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Plant- and Soil-Parameter-Caused Uncertainty of Predicted Surface Fluxes NICOLE MÖLDERS Geophysical Institute, University of Alaska Fairbanks, Fairbanks, Alaska (Manuscript received 8 December 2004, in final form 16 May 2005) ABSTRACT Simulated surface fluxes depend on one or more empirical plant or soil parameters that have a standard deviation (std dev). Thus, simulated fluxes will have a stochastic error (or std dev) resulting from the parameters’ std dev. Gaussian error propagation (GEP) principles are used to calculate the std dev for fluxes predicted by the hydro–thermodynamic soil–vegetation scheme to identify prediction limitations due to stochastic errors, parameterization weaknesses, and critical parameters, and to prioritize which parameters to measure with higher accuracy. Relative errors of net radiation, sensible, latent, and ground heat flux, on average, are 7%, 10%, 6%, and 26%, respectively. The analysis identified the parameterization of thermal conductivity as the dominant influence on the std dev of ground heat flux. For net radiation, critical parameters are vegetation fraction and ground emissivity; for sensible and latent heat fluxes, vegetation fraction. Minimum stomatal resistance and leaf area index dominate the std dev of stomatal resistance for most vegetation and soil types. The empirical parameters with the highest relative error are not necessarily the greatest contributors to the std dev of the predicted flux. Based on the analysis high priority should be given to measurements of vegetation fraction, ground emissivity, minimum stomatal resistance, leaf area index in general, and the permanent wilting point and field capacity for clay and clay loam. Moreover, further specification of clay-type soils and tundra-type vegetation may improve the accuracy of the lower boundary condition in Arctic numerical weather prediction. Since GEP showed itself able to identify critical parameters and (parts of) parameterizations, GEP analysis could form a basis for parameterization intercomparisons and for parameter determination aimed at improving models.

1. Introduction Recent studies showed that major failures in numerical weather prediction (NWP) occur close to the surface (e.g., Hamill and Colucci 1997), the only natural boundary condition in NWP models. These failures may result from inaccurately predicting the fluxes of heat, water vapor, and matter at the surface–atmosphere interface. Systematic errors due to initial conditions, for instance, have been addressed by ensemble modeling techniques or sensitivity studies (e.g., Tracton and Kalnay 1993; Toth et al. 2001). Besides incorrect initial soil conditions, the parameterizations used in land surface models (LSMs) to describe the surface fluxes may cause systematic errors. Such systematic er-

Corresponding author address: Nicole Mölders, Geophysical Institute, University of Alaska Fairbanks, 903 Koyukuk Dr., P.O. Box 757320, Fairbanks, AK 99775-7320. E-mail: [email protected]

© 2005 American Meteorological Society

rors have been examined by comparing results from different LSMs. In this context, the great achievements of the Project for Intercomparison of Land Surface Parameterization Schemes (PILPS) must be mentioned (e.g., Henderson-Sellers et al. 1995; Shao and Henderson-Sellers 1996; Chen et al. 1997; Schlosser et al. 2000; Slater et al. 2001; Luo et al. 2003). PILPS also showed that LSMs strongly differ in accuracy due to empirical parameters (e.g., physiological, phenological, thermal, hydraulic, radiative, etc.) used to represent different vegetation and soil types (e.g., Wilson et al. 1987; Dorman and Sellers 1989). The wide possible parameter range causes a high variability in predicted state variables, energy, and water fluxes (e.g., Avissar 1991; Pollard and Thompson 1995; Mölders 2001) with implications for the structure of the atmospheric boundary layer (e.g., Avissar and Pielke 1989). For example, density of upward shortwave radiation will differ if the albedo value is taken as 0.15 or 0.2. Consequently slightly different temperature conditions and sensible, latent, and ground heat flux densities will

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be established. This phenomenon is known as “parameter interaction.” Various parameter-variation studies have been carried out to examine interaction effects and understand systematic uncertainty arising from parameter choices. Henderson-Sellers (1993), for instance, applied factorial experiments to assess the relative importance of the parameters in an LSM. She found that the most ecologically important parameters are roughness length, porosity, and an empirical factor describing the sensitivity to photosynthetically active radiation, followed by soil and vegetation albedo. Collins and Avissar (1994), for instance, performed a Fourier amplitude sensitivity test to determine the relative contributions of individual parameters to the variance of energy fluxes resulting from a heterogeneous surface. In doing so, they simultaneously varied all parameters according to their individual probability density functions. They found that statistical–dynamical approaches may be simplified by using only the probability density functions of relative stomatal resistance and surface roughness. Parameter-variation methods address systematic errors. They consider the question of whether slightly different parameters will result in significant perturbations of the model result. However, parameter effects or parameterization deficits that accidentally cancel each other out can remain overlooked (HendersonSellers 1993). To minimize parameter interaction, the methods have to be driven by either observation or reanalysis. Systematic errors from parameter choice, initialization, discretization, model assumptions, and physical parameterizations are not only the source of uncertainty. Another source is stochastic error. Usually a mean value for a vegetation or soil characteristic derived from laboratory or field studies is assigned to a grid element of a NWP model, ignoring natural variance given by the standard deviation. The standard deviations of some parameters, however, can be as great as the mean values themselves (e.g., Clapp and Hornberger 1978; Körner et al. 1979; Cosby et al. 1984; Avissar 1991). Stochastic uncertainty results solely from the fact that the aforementioned mean values of empirical soil and plant parameters are in “error” by the amount of the standard deviation. This error propagates in any quantity calculated by use of these mean parameters; that is, each predicted quantity also has a standard deviation due to the standard deviations of its dependent uncertain parameters. Therefore, uncertainty from stochastic errors also should be kept as low as possible. Since parameter-variation methods are deterministic, they cannot determine this stochastic uncertainty.

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Knowledge of a predicted flux’s parameter-induced standard deviation, however, is desirable for model development and improvement. Parameterizations, for instance, can be examined prior to implementation in an LSM with respect to the parameter-induced statistical uncertainty of the predicted fluxes. Parameterizations that lead to fluxes with high standard deviations can be identified and replaced or avoided. An equation to calculate the flux may be very insensitive or sensitive to the parameter’s standard deviation. Therefore, a priori it is unknown whether a huge (small) standard deviation of a parameter will lead to huge or small standard deviation of the predicted flux. It is of interest to identify the parameters that cause huge standard deviation in the fluxes to measure them with high accuracy even if they themselves have low standard deviation. The parameters causing the highest uncertainty are the ones that will guarantee the greatest potential for reduction of uncertainty in predicted fluxes if these parameters are known with higher a degree of certainty. Gaussian error propagation (GEP) principles permit calculating the error (variance, standard deviation, or error bar) of a predicted quantity that results as a consequence of the standard deviation(s) of the parameter(s) it depends on (e.g., Kreyszig 1970; Meyer 1975). Thus, GEP permits the tasks mentioned above to be addressed. This method originally stems from engineering and physics. In these disciplines, it has been applied to determine the statistical uncertainty inherent in a calculated quantity (e.g., current) due to standard deviations of parameters it depends on (e.g., resistances within a Wheatstone bridge). Since, in a mathematical sense, a function to predict a flux is unambiguous; it will always provide the same flux for the same set of state variables, fluxes, and empirical parameters. Thus, for each set of state variables, fluxes, mean parameters, and their standard deviations, GEP can determine the standard deviation of a predicted flux independent of other parameterizations used in an LSM; that is, without parameter interaction or full simulation. Therefore, the standard deviation of predicted surface fluxes can be theoretically analyzed using artificial data for the typical range of environmental conditions. In this study, GEP is applied to the parameterization of surface fluxes in the hydro–thermodynamic soil– vegetation scheme (HTSVS; Kramm et al. 1996; Mölders et al. 2003) used in various NWP (e.g., Mölders and Rühaak 2002; Mölders and Walsh 2004) and chemistry (e.g., Kramm et al. 1994, 1996) models. The standard deviation of predicted surface fluxes that is caused by standard deviation of empirical parameters is examined theoretically for a typical range of environmental

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conditions to identify uncertainty-causing parameters and (parts of) parameter-sensitive parameterizations. To demonstrate the importance of the results for NWP and to understand the parameter-induced limitations of NWP, a simulation is carried out for 0000 UTC 20 July– 1200 UTC 23 July 2001 for a subarctic/Arctic environment with the mesoscale modeling system of the fifthgeneration Pennsylvania State University–National Center for Atmospheric Research (PSU–NCAR) Mesoscale Model (MM5; Dudhia 1993; Grell et al. 1994). Since various other LSMs use parameterizations and parameters similar to those in HTSVS, the results presented here are of interest to a broad NWP community. As a courtesy to the reader, examples of LSMs using similar parameterizations as HTSVS are given.

2. Brief description of the NWP model a. The atmospheric model MM5 is run using the five water classes (cloud water, rainwater, ice, snow, graupel) in the cloud microphysical scheme of Reisner et al. (1998). As the typical horizontal extension of cumulus clouds is considerably smaller than the horizontal grid resolution (45 km), a cumulus scheme (Grell 1993) considers these subgridscale clouds. The parameterization of turbulent transfer processes follows Hong and Pan (1996). Furthermore, Grell et al.’s (1994) simple radiation scheme is applied. The model domain uses a horizontal Arakawa– Lamb-B (Arakawa and Lamb 1977) staggering consisting of 39 ⫻ 39 dot and 38 ⫻ 38 cross grid points with a horizontal grid spacing of 45 km, and 23 vertical layers reaching to 100 hPa. Five snow layers of equal thickness depend on snow depth. Five soil layers are logarithmically spaced between 0.1- and 2.95-m depth. To fulfill the Courant criterion a time step of 120 s is chosen.

b. The surface model HTSVS consists of a multilayer soil model, a singlelayer canopy model (e.g., Kramm et al. 1996; Mölders et al. 2003), and a multilayer snow model (e.g., Mölders and Walsh 2004). HTSVS calculates the fluxes of momentum, heat, and moisture at the vegetation–soil–atmosphere interface. It considers partly vegetation-covered grid cells by a mixture approach (e.g., Deardorff 1978; Kramm et al. 1996). Other LSMs using this approach are, for instance, the Simple Biosphere model (SiB; e.g., Sellers et al. 1986) and the Community Land Model (CLM; Bonan et al. 2002). Jarvis’s (1976) approach for bulkstomatal resistance (see the appendix) serves to calculate transpiration by plants. The Interaction between

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Soil–Biosphere–Atmosphere model (ISBA; e.g., Noilhan and Planton 1989), the Biosphere–Atmosphere Transfer Scheme (BATS; e.g., Dickinson et al. 1993), the Oregon State University LSM (OSULSM; e.g., Chen and Dudhia 2001), and SiB, for instance, also apply the Jarvis approach. Some of them even use the same correction functions (e.g., OSULSM for soil water deficit). Water uptake by plants includes a vertically variable root distribution (e.g., Mölders et al. 2003). The soil model includes heat conduction and water diffusion (including the Richards equation) within the soil, soil freezing and thawing, and the related release of latent heat and consumption of energy, and the effects of frozen ground on the vertical fluxes of heat, moisture, and water vapor (e.g., Kramm et al. 1996; Mölders et al. 2003). Other models using the Richards equation are, for instance, the Canadian Land Surface Scheme (CLASS; e.g., Verseghy 1991) and BATS. Examples of LSMs using a simplified (as compared to HTSVS) formulation of soil heat diffusion are OSULSM, BATS, and CLM. Tables 1–3 summarize the equations of surface fluxes and their dependent empirical parameters, and the soil and plant empirical parameters, respectively. Most modern LSMs use similar parameters.

c. Initialization Initial and boundary conditions stem from the (National Centers for Environmental Prediction) NCEP– NCAR Reanalysis Project (NNRP). A weighted combination of the July and August monthly 5-yr mean green vegetation cover (0.15° resolution) derived from the Advanced Very High Resolution Radiometer (AVHRR) data (Gutman and Ignatov 1998) is used to determine the vegetation fraction of each grid cell. The 1-km resolution U.S. Department of Agriculture (USDA) State Soil Geographic Database (Miller and White 1998) and 10-min resolution U.S. Geological Survey (USGS) terrain and vegetation data are applied to define soil texture, terrain elevation, and vegetation type (Fig. 1). Initial total soil moisture and temperature are interpolated from NNRP data. Partitioning of the total soil moisture between the liquid and solid phase follows Mölders and Walsh (2004). At the bottom of the soil model, soil temperature and total moisture remain constant throughout the simulation.

3. Experimental design a. GEP Every surface flux is a function ⫽ f (1, . . . , n) of one or more empirical parameters i that are the mean

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TABLE 1. Equations for calculation of the surface fluxes and dependency on empirical parameters. Here ⌰␦ and q␦ are the values of potential temperature and specific humidity at the height ␦ close above the surface (see the appendix). The surface temperature and specific humidity of foliage and ground are denoted as Tf, qf, Tg, and qg, respectively. Furthermore, rmt, f and rmt,g are the resistances of the molecular-turbulent layer close to the surfaces of the foliage and ground, which oppose the transfer of heat and matter. Here D, and DT,v are the transfer coefficients with respect to water vapor and heat. Here a and cp are the density and specific heat at constant pressure of air, and , , f, g, ⫺1, Tg, T⫺1, and z⫺1 are the thermal conductivity, the Stefan–Boltzmann constant, the vegetation fraction (expressed as values between 0 and 1), the volumetric water content and temperature at the ground (index g) and in the first soil layer (index ⫺1), and the depth of the first soil layer, respectively. Further, w, s, b, and s are the density of water, porosity, pore size distribution index, and saturated water potential. See Kramm (1995) and Kramm et al. (1996) for calculation of D,[⫽f (b, s, s)], DT,[⫽ f (b, s, s)], rmt, f, rmt,g, and qg; Mölders and Walsh (2004) for ␣g[⫽f (s)] and [⫽f (s, s, b)]; and the appendix for ⌰␦, q␦, qf, and rst. Note that net radiation also depends on astronomical and atmospheric conditions. Uncertainty from these sources is not considered in this study. Surface flux

Equations

Dependent parameters 4 f)[g T g

Q ⫽ [f ␣f ⫹ (1 ⫺ f)␣g]Rsf ⫺ (1 ⫺ ⫺ f [f T 4f ⫹ (1 ⫺ f)R↓lf]

Net radiation Sensible heat flux

H ⫽ ⫺cpa

Latent heat flux

LE ⫽ −aL

冋 冋

⫹ (1 ⫺

g)R↓lf]

Q ⫽ f (f, ␣f, s, f, g)

册 册

1 ⫺ f f 共⌰␦ − Tg兲 ⫹ 共⌰ ⫺ Tf兲 rmt,g rmt, f ␦

f 1 ⫺ f 共q␦ ⫺ qg兲 ⫹ 共q ⫺ qf兲 rmt,g rmt, f ␦

H ⫽ f (f , z0) LE ⫽ f (f , rst,min, LAI, Tmin, Topt, Tmax, s, b1, bst zroot, fc, pwp, z0)

Hs ⫽ ⫺(1/z⫺1)[( ⫹ wLDT)(Tg ⫺ T⫺1) ⫹ wLD(g ⫺ ⫺1)]

Ground heat flux

values obtained from measurements (e.g., Tables 2 and 3). Each calculated flux will be in error by an amount as a consequence of the errors in the measured empirical parameters i, which are typically given by the standard deviation i. GEP principles (e.g., Kreyszig 1970; Meyer 1975) permit determining a predicted flux’s statistical uncertainty (standard deviation) that results from the standard deviations of the empirical plant and soil parameters it depends on. To apply GEP each equation used to predict a flux (e.g., sensible heat flux) is derivated with respect to all its dependent empirical parameters i (e.g., vegetation fraction, roughness length; Table 1). The standard deviation (statistical uncertainty) of the predicted flux can be calculated from these individual derivations (/i) and the standard deviations i of the ith empirical parameters i by (e.g., Kreyszig 1970; Meyer 1975)

Hs ⫽ f (s, s, b, s, cs)

冑兺 冉 n

⫽

i⫽1

⭸ ⭸ i

冊

2

2 i.

共1兲

Here n represents the number of empirical parameters. A standard deviation means that 68% of all values fall within the range of the mean value plus/minus the standard deviation. Note that experimentalists often refer to ⫾ as the error bar. The ratio of the standard deviation to the mean value is the fractional standard deviation or relative error: ⫽ ( /). Remote sensing, field, and/or laboratory measurements have provided the standard deviation for many biological and geophysical characteristics [e.g., albedo, emissivity, leaf area index (LAI), vegetation fraction, porosity; e.g., Clapp and Hornberger 1978; Körner et al. 1979; Cosby et al. 1984; Avissar 1991; Betts and Ball 1997; Jin and Liang 2006]. Standard deviation of the

TABLE 2. Parameters and their std dev as considered in this study. Here, cS and S are the specific heat capacity and density of the dry soil material, respectively. Parameters and std devs for porosity s, pore size distribution index b, and saturated water potential s are from Cosby et al. (1984). Soil type Loamy sand Sandy loam Loam Clay loam Clay a

cS 10 J kg⫺1 K⫺1 3

876 882 896 866 890

⫾ ⫾ ⫾ ⫾ ⫾

Grunwald et al. (2001) Calhoun et al. (2001) c Emissivity of the ground b

69 34 52 72 23

S 10 kg m⫺3 3

1610 1520 1350 1420 1470

⫾ ⫾ ⫾ ⫾ ⫾

100a 140a 110a 80a 140b

s m m⫺3 0.421 0.434 0.439 0.465 0.468

⫾ ⫾ ⫾ ⫾ ⫾

0.072 0.088 0.074 0.054 0.035

4.26 4.74 5.25 8.17 11.55

⫾ ⫾ ⫾ ⫾ ⫾

gc -.-

s m

b -.-

3

1.40 1.72 1.66 3.74 3.93

⫺0.0361 ⫺0.141 ⫺0.355 ⫺0.263 ⫺0.468

⫾ ⫾ ⫾ ⫾ ⫾

0.0537 0.0537 0.0457 0.0525 0.039

0.95 0.95 0.95 0.95 0.95

⫾ ⫾ ⫾ ⫾ ⫾

0.095 0.095 0.095 0.095 0.095

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TABLE 3. Plant-specific parameters (from Pielke 1984; Wilson et al. 1987; Kramm 1995; Jackson et al. 1996) and std dev for vegetation types that occur in the model domain. Here, zroot, z0, rst,min, bst, ␣f, and f are the root length, roughness length, minimum stomatal resistance, an empirical parameter required to determine PAR, the albedo, and the emissivity of foliage, respectively. Here Tmin and Tmax are the temperatures at which stomata close, and Topt is the temperature at which rst reaches its minimum. See text for further details. Vegetation type

zroot m

z0 m

rst,min s m⫺1

bst -.-

␣f -.-

f -.-

Tmin °C

Tmax °C

Topt °C

Grassland Shrubland Deciduous forest Needleleaf forest Mixed forest Sparse vegetation Wooded tundra Mixed tundra

0.26 ⫾ 0.026 0.7 ⫾ 0.07 3 ⫾ 0.3 3 ⫾ 0.3 3 ⫾ 0.3 0.5 ⫾ 0.05 1.81 ⫾ 0.181 1.81 ⫾ 0.181

0.08 ⫾ 0.008 0.03 ⫾ 0.003 0.85 ⫾ 0.085 1.09 ⫾ 0.109 0.8 ⫾ 0.08 0.01 ⫾ 0.001 0.06 ⫾ 0.006 0.05 ⫾ 0.005

70 ⫾ 7 300 ⫾ 30 232 ⫾ 23.2 125 ⫾ 12.5 125 ⫾ 12.5 999 ⫾ 99.9 150 ⫾ 15 150 ⫾ 15

20 ⫾ 2 10 ⫾ 1 22 ⫾ 2.2 25 ⫾ 2.5 23 ⫾ 2.3 20 ⫾ 2 40 ⫾ 4 40 ⫾ 4

0.19 ⫾ 0.018* 0.25 ⫾ 0.025 0.11 ⫾ 0.011* 0.10 ⫾ 0.006* 0.12 ⫾ 0.0085* 0.12 ⫾ 0.012 0.16 ⫾ 0.016 0.16 ⫾ 0.016

0.97 ⫾ 0.097 0.95 ⫾ 0.095 0.95 ⫾ 0.095 0.97 ⫾ 0.097 0.96 ⫾ 0.096 0.91 ⫾ 0.091 0.97 ⫾ 0.097 0.97 ⫾ 0.097

5 ⫾ 0.1 5 ⫾ 0.1 10 ⫾ 0.1 ⫺5 ⫾ 0.1 0 ⫾ 0.1 5 ⫾ 0.1 5 ⫾ 0.1 5 ⫾ 0.1

45 ⫾ 0.1 45 ⫾ 0.1 45 ⫾ 0.1 35 ⫾ 0.1 40 ⫾ 0.1 45 ⫾ 0.1 40 ⫾ 0.1 40 ⫾ 0.1

9 ⫾ 0.1 25 ⫾ 0.1 25 ⫾ 0.1 25 ⫾ 0.1 25 ⫾ 0.1 9 ⫾ 0.1 25 ⫾ 0.1 25 ⫾ 0.1

* Std dev estimates from Betts and Ball (1997); estimates otherwise.

vegetation fraction typically amounts to 10% of the vegetation-fraction value. In NWP models, the distinction is made between plant functional types (e.g., broadleaf forest) rather than between different plants within a biome (e.g., willow, birch, maple, oak, etc.). Thus, no standard deviations exist for the parameters of the “lumped” vegetation classes. Therefore, standard deviations are arbitrarily assumed to be 10% of the respective empirical parameter value except for albedo. Based on the average standard deviation found by Betts and Ball (1997), standard deviation of albedo is set to be 7.8% of the respective mean albedo. Tables 2 and 3 summarize the standard deviations and mean values of empirical parameters used.

b. Uncertainty analysis According to Eq. (1) any set of mean parameters and their standard deviations will always provide the same standard deviation of a predicted flux for the same environmental conditions. Therefore, every surface flux parameterization (Table 1) and its equation for the standard deviation are applied for typical ranges of environmental forcing conditions. To investigate whether some soils, vegetation types, or vegetation fractions provide greater standard deviation than others, this analysis is performed for all soil and vegetation types listed in Tables 2 and 3 and for vegetation fractions in increments of 10%. The latter identifies whether fluxes are more uncertain in sparsely or densely vegetated areas, in spring/fall, summer, or winter, and in midlatitudes/Tropics or deserts. Ideally the terms (/i)i: ⫽{, i}, referred to as contribution (term) hereafter, are of the same order of

magnitude for a balanced (good) parameterization and empirical parameter set. Comparing the magnitude of {, i} identifies critical (unbalanced) parameterizations and the parameters that contribute the most to the standard deviations. 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 LSM. To demonstrate the meaning of the theoretical findings for NWP and to illustrate how parameter-induced uncertainty limits NWP, MM5 is run for 0000 UTC 20 July–1200 UTC 23 July 2001 with the GEP analysis tool. Since MM5 starts without clouds, it takes some time until clouds form in the model. Consequently, insolation is too high during spinup. Therefore, the discussion refers to results after spinup. Note that spinup causes systematic errors.

c. Synoptic situation During 0000 UTC 20 July–1200 UTC 23 July 2001 near-surface air (dewpoint) temperature ranges from ⫺1° to 22°C (⫺1° to 16°C) at low elevation. The beginning was characterized by moderate southern flow over south and southwest Alaska due to cyclonic activity within a trough above the North Pacific and Bering Sea. In the first 12 h, the long-wave disturbance slightly deepened. It propagated east, turning the southern flow on its frontal edge into the dominant feature over Alaska, while the ridge continued building up on the eastern boundary of the domain. In the middle and toward the end of the period, a weak gradient spacious trough influenced the synoptic situation. Cold advection behind the trough reinforced cyclonic activity in the Aleutians. On 23 July, the low dominated Alaska

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FIG. 1. Distribution of (a) vegetation types with vegetation fraction superimposed and (b) soil types with topography (m) superimposed. Names refer to locations mentioned in the text.

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and gradually cut off, whereas the anticyclone over western Canada strengthened. On the surface, the well-developed cyclone in the Bering Sea gradually dissipated and weakened the southern flow over Alaska throughout the first half of the period. Alaska remained on the outer edge of a well-developed eastward-moving surface cyclone located north of Siberia for the rest of the period. During the period, sun rises (sets) around 0300 (2100), 0400 (2330), and 0530 (0100) Alaska Standard Time (AST) in the eastern, central, and western part of the domain. The sun rises (sets) around 500 (2300) AST in the southern central part of the domain, while it barely goes below the horizon in the North. AST is UTC minus 9 h.

4. Results and discussion In addition to empirical soil and plant parameters, net radiation also depends on astronomical and atmospheric conditions that can cause additional uncertainty. Since this study focuses on soil- and plantparameter-induced stochastic errors, uncertainties from astronomic conditions, systematic errors, and aspects of parameter interaction are not addressed. Like HTSVS, most modern LSMs (e.g., BATS, CLM, OSULSM) calculate ground heat flux based on the soil heat diffusion equation. Like HTSVS, all LSMs applying the mixture approach (e.g., SiB, CLM) use similar parameterizations for net radiation, sensible, and latent heat fluxes. HTSVS’ equations will reduce to those applied in LSMs that assume a grid cell to be either totally vegetation covered or vegetation free if vegetation fraction is 1 or 0, respectively. Thus, for these LSMs the uncertainty behavior can be derived from the figures of the theoretical analysis as these two extremes are also shown. LSMs that use a skin temperature Tskin or distinguish between foliage Tf and ground temperature Tg (e.g., SiB, CLM) use similar formulations for upward longwave radiation as HTSVS. HTSVS’ foliage and ground temperature can be related to the skin temperature by Tskin ⫽ fTf ⫹ (1 ⫺ f)Tg where the vegetation fraction values f range between 0 and 1. If Tf is smaller (greater) than Tg and vegetation fraction exceeds 50%, maximum upward longwave radiation will be smaller (greater) than if the foliage and ground have the same temperature. In the following, ground, sensible, and latent heat fluxes are counted positive when directed toward the atmosphere.

a. Overview Standard deviations of net radiation are the smallest for net radiation between 200 and 400 W m⫺2 and in-

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crease quasi-linearly for higher and lower net radiation (Fig. 2). The standard deviations of sensible and latent heat flux quasi-linearly increase with the absolute value of the flux (Figs. 3 and 4). Standard deviation of ground heat flux also increases quasi-linearly with the absolute value of the flux over wide ranges of soil conditions. The increase is nonlinear when the ground is partly frozen (see secondary legs in Fig. 5). The steepness of the increase with increasing absolute value of the flux depends on vegetation and soil type for net radiation and latent heat flux, on vegetation type only for sensible heat flux and on soil type only for ground heat flux. In NWP, the behavior of standard deviations results in a diurnal cycle with a maximum around noon, and for fluxes that change their sign in the diurnal course, a secondary maximum at night (e.g., Fig. 6). Average relative errors of net radiation, sensible, latent, and ground heat fluxes are 7%, 10%, 6%, and 26%, respectively. A sensitivity study with a higher standard deviation for the vegetation fraction provides slightly increased standard deviations of the respective dependent fluxes. Since the standard deviation of vegetation fraction is related to the mean value of vegetation fraction (e.g., mean vegetation fraction of a grid cell) by f ⫽ 0.1f , usage of actual satellite-derived vegetation fraction will also reduce systematic errors in predicted fluxes. A sensitivity study performed with the standard deviations given by Clapp and Hornberger (1978) yields standard deviations in predicted fluxes similar to the studies with Cosby et al.’s (1984) values discussed in the following sections. A sensitivity study carried out for organic soils showed slightly higher standard deviations than for mineral soils. Note that there are no datasets on the distribution of organic soils at the resolution required by NWP models.

b. Net radiation Net radiation Q is the difference between incoming and outgoing irradiance. It depends on vegetation fraction, the emissivity of the foliage and ground, and the albedo of the foliage and ground, where ground albedo is a function of porosity (Table 1). Generally, ground albedo increases when the soil dries and decreases during precipitation (e.g., Charney 1975).

1) THEORETICAL

ANALYSIS

Standard deviations of net radiation Q are calculated for a range of volumetric water content from 0.001 to porosity, vegetation fraction from 0% to 100%, downward shortwave radiation from 0 to 750W m⫺2,

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FIG. 2. (a) Net radiation fluxes Q vs std dev Q as obtained by the theoretical analysis using the data given in section 4b(1). Results shown are for loam covered by mixed forest. Note that scatterplots for other soil and/or vegetation combinations are distinguished by the steepness of the relationship between the flux and its standard deviation. (b) Net radiation, its std dev, and relative error as obtained for typical ranges of atmospheric conditions. Example shown for shrubland on loam for ⫽ 0.211 m3 m⫺3, Rs ↓⫽ 360 W m⫺2, and Tf ⫽ Tg ⫺ 5 K. Horizontal distribution of (c) net radiation flux and (d) its std dev after 48 h of simulation. Note that simulation starts at 0000 UTC 20 Jul (1500 AST 19 Jul) 2001.

downward longwave radiation from 100 to 475 W m⫺2, and foliage and ground temperatures from ⫺30° to 40°C. For these environmental conditions, the standard deviation and relative error of net radiation are, on average, less than 13.67 W m⫺2 and 13.69% (sparse vegetation on sandy loam). The smallest standard deviation and relative error are 11.9 W m⫺2 (mixed forest on clay) and 1% (wooded and mixed tundra on loamy sand), respectively. Table 4 summarizes the relative errors for various soil and vegetation combinations. Standard deviations Q are the highest for great radiation deficits (corresponding to winter or Arctic con-

ditions), and the smallest for net radiation between 200 and 500 W m⫺2. They increase quasi-linearly for higher and lower net radiation (Fig. 2). They are minimal (⬍10 W m⫺2) for intermediate vegetation fraction and net radiation of 300–500 W m⫺2. They grow as vegetation fraction decreases/increases and incoming shortwave radiation increases/decreases. The position of the minimum varies slightly with soil type because of sensitivity to ground emissivity and albedo. Standard deviations are typically the lowest at low temperatures and relatively dry ( ⬍ 0.15 m3 m⫺3) or wet ( ⬎ 0.25 m3 m⫺3) soil conditions. If temperature

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FIG. 3. (a) Same as in Fig. 2a, but for sensible heat flux H vs std dev H using the data in section 4c(1) A friction velocity of 0.25 m s⫺1 is assumed. To show positive and negative fluxes, Tf ⫽ TR ⫾ 2 K, Tg ⫽ TR ⫿ 2 K. (b) Same as in Fig. 2b, but sensible heat flux. Example shown for mixed forest on loam for Tg ⫽ TR ⫹ 2 K, Tf ⫽ TR ⫹ 5 K, p ⫽ 1013.25 hP, u* ⫽ 0.5 m s⫺1. (c), (d) Same as in Figs. 2c,d, but for sensible heat flux and its std dev after 48 h of simulation.

exceeds a critical threshold value (which differs with soil type), Q becomes nearly independent of volumetric water content. The qualitative behavior described above differs only slightly with vegetation type. On average, the term related to ground emissivity, {Q, g}, dominates the standard deviation of net radiation independent of soil type. For most soil and vegetation combinations, the contribution of {Q, g} to Q is twice as high as the contribution related to foliage albedo, porosity, or vegetation fraction. On average, standard deviation of vegetation fraction and porosity contribute the least of all parameters to Q except for sandy loam and mixed forest. For

mixed forest, the contribution of {Q, f } (the term related to vegetation fraction) to Q is twice as high as {Q, f } is for the other vegetation types. The terms related to foliage albedo {Q, ␣f } and foliage emissivity {Q, f } only slightly differ in value except for tundra. Here, {Q, ␣f } is about 1.4 times higher and {Q, f } is slightly lower than for other vegetation types. For mixed forest, {Q, ␣f } contributes 0.3 times less than for all other vegetation types. Since vegetation fraction, emissivity, and albedo strongly change with season, using actual values derived from satellite data provides potential for improved NWP in general by reducing the stochastic and systematic errors.

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FIG. 4. (a) Same as in Fig. 2a, but for latent heat flux LE vs std dev LE. A friction velocity of 0.25 m s⫺1 is assumed. To show positive and negative fluxes, Tf ⫽ TR ⫾ 1 K, Tg ⫽ TR ⫿ 2 K, and qg ⫽ qR ⫿ 0.0005 kg/kg. (b) Same as in Fig. 2b, but for latent heat flux. Example shown for shrubland on clay loam with ⫽ 0.2325 m 3 m ⫺3 , T g ⫽ T R ⫺ 4 K, T f ⫽ T R ⫺ 6 K, q R ⫽ 0.005 kg/kg, qg ⫽ 0.01 kg/kg, p ⫽ 1013.25 hPa, u* ⫽ 0.3 m s⫺1, and Rs ↓⫽ 400 W m⫺2. (c), (d) Same as in Figs. 2c,d, but for latent heat flux and its std dev after 48 h of simulation.

2) NWP

EXAMPLE

The Yukon Territory and the ocean are cloud free most of the time. Except for the coastal areas Alaska is cloud covered most of the time. Net radiation locally exceeds 600 W m⫺2 (around noon) on the North Slope, in the Yukon Territory and along the west coast of Alaska, while it remains less than 200 W m⫺2 in the cloud-covered areas (e.g., Fig. 2). Due to the behavior described above, standard deviation of net radiation is greater in cloud-free than cloudy areas, increases toward noon, and decreases toward midnight (e.g., Fig. 6), and its spatial distribution shows no

obvious correlation to soil or vegetation types (e.g., Figs. 1 and 2).

c. Sensible heat flux Sensible heat flux depends on vegetation fraction and roughness length via the turbulent resistances (Table 1).

1) THEORETICAL

ANALYSIS

Sensible heat fluxes H and their standard deviations H are determined by soil, air, and foliage temperatures

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FIG. 5. (a) Same as in Fig. 2a, but for ground heat flux Hs vs std dev. (b) Same as in Fig. 2b, but for ground heat flux. Example shown for sandy loam with (b) (g ⫺ ⫺1)/z⫺1 ⫽ 0.01 m3 m⫺3/0.05 m) and (Tg ⫺ T⫺1)/z⫺1 ⫽ 1 K/0.05 m, and (c) ⫽ 0.3255 m3 m⫺3 and (g ⫺ ⫺1)/z⫺1) ⫽ 0.01 m3 m⫺3/0.05 m). Note that plots for (g ⫺ ⫺1)/z⫺1) ⫽ 0 m3 m⫺3/m (as used in many LSMs) look similar. (d) Same as in Fig. 2c, but for ground heat flux and (e) its std dev after 48 h of simulation.

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2) NWP

FIG. 6. Temporal evolution since 0000 UTC 20 Jul (1500 AST 19 Jul) 2001 of net radiation, sensible heat flux, latent heat flux, and ground heat flux as well as their std devs, shown as averages for the entire land surface in the model domain.

between ⫺30° and 40°C, and friction velocity between 0 and 1 m s⫺1. Except for sparse vegetation (22.2%), the relative error of sensible heat flux typically remains less than 9.2% (grassland), 11.9% (shrubland), 5.2% (deciduous forest), 5.3% (coniferous forest), 5.2% (mixed forest), 9.1% (wooded tundra), and 7.8% (mixed tundra). The absolute value of sensible heat flux grows as wind speed increases, as does the standard deviation of sensible heat flux. The only marginal increase of standard deviation with increasing temperature at low wind speed becomes a slightly stronger increase for high wind speed as the flux grows. For most conditions, H exhibits a slight, quasi-linear increase with increasing absolute value of the sensible heat flux (Fig. 3). Standard deviations of sensible heat flux typically increase with increasing vegetation fraction. This increase is stronger at high than at low temperatures (e.g., Fig. 3). The general behavior described above changes marginally with vegetation types via roughness length (therefore not shown).

EXAMPLE

During the day, sensible heat fluxes are downward (up to ⫺180 W m⫺2) nearly everywhere in the mountains (e.g., Fig. 3). In the interior of Alaska and along the lower Yukon River, sensible heat fluxes are upward (up to 50 W m⫺2). At night, sensible heat fluxes are negative except over water. In mountainous areas, they are as low as ⫺208 W m⫺2, while they seldom exceed ⫺5 W m⫺2 at low elevation. Areas of high vegetation fraction are apparent in the horizontal distribution of standard deviation as higher values than those obtained for low vegetation fraction (e.g., Fig. 3). The lowest overall standard deviation of sensible heat flux exists at high elevation with low vegetation fraction. At this time of the year, the atmosphere cools at night and heats up during the day. Therefore, standard deviation of sensible heat flux shows a noon maximum (e.g., Fig. 6), locally often a secondary minor maximum at midnight, and minima around sunrise/sunset.

d. Latent heat flux Latent heat flux depends on vegetation fraction f , roughness length z0, and bulk-stomatal resistance rst (Table 1). The latter depends on photosynthetically active radiation (PAR), specific humidity deficit between leaf and ambient air, leaf temperature, and soil moisture deficit (see the appendix). These dependencies are considered by the so-called empirical correction functions that range between 0 and 1 (e.g., Jarvis 1976; Sellers et al. 1986; Hicks et al. 1987; Dingman 1994). Empirical parameters used in these correction functions are minimum stomatal resistance rst,min, the temperatures at which stomata close, Tmin and Tmax, the temperature Topt at which rst reaches its minimum, the albedo ␣f and emissivity f of foliage, a parameter bst used to determine PAR, field capacity fc, permanent wilting point pwp, and root length zroot (Table 1).

1) THEORETICAL TABLE 4. Relative error of net radiation (in % absolute) for various soil- and vegetation-type combinations.

Grassland Shrubland Deciduous forest Needleleaf forest Mixed forest Sparse vegetation Wooded tundra Mixed tundra

Loamy sand

Sandy loam

Loam

Clay loam

Clay

9.26 12.57 4.71 10.69 2.75 13.60 0.95 0.95

9.45 12.70 5.05 10.96 2.80 13.69 1.11 1.11

9.24 12.56 5.08 10.94 2.76 13.58 1.11 1.11

8.98 12.36 7.76 11.05 2.73 13.67 1.21 1.21

8.85 12.24 11.01 11.01 2.70 13.58 1.19 1.19

ANALYSIS

Latent heat fluxes LE and their standard deviations LE are determined by soil, air, and foliage temperatures between ⫺20° and 30°C, specific humidity of the air and at the ground between 0.00045 and 0.005 kg kg⫺1, friction velocity between 0 and 1 m s⫺1, shortwave radiation between 0 and 750 W m⫺2, and volumetric water content between permanent wilting point and porosity. For the same soil type, LE strongly differs for high and low vegetation. On average, the relative error is less for high than for low vegetation. It is the highest for wooded tundra (up to 25.27% on average; Table 5).

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TABLE 5. Same as in Table 4, but for latent heat flux.

Grassland Shrubland Deciduous forest Needleleaf forest Mixed forest Sparse vegetation Wooded tundra Mixed tundra

Loamy sand

Sandy loam

Loam

Clay loam

Clay

1.31 12.22 5.19 5.62 2.70 0.02 7.15 8.21

4.69 6.74 6.16 8.53 3.74 0.02 11.07 7.38

0.75 3.14 6.01 7.34 4.97 0.02 4.51 2.20

7.31 5.15 5.60 6.90 4.17 0.02 6.80 0.46

19.04 4.38 6.39 6.89 5.96 0.02 25.27 4.36

This means that distinguishing more types of tundra may yield improvement of latent heat flux prediction in the Arctic. Generally, LE increases nonlinearly with vegetation fraction. This increase is usually (slightly) greater under warmer than cooler conditions (Fig. 4). Superimposed is a nonlinear increase of standard deviation as wind speed (and hence latent heat flux) increases. Overall, LE increases quasi-linearly with the absolute value of the latent heat flux. The increase in LE with increasing vegetation fraction, among other things, results from the higher weight given to the standard deviation of empirical parameters when calculating stomatal resistance as vegetation cover extends. This strong impact of vegetation fraction on the standard deviation of latent heat flux demands the incorporation of the actual satellite-derived vegetation fraction in NWP. Despite the strong difference between standard deviations of permanent wilting point for the various soil types, standard deviation of latent heat flux differs only slightly with soil type. Relative error, however, strongly differs with soil type (Table 5). The same is true for field capacity. On average, the contribution of stomatal resistance {LE, rst} and vegetation fraction {LE, f } are of a similar order of magnitude. Generally, {LE, rst} contributes more than {LE, f } for all kinds of forest. The opposite is true for the other vegetation types. The contributions vary slightly with soil type for constant vegetation type. This means that increasing the accuracy of the parameters on which stomatal resistance depends provides the potential to reduce the standard deviation in stomatal resistance rst and finally in latent heat flux. To identify which parameters provide the greatest potential for reducing standard deviation of stomatal resistance, standard deviation of calculated stomatal resistance is examined. Standard deviation of stomatal resistance rst amounts to 15% on average. It may be much greater especially as the correction functions [Eqs. (A7)–(A10)] provide values close to 0 or 1. If air

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temperature, for instance, approaches the minimum or maximum temperature at which stomata close, the standard deviation of calculated stomatal resistance will increase. On average, the parameters used in the correction functions for PAR, ambient temperature and moisture contribute the least to the standard deviation in stomatal resistance. Contributions related to minimum stomatal resistance rst,min and LAI dominate the standard deviation of stomatal resistance for all vegetation types. Unfortunately, measurements show that these quantities vary by an order of magnitude even for the same plant type (e.g., Körner et al. 1979; Avissar 1991). Thus, standard deviation in minimum stomatal resistance limits the prediction of latent heat flux over vegetated areas. For shrubland the terms related to permanent wilting point {rst, pwp} and field capacity {rst, fc} contribute to the standard deviation of stomatal resistance rst by the same order of magnitude (but usually with lower values) as those of LAI and rst,min. The terms {rst, pwp} and {rst, fc} gain similar importance to rst,min and LAI for clay and clay loam. This means that further distinction of clay-type soils provides potential for reducing the standard deviations in stomatal resistance and hence latent heat flux. Also, model improvement may be gained by finding/developing a correction function for soil water deficit that uses parameters that cause a lower standard deviation in calculated stomatal resistance.

2) NWP

EXAMPLE

During the day, latent heat fluxes are positive except for some coastal and mountainous areas and the ocean (Fig. 4). In cloudy areas, latent heat fluxes typically range between 10 and 120 W m⫺2, while they reach up to 360 W m⫺2 in cloud-free areas. The long daylight hours in combination with the great water availability results in high latent heat fluxes in the north. Latent heat fluxes are close to zero or negative in the southern model domain when the sun gets below the horizon. At that time, latent heat fluxes still reach 70 W m⫺2 on the North Slope and in the northern Brooks Range, and 40 W m⫺2 in the Interior. As expected from the theoretical analysis, standard deviation of latent heat flux is usually greater in cloudfree (e.g., Yukon Territory, western Alaska in Fig. 4) than cloudy areas (e.g., central Alaska in Fig. 4). After precipitation, LE increases slightly as LE grows in response to the wet soil. Thus, as expected form the theoretical analysis, areas of high precipitation occasionally become visible in the distribution of standard deviation (not shown).

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Mixed forest is easily visible in the LE distribution by its lower standard deviation, and shrubland and wooded tundra by their higher standard deviations than adjacent vegetation (e.g., Fig. 4). This confirms the potential for improvement in Arctic NWP by finer distinctions for tundra, especially because of the widespread presence of tundra in the Arctic (Fig. 1). The standard deviation follows a diurnal cycle with a primary peak around noon (e.g., Fig. 6) and locally a secondary weaker peak at night. South of the North Slope, stomata already close at night, leading to lower standard deviations in the southern than northern part of the domain.

e. Ground heat flux Calculation of ground heat flux involves pore-size distribution index, porosity, saturated water potential, and thermal conductivity (Table 1), where the latter is a function of the former parameters (e.g., McCumber and Pielke 1981; Kramm et al. 1996).

1) THEORETICAL

ANALYSIS

Ground heat fluxes Hs and their standard deviations Hs are determined by soil temperatures and volumetric water content from ⫺20° to 30°C and 0.001 to porosity, respectively. The gradients of soil temperature and total volumetric water content vary from ⫺187.5 to 187.5 K m⫺1, and ⫺187.5 to 187.5 m3 m⫺3 m⫺1, respectively. Standard deviation in ground heat flux increases slightly with increasing absolute value of ground heat flux. Generally, errors will be greater for great than small absolute values of temperature gradients except if the flux is close to zero (Fig. 5). Overall, the averaged relative error remains below 26%. It is the lowest for loamy sand (about 14%), followed by sandy loam (16%), loam (25%), clay loam (35%), and clay (40%), but for some conditions of soil temperature and soil temperature gradient, individual relative errors may exceed 50% (Fig. 5). This means that the standard deviation of ground heat flux may limit agricultural applications. The impact for NWP usually will not be dramatic as ground heat fluxes are typically notably smaller than the net radiation, latent, and sensible heat fluxes. Standard deviation of ground heat flux is independent of soil temperature for soil temperatures above 0°C (e.g., Fig. 5). At soil temperatures below 0°C it exhibits a slight nonlinear decrease with decreasing soil temperature for negative and positive temperature gradients (Tg ⫺ T⫺1)/z⫺1. The steepness of the decrease (increase) depends on soil type. Relative errors are maximal where typically most of the soil water freezes (Fig. 5). This temperature differs with soil type. The

FIG. 7. Thermal conductivity and its std dev . Example shown for sandy loam with (g ⫺ ⫺1)/z⫺1 ⫽ ⫺0.01 m3 m⫺3/0.05 m and (Tg ⫺ T⫺1)/z⫺1 ⫽ ⫺1 K/0.05 m. Note that plots for (g ⫺ ⫺1)/z⫺1 ⫽ 0 m3 m⫺3/m (as used in many LSMs) look similar.

absolute value of ground heat flux and its relative error increase with decreasing relative volumetric water content. This means that ground heat flux is more reliable after precipitation or in the Tropics than during drought or in deserts. All soil types show similar qualitative behavior. Since most modern LSMs use the soil heat diffusion equation for ground heat flux, the contribution terms are analyzed. Excluding the cross effects [i.e., wLD(g ⫺ ⫺1)/z⫺1 ⫽ 0 as is done in most other LSMs] shows that this term does not cause the great standard deviation of ground heat flux. Instead, the contribution of the standard deviation of thermal conductivity is often more than an order of magnitude greater than the contribution of all other terms. The parameterization of thermal conductivity has already been pointed out as problematic by P. Viterbo (2003, personal communication). As discussed above, thermal conductivity depends on pore size distribution index, saturated water potential and porosity, and shows a nonlinear behavior (Fig. 7). Below 0°C, thermal conductivity decreases less strongly with decreasing volumetric water content than above 0°C, and it decreases when soil temperature decreases. This decrease is stronger for high than low relative volumetric water content. Thermal conductivity becomes independent of temperature for soils warmer than 0°C. Thermal conductivity is the highest in loamy sand at high relative volumetric water content, while it is the lowest in loam. Standard deviation of thermal conductivity is the greatest for wet hot soils (e.g., the

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Tropics), and the lowest for dry frozen ground (e.g., midlatitudes in winter). Note that permafrost soils typically fall in the saturated cold range (Fig. 7). All soils show similar general behavior. A parameterization for thermal conductivity that causes less uncertainty has to be identified and implemented in the future. Despite the huge standard deviation of saturated water potential (relative error up to 149%: cf. Table 2), standard deviations of porosity and pore size distribution contribute more strongly to the standard deviation of ground heat flux than does that of saturated water potential. Consequently, increasing the accuracy of porosity and pore size distribution provides more potential for improving the lower boundary condition in NWP models than increasing the accuracy of the parameters for saturated water potential.

2) NWP

EXAMPLE

Generally, ground heat flux is directed toward the atmosphere in the interior of Alaska, lower Yukon River, Coastal Mountains, most of the Yukon Territory, and the western North Slope during the night, and toward the soil everywhere during the day (e.g., Fig. 5). Daytime soil heating (locally less than ⫺100 W m⫺2) exceeds the nighttime cooling (locally about 25 W m⫺2). Standard deviations are less for cloudy than cloudfree areas as cloudiness and cloud thickness reduces soil heating/cooling. At high elevation, local soil freezing/ thawing causes great standard deviations in simulated ground heat flux. Standard deviation grows as soils dry, leading to the high values in the Yukon Territory (e.g., Fig. 5). Since ground heat flux changes direction in the diurnal course, the diurnal courses of standard deviations show maxima at noon (e.g., Fig. 6) on average, and locally also at night. No spatial relationship between the uncertainty in Hs and soil type is obvious.

5. Conclusions All modern NWP models use parameterizations for the land surface processes that rely on empirical plant and soil parameters to represent different vegetation and soil types. Usually, mean values of the vegetation or soil characteristic derived from laboratory or field studies are assigned, ignoring their standard deviations. The standard deviations of some parameters, however, can be even greater than the mean value itself. Thus, every predicted flux will be in “error” by some amount (known as the standard deviation) as a consequence of the dependent parameters’ standard deviations. The aims of this study are to 1) determine the stochastic

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errors (standard deviations) of predicted surface fluxes that are caused by the standard deviations of empirical parameters, 2) identify critical parameters and (parts of) parameterizations, and 3) demonstrate how stochastic errors may limit Arctic NWP. GEP principles (e.g., Kreyszig 1970; Meyer 1975) are introduced to calculate the standard deviations of the simulated fluxes. On average, the relative errors in net radiation, sensible, latent, and ground heat fluxes amount to about 7%, 10%, 6%, and 26%, respectively. This means that ground heat flux prediction currently limits the accuracy with which the lower boundary condition can be determined in NWP. However, typically ground heat flux is much smaller than net radiation, sensible, and latent heat fluxes and can therefore be considered of less importance for NWP as compared to the other fluxes. The statistical uncertainty in predicted net radiation, sensible, and latent heat fluxes is less than the measurement errors typical for these fluxes (e.g., Panin et al. 1998; Willson et al. 2002). Field experiments show imbalances of about 100–250 W m⫺2 or 20%–50% of net radiation due to conceptual deficiency (i.e., systematic errors; Panin et al. 1998). Therefore, if the parameters are chosen reasonably and the parameterizations capture the governing physical processes, measured fluxes alone cannot serve to tell us which parameters/parameterizations provide better results in predicted fluxes. For the same environmental conditions, the standard deviation and relative error of net radiation are, on average, less than 13.67 W m⫺2 and 13.69% (sparse vegetation on sandy loam). The smallest standard deviation and relative error are 11.9 W m⫺2 (mixed forest on clay) and 1% (wooded and mixed tundra on loamy sand), respectively. The parameters of wooded and mixed tundra that are required to calculate stomatal resistance, however, cause the highest standard deviations in latent heat flux. Note that net radiation and latent heat fluxes do not depend on exactly the same parameters (Table 1). Ground heat fluxes have their highest relative errors for clay loam (35%) and clay (40%), on average. Introducing subclasses of clay-type soils and tundra (to obtain less uncertain parameters) may improve the accuracy at the lower boundary in NWP models in regions with large coverage by such soils and vegetation. Standard deviation of predicted net radiation is greater for low and high fractions than for intermediate vegetation fractions. Standard deviation of predicted latent heat flux increases with increasing vegetation fraction, among other things, because uncertain empirical parameters used to calculate stomatal resistance have a higher fractional weight as vegetation cover ex-

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tends. Standard deviation of sensible heat flux also increases as vegetation fractions increase. Since vegetation fraction plays a large role in the statistical uncertainty of all vegetation-fraction-dependent fluxes, and since the standard deviation of the vegetation fraction also depends on the mean vegetation fraction, utilizing actual satellite-derived vegetation fraction has great potential for improving NWP, in general. The analysis identified the parameterization of thermal conductivity as the dominant influence on the standard deviation of ground heat flux. Thus, this parameterization is to be replaced in the future by one that will cause less uncertainty. These findings also demonstrate that GEP can be used for identification of critical (parts of) parameterizations. The analysis by GEP also showed that the empirical parameters with the highest relative error are not necessarily the greatest contributors to the standard deviation of the predicted flux. Saturated water potential, for instance, has a high relative error, but contributes less to the standard deviation of ground heat flux than does porosity or the pore size distribution index. The same relative error can even propagate to different standard deviations. Despite ground emissivity relative error amounts of 10%, which are similar to relative error amounts of the foliage emissivity or vegetation fraction, emissivity of the ground usually dominates the standard deviation of net radiation. Contributions related to minimum stomatal resistance and LAI govern the standard deviation of stomatal resistance despite the fact that the relative error of LAI and minimum stomatal resistance is 10%, similar to the relative error values of other empirical parameters that stomatal resistance depends on (Table 3). Based on these findings, focus should be on obtaining parameter sets with reduced standard deviation for ground emissivity, LAI, and minimum stomatal resistance, and on further specification of tundra and clay types. Reducing the standard deviation of stomatal resistance, and, hence, latent heat flux requires considering more plant and soil types. A high priority should be placed on compiling such diversified parameter sets in the future. The findings also elucidate the usefulness of GEP for systematically prioritizing which parameter to measure first. Since all modern LSMs used in NWP models require the same kind of empirical parameters as those used in HTSVS, similar standard deviations in predicted fluxes are expected for other LSMs using similar parameterizations as HTSVS. The standard deviations found for the predicted fluxes have to be considered as the current limit of predictability related to the statistical un-

certainty of empirical parameters. Additional limitations exist due to systematic errors (e.g., from initialization or parameterizations) that have already been discussed elsewhere (e.g., Toth and Kalnay 1993; Henderson-Sellers et al. 1995; Shao and Henderson-Sellers 1996; Chen et al. 1997; Schlosser et al. 2000; Slater et al. 2001; Toth et al. 2001; Luo et al. 2003). Investigations of different LSMs by GEP may identify physical parameterizations that allow surface fluxes to be predicted with low uncertainty even while using the same uncertain empirical parameters. Model improvement needs the combined efforts of better parameter sets at higher resolution and more physical parameterizations. Results of GEP analysis could form a basis for intercomparison of LSMs, and a guide for parameter determination aimed at improving LSMs and NWP. Acknowledgments. I thank G. Kramm, A. Rulo, M. Jankov, and the anonymous reviewers for their helpful discussions and fruitful comments, and C. O’Connor for her careful editing. BMBF and NSF financially supported this study under Contracts 07ATF30, OPP0327664, and ATM0232198.

APPENDIX Parameter-Dependent Functions Relevant for Latent Heat Flux Calculation Potential temperature ⌰␦ and specific humidity q␦, both at height ␦ close above the foliage, and specific humidity of the foliage qf are given by (Mölders et al. 2003) ⌰␦ ⫽

冋

册

rmt,Hrt ⌰R 1 ⫺ f f T ⫹ T ⫹ , rmt,H ⫹ rt rmt,g g rmt,f f rt

rmt,Ert q␦ ⫽ rmt,E ⫹ rt

⫹

冉

f 1 ⫺ f rmt,fg ⫹ qg 1 1 rmt,g 1 ⫹ ⫹ rmt, f rst rmt,fg rmt, f

冦冤

冉

f rst

冊

1 1 1 ⫹ ⫹ r rst rmt,fg rmt, f mt, f

and qf ⫽

共A1兲

1 1

1 1 ⫹ ⫹ rst rmt,fg rmt, f

冋

冊

qi ⫹

qR rt

冧

冥

共A2兲

,

册

qg q␦ qi ⫹ ⫹ , rst rmt,fg rmt, f

共A3兲

where Tf , Tg, and qg are the surface temperature of the foliage and ground and the specific water vapor at the

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MONTHLY WEATHER REVIEW—SPECIAL SECTION

ground. Furthermore, rmt, f , rmt,g, and rmt, fg are the resistances of the molecular-turbulent layer close to the surfaces of the foliage and the ground, and in the environment between the foliage and ground surface, which inhibit the transfer of heat and matter (e.g., Kramm et al. 1996). Here ⌰R and qR denote the potential temperature and specific humidity at reference height. Here qi ⫽ qi(Tf) is the specific humidity within the stomatal cavities, assumed to be at saturation. The quantities rmt,H and rmt,E are defined by (Kramm 1995) 1 ⫺ f f ⫽ ⫹ , rmt,H rmt,g rmt, f 1

g␦ ⫽

再

共A4兲

and 1 ⫺ f ⫽ ⫹ rmt,E rmt,g 1

冉

1 1 ⫹ rst rmt,fg f . 1 1 1 ⫹ ⫹ r rst rmt,fg rmt, f mt, f

rst ⫽

rst,min , LAIg␦ gPARgT g

a关qi共Tf兲 ⫺ q␦兴 ⬎ 0.01152 kg m⫺3

冊 再

⫺1

,

共A7兲 共A8兲

,

Tf ⫺ Tmin Tmax ⫺ Tf gT ⫽ Topt ⫺ Tmin Tmax ⫺ Topt

共A6兲

where the correction functions

0.233 bst PAR

冊

共A5兲

Bulk-stomatal resistance is given by (e.g., Sellers et al. 1986; Hicks et al. 1987; Dingman 1994)

a关qi共Tf兲 ⫺ q␦兴 ⱕ 0.01152 kg m⫺3

冉

冊

冉

1 ⫺ 66.6a关qi共Tf兲 ⫺ q␦兴

gPAR ⫽ 1 ⫹

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冎

Tmax⫺Topt Topt⫺Tmin

,

共A9兲

and n

g ⫽

共j ⫺ pwp兲⌬zj fc ⫺ pwp 兲zroot

兺 共 j⫽1

consider the sensitivity to the water vapor deficit, photosynthetically active radiation, ambient air temperature, and water availability. Herein, n is the deepest layer where roots still exist, and j and ⌬zj are the volumetric water content and thickness of layer j. REFERENCES Arakawa, A., and V. R. Lamb, 1977: Computational design of the basic dynamical processes of the UCLA general circulation model. Methods of Computational Physics, Vol. 17, Academic Press 174–265. Avissar, R., 1991: A statistical-dynamical approach to parameterize subgrid-scale land-surface heterogeneity in climate models. Surv. Geophys., 12, 155–178. ——, and R. A. Pielke, 1989: A parameterization of heterogeneous land surface for atmospheric numerical models and its impact on regional meteorology. Mon. Wea. Rev., 117, 2113–2136. Betts, A. K., and J. H. Ball, 1997: Albedo over the boreal forest. J. Geophys. Res., 102D, 28 901–28 909. Bonan, G. B., K. W. Oleson, M. Vertenstein, S. Levis, X. Zeng, Y. Dai, R. E. Dickinson, and Z.-L. Yang, 2002: The land surface climatology of the community land model coupled to the NCAR Community Climate Model. J. Climate, 15, 1115–1130. Calhoun, F. G., N. E. Smeck, B. L. Slater, J. M. Bigham, and G. F. Hall, 2001: Predicting bulk density of Ohio soils from mor-

共A10兲

phology, genetic principles, and laboratory characterization data. Soil Sci. Soc. Amer. J., 65, 811–819. Charney, J., 1975: Dynamics of desert and droughts in the Sahel. Quart. J. Roy. Meteor. Soc., 101, 193–202. Chen, F., and J. Dudhia, 2001: Coupling an advanced land surface hydrology model with the Penn State/NCAR MM5 modeling system. Part I: Model description and implementation. Mon. Wea. Rev., 129, 569–585. Chen, T. H., and Coauthors, 1997: Cabauw experimental results from the Project of Intercomparison of Land Surface Schemes (PILPS). J. Climate, 10, 1194–1215. Clapp, R. B., and G. M. Hornberger, 1978: Empirical equations for some soil hydraulic properties. Water Resour. Res., 14, 601–604. Collins, D. C., and R. Avissar, 1994: An evaluation with the Fourier amplitude sensitivity test (FAST) of which land surface parameters are of greatest importance in atmospheric modeling. J. Climate, 7, 681–703. Cosby, B. J., G. M. Hornberger, R. B. Clapp, and T. R. Ginn, 1984: A statistical exploration of the relationships of soil moisture characteristics to the physical properties of soils. Water Resour. Res., 20, 682–690. Deardorff, J. W., 1978: Efficient prediction of ground surface temperature and moisture, with inclusion of a layer of vegetation. J. Geophys. Res., 83, 1889–1903. Dickinson, R. E., A. Henderson-Sellers, and P. J. Kennedy, 1993: Biosphere Atmosphere Transfer Scheme (BATS) version 1e

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as coupled to the NCAR Community Climate Model. NCAR Tech. Note NCAR/TN-378⫹STR, 72 pp. Dingman, S. L., 1994: Physical Hydrology. Macmillan Publishing Company, 575 pp. Dorman, J. L., and P. J. Sellers, 1989: A global climatology of albedo, roughness length, and stomatal resistance for atmospheric general circulation models as represented by the simple biosphere model (SiB). J. Appl. Meteor., 28, 833–855. Dudhia, J., 1993: A nonhydrostatic version of the Penn State– NCAR Mesoscale Model: Validation tests and simulation of an Atlantic cyclone and cold front. Mon. Wea. Rev., 121, 1493–1513. Grell, G., 1993: Prognostic evaluation of assumptions used by cumulus parameterizations. Mon. Wea. Rev., 121, 764–787. ——, J. Dudhia, and D. Stauffer, 1994: A description of the fifthgeneration Penn State/NCAR Mesoscale Model (MM5). NCAR Tech. Note NCAR/TN-398⫹STR, 117 pp. Grunwald, S., D. J. Rooney, K. McSweeney, and B. Lowery, 2001: Development of pedotransfer functions for a profile cone penetrometer. Geoderma, 100, 25–47. Gutman, G., and A. Ignatov, 1998: The derivation of green vegetation from NOAA/AVHHRR data for use in numerical weather prediction models. Int. J. Remote Sens., 19, 1533– 1543. Hamill, T. M., and S. J. Colucci, 1997: Verification of Eta-RSM short-range ensemble forecasts. Mon. Wea. Rev., 125, 1312– 1327. Henderson-Sellers, A., 1993: A factorial assessment of the sensitivity of the BATS land surface parameterization. J. Climate, 6, 227–247. ——, A. J. Pitman, P. K. Love, P. Irannejad, and T. H. Chen, 1995: The Project for Intercomparison of Land Surface Parameterization Schemes (PILPS): Phases 2 and 3. Bull. Amer. Meteor. Soc., 76, 489–503. Hicks, B. B., D. D. Baldocchi, T.-P. Meyers, R. P. Hosker Jr., and R. P. Matt, 1987: A preliminary multiple resistance routine for deriving dry deposition velocities from measured quantities. Water Air Soil Pollut., 36, 311–330. Hong, S.-Y., and H.-L. Pan, 1996: Nonlocal boundary layer vertical diffusion in a medium-range forecast model. Mon. Wea. Rev., 124, 2322–2339. Jackson, R. B., J. Canadell, J. R. Ehleringer, H. A. Mooney, O. E. Sala, and E. D. Schulze, 1996: A global analysis of root distributions for terrestrial biomes. Oecologia, 108, 389–411. Jarvis, P. G., 1976: The interpretation of the variations in leaf water potential and stomatal conductance found in canopies in the field. Philos. Trans. Roy. Soc. London, 273, 593–610. Jin, M., and S. Liang, 2006: Improving land surface emissivity parameter of land surface models using global remote sensing observations. J. Climate, in press. Körner, C., J. A. Scheel, and H. Bauer, 1979: Maximum leaf diffusive conductance in vascular plants. Photosynhetica, 13, 45– 82. Kramm, G., 1995: Zum Austausch von Ozon und reaktiven Stickstoffverbindungen zwischen Atmosphäre und Biosphäre. Maraun-Verlag, 268 pp. ——, R. Dlugi, N. Mölders, and H. Müller, 1994: Numerical investigations of the dry deposition of reactive trace gases. Air Pollution II, Vol. 1: Computer Simulation, J. M. Baldasano et al., Eds., Computational Mechanics Publications, 285–307. ——, N. Beier, T. Foken, H. Müller, P. Schröder, and W. Seiler, 1996: A SVAT scheme for NO, NO2, and O3—Model description. Meteor. Atmos. Phys., 61, 89–106.

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Kreyszig, E., 1970: Statistische Methoden und ihre Anwendung. Vanden Hoeck & Ruprecht, 422 pp. Luo, L., and Coauthors, 2003: Effects of frozen soil on soil temperature, spring infiltration, and runoff: Results from the PILPS 2(d) experiment at Valdai, Russia. J. Hydrometeor., 4, 334–351. McCumber, M., and R. A. Pielke, 1981: Simulation of the effects of surface fluxes of heat and moisture in a mesoscale numerical model 1. Soil layer. J. Geophys. Res., 86C, 9929– 9938. Meyer, S. L., 1975: Data Analysis for Scientists and Engineers. J. Wiley & Sons, 513 pp. Miller, D. A., and R. A. White, 1998: A conterminous United States multilayer soil characteristics dataset for regional climate and hydrological modeling. Earth Interactions, 2. [Available online at http://EarthInteractions.org.] Mölders, N., 2001: On the uncertainty in mesoscale modeling caused by surface parameters. Meteor. Atmos. Phys., 76, 119– 141. ——, and W. Rühaak, 2002: On the impact of explicitly predicted runoff on the simulated atmospheric response to small-scale land-use changes—An integrative modeling approach. Atmos. Res., 63, 3–38. ——, and J. E. Walsh, 2004: Atmospheric response to soil-frost and snow in Alaska in March. Theor. Appl. Climatol., 77, doi:10.1007/s00704-0032-5. ——, U. Haferkorn, J. Döring, and G. Kramm, 2003: Long-term numerical investigations on the water budget quantities predicted by the hydro-thermodynamic soil vegetation scheme (HTSVS)—Part I: Description of the model and impact of long-wave radiation, roots, snow, and soil frost. Meteor. Atmos. Phys., 84, 115–135. Noilhan, J., and S. Planton, 1989: A simple parameterization of land surface processes for meteorological models. Mon. Wea. Rev., 117, 536–549. Panin, G. N., G. Tetzlaff, and A. Raabe, 1998: Inhomogeneity of the land surface and problems in the parameterization of surface fluxes in natural conditions. Theor. Appl. Climatol., 60, 163–178. Pielke, R. A., 1984: Mesoscale Meteorological Modelling. Academic Press, Inc., 612 pp. Pollard, D., and S. L. Thompson, 1995: Use of a land-surfacetransfer scheme (LSX) in a global climate model: The response to doubling stomatal resistance. Global Planet. Change, 10, 129–161. Reisner, J., R. M. Rasmussen, and R. T. Bruintjes, 1998: Explicit forecasting of supercooled liquid water in winter storms using the MM5 mesoscale model. Quart. J. Roy. Meteor. Soc., 124B, 1071–1107. Schlosser, C.A., and Coauthors, 2000: Simulations of a boreal grassland hydrology at Valdai, Russia: PILPS phase 2(d). Mon. Wea. Rev., 128, 301–321. Sellers, P. J., Y. Mintz, Y. C. Sud, and A. Dalcher, 1986: A Simple Biosphere model (SIB) for use within general circulation models. J. Atmos. Sci., 43, 505–531. Shao, Y., and A. Henderson-Sellers, 1996: Modeling soil moisture: A project for intercomparison of land surface parameterization schemes phase 2 (b). J. Geophys. Res., 101D, 7227–7250. Slater, A. G., and Coauthors, 2001: The representation of snow in land surface schemes: Results from PILPS 2(d). J. Hydrometeor., 2, 7–25. Toth, Z., and E. Kalnay, 1993: Ensemble forecasting at the NMC:

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The generation of perturbations. Bull. Amer. Meteor. Soc., 74, 2317–2330. ——, Y. Zhu, and T. Marchok, 2001: The use of ensembles to identify forecasts with small and large uncertainty. Wea. Forecasting, 16, 463–477. Tracton, M. S., and E. Kalnay, 1993: Operational ensemble prediction at the National Meteorological Center: Practical aspects. Wea. Forecasting, 8, 379–398.

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Verseghy, D. L., 1991: CLASS—A Canadian land surface scheme for GCMs. 1. Soil model. Int. J. Climatol., 11, 111–133. Willson, K., and Coauthors, 2002: Energy balance closure of FLUXNET sites. Agric. For. Meteor., 113, 223–243. Wilson, M. F., A. Henderson-Sellers, R. E. Dickinson, and P. J. Kennedy, 1987: Sensitivity of the biosphere–atmosphere transfer scheme (BATS) to the inclusion of variable soil characteristics. J. Climate Appl. Meteor., 26, 341–362.

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Plant- and Soil-Parameter-Caused Uncertainty of Predicted Surface Fluxes NICOLE MÖLDERS Geophysical Institute, University of Alaska Fairbanks, Fairbanks, Alaska (Manuscript received 8 December 2004, in final form 16 May 2005) ABSTRACT Simulated surface fluxes depend on one or more empirical plant or soil parameters that have a standard deviation (std dev). Thus, simulated fluxes will have a stochastic error (or std dev) resulting from the parameters’ std dev. Gaussian error propagation (GEP) principles are used to calculate the std dev for fluxes predicted by the hydro–thermodynamic soil–vegetation scheme to identify prediction limitations due to stochastic errors, parameterization weaknesses, and critical parameters, and to prioritize which parameters to measure with higher accuracy. Relative errors of net radiation, sensible, latent, and ground heat flux, on average, are 7%, 10%, 6%, and 26%, respectively. The analysis identified the parameterization of thermal conductivity as the dominant influence on the std dev of ground heat flux. For net radiation, critical parameters are vegetation fraction and ground emissivity; for sensible and latent heat fluxes, vegetation fraction. Minimum stomatal resistance and leaf area index dominate the std dev of stomatal resistance for most vegetation and soil types. The empirical parameters with the highest relative error are not necessarily the greatest contributors to the std dev of the predicted flux. Based on the analysis high priority should be given to measurements of vegetation fraction, ground emissivity, minimum stomatal resistance, leaf area index in general, and the permanent wilting point and field capacity for clay and clay loam. Moreover, further specification of clay-type soils and tundra-type vegetation may improve the accuracy of the lower boundary condition in Arctic numerical weather prediction. Since GEP showed itself able to identify critical parameters and (parts of) parameterizations, GEP analysis could form a basis for parameterization intercomparisons and for parameter determination aimed at improving models.

1. Introduction Recent studies showed that major failures in numerical weather prediction (NWP) occur close to the surface (e.g., Hamill and Colucci 1997), the only natural boundary condition in NWP models. These failures may result from inaccurately predicting the fluxes of heat, water vapor, and matter at the surface–atmosphere interface. Systematic errors due to initial conditions, for instance, have been addressed by ensemble modeling techniques or sensitivity studies (e.g., Tracton and Kalnay 1993; Toth et al. 2001). Besides incorrect initial soil conditions, the parameterizations used in land surface models (LSMs) to describe the surface fluxes may cause systematic errors. Such systematic er-

Corresponding author address: Nicole Mölders, Geophysical Institute, University of Alaska Fairbanks, 903 Koyukuk Dr., P.O. Box 757320, Fairbanks, AK 99775-7320. E-mail: [email protected]

© 2005 American Meteorological Society

rors have been examined by comparing results from different LSMs. In this context, the great achievements of the Project for Intercomparison of Land Surface Parameterization Schemes (PILPS) must be mentioned (e.g., Henderson-Sellers et al. 1995; Shao and Henderson-Sellers 1996; Chen et al. 1997; Schlosser et al. 2000; Slater et al. 2001; Luo et al. 2003). PILPS also showed that LSMs strongly differ in accuracy due to empirical parameters (e.g., physiological, phenological, thermal, hydraulic, radiative, etc.) used to represent different vegetation and soil types (e.g., Wilson et al. 1987; Dorman and Sellers 1989). The wide possible parameter range causes a high variability in predicted state variables, energy, and water fluxes (e.g., Avissar 1991; Pollard and Thompson 1995; Mölders 2001) with implications for the structure of the atmospheric boundary layer (e.g., Avissar and Pielke 1989). For example, density of upward shortwave radiation will differ if the albedo value is taken as 0.15 or 0.2. Consequently slightly different temperature conditions and sensible, latent, and ground heat flux densities will

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be established. This phenomenon is known as “parameter interaction.” Various parameter-variation studies have been carried out to examine interaction effects and understand systematic uncertainty arising from parameter choices. Henderson-Sellers (1993), for instance, applied factorial experiments to assess the relative importance of the parameters in an LSM. She found that the most ecologically important parameters are roughness length, porosity, and an empirical factor describing the sensitivity to photosynthetically active radiation, followed by soil and vegetation albedo. Collins and Avissar (1994), for instance, performed a Fourier amplitude sensitivity test to determine the relative contributions of individual parameters to the variance of energy fluxes resulting from a heterogeneous surface. In doing so, they simultaneously varied all parameters according to their individual probability density functions. They found that statistical–dynamical approaches may be simplified by using only the probability density functions of relative stomatal resistance and surface roughness. Parameter-variation methods address systematic errors. They consider the question of whether slightly different parameters will result in significant perturbations of the model result. However, parameter effects or parameterization deficits that accidentally cancel each other out can remain overlooked (HendersonSellers 1993). To minimize parameter interaction, the methods have to be driven by either observation or reanalysis. Systematic errors from parameter choice, initialization, discretization, model assumptions, and physical parameterizations are not only the source of uncertainty. Another source is stochastic error. Usually a mean value for a vegetation or soil characteristic derived from laboratory or field studies is assigned to a grid element of a NWP model, ignoring natural variance given by the standard deviation. The standard deviations of some parameters, however, can be as great as the mean values themselves (e.g., Clapp and Hornberger 1978; Körner et al. 1979; Cosby et al. 1984; Avissar 1991). Stochastic uncertainty results solely from the fact that the aforementioned mean values of empirical soil and plant parameters are in “error” by the amount of the standard deviation. This error propagates in any quantity calculated by use of these mean parameters; that is, each predicted quantity also has a standard deviation due to the standard deviations of its dependent uncertain parameters. Therefore, uncertainty from stochastic errors also should be kept as low as possible. Since parameter-variation methods are deterministic, they cannot determine this stochastic uncertainty.

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Knowledge of a predicted flux’s parameter-induced standard deviation, however, is desirable for model development and improvement. Parameterizations, for instance, can be examined prior to implementation in an LSM with respect to the parameter-induced statistical uncertainty of the predicted fluxes. Parameterizations that lead to fluxes with high standard deviations can be identified and replaced or avoided. An equation to calculate the flux may be very insensitive or sensitive to the parameter’s standard deviation. Therefore, a priori it is unknown whether a huge (small) standard deviation of a parameter will lead to huge or small standard deviation of the predicted flux. It is of interest to identify the parameters that cause huge standard deviation in the fluxes to measure them with high accuracy even if they themselves have low standard deviation. The parameters causing the highest uncertainty are the ones that will guarantee the greatest potential for reduction of uncertainty in predicted fluxes if these parameters are known with higher a degree of certainty. Gaussian error propagation (GEP) principles permit calculating the error (variance, standard deviation, or error bar) of a predicted quantity that results as a consequence of the standard deviation(s) of the parameter(s) it depends on (e.g., Kreyszig 1970; Meyer 1975). Thus, GEP permits the tasks mentioned above to be addressed. This method originally stems from engineering and physics. In these disciplines, it has been applied to determine the statistical uncertainty inherent in a calculated quantity (e.g., current) due to standard deviations of parameters it depends on (e.g., resistances within a Wheatstone bridge). Since, in a mathematical sense, a function to predict a flux is unambiguous; it will always provide the same flux for the same set of state variables, fluxes, and empirical parameters. Thus, for each set of state variables, fluxes, mean parameters, and their standard deviations, GEP can determine the standard deviation of a predicted flux independent of other parameterizations used in an LSM; that is, without parameter interaction or full simulation. Therefore, the standard deviation of predicted surface fluxes can be theoretically analyzed using artificial data for the typical range of environmental conditions. In this study, GEP is applied to the parameterization of surface fluxes in the hydro–thermodynamic soil– vegetation scheme (HTSVS; Kramm et al. 1996; Mölders et al. 2003) used in various NWP (e.g., Mölders and Rühaak 2002; Mölders and Walsh 2004) and chemistry (e.g., Kramm et al. 1994, 1996) models. The standard deviation of predicted surface fluxes that is caused by standard deviation of empirical parameters is examined theoretically for a typical range of environmental

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conditions to identify uncertainty-causing parameters and (parts of) parameter-sensitive parameterizations. To demonstrate the importance of the results for NWP and to understand the parameter-induced limitations of NWP, a simulation is carried out for 0000 UTC 20 July– 1200 UTC 23 July 2001 for a subarctic/Arctic environment with the mesoscale modeling system of the fifthgeneration Pennsylvania State University–National Center for Atmospheric Research (PSU–NCAR) Mesoscale Model (MM5; Dudhia 1993; Grell et al. 1994). Since various other LSMs use parameterizations and parameters similar to those in HTSVS, the results presented here are of interest to a broad NWP community. As a courtesy to the reader, examples of LSMs using similar parameterizations as HTSVS are given.

2. Brief description of the NWP model a. The atmospheric model MM5 is run using the five water classes (cloud water, rainwater, ice, snow, graupel) in the cloud microphysical scheme of Reisner et al. (1998). As the typical horizontal extension of cumulus clouds is considerably smaller than the horizontal grid resolution (45 km), a cumulus scheme (Grell 1993) considers these subgridscale clouds. The parameterization of turbulent transfer processes follows Hong and Pan (1996). Furthermore, Grell et al.’s (1994) simple radiation scheme is applied. The model domain uses a horizontal Arakawa– Lamb-B (Arakawa and Lamb 1977) staggering consisting of 39 ⫻ 39 dot and 38 ⫻ 38 cross grid points with a horizontal grid spacing of 45 km, and 23 vertical layers reaching to 100 hPa. Five snow layers of equal thickness depend on snow depth. Five soil layers are logarithmically spaced between 0.1- and 2.95-m depth. To fulfill the Courant criterion a time step of 120 s is chosen.

b. The surface model HTSVS consists of a multilayer soil model, a singlelayer canopy model (e.g., Kramm et al. 1996; Mölders et al. 2003), and a multilayer snow model (e.g., Mölders and Walsh 2004). HTSVS calculates the fluxes of momentum, heat, and moisture at the vegetation–soil–atmosphere interface. It considers partly vegetation-covered grid cells by a mixture approach (e.g., Deardorff 1978; Kramm et al. 1996). Other LSMs using this approach are, for instance, the Simple Biosphere model (SiB; e.g., Sellers et al. 1986) and the Community Land Model (CLM; Bonan et al. 2002). Jarvis’s (1976) approach for bulkstomatal resistance (see the appendix) serves to calculate transpiration by plants. The Interaction between

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Soil–Biosphere–Atmosphere model (ISBA; e.g., Noilhan and Planton 1989), the Biosphere–Atmosphere Transfer Scheme (BATS; e.g., Dickinson et al. 1993), the Oregon State University LSM (OSULSM; e.g., Chen and Dudhia 2001), and SiB, for instance, also apply the Jarvis approach. Some of them even use the same correction functions (e.g., OSULSM for soil water deficit). Water uptake by plants includes a vertically variable root distribution (e.g., Mölders et al. 2003). The soil model includes heat conduction and water diffusion (including the Richards equation) within the soil, soil freezing and thawing, and the related release of latent heat and consumption of energy, and the effects of frozen ground on the vertical fluxes of heat, moisture, and water vapor (e.g., Kramm et al. 1996; Mölders et al. 2003). Other models using the Richards equation are, for instance, the Canadian Land Surface Scheme (CLASS; e.g., Verseghy 1991) and BATS. Examples of LSMs using a simplified (as compared to HTSVS) formulation of soil heat diffusion are OSULSM, BATS, and CLM. Tables 1–3 summarize the equations of surface fluxes and their dependent empirical parameters, and the soil and plant empirical parameters, respectively. Most modern LSMs use similar parameters.

c. Initialization Initial and boundary conditions stem from the (National Centers for Environmental Prediction) NCEP– NCAR Reanalysis Project (NNRP). A weighted combination of the July and August monthly 5-yr mean green vegetation cover (0.15° resolution) derived from the Advanced Very High Resolution Radiometer (AVHRR) data (Gutman and Ignatov 1998) is used to determine the vegetation fraction of each grid cell. The 1-km resolution U.S. Department of Agriculture (USDA) State Soil Geographic Database (Miller and White 1998) and 10-min resolution U.S. Geological Survey (USGS) terrain and vegetation data are applied to define soil texture, terrain elevation, and vegetation type (Fig. 1). Initial total soil moisture and temperature are interpolated from NNRP data. Partitioning of the total soil moisture between the liquid and solid phase follows Mölders and Walsh (2004). At the bottom of the soil model, soil temperature and total moisture remain constant throughout the simulation.

3. Experimental design a. GEP Every surface flux is a function ⫽ f (1, . . . , n) of one or more empirical parameters i that are the mean

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TABLE 1. Equations for calculation of the surface fluxes and dependency on empirical parameters. Here ⌰␦ and q␦ are the values of potential temperature and specific humidity at the height ␦ close above the surface (see the appendix). The surface temperature and specific humidity of foliage and ground are denoted as Tf, qf, Tg, and qg, respectively. Furthermore, rmt, f and rmt,g are the resistances of the molecular-turbulent layer close to the surfaces of the foliage and ground, which oppose the transfer of heat and matter. Here D, and DT,v are the transfer coefficients with respect to water vapor and heat. Here a and cp are the density and specific heat at constant pressure of air, and , , f, g, ⫺1, Tg, T⫺1, and z⫺1 are the thermal conductivity, the Stefan–Boltzmann constant, the vegetation fraction (expressed as values between 0 and 1), the volumetric water content and temperature at the ground (index g) and in the first soil layer (index ⫺1), and the depth of the first soil layer, respectively. Further, w, s, b, and s are the density of water, porosity, pore size distribution index, and saturated water potential. See Kramm (1995) and Kramm et al. (1996) for calculation of D,[⫽f (b, s, s)], DT,[⫽ f (b, s, s)], rmt, f, rmt,g, and qg; Mölders and Walsh (2004) for ␣g[⫽f (s)] and [⫽f (s, s, b)]; and the appendix for ⌰␦, q␦, qf, and rst. Note that net radiation also depends on astronomical and atmospheric conditions. Uncertainty from these sources is not considered in this study. Surface flux

Equations

Dependent parameters 4 f)[g T g

Q ⫽ [f ␣f ⫹ (1 ⫺ f)␣g]Rsf ⫺ (1 ⫺ ⫺ f [f T 4f ⫹ (1 ⫺ f)R↓lf]

Net radiation Sensible heat flux

H ⫽ ⫺cpa

Latent heat flux

LE ⫽ −aL

冋 冋

⫹ (1 ⫺

g)R↓lf]

Q ⫽ f (f, ␣f, s, f, g)

册 册

1 ⫺ f f 共⌰␦ − Tg兲 ⫹ 共⌰ ⫺ Tf兲 rmt,g rmt, f ␦

f 1 ⫺ f 共q␦ ⫺ qg兲 ⫹ 共q ⫺ qf兲 rmt,g rmt, f ␦

H ⫽ f (f , z0) LE ⫽ f (f , rst,min, LAI, Tmin, Topt, Tmax, s, b1, bst zroot, fc, pwp, z0)

Hs ⫽ ⫺(1/z⫺1)[( ⫹ wLDT)(Tg ⫺ T⫺1) ⫹ wLD(g ⫺ ⫺1)]

Ground heat flux

values obtained from measurements (e.g., Tables 2 and 3). Each calculated flux will be in error by an amount as a consequence of the errors in the measured empirical parameters i, which are typically given by the standard deviation i. GEP principles (e.g., Kreyszig 1970; Meyer 1975) permit determining a predicted flux’s statistical uncertainty (standard deviation) that results from the standard deviations of the empirical plant and soil parameters it depends on. To apply GEP each equation used to predict a flux (e.g., sensible heat flux) is derivated with respect to all its dependent empirical parameters i (e.g., vegetation fraction, roughness length; Table 1). The standard deviation (statistical uncertainty) of the predicted flux can be calculated from these individual derivations (/i) and the standard deviations i of the ith empirical parameters i by (e.g., Kreyszig 1970; Meyer 1975)

Hs ⫽ f (s, s, b, s, cs)

冑兺 冉 n

⫽

i⫽1

⭸ ⭸ i

冊

2

2 i.

共1兲

Here n represents the number of empirical parameters. A standard deviation means that 68% of all values fall within the range of the mean value plus/minus the standard deviation. Note that experimentalists often refer to ⫾ as the error bar. The ratio of the standard deviation to the mean value is the fractional standard deviation or relative error: ⫽ ( /). Remote sensing, field, and/or laboratory measurements have provided the standard deviation for many biological and geophysical characteristics [e.g., albedo, emissivity, leaf area index (LAI), vegetation fraction, porosity; e.g., Clapp and Hornberger 1978; Körner et al. 1979; Cosby et al. 1984; Avissar 1991; Betts and Ball 1997; Jin and Liang 2006]. Standard deviation of the

TABLE 2. Parameters and their std dev as considered in this study. Here, cS and S are the specific heat capacity and density of the dry soil material, respectively. Parameters and std devs for porosity s, pore size distribution index b, and saturated water potential s are from Cosby et al. (1984). Soil type Loamy sand Sandy loam Loam Clay loam Clay a

cS 10 J kg⫺1 K⫺1 3

876 882 896 866 890

⫾ ⫾ ⫾ ⫾ ⫾

Grunwald et al. (2001) Calhoun et al. (2001) c Emissivity of the ground b

69 34 52 72 23

S 10 kg m⫺3 3

1610 1520 1350 1420 1470

⫾ ⫾ ⫾ ⫾ ⫾

100a 140a 110a 80a 140b

s m m⫺3 0.421 0.434 0.439 0.465 0.468

⫾ ⫾ ⫾ ⫾ ⫾

0.072 0.088 0.074 0.054 0.035

4.26 4.74 5.25 8.17 11.55

⫾ ⫾ ⫾ ⫾ ⫾

gc -.-

s m

b -.-

3

1.40 1.72 1.66 3.74 3.93

⫺0.0361 ⫺0.141 ⫺0.355 ⫺0.263 ⫺0.468

⫾ ⫾ ⫾ ⫾ ⫾

0.0537 0.0537 0.0457 0.0525 0.039

0.95 0.95 0.95 0.95 0.95

⫾ ⫾ ⫾ ⫾ ⫾

0.095 0.095 0.095 0.095 0.095

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TABLE 3. Plant-specific parameters (from Pielke 1984; Wilson et al. 1987; Kramm 1995; Jackson et al. 1996) and std dev for vegetation types that occur in the model domain. Here, zroot, z0, rst,min, bst, ␣f, and f are the root length, roughness length, minimum stomatal resistance, an empirical parameter required to determine PAR, the albedo, and the emissivity of foliage, respectively. Here Tmin and Tmax are the temperatures at which stomata close, and Topt is the temperature at which rst reaches its minimum. See text for further details. Vegetation type

zroot m

z0 m

rst,min s m⫺1

bst -.-

␣f -.-

f -.-

Tmin °C

Tmax °C

Topt °C

Grassland Shrubland Deciduous forest Needleleaf forest Mixed forest Sparse vegetation Wooded tundra Mixed tundra

0.26 ⫾ 0.026 0.7 ⫾ 0.07 3 ⫾ 0.3 3 ⫾ 0.3 3 ⫾ 0.3 0.5 ⫾ 0.05 1.81 ⫾ 0.181 1.81 ⫾ 0.181

0.08 ⫾ 0.008 0.03 ⫾ 0.003 0.85 ⫾ 0.085 1.09 ⫾ 0.109 0.8 ⫾ 0.08 0.01 ⫾ 0.001 0.06 ⫾ 0.006 0.05 ⫾ 0.005

70 ⫾ 7 300 ⫾ 30 232 ⫾ 23.2 125 ⫾ 12.5 125 ⫾ 12.5 999 ⫾ 99.9 150 ⫾ 15 150 ⫾ 15

20 ⫾ 2 10 ⫾ 1 22 ⫾ 2.2 25 ⫾ 2.5 23 ⫾ 2.3 20 ⫾ 2 40 ⫾ 4 40 ⫾ 4

0.19 ⫾ 0.018* 0.25 ⫾ 0.025 0.11 ⫾ 0.011* 0.10 ⫾ 0.006* 0.12 ⫾ 0.0085* 0.12 ⫾ 0.012 0.16 ⫾ 0.016 0.16 ⫾ 0.016

0.97 ⫾ 0.097 0.95 ⫾ 0.095 0.95 ⫾ 0.095 0.97 ⫾ 0.097 0.96 ⫾ 0.096 0.91 ⫾ 0.091 0.97 ⫾ 0.097 0.97 ⫾ 0.097

5 ⫾ 0.1 5 ⫾ 0.1 10 ⫾ 0.1 ⫺5 ⫾ 0.1 0 ⫾ 0.1 5 ⫾ 0.1 5 ⫾ 0.1 5 ⫾ 0.1

45 ⫾ 0.1 45 ⫾ 0.1 45 ⫾ 0.1 35 ⫾ 0.1 40 ⫾ 0.1 45 ⫾ 0.1 40 ⫾ 0.1 40 ⫾ 0.1

9 ⫾ 0.1 25 ⫾ 0.1 25 ⫾ 0.1 25 ⫾ 0.1 25 ⫾ 0.1 9 ⫾ 0.1 25 ⫾ 0.1 25 ⫾ 0.1

* Std dev estimates from Betts and Ball (1997); estimates otherwise.

vegetation fraction typically amounts to 10% of the vegetation-fraction value. In NWP models, the distinction is made between plant functional types (e.g., broadleaf forest) rather than between different plants within a biome (e.g., willow, birch, maple, oak, etc.). Thus, no standard deviations exist for the parameters of the “lumped” vegetation classes. Therefore, standard deviations are arbitrarily assumed to be 10% of the respective empirical parameter value except for albedo. Based on the average standard deviation found by Betts and Ball (1997), standard deviation of albedo is set to be 7.8% of the respective mean albedo. Tables 2 and 3 summarize the standard deviations and mean values of empirical parameters used.

b. Uncertainty analysis According to Eq. (1) any set of mean parameters and their standard deviations will always provide the same standard deviation of a predicted flux for the same environmental conditions. Therefore, every surface flux parameterization (Table 1) and its equation for the standard deviation are applied for typical ranges of environmental forcing conditions. To investigate whether some soils, vegetation types, or vegetation fractions provide greater standard deviation than others, this analysis is performed for all soil and vegetation types listed in Tables 2 and 3 and for vegetation fractions in increments of 10%. The latter identifies whether fluxes are more uncertain in sparsely or densely vegetated areas, in spring/fall, summer, or winter, and in midlatitudes/Tropics or deserts. Ideally the terms (/i)i: ⫽{, i}, referred to as contribution (term) hereafter, are of the same order of

magnitude for a balanced (good) parameterization and empirical parameter set. Comparing the magnitude of {, i} identifies critical (unbalanced) parameterizations and the parameters that contribute the most to the standard deviations. 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 LSM. To demonstrate the meaning of the theoretical findings for NWP and to illustrate how parameter-induced uncertainty limits NWP, MM5 is run for 0000 UTC 20 July–1200 UTC 23 July 2001 with the GEP analysis tool. Since MM5 starts without clouds, it takes some time until clouds form in the model. Consequently, insolation is too high during spinup. Therefore, the discussion refers to results after spinup. Note that spinup causes systematic errors.

c. Synoptic situation During 0000 UTC 20 July–1200 UTC 23 July 2001 near-surface air (dewpoint) temperature ranges from ⫺1° to 22°C (⫺1° to 16°C) at low elevation. The beginning was characterized by moderate southern flow over south and southwest Alaska due to cyclonic activity within a trough above the North Pacific and Bering Sea. In the first 12 h, the long-wave disturbance slightly deepened. It propagated east, turning the southern flow on its frontal edge into the dominant feature over Alaska, while the ridge continued building up on the eastern boundary of the domain. In the middle and toward the end of the period, a weak gradient spacious trough influenced the synoptic situation. Cold advection behind the trough reinforced cyclonic activity in the Aleutians. On 23 July, the low dominated Alaska

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FIG. 1. Distribution of (a) vegetation types with vegetation fraction superimposed and (b) soil types with topography (m) superimposed. Names refer to locations mentioned in the text.

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and gradually cut off, whereas the anticyclone over western Canada strengthened. On the surface, the well-developed cyclone in the Bering Sea gradually dissipated and weakened the southern flow over Alaska throughout the first half of the period. Alaska remained on the outer edge of a well-developed eastward-moving surface cyclone located north of Siberia for the rest of the period. During the period, sun rises (sets) around 0300 (2100), 0400 (2330), and 0530 (0100) Alaska Standard Time (AST) in the eastern, central, and western part of the domain. The sun rises (sets) around 500 (2300) AST in the southern central part of the domain, while it barely goes below the horizon in the North. AST is UTC minus 9 h.

4. Results and discussion In addition to empirical soil and plant parameters, net radiation also depends on astronomical and atmospheric conditions that can cause additional uncertainty. Since this study focuses on soil- and plantparameter-induced stochastic errors, uncertainties from astronomic conditions, systematic errors, and aspects of parameter interaction are not addressed. Like HTSVS, most modern LSMs (e.g., BATS, CLM, OSULSM) calculate ground heat flux based on the soil heat diffusion equation. Like HTSVS, all LSMs applying the mixture approach (e.g., SiB, CLM) use similar parameterizations for net radiation, sensible, and latent heat fluxes. HTSVS’ equations will reduce to those applied in LSMs that assume a grid cell to be either totally vegetation covered or vegetation free if vegetation fraction is 1 or 0, respectively. Thus, for these LSMs the uncertainty behavior can be derived from the figures of the theoretical analysis as these two extremes are also shown. LSMs that use a skin temperature Tskin or distinguish between foliage Tf and ground temperature Tg (e.g., SiB, CLM) use similar formulations for upward longwave radiation as HTSVS. HTSVS’ foliage and ground temperature can be related to the skin temperature by Tskin ⫽ fTf ⫹ (1 ⫺ f)Tg where the vegetation fraction values f range between 0 and 1. If Tf is smaller (greater) than Tg and vegetation fraction exceeds 50%, maximum upward longwave radiation will be smaller (greater) than if the foliage and ground have the same temperature. In the following, ground, sensible, and latent heat fluxes are counted positive when directed toward the atmosphere.

a. Overview Standard deviations of net radiation are the smallest for net radiation between 200 and 400 W m⫺2 and in-

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crease quasi-linearly for higher and lower net radiation (Fig. 2). The standard deviations of sensible and latent heat flux quasi-linearly increase with the absolute value of the flux (Figs. 3 and 4). Standard deviation of ground heat flux also increases quasi-linearly with the absolute value of the flux over wide ranges of soil conditions. The increase is nonlinear when the ground is partly frozen (see secondary legs in Fig. 5). The steepness of the increase with increasing absolute value of the flux depends on vegetation and soil type for net radiation and latent heat flux, on vegetation type only for sensible heat flux and on soil type only for ground heat flux. In NWP, the behavior of standard deviations results in a diurnal cycle with a maximum around noon, and for fluxes that change their sign in the diurnal course, a secondary maximum at night (e.g., Fig. 6). Average relative errors of net radiation, sensible, latent, and ground heat fluxes are 7%, 10%, 6%, and 26%, respectively. A sensitivity study with a higher standard deviation for the vegetation fraction provides slightly increased standard deviations of the respective dependent fluxes. Since the standard deviation of vegetation fraction is related to the mean value of vegetation fraction (e.g., mean vegetation fraction of a grid cell) by f ⫽ 0.1f , usage of actual satellite-derived vegetation fraction will also reduce systematic errors in predicted fluxes. A sensitivity study performed with the standard deviations given by Clapp and Hornberger (1978) yields standard deviations in predicted fluxes similar to the studies with Cosby et al.’s (1984) values discussed in the following sections. A sensitivity study carried out for organic soils showed slightly higher standard deviations than for mineral soils. Note that there are no datasets on the distribution of organic soils at the resolution required by NWP models.

b. Net radiation Net radiation Q is the difference between incoming and outgoing irradiance. It depends on vegetation fraction, the emissivity of the foliage and ground, and the albedo of the foliage and ground, where ground albedo is a function of porosity (Table 1). Generally, ground albedo increases when the soil dries and decreases during precipitation (e.g., Charney 1975).

1) THEORETICAL

ANALYSIS

Standard deviations of net radiation Q are calculated for a range of volumetric water content from 0.001 to porosity, vegetation fraction from 0% to 100%, downward shortwave radiation from 0 to 750W m⫺2,

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FIG. 2. (a) Net radiation fluxes Q vs std dev Q as obtained by the theoretical analysis using the data given in section 4b(1). Results shown are for loam covered by mixed forest. Note that scatterplots for other soil and/or vegetation combinations are distinguished by the steepness of the relationship between the flux and its standard deviation. (b) Net radiation, its std dev, and relative error as obtained for typical ranges of atmospheric conditions. Example shown for shrubland on loam for ⫽ 0.211 m3 m⫺3, Rs ↓⫽ 360 W m⫺2, and Tf ⫽ Tg ⫺ 5 K. Horizontal distribution of (c) net radiation flux and (d) its std dev after 48 h of simulation. Note that simulation starts at 0000 UTC 20 Jul (1500 AST 19 Jul) 2001.

downward longwave radiation from 100 to 475 W m⫺2, and foliage and ground temperatures from ⫺30° to 40°C. For these environmental conditions, the standard deviation and relative error of net radiation are, on average, less than 13.67 W m⫺2 and 13.69% (sparse vegetation on sandy loam). The smallest standard deviation and relative error are 11.9 W m⫺2 (mixed forest on clay) and 1% (wooded and mixed tundra on loamy sand), respectively. Table 4 summarizes the relative errors for various soil and vegetation combinations. Standard deviations Q are the highest for great radiation deficits (corresponding to winter or Arctic con-

ditions), and the smallest for net radiation between 200 and 500 W m⫺2. They increase quasi-linearly for higher and lower net radiation (Fig. 2). They are minimal (⬍10 W m⫺2) for intermediate vegetation fraction and net radiation of 300–500 W m⫺2. They grow as vegetation fraction decreases/increases and incoming shortwave radiation increases/decreases. The position of the minimum varies slightly with soil type because of sensitivity to ground emissivity and albedo. Standard deviations are typically the lowest at low temperatures and relatively dry ( ⬍ 0.15 m3 m⫺3) or wet ( ⬎ 0.25 m3 m⫺3) soil conditions. If temperature

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FIG. 3. (a) Same as in Fig. 2a, but for sensible heat flux H vs std dev H using the data in section 4c(1) A friction velocity of 0.25 m s⫺1 is assumed. To show positive and negative fluxes, Tf ⫽ TR ⫾ 2 K, Tg ⫽ TR ⫿ 2 K. (b) Same as in Fig. 2b, but sensible heat flux. Example shown for mixed forest on loam for Tg ⫽ TR ⫹ 2 K, Tf ⫽ TR ⫹ 5 K, p ⫽ 1013.25 hP, u* ⫽ 0.5 m s⫺1. (c), (d) Same as in Figs. 2c,d, but for sensible heat flux and its std dev after 48 h of simulation.

exceeds a critical threshold value (which differs with soil type), Q becomes nearly independent of volumetric water content. The qualitative behavior described above differs only slightly with vegetation type. On average, the term related to ground emissivity, {Q, g}, dominates the standard deviation of net radiation independent of soil type. For most soil and vegetation combinations, the contribution of {Q, g} to Q is twice as high as the contribution related to foliage albedo, porosity, or vegetation fraction. On average, standard deviation of vegetation fraction and porosity contribute the least of all parameters to Q except for sandy loam and mixed forest. For

mixed forest, the contribution of {Q, f } (the term related to vegetation fraction) to Q is twice as high as {Q, f } is for the other vegetation types. The terms related to foliage albedo {Q, ␣f } and foliage emissivity {Q, f } only slightly differ in value except for tundra. Here, {Q, ␣f } is about 1.4 times higher and {Q, f } is slightly lower than for other vegetation types. For mixed forest, {Q, ␣f } contributes 0.3 times less than for all other vegetation types. Since vegetation fraction, emissivity, and albedo strongly change with season, using actual values derived from satellite data provides potential for improved NWP in general by reducing the stochastic and systematic errors.

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FIG. 4. (a) Same as in Fig. 2a, but for latent heat flux LE vs std dev LE. A friction velocity of 0.25 m s⫺1 is assumed. To show positive and negative fluxes, Tf ⫽ TR ⫾ 1 K, Tg ⫽ TR ⫿ 2 K, and qg ⫽ qR ⫿ 0.0005 kg/kg. (b) Same as in Fig. 2b, but for latent heat flux. Example shown for shrubland on clay loam with ⫽ 0.2325 m 3 m ⫺3 , T g ⫽ T R ⫺ 4 K, T f ⫽ T R ⫺ 6 K, q R ⫽ 0.005 kg/kg, qg ⫽ 0.01 kg/kg, p ⫽ 1013.25 hPa, u* ⫽ 0.3 m s⫺1, and Rs ↓⫽ 400 W m⫺2. (c), (d) Same as in Figs. 2c,d, but for latent heat flux and its std dev after 48 h of simulation.

2) NWP

EXAMPLE

The Yukon Territory and the ocean are cloud free most of the time. Except for the coastal areas Alaska is cloud covered most of the time. Net radiation locally exceeds 600 W m⫺2 (around noon) on the North Slope, in the Yukon Territory and along the west coast of Alaska, while it remains less than 200 W m⫺2 in the cloud-covered areas (e.g., Fig. 2). Due to the behavior described above, standard deviation of net radiation is greater in cloud-free than cloudy areas, increases toward noon, and decreases toward midnight (e.g., Fig. 6), and its spatial distribution shows no

obvious correlation to soil or vegetation types (e.g., Figs. 1 and 2).

c. Sensible heat flux Sensible heat flux depends on vegetation fraction and roughness length via the turbulent resistances (Table 1).

1) THEORETICAL

ANALYSIS

Sensible heat fluxes H and their standard deviations H are determined by soil, air, and foliage temperatures

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FIG. 5. (a) Same as in Fig. 2a, but for ground heat flux Hs vs std dev. (b) Same as in Fig. 2b, but for ground heat flux. Example shown for sandy loam with (b) (g ⫺ ⫺1)/z⫺1 ⫽ 0.01 m3 m⫺3/0.05 m) and (Tg ⫺ T⫺1)/z⫺1 ⫽ 1 K/0.05 m, and (c) ⫽ 0.3255 m3 m⫺3 and (g ⫺ ⫺1)/z⫺1) ⫽ 0.01 m3 m⫺3/0.05 m). Note that plots for (g ⫺ ⫺1)/z⫺1) ⫽ 0 m3 m⫺3/m (as used in many LSMs) look similar. (d) Same as in Fig. 2c, but for ground heat flux and (e) its std dev after 48 h of simulation.

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2) NWP

FIG. 6. Temporal evolution since 0000 UTC 20 Jul (1500 AST 19 Jul) 2001 of net radiation, sensible heat flux, latent heat flux, and ground heat flux as well as their std devs, shown as averages for the entire land surface in the model domain.

between ⫺30° and 40°C, and friction velocity between 0 and 1 m s⫺1. Except for sparse vegetation (22.2%), the relative error of sensible heat flux typically remains less than 9.2% (grassland), 11.9% (shrubland), 5.2% (deciduous forest), 5.3% (coniferous forest), 5.2% (mixed forest), 9.1% (wooded tundra), and 7.8% (mixed tundra). The absolute value of sensible heat flux grows as wind speed increases, as does the standard deviation of sensible heat flux. The only marginal increase of standard deviation with increasing temperature at low wind speed becomes a slightly stronger increase for high wind speed as the flux grows. For most conditions, H exhibits a slight, quasi-linear increase with increasing absolute value of the sensible heat flux (Fig. 3). Standard deviations of sensible heat flux typically increase with increasing vegetation fraction. This increase is stronger at high than at low temperatures (e.g., Fig. 3). The general behavior described above changes marginally with vegetation types via roughness length (therefore not shown).

EXAMPLE

During the day, sensible heat fluxes are downward (up to ⫺180 W m⫺2) nearly everywhere in the mountains (e.g., Fig. 3). In the interior of Alaska and along the lower Yukon River, sensible heat fluxes are upward (up to 50 W m⫺2). At night, sensible heat fluxes are negative except over water. In mountainous areas, they are as low as ⫺208 W m⫺2, while they seldom exceed ⫺5 W m⫺2 at low elevation. Areas of high vegetation fraction are apparent in the horizontal distribution of standard deviation as higher values than those obtained for low vegetation fraction (e.g., Fig. 3). The lowest overall standard deviation of sensible heat flux exists at high elevation with low vegetation fraction. At this time of the year, the atmosphere cools at night and heats up during the day. Therefore, standard deviation of sensible heat flux shows a noon maximum (e.g., Fig. 6), locally often a secondary minor maximum at midnight, and minima around sunrise/sunset.

d. Latent heat flux Latent heat flux depends on vegetation fraction f , roughness length z0, and bulk-stomatal resistance rst (Table 1). The latter depends on photosynthetically active radiation (PAR), specific humidity deficit between leaf and ambient air, leaf temperature, and soil moisture deficit (see the appendix). These dependencies are considered by the so-called empirical correction functions that range between 0 and 1 (e.g., Jarvis 1976; Sellers et al. 1986; Hicks et al. 1987; Dingman 1994). Empirical parameters used in these correction functions are minimum stomatal resistance rst,min, the temperatures at which stomata close, Tmin and Tmax, the temperature Topt at which rst reaches its minimum, the albedo ␣f and emissivity f of foliage, a parameter bst used to determine PAR, field capacity fc, permanent wilting point pwp, and root length zroot (Table 1).

1) THEORETICAL TABLE 4. Relative error of net radiation (in % absolute) for various soil- and vegetation-type combinations.

Grassland Shrubland Deciduous forest Needleleaf forest Mixed forest Sparse vegetation Wooded tundra Mixed tundra

Loamy sand

Sandy loam

Loam

Clay loam

Clay

9.26 12.57 4.71 10.69 2.75 13.60 0.95 0.95

9.45 12.70 5.05 10.96 2.80 13.69 1.11 1.11

9.24 12.56 5.08 10.94 2.76 13.58 1.11 1.11

8.98 12.36 7.76 11.05 2.73 13.67 1.21 1.21

8.85 12.24 11.01 11.01 2.70 13.58 1.19 1.19

ANALYSIS

Latent heat fluxes LE and their standard deviations LE are determined by soil, air, and foliage temperatures between ⫺20° and 30°C, specific humidity of the air and at the ground between 0.00045 and 0.005 kg kg⫺1, friction velocity between 0 and 1 m s⫺1, shortwave radiation between 0 and 750 W m⫺2, and volumetric water content between permanent wilting point and porosity. For the same soil type, LE strongly differs for high and low vegetation. On average, the relative error is less for high than for low vegetation. It is the highest for wooded tundra (up to 25.27% on average; Table 5).

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TABLE 5. Same as in Table 4, but for latent heat flux.

Grassland Shrubland Deciduous forest Needleleaf forest Mixed forest Sparse vegetation Wooded tundra Mixed tundra

Loamy sand

Sandy loam

Loam

Clay loam

Clay

1.31 12.22 5.19 5.62 2.70 0.02 7.15 8.21

4.69 6.74 6.16 8.53 3.74 0.02 11.07 7.38

0.75 3.14 6.01 7.34 4.97 0.02 4.51 2.20

7.31 5.15 5.60 6.90 4.17 0.02 6.80 0.46

19.04 4.38 6.39 6.89 5.96 0.02 25.27 4.36

This means that distinguishing more types of tundra may yield improvement of latent heat flux prediction in the Arctic. Generally, LE increases nonlinearly with vegetation fraction. This increase is usually (slightly) greater under warmer than cooler conditions (Fig. 4). Superimposed is a nonlinear increase of standard deviation as wind speed (and hence latent heat flux) increases. Overall, LE increases quasi-linearly with the absolute value of the latent heat flux. The increase in LE with increasing vegetation fraction, among other things, results from the higher weight given to the standard deviation of empirical parameters when calculating stomatal resistance as vegetation cover extends. This strong impact of vegetation fraction on the standard deviation of latent heat flux demands the incorporation of the actual satellite-derived vegetation fraction in NWP. Despite the strong difference between standard deviations of permanent wilting point for the various soil types, standard deviation of latent heat flux differs only slightly with soil type. Relative error, however, strongly differs with soil type (Table 5). The same is true for field capacity. On average, the contribution of stomatal resistance {LE, rst} and vegetation fraction {LE, f } are of a similar order of magnitude. Generally, {LE, rst} contributes more than {LE, f } for all kinds of forest. The opposite is true for the other vegetation types. The contributions vary slightly with soil type for constant vegetation type. This means that increasing the accuracy of the parameters on which stomatal resistance depends provides the potential to reduce the standard deviation in stomatal resistance rst and finally in latent heat flux. To identify which parameters provide the greatest potential for reducing standard deviation of stomatal resistance, standard deviation of calculated stomatal resistance is examined. Standard deviation of stomatal resistance rst amounts to 15% on average. It may be much greater especially as the correction functions [Eqs. (A7)–(A10)] provide values close to 0 or 1. If air

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temperature, for instance, approaches the minimum or maximum temperature at which stomata close, the standard deviation of calculated stomatal resistance will increase. On average, the parameters used in the correction functions for PAR, ambient temperature and moisture contribute the least to the standard deviation in stomatal resistance. Contributions related to minimum stomatal resistance rst,min and LAI dominate the standard deviation of stomatal resistance for all vegetation types. Unfortunately, measurements show that these quantities vary by an order of magnitude even for the same plant type (e.g., Körner et al. 1979; Avissar 1991). Thus, standard deviation in minimum stomatal resistance limits the prediction of latent heat flux over vegetated areas. For shrubland the terms related to permanent wilting point {rst, pwp} and field capacity {rst, fc} contribute to the standard deviation of stomatal resistance rst by the same order of magnitude (but usually with lower values) as those of LAI and rst,min. The terms {rst, pwp} and {rst, fc} gain similar importance to rst,min and LAI for clay and clay loam. This means that further distinction of clay-type soils provides potential for reducing the standard deviations in stomatal resistance and hence latent heat flux. Also, model improvement may be gained by finding/developing a correction function for soil water deficit that uses parameters that cause a lower standard deviation in calculated stomatal resistance.

2) NWP

EXAMPLE

During the day, latent heat fluxes are positive except for some coastal and mountainous areas and the ocean (Fig. 4). In cloudy areas, latent heat fluxes typically range between 10 and 120 W m⫺2, while they reach up to 360 W m⫺2 in cloud-free areas. The long daylight hours in combination with the great water availability results in high latent heat fluxes in the north. Latent heat fluxes are close to zero or negative in the southern model domain when the sun gets below the horizon. At that time, latent heat fluxes still reach 70 W m⫺2 on the North Slope and in the northern Brooks Range, and 40 W m⫺2 in the Interior. As expected from the theoretical analysis, standard deviation of latent heat flux is usually greater in cloudfree (e.g., Yukon Territory, western Alaska in Fig. 4) than cloudy areas (e.g., central Alaska in Fig. 4). After precipitation, LE increases slightly as LE grows in response to the wet soil. Thus, as expected form the theoretical analysis, areas of high precipitation occasionally become visible in the distribution of standard deviation (not shown).

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Mixed forest is easily visible in the LE distribution by its lower standard deviation, and shrubland and wooded tundra by their higher standard deviations than adjacent vegetation (e.g., Fig. 4). This confirms the potential for improvement in Arctic NWP by finer distinctions for tundra, especially because of the widespread presence of tundra in the Arctic (Fig. 1). The standard deviation follows a diurnal cycle with a primary peak around noon (e.g., Fig. 6) and locally a secondary weaker peak at night. South of the North Slope, stomata already close at night, leading to lower standard deviations in the southern than northern part of the domain.

e. Ground heat flux Calculation of ground heat flux involves pore-size distribution index, porosity, saturated water potential, and thermal conductivity (Table 1), where the latter is a function of the former parameters (e.g., McCumber and Pielke 1981; Kramm et al. 1996).

1) THEORETICAL

ANALYSIS

Ground heat fluxes Hs and their standard deviations Hs are determined by soil temperatures and volumetric water content from ⫺20° to 30°C and 0.001 to porosity, respectively. The gradients of soil temperature and total volumetric water content vary from ⫺187.5 to 187.5 K m⫺1, and ⫺187.5 to 187.5 m3 m⫺3 m⫺1, respectively. Standard deviation in ground heat flux increases slightly with increasing absolute value of ground heat flux. Generally, errors will be greater for great than small absolute values of temperature gradients except if the flux is close to zero (Fig. 5). Overall, the averaged relative error remains below 26%. It is the lowest for loamy sand (about 14%), followed by sandy loam (16%), loam (25%), clay loam (35%), and clay (40%), but for some conditions of soil temperature and soil temperature gradient, individual relative errors may exceed 50% (Fig. 5). This means that the standard deviation of ground heat flux may limit agricultural applications. The impact for NWP usually will not be dramatic as ground heat fluxes are typically notably smaller than the net radiation, latent, and sensible heat fluxes. Standard deviation of ground heat flux is independent of soil temperature for soil temperatures above 0°C (e.g., Fig. 5). At soil temperatures below 0°C it exhibits a slight nonlinear decrease with decreasing soil temperature for negative and positive temperature gradients (Tg ⫺ T⫺1)/z⫺1. The steepness of the decrease (increase) depends on soil type. Relative errors are maximal where typically most of the soil water freezes (Fig. 5). This temperature differs with soil type. The

FIG. 7. Thermal conductivity and its std dev . Example shown for sandy loam with (g ⫺ ⫺1)/z⫺1 ⫽ ⫺0.01 m3 m⫺3/0.05 m and (Tg ⫺ T⫺1)/z⫺1 ⫽ ⫺1 K/0.05 m. Note that plots for (g ⫺ ⫺1)/z⫺1 ⫽ 0 m3 m⫺3/m (as used in many LSMs) look similar.

absolute value of ground heat flux and its relative error increase with decreasing relative volumetric water content. This means that ground heat flux is more reliable after precipitation or in the Tropics than during drought or in deserts. All soil types show similar qualitative behavior. Since most modern LSMs use the soil heat diffusion equation for ground heat flux, the contribution terms are analyzed. Excluding the cross effects [i.e., wLD(g ⫺ ⫺1)/z⫺1 ⫽ 0 as is done in most other LSMs] shows that this term does not cause the great standard deviation of ground heat flux. Instead, the contribution of the standard deviation of thermal conductivity is often more than an order of magnitude greater than the contribution of all other terms. The parameterization of thermal conductivity has already been pointed out as problematic by P. Viterbo (2003, personal communication). As discussed above, thermal conductivity depends on pore size distribution index, saturated water potential and porosity, and shows a nonlinear behavior (Fig. 7). Below 0°C, thermal conductivity decreases less strongly with decreasing volumetric water content than above 0°C, and it decreases when soil temperature decreases. This decrease is stronger for high than low relative volumetric water content. Thermal conductivity becomes independent of temperature for soils warmer than 0°C. Thermal conductivity is the highest in loamy sand at high relative volumetric water content, while it is the lowest in loam. Standard deviation of thermal conductivity is the greatest for wet hot soils (e.g., the

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Tropics), and the lowest for dry frozen ground (e.g., midlatitudes in winter). Note that permafrost soils typically fall in the saturated cold range (Fig. 7). All soils show similar general behavior. A parameterization for thermal conductivity that causes less uncertainty has to be identified and implemented in the future. Despite the huge standard deviation of saturated water potential (relative error up to 149%: cf. Table 2), standard deviations of porosity and pore size distribution contribute more strongly to the standard deviation of ground heat flux than does that of saturated water potential. Consequently, increasing the accuracy of porosity and pore size distribution provides more potential for improving the lower boundary condition in NWP models than increasing the accuracy of the parameters for saturated water potential.

2) NWP

EXAMPLE

Generally, ground heat flux is directed toward the atmosphere in the interior of Alaska, lower Yukon River, Coastal Mountains, most of the Yukon Territory, and the western North Slope during the night, and toward the soil everywhere during the day (e.g., Fig. 5). Daytime soil heating (locally less than ⫺100 W m⫺2) exceeds the nighttime cooling (locally about 25 W m⫺2). Standard deviations are less for cloudy than cloudfree areas as cloudiness and cloud thickness reduces soil heating/cooling. At high elevation, local soil freezing/ thawing causes great standard deviations in simulated ground heat flux. Standard deviation grows as soils dry, leading to the high values in the Yukon Territory (e.g., Fig. 5). Since ground heat flux changes direction in the diurnal course, the diurnal courses of standard deviations show maxima at noon (e.g., Fig. 6) on average, and locally also at night. No spatial relationship between the uncertainty in Hs and soil type is obvious.

5. Conclusions All modern NWP models use parameterizations for the land surface processes that rely on empirical plant and soil parameters to represent different vegetation and soil types. Usually, mean values of the vegetation or soil characteristic derived from laboratory or field studies are assigned, ignoring their standard deviations. The standard deviations of some parameters, however, can be even greater than the mean value itself. Thus, every predicted flux will be in “error” by some amount (known as the standard deviation) as a consequence of the dependent parameters’ standard deviations. The aims of this study are to 1) determine the stochastic

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errors (standard deviations) of predicted surface fluxes that are caused by the standard deviations of empirical parameters, 2) identify critical parameters and (parts of) parameterizations, and 3) demonstrate how stochastic errors may limit Arctic NWP. GEP principles (e.g., Kreyszig 1970; Meyer 1975) are introduced to calculate the standard deviations of the simulated fluxes. On average, the relative errors in net radiation, sensible, latent, and ground heat fluxes amount to about 7%, 10%, 6%, and 26%, respectively. This means that ground heat flux prediction currently limits the accuracy with which the lower boundary condition can be determined in NWP. However, typically ground heat flux is much smaller than net radiation, sensible, and latent heat fluxes and can therefore be considered of less importance for NWP as compared to the other fluxes. The statistical uncertainty in predicted net radiation, sensible, and latent heat fluxes is less than the measurement errors typical for these fluxes (e.g., Panin et al. 1998; Willson et al. 2002). Field experiments show imbalances of about 100–250 W m⫺2 or 20%–50% of net radiation due to conceptual deficiency (i.e., systematic errors; Panin et al. 1998). Therefore, if the parameters are chosen reasonably and the parameterizations capture the governing physical processes, measured fluxes alone cannot serve to tell us which parameters/parameterizations provide better results in predicted fluxes. For the same environmental conditions, the standard deviation and relative error of net radiation are, on average, less than 13.67 W m⫺2 and 13.69% (sparse vegetation on sandy loam). The smallest standard deviation and relative error are 11.9 W m⫺2 (mixed forest on clay) and 1% (wooded and mixed tundra on loamy sand), respectively. The parameters of wooded and mixed tundra that are required to calculate stomatal resistance, however, cause the highest standard deviations in latent heat flux. Note that net radiation and latent heat fluxes do not depend on exactly the same parameters (Table 1). Ground heat fluxes have their highest relative errors for clay loam (35%) and clay (40%), on average. Introducing subclasses of clay-type soils and tundra (to obtain less uncertain parameters) may improve the accuracy at the lower boundary in NWP models in regions with large coverage by such soils and vegetation. Standard deviation of predicted net radiation is greater for low and high fractions than for intermediate vegetation fractions. Standard deviation of predicted latent heat flux increases with increasing vegetation fraction, among other things, because uncertain empirical parameters used to calculate stomatal resistance have a higher fractional weight as vegetation cover ex-

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tends. Standard deviation of sensible heat flux also increases as vegetation fractions increase. Since vegetation fraction plays a large role in the statistical uncertainty of all vegetation-fraction-dependent fluxes, and since the standard deviation of the vegetation fraction also depends on the mean vegetation fraction, utilizing actual satellite-derived vegetation fraction has great potential for improving NWP, in general. The analysis identified the parameterization of thermal conductivity as the dominant influence on the standard deviation of ground heat flux. Thus, this parameterization is to be replaced in the future by one that will cause less uncertainty. These findings also demonstrate that GEP can be used for identification of critical (parts of) parameterizations. The analysis by GEP also showed that the empirical parameters with the highest relative error are not necessarily the greatest contributors to the standard deviation of the predicted flux. Saturated water potential, for instance, has a high relative error, but contributes less to the standard deviation of ground heat flux than does porosity or the pore size distribution index. The same relative error can even propagate to different standard deviations. Despite ground emissivity relative error amounts of 10%, which are similar to relative error amounts of the foliage emissivity or vegetation fraction, emissivity of the ground usually dominates the standard deviation of net radiation. Contributions related to minimum stomatal resistance and LAI govern the standard deviation of stomatal resistance despite the fact that the relative error of LAI and minimum stomatal resistance is 10%, similar to the relative error values of other empirical parameters that stomatal resistance depends on (Table 3). Based on these findings, focus should be on obtaining parameter sets with reduced standard deviation for ground emissivity, LAI, and minimum stomatal resistance, and on further specification of tundra and clay types. Reducing the standard deviation of stomatal resistance, and, hence, latent heat flux requires considering more plant and soil types. A high priority should be placed on compiling such diversified parameter sets in the future. The findings also elucidate the usefulness of GEP for systematically prioritizing which parameter to measure first. Since all modern LSMs used in NWP models require the same kind of empirical parameters as those used in HTSVS, similar standard deviations in predicted fluxes are expected for other LSMs using similar parameterizations as HTSVS. The standard deviations found for the predicted fluxes have to be considered as the current limit of predictability related to the statistical un-

certainty of empirical parameters. Additional limitations exist due to systematic errors (e.g., from initialization or parameterizations) that have already been discussed elsewhere (e.g., Toth and Kalnay 1993; Henderson-Sellers et al. 1995; Shao and Henderson-Sellers 1996; Chen et al. 1997; Schlosser et al. 2000; Slater et al. 2001; Toth et al. 2001; Luo et al. 2003). Investigations of different LSMs by GEP may identify physical parameterizations that allow surface fluxes to be predicted with low uncertainty even while using the same uncertain empirical parameters. Model improvement needs the combined efforts of better parameter sets at higher resolution and more physical parameterizations. Results of GEP analysis could form a basis for intercomparison of LSMs, and a guide for parameter determination aimed at improving LSMs and NWP. Acknowledgments. I thank G. Kramm, A. Rulo, M. Jankov, and the anonymous reviewers for their helpful discussions and fruitful comments, and C. O’Connor for her careful editing. BMBF and NSF financially supported this study under Contracts 07ATF30, OPP0327664, and ATM0232198.

APPENDIX Parameter-Dependent Functions Relevant for Latent Heat Flux Calculation Potential temperature ⌰␦ and specific humidity q␦, both at height ␦ close above the foliage, and specific humidity of the foliage qf are given by (Mölders et al. 2003) ⌰␦ ⫽

冋

册

rmt,Hrt ⌰R 1 ⫺ f f T ⫹ T ⫹ , rmt,H ⫹ rt rmt,g g rmt,f f rt

rmt,Ert q␦ ⫽ rmt,E ⫹ rt

⫹

冉

f 1 ⫺ f rmt,fg ⫹ qg 1 1 rmt,g 1 ⫹ ⫹ rmt, f rst rmt,fg rmt, f

冦冤

冉

f rst

冊

1 1 1 ⫹ ⫹ r rst rmt,fg rmt, f mt, f

and qf ⫽

共A1兲

1 1

1 1 ⫹ ⫹ rst rmt,fg rmt, f

冋

冊

qi ⫹

qR rt

冧

冥

共A2兲

,

册

qg q␦ qi ⫹ ⫹ , rst rmt,fg rmt, f

共A3兲

where Tf , Tg, and qg are the surface temperature of the foliage and ground and the specific water vapor at the

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ground. Furthermore, rmt, f , rmt,g, and rmt, fg are the resistances of the molecular-turbulent layer close to the surfaces of the foliage and the ground, and in the environment between the foliage and ground surface, which inhibit the transfer of heat and matter (e.g., Kramm et al. 1996). Here ⌰R and qR denote the potential temperature and specific humidity at reference height. Here qi ⫽ qi(Tf) is the specific humidity within the stomatal cavities, assumed to be at saturation. The quantities rmt,H and rmt,E are defined by (Kramm 1995) 1 ⫺ f f ⫽ ⫹ , rmt,H rmt,g rmt, f 1

g␦ ⫽

再

共A4兲

and 1 ⫺ f ⫽ ⫹ rmt,E rmt,g 1

冉

1 1 ⫹ rst rmt,fg f . 1 1 1 ⫹ ⫹ r rst rmt,fg rmt, f mt, f

rst ⫽

rst,min , LAIg␦ gPARgT g

a关qi共Tf兲 ⫺ q␦兴 ⬎ 0.01152 kg m⫺3

冊 再

⫺1

,

共A7兲 共A8兲

,

Tf ⫺ Tmin Tmax ⫺ Tf gT ⫽ Topt ⫺ Tmin Tmax ⫺ Topt

共A6兲

where the correction functions

0.233 bst PAR

冊

共A5兲

Bulk-stomatal resistance is given by (e.g., Sellers et al. 1986; Hicks et al. 1987; Dingman 1994)

a关qi共Tf兲 ⫺ q␦兴 ⱕ 0.01152 kg m⫺3

冉

冊

冉

1 ⫺ 66.6a关qi共Tf兲 ⫺ q␦兴

gPAR ⫽ 1 ⫹

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冎

Tmax⫺Topt Topt⫺Tmin

,

共A9兲

and n

g ⫽

共j ⫺ pwp兲⌬zj fc ⫺ pwp 兲zroot

兺 共 j⫽1

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Verseghy, D. L., 1991: CLASS—A Canadian land surface scheme for GCMs. 1. Soil model. Int. J. Climatol., 11, 111–133. Willson, K., and Coauthors, 2002: Energy balance closure of FLUXNET sites. Agric. For. Meteor., 113, 223–243. Wilson, M. F., A. Henderson-Sellers, R. E. Dickinson, and P. J. Kennedy, 1987: Sensitivity of the biosphere–atmosphere transfer scheme (BATS) to the inclusion of variable soil characteristics. J. Climate Appl. Meteor., 26, 341–362.