquid ater content, absorption of radiation, diffusion and constant change of liquid ater content ith time, the absolute temperature changes increase ith increasing soil temperature gradient (Fig. 17). These absolute changes are higher for high positive temperature gradients ith sloly floing meltater than at negative meltater temperature gradients. There are combinations of meltater speed and soil temperature gradients at hich the change in temperature is zero. At high negative meltater temperature gradients a zero change in temperature requires a higher meltater speed than at high positive temperature gradients. The uncertainty is constant for the aforementioned constant condition, but the relative error follos the pattern of the temperature changes, but increases as temperature changes decrease and vice versa. At constant liquid ater content, diffusion, meltater speed and constant change of liquid ater content ith time, the temperature changes decrease ith increasing gradient of shortave radiation ith depth (Fig. 18). The temperature changes hardly increase ith increasing meltater temperature gradient. Thus, under the given conditions, the uncertainty decreases ith 26

27 increasing temperature gradient and hardly depends on the radiation gradient. The relative error increases to 70% ith increasing radiation absorption to a critical value and then increases. For thin packs the uncertainty remains less than 10%. Fig. 18. Temporal change of temperature, its uncertainty and relative error for various conditions. Variables hold constant are given at the top of the figure. At constant liquid ater content, diffusion, meltater speed and constant radiation absorption, the temperature changes decrease ith increasing changes in liquid ater content and slightly increase ith increasing meltater temperature gradient (Fig. 19). The uncertainty hardly depends on the liquid ater content changes and decreases ith meltater temperature gradient. The relative error ill remain belo 20% at all meltater temperature gradients if the ater content decreases. Such decreases are either associated ith percolation or freezing. During melt, hen meltater or rain percolate from above, the liquid ater content of increases and relative errors exceed 20%. This means that uncertainty in predicted temperatures is higher during thaing than during freezing. At constant liquid ater content, meltater speed, constant radiation absorption, and change in liquid ater content, the temperature changes increase ith increasing meltater temperature gradient and decreasing thermal diffusion. The uncertainty mirrors around the zero temperature gradient. Relative errors range from 10 to 40%. 27

28 At constant liquid ater content, radiation absorption, meltater temperature gradient and change in liquid ater content, the temperature changes increase ith increasing meltater speed and absorption. The uncertainty hardly depends on the meltater speed and increases marginally ith increasing absorption. Relative errors are 39.5% for no movement of meltater and decrease as the speed of the meltater increases. Fig. 19. Temporal change of temperature, its uncertainty and relative error for various conditions. Variables hold constant are given at the top of the figure. At constant liquid ater content, thermal diffusion, radiation absorption, and meltater temperature gradient, the temperature change decreases ith increasing change in liquid ater content independent of the meltater speed. The relative error remains belo 15% for most combinations of liquid ater content change and meltater speed. At constant liquid ater content, radiation absorption, and meltater temperature gradient, and change in liquid ater content, the temperature change increases ith increasing thermal diffusion gradient and decreases ith increasing speed of the meltater. The uncertainty mirrors around the zero diffusion. Herein uncertainty increases ith increasing absolute value of thermal diffusion. The relative errors range beteen 6 and 40%. Sno temperature changes have an uncertainty less than 15% for all liquid ater content and absorption combinations and for all liquid ater content values and their temporal changes hen the other variables are held constant. Sno temperature changes have an uncertainty less than 20% for (1) all 28

29 meltater speed and meltater temperature, (2) all temporal changes in liquid ater content and meltater speed, (3) temporal changes in liquid ater content and absorption, (4) thermal diffusion and meltater speeds, and (5) all liquid ater content and meltater speed hen the respective other variables are held constant. Uncertainty exceeds 40% in many cases for certain ranges of absorption at all gradients in meltater temperature. It also exceeds 40% for high liquid ater content at all gradients of meltater temperature. These to mentioned cases, hoever, seldom occur in nature. For all other combinations not explicitly mentioned the uncertainty remains belo 40%. Note that the relative higher percentage errors (>20%) occur for situations that seldom occur in nature. Thus, one may conclude that for ide ranges of typical combinations the relative error remains ithin the 20% percentage. 15. Incoming Solar Energy Scattered and Absorbed ithin the Snopack Solar energy scattered and absorbed ithin the pack is given by (e.g., Dunkle and Bevans 1956) R ss,z = R ss (1 α )exp( k ext,z (h z)) (17) Where k ext,z is the extinction coefficient for the -layer from the surface at h to the level z in the pack (Mölders and Walsh 2004). The empirical parameters include the exctinction coefficient, albedo, and porosity. The extinction coefficient as assumed to be the average of the typical values for dry and et, m -1 and its standard deviation as taken to be 10%. The value for the albedo as an average of albedos integrated over the visible and solar IR (Grenfell, 2004), , and its standard deviation is assumed to be The standard value for porosity and its standard deviation used as the same as the equations above. The incoming shortave radiation as varied beteen 0 and 200 W/m 2, the depth as varied beteen 0.01 and 3m, and the level ithin the pack as varied from 0.01 to 1m. At constant solar radiation, the scattered and absorbed solar radiation decreases ith the thickness of the pack and increases ith depth traveled through the pack (Fig. 20). Although the total uncertainty slightly nonlinearly decreases ith increasing thickness and increases ith depth traveled through the pack, the total relative uncertainty increases. The total relative uncertainty follos a nonlinear for lo values of thickness, but as the depth increases, this behavior becomes more nonlinear, ranging from 7.4 to 8.5%. 29

30 Fig. 20. Absorbed solar radiation in the pack, its uncertainty and relative error for various conditions at constant solar donard radiation. Fig. 21. Absorbed solar radiation in the pack, its uncertainty and relative error for various conditions at constant depth. 30

31 At constant pack thickness, the scattered and absorbed solar radiation increases linearly ith a strong dependence on increasing solar radiation and the depth traveled through the pack (Fig. 21). This is because the more solar radiation is incident on the layer, the more it ill be scattered and absorbed ithin the layer. The total uncertainty follos a similar pattern. Total relative uncertainty, hoever, increases ith the depth traveled through the pack, independent of the solar radiation. Fig. 22. Absorbed solar radiation in the pack, its uncertainty and relative error for various conditions at a given depth in the pack. At constant depth traveled through the pack, the scattered and absorbed radiation increases ith the solar radiation and decreases ith the thickness of the pack (Fig. 22). The total uncertainty increases ith increasing solar radiation, and decreases ith thickness until a certain critical value is reached, after hich it increases. The total relative uncertainty increases ith decreasing solar radiation and increasing depth traveled through the pack. 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 pack thickness and loer values of depth traveled through the pack. At constant thickness, uncertainty nonlinearly increases ith solar radiation, but decreases ith depth traveled ithin the pack until a certain critical value of about 0.7m is reached, after hich it increases. Relative uncertainty shos similar behavior, but follos a linear trend instead. At constant depth traveled through the pack, uncertainty nonlinearly increases ith pack thickness and solar radiation. The relative uncertainty caused by the albedo ranges 31

32 from about 4 to 7.5%. It is equivalent to a constant value throughout the different combinations of holding one value constant, hile varying another. The relative uncertainty caused by porosity ranges from about 0 to 3%. At constant solar radiation, uncertainty increases linearly (nonlinearly) for lo (high) values of pack thickness ith increasing depth traveled through the pack. At constant pack thickness, relative uncertainty is equivalent to 2.28%. At constant depth traveled through the pack, uncertainty increases nonlinearly ith pack thickness and solar radiation. 16. Sno Volumetric Heat Capacity The volumetric heat capacity of (Fröhlich and Mölders, 2002) C = (1 φ)c ρ + θc ρ + ( φ θ) c ρ (18) ice ice p a depends on the composition of, here c ice = 2105 Jkg -1 K -1 is the specific heat capacity of ice, φ-θ is the air-filled pore-space, and c p = 1004 Jkg -1 K -1 is the specific heat capacity of air at constant pressure. Sno porosity served as the critical parameter, and the volumetric ater content as varied beteen 0 and m 3 m -3. The volumetric heat capacity of linearly increases ith increasing liquid ater content from 608,577 to Jkg -1 K -1 m -3 hen all pores are filled. The uncertainty in volumetric ater content amounts to 134,888 Jkg -1 K -1 m -3. The relative error nonlinearly decreases from less than 25% to less than 5% as the pack melts. This means that simulated volumetric heat capacity of dry packs is higher than for mature melting packs. 17. Rate of Change of the Volumetric Water Content The change in the volumetric ater content in a model-layer is given by (Mölders and Walsh 2004) θ J ciceρ = + t z L ρ f T t The first term describes the change of percolation through the matured -layer ith depth, and the second term describes freezing and thaing. The critical parameter in this equation is porosity. The ater volumetric content as varied beteen 0 and m 3 m -3. The percolation gradient, J/ z depends on the volumetric ater content gradient, 4 θ/ z 4, hich as varied beteen and m 12 m -12 m -4, and the rate of change of the temperature, T / t hich as varied beteen and K/s. The temporal change in volumetric ater content is a function of the change of the ater flux ith depth and freezing/thaing (Fig. 23). In a et pack of given ater content, it increases 32 (19)

33 ith increasing value of the temperature change and slightly increases ith increasing percolation from the layer above. Herein a temperature decrease denotes to a melting layer and an increase denotes to a freezing layer. In a completely dry pack, the volumetric liquid ater change in is loer than the pack that contains some ater. The percolation from the layer above (into the layer belo) contributes ith a positive increase (reduction). The empirical parameter causing uncertainty in the prediction of the change in volumetric ater content is porosity. For given volumetric ater content, the uncertainty in predicted liquid ater content changes ill be the loest if a melting layer gains appreciably ater from above or if a freezing layer looses appreciable amounts of ater to the layer belo. The relative uncertainty remains quite high in these cases, reaching up to 9000%. The reason for this high error is because relatively lo uncertainty exists in the effective porosity hich is located in the denominator of the percolation gradient of the equation. In a completely dry pack, the error is smallest, hich leads to the conclusion that later in the melt, the predictions have loer accuracies. Fig. 23. Temporal change in ater content, its uncertainty and relative error for various conditions for a dry (left) and et (right) pack. 4. Conclusion We used the Gaussian Propagation Techniques to study model uncertainty in the predicted values of R ls, G, S F, t p, h, ρ, π, θ ret, dh /dt, ρ,surf, p / t (change in density caused by compaction), p / t_destructive (change in density caused by destructive metamorphism), J, k, T / t, R ss,z, C, θ/ t caused by statistical uncertainty of empirical and soil parameters. The uncertainty in the upard directed longave radiation flux caused by the emissivity of remains belo 10% for most climates (polar and midlatitude). The uncertainty in infiltration is dominated by the ater potential at saturation parameter, hile the ponding time is dominated by soil porosity. The soil 33