This paper discusses a series of sensitivity experiments aimed at testing the surface heterogeneity representation proposed in the companion paper by Giorgi. When driven by observed climatic forcings at three locations and run in point mode, the model shows good performance in reproducing observed surface fluxes. The temperature heterogeneity representation mostly affects the process of snow formation and, therefore, the winter and spring energy and water budgets. The soil water heterogeneity primarily influences the processes of soil water movement and runoff generation, thereby modifying the surface hydrologic budget. In addition, the heterogeneous model results compare reasonably well with aggregated results from point-mode experiments. Model sensitivity to the presence of impermeable surface fractional cover, fractional precipitation area, and a crude delayed runoff formulation is also discussed. As a next phase of model evaluation, it is planned to include this heterogeneity representation within a regional climate model and assess its effect on atmospheric circulations.
In the companion paper by Giorgi (1997; hereafter referred to as G97), the theoretical framework of a heterogeneous land surface representation has been presented and implemented within a complex land surface scheme. The approach makes use of aspects of the mosaic (Avissar and Pielke 1989; Koster and Suarez 1992) and statistical–dynamical (Avissar 1992; Sivapalan and Woods 1995) methodologies. Currently, a stand-alone version of the scheme has been developed. Before proceeding to its inclusion within a global or regional climate model, a series of validation and sensitivity experiments have been completed to test its performance. Discussion of the results from such experiments is presented in this paper.
In the first portion of the paper (section 3), the model of G97 is driven by observed meteorological fields at three different locations, and the fluxes produced by the model are compared with available observations. The observed data were taken at Cabauw, the Netherlands, (Beljaars and Viterbo 1994), at a location in southwestern France, as part of the HAPEX–MOBILHY field experiment (Andre et al. 1986; Goutourbe and Tarrieu 1991); and in central Amazonia, as part of the Amazonian Rainforest Meteorological Experiment (ARME; Shuttleworth et al. 1984a,b; Shuttleworth 1988). In these experiments, the model is run in point mode, that is, without inclusion of heterogeneity effects, and the purpose of this preliminary set of experiments is to evaluate the physics processes included in the base land surface scheme used by G97.
In the second set of experiments (sections4–6), the model is driven by the climatology of Seth et al. (1994; hereafter referred to as SGD). This climatology was derived from observed monthly climate data for the southeastern United States. Results from point-mode runs are first compared with those of SGD, who used the Biosphere–Atmosphere Transfer Scheme (or BATS, Dickinson et al. 1993). The sensitivity of the proposed subgrid-scale heterogeneity formulation to relevant parameters is then assessed. In addition, as a limited validation analysis of the heterogeneity representation, results from heterogeneous runs are compared with those from aggregated point-mode experiments.
In the next section, a brief summary of the main characteristics of the model is first given. A detailed description of it, as well as the definition of all quantities used in this paper, can be found in G97; therefore it is not repeated here.
2. Summary description of a surface process scheme including heterogeneity representation
The scheme used here incorporates five model subcomponents.
A vertical soil-layer model that calculates soil temperature, liquid water content relative to saturation, ice water content relative to saturation, and fraction of frozen soil. The soil model essentially solves the vertical thermal diffusion equation and an equation for liquid water diffusion and gravitational drainage. Liquid and frozen water do not coexist, that is, they occupy the unfrozen and frozen portions of a soil layer, respectively.
A one-layer vegetation model that calculates the temperature of canopy foliage, canopy air, and ground surface and the water vapor mixing ratio of canopy air. These quantities are computed through a coupled set of four equations of energy and water budget for the canopy foliage, canopy air, and the ground skin surface. Water interception and reevaporation by foliage, as well as transpiration from stomatal pores, is included.
A vertical snow model that employs an adaptive grid to calculate snow temperatures by solving the vertical heat diffusion equation. Snow fractional cover and total depth are explicitly calculated from snow fall, snowmelt, and sublimation.
An impermeable (urban) surface model that calculates skin and deep soil temperatures using a force–restore formulation for a nonevaporating surface.
A surface hydrology model that calculates surface water from surface runoff generation and exchange between surface and soil water (delayed runoff).
At a given grid box, the fraction of vegetated, bare soil, snow-covered, and urban surface are specified or calculated, and the gridbox-averaged fluxes are calculated by assuming that each subgrid fraction separately exchanges radiant energy, sensible heat, and water vapor with the atmosphere. Precipitation can be assumed to occur only on a fraction of the grid box and fractional areas of rainfall and snowfall are calculated.
Surface “intrapatch” heterogeneity is described by assuming that the soil water content relative to saturation and the temperature of soil, vegetation, ground skin surface, canopy air, near-surface atmosphere, and snow are distributed according to analytical probability density functions (PDFs). The PDFs are linear and symmetrical around the gridpoint average values [Eqs. (11a) and (11b) and Fig. 2 in G97], and nonlinear terms appearing in model equations that involve soil water and temperature are analytically integrated over the corresponding PDFs. This is accomplished by applying heterogeneity operators of Eqs. (8) and (15) in G97. The two independent parameters defining the PDF are the half-width αpdf and the height ratio (ratio of the lowest to the highest value of the distribution) γpdf.
3. Model validation against observed datasets
As a first validation step, three observed datasets were used for testing of the model physics in its basic, point-mode configuration (i.e., without heterogeneous representation). For each dataset, the forcing meteorological fields consist of incoming solar and longwave radiation, precipitation, surface air pressure and air temperature, wind speed, and water vapor mixing ratio at a level near the surface. The data are provided at intervals of 30 min for a full calendar year. All simulations start on 1 January and end on 31 December. Some relevant parameter values used for the base experiments in each of the three cases, most of which were suggested by the originators of the datasets (P. Viterbo 1994, personal communication; J. F. Mahfouf 1994, personal communication), are summarized in Table 1. Forcing solar radiative flux and precipitation are presented in Figs. 1a,b.
Since the observations are point measurements, for all of the validation runs presented in this section the half width of the temperature and soil water PDFs was kept small (0.01) so that the scheme effectively behaved as a point model. Also, in all experiments the soil model utilized six layers of thickness 0.05, 0.10, 0.25, 0.60, 1.5, 3.0 m from the surface to the bottom of a 5.5-m total model soil region and the snow model utilized three adaptive vertical layers of 5-cm minimum depth (see G97). Both these configurations are considered baseline in LSX (Pollard and Thompson 1995), the scheme from which the present soil and snow components are derived. Some sensitivity experiments to the number of soil layers is presented in section 3b.
a. Cabauw experiment
The Cabauw data are described in detail by Beljaars and Viterbo (1994). The data were collected at the 200-m meteorological tower in Cabauw, the Netherlands, over flat terrain consisting mainly of grassland with no relevant obstacles up to a distance of 200 m from the tower. The soil consists of a 1-m-deep layer of clay overlying a 10-m-deep layer of peat that is artificially kept at saturation. The forcing meteorological data for the model are taken at a height of 20 m from the surface and the dataset spans the whole year 1987. Figures 1a,b show the averaged monthly solar radiation and precipitation forcings. Solar radiation goes from a winter minimum of approximately 20 W m−2 to a summer maximum of 210 W m−2, while the precipitation forcing does not show a pronounced seasonal cycle.
Data for verification of surface net radiation, sensible heat flux, latent heat flux, and skin temperature (which is hereafter obtained from the upward longwave radiation flux) have been assembled as described by Beljaars and Viterbo (1994) and Viterbo and Beljaars (1995) for the same period. In the base run, initial soil water content is at its saturation value and zero permeability (bperm) is assumed as bottom boundary condition. This results in the bottom two model layers being always close to saturation and therefore mimics the saturated deep peat layer present at Cabauw.
Figures 2a–d compare model and observed monthly averaged net radiation, surface sensible and latent heat fluxes, and skin temperature. The observed seasonal and intermonth variations are well reproduced by the model, and differences between model and observed monthly fluxes and temperatures are mostly less than 5–10 W m−2 and 1–2 K, respectively. The corresponding yearly averages are reported in Table 2, which shows a close agreement between model results and observations. Also shown inTable 2 is the standard deviation of the daily values, which is a measure of daily variability. As illustrative examples, observed and simulated daily values of surface sensible and latent heat flux are shown in Figs. 3a,b. The correlation coefficients between observed and simulated daily values for surface fluxes and temperatures were in the range 0.83–0.99. Therefore, the model in its base configuration was able to reproduce the observed surface fluxes and skin temperature on the daily, monthly, and seasonal timescales.
As an example of the model performance in simulating diurnal cycles of temperature and surface fluxes, Figs. 4a–d show model and observed half-hourly data for the month of July. The half-hourly values were obtained by averaging the data at a given time of the day over all the days of the month. The observed diurnal cycles of skin temperature and net radiation are well reproduced by the model, although during nighttime the model net radiative fluxes are greater in magnitude than observed. The peak daytime sensible heat fluxes are somewhat larger, and the peak latent heat fluxes lower, in the model than in the observations, but overall the observed diurnal cycle of the variables in Fig. 4 is reproduced by the model. Similar results were found for other months when differences between nighttime and peak daytime observed and simulated fluxes were a few to a few tens of watts per square meter.
In summary, the present scheme in its base configuration reproduced the characteristics of observed fluxes and skin temperature for this experiment. The following experiments were carried out to test the sensitivity of the results to relevant parameters. 1) The minimum stomatal resistance rs, min was set at 200 s m−1, that is, the value used in BATS for grassland. 2) The leaf area index LAI was doubled compared to its base values. 3) The vegetation roughness length z0, veg was increased from 0.15 to 0.4 m (value used by Viterbo and Beljaars 1995). 4) The soil texture type itext was changed from 10 (mostly clay) to 3 (mostly sand) in the BATS scale (see G97). 5) The bottom soil permeability was changed from 0 to 1 (i.e., free drainage). 6) The initial soil water content wl, in was changed from saturated to one-half of the saturation value.
Table 2 reports the yearly averages and standard deviations for fluxes and skin temperature in the Cabauw sensitivity experiments. Of all parameters tested, the model is most sensitive to the minimum stomatal resistance. In particular, the minimum resistance used in BATS produces too low evaporation and too high sensible heat flux compared to observations. Doubling of the LAI induces an average increase in latent heat flux of about 7 W m−2, with a corresponding decrease in sensible heat flux. The latent heat flux sensitivity to the increase in roughness length tested is only about 3 W m−2. The model did not show great sensitivity to soil texture and bottom permeability, while halving the initial soil water content produced a decrease in latent heat flux of about 3 W m−2. Overall, the sensitivity of yearly averaged net radiation and surface skin temperature was in the range of a few watts per square meter and less than 1 K, respectively.
b. HAPEX experiment
The HAPEX data are at the site of Caumont (SAMER station number 3) of HAPEX–MOBILHY (Andre et al. 1986; Goutourbe 1991) for the entire year 1986. The vegetation at the site consisted of a soya crop with a growing season beginning in May and extending through the end of September. The soil wasmostly a mixture of silt and sand. Note that, based on available soil moisture data (J. F. Mahfouf 1994, personal communication), a lower value of wilting point was used in the base experiment than suggested in BATS for the texture type 6. Forcing solar radiation and precipitation are presented in Figs. 1a,b, which show that precipitation exhibited maxima in April and December. For verification of surface variables, the model-simulated sensible and latent heat fluxes are compared with estimates obtained using a Penman–Monteith model tuned for the location (J. F. Mahfouf 1994, personal communication). The Penman–Monteith formulation derives the surface sensible and latent heat flux from observations of net radiation, mean soil moisture, precipitation, screen height temperature, specific humidity, and wind speed.
Figures 5a–c present monthly net radiation, sensible heat and latent heat flux produced by the model base run, and the corresponding observed net radiation data and Penman–Monteith-derived sensible and latent heat flux. Observed and simulated monthly net radiative fluxes differ by less 10 W m−2. Differences between the Penman–Monteith latent heat and sensible heat estimates and the present model results are also mostly within 10 W m−2, except for sensible heat in April, May, July, and September and latent heat in May, July, and August. The yearly averages of radiation, sensible, and latent heat flux in the present experiment are very close to those found in the observed and Penman–Monteith data (see Table 3); however, our model shows a somewhat lower daily variability in sensible heat flux. Excellent correlation was found between model and observed daily net radiation, with a correlation coefficient of about 0.97. The correlation coefficients between model daily sensible and latent heat fluxes and Penman–Monteith estimates were 0.72 and 0.87, respectively.
A number of experiments were conducted to assess the sensitivity of surface fluxes to parameters in the soil model configuration: 1) inclusion of a free bottom drainage condition (bperm = 1); 2) coarse (sandy) soil texture class (itext = 3); 3) fine (clay) soil texture class (itext = 10); 4) root fraction, froot, of 0.25 in each of the top four layers; 5) use of 55 soil layers of 0.1-m depth and no bottom drainage condition (root fraction of 0.2 in the top 5 layers); and 6) use of 55 soil layers of 0.1-m depth and free bottom drainage condition (root fraction of 0.2 in the top five layers). Table 3 shows the yearly averaged net radiative, sensible, and latent heat fluxes. The use of a finer vertical level configuration tends to increase the sensible heat flux and decrease the latent heat flux (evaporation) by about 10%–15%. The bottom drainage condition modifies the fluxes by about 10%, while the flux sensitivity to soil texture specification is 2–7 W m−2.
The results of Table 3 can be interpreted as follows. In the presence of coarser soil texture (EXP2), soil water infiltrates downward more efficiently. As a consequence, the top model layers are drier than in the base run. This results in lower evaporation, higher sensible heat flux, higher surface albedoes, and lower surface absorbed radiation. In the experiment utilizing fine soil texture (EXP3), the decrease in evaporation compared to the base experiment is mostly caused by reduced summer transpiration. This is due to the fact that the summer soil moisture is relatively close to the wilting point (∼0.5). By comparison, the Cabauw experiment showed less sensitivity to soil texture because the soil water content was always above the wilting point.
Also available for model validation are soil water content measurementsat 10-cm intervals down to a depth of 1.6 m on 40 days throughout the year. For comparison with the model results, these soil moisture data were interpolated onto the model soil layers and averaged by month. Figures 6a,b compare observed and simulated monthly soil water content relative to saturation in the top 15 cm of soil (top two model layers) and in the top meter of soil (top four model layers) for four experiments, that is, the base run and the sensitivity runs EXP1, EXP5, and EXP6 in Table 3.
Near the surface (top 15 cm) the observations show a pronounced seasonal cycle characterized by a winter maximum of 0.80 and a late summer–early fall minimum of 0.25–0.30. In the whole 1-m top layer, the seasonal cycle is less marked, from a winter maximum of 0.65 to a late summer–early fall minimum of about 0.35. The model in its base configuration roughly reproduces the soil water cycle in the top 15 cm of soil but overpredicts the summer minimum in the whole top 1-m layer. Use of a finer vertical grid and of the free drainage bottom condition decreases the summer water content for the 1-m soil layer, which becomes more in line with observations. Near the surface, the base configuration, which adopts a 5-cm soil layer, yields better results than when using uniform 10-cm soil layers. In general, the greatest sensitivity to vertical discretization occurs in the summer, perhaps also as a result of the rooting zone specification, and at greater depths, when the resolution in the base model configuration is low. Overall, however, the sensitivity to vertical discretization is not pronounced, and the base model configuration yields a fairly accurate approximation to the results obtained with finer vertical resolution, especially in the top 0.40-m soil zone. A few extra layers could be added in the 0.5–1.5-m soil zone to further improve the soil water content simulation there.
c. ARME experiment
The ARME site is located in the Reserva Florestal Ducke, 25 km from Manaus, Brazil, and is covered by a dense tropical forest of about 30-m height overlying a loam-type soil (Shuttleworth et al. 1984a,b; Shuttleworth 1988). Meteorological observations were taken at a height of approximately 45 m (15 m above the forest canopy) for the period 1 September 1983–30 September 1985 with a 1-h time interval. Data for the full year 1984 was extracted from this 2-yr period and used to drive the model runs. The solar radiative forcing peaks in the late fall, but it does not show a strong seasonal cycle, while precipitation peaks in the winter and early spring months (Figs. 1a,b). Unfortunately, data for model validation were not available, so the present model results are mostly compared with estimates from a Penman–Monteith model tuned for the site as reported in Viterbo and Beljaars (1995).
Figure 7 presents the monthly averaged surface net radiation, latent heat, and sensible heat flux for the base model run. For all quantities, especially latent heat flux, the seasonal cycle is not pronounced. Net radiation and sensible heat flux show a maximum in July–September, while the latent heat flux is between 95 and 105 W m−2 throughout the year. Both the latent heat flux and the net radiation flux are in line with the Penman–Monteith estimates given by Viterbo and Beljaars (1995). Average daily maximum and minimum latent heat flux in our experiment were approximately 220 and 30 W m−2 in both January and July (not shown). Also, the daily variability of sensible heat flux, as measured by the standard deviation of the daily values, was similar to that of the Cabauwand HAPEX experiments (see Table 4), about 30 W m−2, while the daily variability of latent heat flux was lower than in the other experiments, only about 21 W m−2.
A number of experiments were carried out to test the model sensitivity to soil and vegetation characteristics: 1) free drainage bottom condition; 2) soil texture class 1 (sandy soil); 3) soil texture class 1 (sandy soil) and free drainage bottom condition; 4) initial soil water content equal to 0.4; 5) initial soil water content equal to 0.9; 6) roughness length increased to 3 m; 7) leaf area index decreased to 3; 8) minimum stomatal resistance decreased to 150 s m−1 (value used by BATS); 9) minimum stomatal resistance decreased to 40 s m−1 (value used in the Cabauw and HAPEX experiments); and 10) no precipitation interception/reevaporation by the canopy foliage.
Table 4 shows the yearly average net radiative, sensible heat, and latent heat fluxes for all sensitivity experiments. The results from the first six runs indicate that the ARME experiment is not sensitive to modifications in soil characteristics, that is, over the Amazon region vegetation dominates the surface–atmosphere exchanges. Increasing roughness length and decreasing the minimum stomatal resistance lead to excessive evapotranspiration, while halving the leaf area index yields a decrease in evapotranspiration of about 25%. Finally, comparison of the base run with EXP10 indicates that reevaporation of intercepted precipitation accounts for about 15% of the yearly averaged evaporation.
4. Model sensitivity to surface heterogeneity representation
To test the parameterization of surface heterogeneity proposed in G97 we use the idealized experiment of SGD. The primary reason for this is that the forcing climatology for this experiment did not refer to a specific point location but was designed to be characteristic of a region of size comparable to a general circulation model (GCM) grid box. In particular, SGD developed a forcing climatology typical of an eastern-central United States region using monthly precipitation and temperature data from Legates and Willmott (1990a,b) and monthly solar radiation from the Solar Energy Research Institute (1981). The precipitation and air temperature forcing are reported in Figs. 1a,b. Atmospheric water vapor was obtained assuming a 75% relative humidity, and downward infrared radiation was expressed in terms of atmospheric emissivity and temperature. A half-hourly climatology was obtained by SGD from the monthly climatology by superposing a sinusoidal diurnal cycle to temperature and a half sinusoidal cycle with zero value at night for solar radiation.
For precipitation, two regimes were specified. During the cold season months (October–March) precipitation was assumed to take the form of large-scale events of 12-h duration occurring every 6 days. Warm season precipitation (April through September) was defined by daily afternoon convection events of 2-h duration. Intensity of rainfall was determined such that the average of precipitation from these events was equal to the monthly average. Surface pressure and wind speed were set constant throughout the year with values of 1000 mb and 3 m s−1, respectively. Cloudiness occurred only during precipitation events, yielding a decrease in solar radiation and an increase in atmospheric infrared emissivity. The only modification to the SGD climatology introduced in this work consists of a decrease in forcing air temperature of 2.5° during precipitation events. This was done in order to increase snowfall (and thus better test the snow formulation), which is assumed to occur when the air temperatureis lower than 0°C.
SGD conducted a series of BATS sensitivity experiments to surface-type specification. The experiment specifying surface grass cover is chosen for use in this work. The set of relevant model parameters for this experiment is given in Table 1. A bottom model permeability of 0.05 was chosen so that the soil water content in the present model was close to that produced by BATS (monthly values throughout the year of 0.65–0.70 in the top 10 cm and 0.69–0.73 in the top meter). Figures 8a–c compare the monthly net radiation, sensible heat, and latent heat flux as simulated by BATS and by the present model run in point mode (i.e., with no heterogeneity representation). The main differences between the two schemes occur during the winter months when the present model produces more extensive snow cover than BATS. This results in lower net radiative as well as sensible and latent heat flux. It should be noted that the temperature threshold for snowfall in the standard version of BATS, 2.2°C, was lowered to 0°C for consistency with the present model and this change might have contributed to the relatively low snow amounts in the BATS run. Except for January and February, the present scheme produces somewhat higher net radiative, sensible, and latent heat flux than BATS, but the differences between the two schemes are generally less than 10 W m−2.
In the previous sections we discussed the model sensitivity to vegetation and soil characteristics, so we do not present a similar discussion for the experiment used in this section. Here we mostly focus on the effects of the surface heterogeneity representation described in G97. Many factors can contribute to surface heterogeneity, especially at the scale of atmospheric model grid boxes of a few hundred kilometers size. Physiography is a critical one. In mountainous regions, topographical elevation can vary by a few thousand meters within a distance of a few hundred kilometers. Assuming an average lapse rate of 6.5°C km−1, this would lead to temperature variations in excess of 10 K within such distance. In an atmospheric model grid box, however, elevation and temperatures are represented by single average values. As a consequence, for example, snow simulation in mountainous regions by GCMs or even regional atmospheric models can be poor. Additional spatial temperature variability can be caused by surface aspect and orientation, soil properties, variations in vegetation characteristics, and microclimatic conditions.
Soil water content can also exhibit substantial spatial high-frequency variability in response to topographic forcing and variations in soil and vegetation characteristics (e.g., Kalma and Sivapalan 1995). For example, within distances of a few to a few hundred kilometers, saturated drainage areas can easily coexist with very dry regions, for example, in the presence of steeply sloping terrain. Finally, climate and hydrologic forcing associated with precipitation and snowmelt can be highly variable in space and therefore can further enhance the heterogeneous nature of surface temperature and soil moisture.
In our model, the basic characteristics of the temperature and soil moisture distributions are defined in terms of two independent parameters of the PDF, the half-width αpdf and the height ratio γpdf (see G97). The greater αpdf, the wider the distribution and the more pronounced the variability; the closer γpdf to 1, the more evenly distributed the PDF. It can therefore be expected that the effects of heterogeneity increase as αpdf increases and as γpdf becomes closer to 1. In this section, the effects of this heterogeneity representation is tested through a set of sensitivity experiments tovalues of the PDF parameters.
Before discussing these experiments, it is useful to remember that the heterogeneity representation is used for water content relative to saturation and temperature at all soil layers and for temperature of skin ground, snow, canopy air, canopy foliage, and anemometer (atmospheric) height. For simplicity, it is assumed that the PDFs for all these temperature variables are characterized by the same (time varying) parameter values, except for snow and the two bottom soil layers. In the presence of snow with fractional cover fsn, the half-width for the snow temperature distribution is given by
where α̃pdf is a base value (e.g., determined by the topography variability). Over the snow-free area
Also, condition applies Tsn + αTsnpdf ≤ 273.15 K (see G97). In the two bottom soil layers, a constant value of αTpdf = 0.5 K is used.
For soil water content, the half-width parameter is given by
where α̃wlpdf is a base value. The condition (3) ensures that the distributed water content is never lower than 0 or greater than 1.
In the experiments discussed in this section, unless otherwise specified, we assume that precipitation falls uniformly over the grid box. Sensitivity to the fractional precipitation area is examined in section 6a. Note, however, that precipitation is in the form of snow over the fractional area in which the anemometer temperature is below the freezing point (see section 3c of G97).
a. Temperature heterogeneity
In this section results are discussed in which only the model temperatures are assumed to follow continuous PDFs, while gridbox average, or point, values are assumed for soil water content. The focus of the discussion is on a limited number of variables that are most likely affected by the temperature heterogeneous representation: latent and sensible heat flux, skin temperature, and snow amount (the product of snow fractional cover times snow depth). Table 5 presents cold season (October–March) and warm season (April–September) averages for the base run and 10 sensitivity experiments: runs with αTpdf equal to 1, 3, 5, and 7 K and γTpdf equal to 0.5 (EXP1–EXP4, respectively); runs with αTpdf = 7 and γTpdf = 0.1 and 0.95 (EXP5 and EXP6, respectively); and runsincluding the atmospheric humidity, infrared radiation, precipitation, and solar radiation corrections for snow-covered areas described in section 3c of G97 (EXP7 through EXP10, respectively).
First note that the the warm season simulation is little sensitive to the inclusion of temperature heterogeneity. During summer, temperature heterogeneity in our model enters the infrared radiative term, the calculation of saturation vapor pressure, and the temperature dependence of stomatal resistance (see G97). Integration of these terms over the temperature PDFs tested here does not strongly affect the simulated fluxes for a grass surface. Conversely, the heterogeneity effect during the winter is substantial, mostly because, as αTpdf increases, larger fractional snowfall areas and snow amounts occur. Between the base run and the run with αTpdf = 7 (EXP4) the snow amount during the cold season increases by about 50%. Correspondingly, the averaged cold season sensible heat flux decreases by about 3 W m−2. In the runs with αTpdf = 7, the snowpack persists also during the early spring months, which leads to the relatively small warm season average snow values of Table 5. The two experiments denoted with EXP5 and EXP6 illustrate that although the heterogeneity effect increases with γTpdf, the sensitivity to this parameter is not pronounced. Even with γTpdf = 0.1, the effects of heterogeneity are substantial and the differences between EXP5 and EXP6 are relatively small.
As discussed in G97, if it is assumed that the temperature heterogeneity is due to topographic variability, in addition to temperature, other components of the average forcing climate can be subdivided into values over snow-covered (cold portion) and snow-free (warm portion) areas. For example, it is known that precipitation exhibits a topographical dependence (e.g., Sevruk 1989). Therefore, if we assume that snow occurs at higher elevations than rain, the actual precipitation rate will be higher than the gridbox average over the snow-covered area and lower than the average over the snow-free area. Similarly, atmospheric humidity and downward infrared flux generally decrease with increasing elevation. As a final example, snow cover may persist because of relatively low insolation due to topographic shading and orientation. All these processes can affect snow simulation as well as the fluxes over the snow-free areas.
G97 presents ways to calculate elevation-induced corrections to atmospheric humidity, infrared radiation, precipitation, and solar radiation over snow-covered and snow-free areas. To assess the model sensitivity to the inclusion of these effects, EXP7 through EXP10 in Table 5 were completed: in EXP7 and EXP8 the humidity and infrared corrections where applied as described in section 3c of G97; in EXP9 and EXP10, it was assumed that precipitation over snow-covered areas was twice as much as, and solar radiation was 50% lower than, over snow-free areas. All these experiments use as reference the model configuration of EXP4 (i.e., αTpdf = 7 K).
Inclusion of the atmospheric moisture correction term, which decreases the water vapor mixing ratio over snow and increases it over snow-free areas, increased the simulated snow cover by less than 10%. A decrease in grid-average cold season latent heat fluxwas found, about 1.5 W m−2, mostly as a result of decreased evaporation over snow-free areas in response to higher atmospheric moisture. The model sensitivity to the infrared correction term is more pronounced, with increases in snow amount compared to the reference run of about 28%. The precipitation and solar radiation corrections produced even greater effects, with an increase in cold season snow amounts of about 40% and 60%, respectively. These changes in snow amounts also produced corresponding changes in average surface sensible and latent fluxes of up to 5 W m−2.
Summarizing the main results of this section, the temperature heterogeneity representation mostly affected the winter and early spring surface energy and water budget of the model, primarily as a result of modifications in the snowpack evolution.
b. Soil water heterogeneity
Soil water heterogeneity is described by assigning a PDF to the soil water content relative to saturation, wl, at each layer and integrating the vertical soil water diffusion, runoff production, and soil water supply terms as described in G97. A number of sensitivity experiments were conducted to assess the model sensitivity to values of αwlpdf and γwlpdf and other parameters: γwlpdf = 0.5 and αwlpdf = 0.1, 0.2, 0.3, 0.4 (EXP1–EXP4, respectively); αwlpdf = 0.4 and γwlpdf = 0.95 and 0.1 (EXP5 and EXP6, respectively); γwlpdf = 0.5 and αwlpdf varying from 0.4 at the top soil layer to 0.1 at the bottom soil layer (EXP7), two experiments similar to EXP1 and EXP4 but including the no drainage bottom boundary condition (EXP8 and EXP9, respectively); and one experiment with αwlpdf = 0.4 but no heterogeneity representation in the soil water diffusion formulation.
Table 6 shows cold season and warm season average sensible heat flux, latent heat flux, skin surface temperature, surface runoff, drainage below the model soil zone, and soil water content in the top and bottom soil layers. As the value of αwlpdf increases, surface runoff, drainage and sensible heat flux increase, and evaporation and soil water content near the surface correspondingly decrease. Basically, infiltration and surface runoff efficiencies increase with αwlpdf as a result of their nonlinear nature, thereby driving the changes in the surface hydrologic and energy cycle. The soil water content in the bottom model level is regulated by the relative efficiency of water infiltration from above and drainage below.
The soil water heterogeneity effect on the surface fluxes is quite large in both seasons, up to a few tens of watts per square meter, with the greatest contribution given by the water infiltration process. This last result is illustrated by the comparison ofEXP10, in which the water movement formulation does not include heterogeneity, with EXP1 and EXP4. The sensible and latent heat fluxes and runoff in EXP10 are closer to those of EXP1 than to those of EXP4. Similarly to the case of temperature heterogeneity, the model is not very sensitive to the value of γwlpdf (see results from EXP5 and EXP6).
Note that for the present experiment, surface runoff production is small, and in fact is zero during the summer when drainage is allowed. This is because runoff production requires either saturated top soils or high precipitation rates (see section 3e of G97), and our experiments are characterized by relatively low (gridbox average) precipitation rates and upper soil water contents. When no bottom drainage is allowed, the top soil becomes much closer to saturation and surface runoff increases. Overall, the bottom drainage condition greatly affects the soil water cycle as well as the surface fluxes and thus appears to be a critical parameter of model setup.
EXP7 is designed to simulate conditions in which the soil water heterogeneity is large near the surface and decreases with depth. The results from this experiment are generally intermediate compared to those produced in homogeneous soil conditions. Surface fluxes and top soil water are close to those of EXP4, while drainage and bottom soil water are closer to those of EXP1.
Finally, an experiment was completed that included both temperature and soil water content heterogeneity (EXP11 in Table 6). The parameters αTpdf and αwlpdf had values of 7 K and 0.4, respectively, that is, the largest values tested in the previous runs, and γTpdf, γwlpdf were both equal to 0.5. Comparison of EXP11 with EXP4 of Tables 5 and 6 shows that the temperature heterogeneity effects on cold season surface conditions and the soil water heterogeneity effects on the simulated surface hydrology are basically similar to those in the runs where these effects are treated separately (the cold season snow amount for EXP11 in Table 6 was equal to 0.094 m). This is essentially because the temperature and soil water heterogeneity effects act primarily on different processes.
5. Comparison of heterogeneous model simulations with aggregated point-mode model simulations
One way to verify the surface heterogeneity parameterization presented in this work is to compare heterogeneous runs with aggregated results from a series of homogeneous runs. This can be done in a relatively straightforward way for temperature, while it is more difficult for soil water content. In the next two sections, attempts at verification of the temperature and soil water content heterogeneous representation are discussed separately.
a. Temperature heterogeneity
The following experiment was designed for verification of the temperature heterogeneity representation. A set of 19 simulations was conducted with the model running in point mode (i.e., no heterogeneity) and with forcing atmospheric temperatures modified compared to the base run (which is here designated as the T̄ case). For each run, a constant perturbation to the forcing atmospheric temperature was applied at every time step, where the perturbation varies by 1 K between successive experiments. The set of experiments thus covers therange of T̄ − 9 K for, say, experiment 1 to T̄ + 9 K for experiment 19. Initial soil temperatures are correspondingly modified and water vapor mixing ratio is adjusted to maintain the same relative humidity as in the base experiment. Finally, the downward incident longwave radiation flux is modified to reflect its dependency on the fourth power of temperature. These experiments might, for example, simulate a situation in which the 19 grid points were located at elevation intervals of 150 m (e.g., assuming a lapse rate of 6.5 K km−1).
A series of three experiments with αTpdf = 7 K and forcing atmospheric temperatures at T̄ − 2 K, T̄, and T̄ + 2 K, and five experiments with αTpdf = 5 K and forcing atmospheric temperatures of T̄ − 4 K to T̄ + 4 K (at intervals of 2 K) were then completed. All these experiments assumed γTpdf = 1 for simplicity. Finally, the results from the distributed runs were compared with the aggregated results from the point runs with forcing temperatures within the range of the corresponding distributed run. This allows an evaluation of how the heterogeneous temperature formulation describes the aggregated characteristics of the point runs. Since the greatest effects of the surface heterogeneous representation occur in the winter due to snowpack dynamics, the analysis in this section is limited to the cold season averages. In the warm season, sensible and latent heat flux as well as surface temperature did not show a strong dependency on forcing temperature.
Figures 9a–d show the cold season surface skin temperature, sensible heat flux, latent heat flux, and snow amount for all the point-mode and distributed runs. Also shown are the results aggregated from all point-mode runs with forcing temperatures within the ranges covered by the distributed runs. Analysis of the point runs shows that most of the nonlinear model dependency on forcing temperature occurs in correspondence of the passage from snow-dominated to snow-free regimes. This introduces a strong nonlinearity in the system, similar to a threshold function.
Comparison of the results from the heterogeneous runs with those aggregated from the point runs indicate that the heterogeneous formulation is close to, but somewhat underpredicts, the snow amounts aggregated from the corresponding point-mode run ensembles, both for αTpdf = 5 and 7 K. The snow amounts produced by the heterogeneous runs are, however, much closer to the aggregated point-run amounts than the corresponding point runs that use average temperature. A similar result is found for the latent heat flux. The variable for which the poorest agreement is found between heterogeneous runs and aggregated point-mode runs is sensible heat flux for the experiments at lower temperatures, when differences between heterogeneous and aggregated point-mode runs can be of the order of 5 W m−2. This may be due to the absence of the application of the heterogeneity operation to the stability-dependent drag coefficient for surface–atmosphere exchanges (see G97). Skin radiative temperature does not show a pronounced nonlinear trend, however, the heterogeneous runs are always within 0.3 K of the aggregated point-mode runs and closer to them than the corresponding individual point-mode runs.
Overall, Figs. 9a–d show that the heterogeneous runs reproduce the basic characteristics of snow, temperature, and surface fluxes aggregated from the point-mode runs. Inparticular, they show that the heterogeneous runs reproduce the aggregated point-mode results much better than the corresponding point-mode runs forced by the averaged temperatures. This indicates that the heterogeneous representation would provide a better description of the surface water and energy budget during the cold season than an average point representation in areas characterized by marked temperature variability, for example, areas characterized by pronounced topographical features.
b. Soil water content heterogeneity
The comparison between heterogeneous and point-mode soil water simulations focuses on the bottom drainage term since this provides an evaluation of the highly nonlinear soil water movement formulation. In order to compare point and heterogeneous simulations, we first conducted a set of 13 point-mode runs in which the initial soil water content relative to saturation was set to 0.35, 0.40, 0.45, 0.5, . . . , 0.90, 0.95. The model parameter set up is the same as in the base run of section 4 and in all experiments we set bperm = 0.05. Three sets of heterogeneous simulations were then conducted, with αwlpdf = 0.1, 0.2, and 0.3 and initial soil water content equal to 0.35, 0.45, . . . , 0.95. The value of γwlpdf was set to 1 for all runs.
Since the soil water content in the bottom model layer during the first month of simulation (January) does not change much from its initial value, comparison of the January drainage from the heterogeneous runs with the aggregated drainage from the point-mode runs in which initial soil moisture is within the range of the corresponding heterogeneous PDFs can provide a rough evaluation of the soil water movement formulation. Table 7 reports the drainage for the experiments with αwlpdf = 0.3, wl, in = 0.65, and αwlpdf = 0.2 wl, in = 0.55, 0.65, and 0.75, along with the drainage aggregated from the point-mode runs in which wl, in is within the range of the PDFs in the heterogeneous runs. Also shown are the corresponding drainage terms for the individual point-mode runs, with wl, in the same as in the heterogeneous runs. Drainage from the individual point-mode runs differs from that of the aggregated point-mode runs by up to an order of magnitude or more. The agreement of the heterogeneous runs with the aggregated point-mode runs is much better in all cases, although differences between aggregated point-mode and heterogeneous runs are between 25% and a factor of 2. In particular, the heterogeneous runs appear to underpredict drainage for relatively low water contents and overpredict it for relatively high water contents.
Figures 10a,b show the yearly averaged soil water content relative to saturation in the bottom soil model level, wl, bot, and the drainage rates Dr as a function of the initial soil moisture wl, in. In the point-mode runs, wl, bot increases with wl, in until it asymptotes after wl, in = 0.8. Following the soil water content, drainage increases with wl, in. Consistent with the nonlinear nature of the soil water diffusion process, drainage increases withαwlpdf and in the heterogeneous runs has a trend similar to that of the point-mode runs. The soil moisture in the bottom model level tends to increase with αwlpdf for the initially drier runs and to decrease with αwlpdf for the initially wetter runs. This is because in the drier runs, the upper soil levels are increasingly wetted by precipitation and water infiltrates toward the lower levels. Conversely, in the initially wetter runs the dominant process is water drainage from the lowest soil levels.
The results of Table 7 thus illustrate how, in the presence of soil moisture heterogeneity, the heterogenous representation tends to improve the description of soil water movement compared to the homogeneous point-mode representation. However, still significant differences are found between the heterogeneous and aggregated point-mode results. One reason is that, as we already discussed, our heterogeneous representation only partially takes into account nonlinear processes. Another may be associated with the assumption of a constant αwlpdf. A process such as drainage will mostly tend to eliminate areas with high water content, so that the assumption of a fixed value of αwlpdf for a drainage-dominated situation may not be realistic. Rather, the value of αwlpdf should likely decrease due to the fact that drainage is very efficient at high water content and becomes rapidly negligible for low water contents.
Finally, note that for the experiments discussed in this section, sensible and latent heat fluxes were not sensitive to initial soil water content since precipitation was sufficient to keep the near-surface soil water content at relatively high levels.
6. Additional sensitivity experiments
a. Model sensitivity to precipitation characteristics
In most experiments discussed in the previous sections, surface runoff production was a relatively small component of the surface hydrologic cycle, especially during the summer months. As already discussed, this is mostly because the relatively low instantaneous precipitation rates in the SGD experiment did not cause saturation of the top model level. This section examines the model sensitivity to precipitation characteristics through a set of experiments in which the precipitation rates and/or the fractional precipitation area are modified.
Pitman et al. (1992) have shown that the inclusion of fractional precipitation in BATS can modify the surface hydrologic regime from evaporation dominated to runoff dominated. This result, however, depends critically on the parameterization used for runoff generation. In BATS, the runoff rate is explicitly proportional to the precipitation rate and the soil water content, so that, for example, runoff can occur also if the soil is not saturated. Therefore, a decrease in fractional precipitation area that results in a corresponding increase in precipitation rate over the area will lead to an increase in runoff. A parameterization solely based on soil water content and infiltration, such as the one used here, can be expected to be less sensitive to the fractional precipitation area.
The following series of experiments was completed: 1) precipitation rates multiplied by 2 throughout the year; 2) fractional precipitation area equal to 0.1 in the absence ofsnow fall; 3) conditions 1 and 2 together; and 4) fractional precipitation area of 0.1 and αwlpdf varying from 0.4 at the surface to 0.1 at the bottom of the model soil zone. Results for cold and warm season latent heat, sensible heat, surface runoff, and near surface water content are shown in Table 8. In all the runs, a no-drainage bottom condition is assumed since preliminary experiments showed that in the presence of bottom permeability, precipitation changes mostly resulted in increases of drainage rates and evaporation, with little effect on surface runoff production.
Doubling of precipitation (EXP2) produces the greatest effects. Compared to the base run, cold season runoff increases by over a factor of 2 and warm season runoff becomes large. Sensible heat flux decreases in both seasons, while latent heat flux increases in the warm season but decreases in the cold season. This is mostly because of much greater snow amounts in EXP2 compared to BASE. In the absence of surface heterogeneity, the model sensitivity to reduced fractional rain area is not marked, although it leads to an increase in runoff and sensible heat flux and a decrease in latent heat flux (and evaporation). Among the contributions to a decrease in evaporation associated with reduced fractional precipitation area is the decrease in interception and reevaporation by the vegetative canopy. The amount of water intercepted by the canopy was in fact lower in EXP2 than in BASE by about a factor of 2.
When surface heterogeneity is included (EXP4), the effect of reduced fractional precipitation area becomes more pronounced. An increase in αwlpdf causes greater portions of the model grid box to become saturated and produce runoff. However, infiltration also increases with αwlpdf, thereby decreasing the soil water content near the surface and, as a consequence, the efficiency in runoff generation. Surface runoff production in EXP4 is significant also in the warm season, even though the soil water content near the surface (and thus evaporation) decreases compared to BASE due to more efficient downward infiltration. Note that, in general, it can be expected that soil water heterogeneity would increase with decreasing fractional rain area so that these two factors should probably be related in the model setup. Overall, the results of this section indicate that the model sensitivity to fractional rain area is less pronounced than found by Pitman et al. (1992), which, as previously discussed, can be ascribed to the different descriptions of surface runoff production processes in BATS and in the present scheme.
b. Model sensitivity to impermeable surface
The presence of an impermeable, nonevaporating surface has a twofold effect on the gridbox average fluxes: it increases the sensible heat flux while decreasing evapotranspiration, and it increases runoff generation, since precipitation falling on the impermeable surface fraction does not infiltrate into the soil but flows into surface runoff. The magnitude of both these effects is illustrated in Table 9, which presents gridbox-averaged surface fluxes, runoff, and surface skin temperature for experiments with different values of fractional impermeable surface cover.
Two sets of experiments are presented, one with a relatively high value of roughness length, 1 m, to illustrate the effects of urbanlike environments, and one with a low roughness length value, 0.1 m, more characteristic of flatter terrain. In both cases, the increase in sensible heat flux and runoff with the fractional impermeable surface cover isquasi-linear. It is interesting to note that a decrease in roughness length causes a decrease in temperature and an increase in (upward) sensible heat flux during the cold season and opposite responses during the warm season. This is because in the cold season during nighttime lower roughness length implies lower downward heat flux and thus lower surface temperatures. Conversely, in the warm season the effect of lower daytime (upward) sensible heat flux associated with the decrease in roughness length dominates. It is evident from Table 9 that the presence of impermeable surfaces can greatly affect the partitioning of incoming energy into sensible and latent heat flux and the partitioning of precipitation into runoff and evaporation. The average skin temperature over the impermeable surface fraction was greater than the gridpoint average by several tenths of a degree in both seasons.
c. Model sensitivity to the inclusion of a simple delayed runoff formulation
The surface model used in this work includes a water exchange term between near-surface soil and surface water, which crudely simulates delayed runoff. This formulation is described in section 3e(3) of G97 and basically consists of a linear term by which delayed runoff occurs at a specified rate τex if the soil water content at a certain model layer exceeds a threshold value. If the soil water heterogeneity representation is present, delayed runoff occurs only for the portion of the grid box for which soil water content exceeds the given threshold. In order to test the model sensitivity to this exchange term, a series of experiments was conducted assuming a soil water threshold of 0.7 in the top two model layers (top 15 cm) and 0.8 in the third (15 to 40 cm) and exchange times, τex, of about 6 days in the top two layers and 12 days in the third. Four experiments were completed with combination of drainage and no-drainage conditions and with and without soil water heterogeneity. Two additional experiments were then completed with increased value of τex by a factor of 5. In the presence of surface heterogeneity, the calculation of αwlpdf was somewhat modified to account for the fact that the exchange term only removes soil water from regions where the water content is high, thereby narrowing the PDF. This effect is roughly accounted for by assuming that a base value of αwlpdf = 0.4 decreases when the exchange term is active with the same timescale used in the soil water exchange, that is, τex.
Results of surface sensible and latent heat fluxes, near-surface soil water content, and runoff (surface plus delayed) for the six experiments are summarized in Table 10. Comparison with the base run of Table 5 shows that the inclusion of the exchange term substantially increases runoff production, which becomes comparable or even greater than the drainage term. Inclusion of surface heterogeneity increases downward infiltration and therefore decreases the near-surface water contents. This, in turn, induces a decrease in evaporation and in the delayed runoff term, which, however, remains significant. Finally, the experiments with increased exchange times show that surface runoff is sensitive to the value of this parameter, as runoff decreases by a factor of 1.5–2 in both seasons in EXP5 and EXP6 compared to EXP1 and EXP2 (Table 10).
7. Summary and conclusions
In a companion paper, G97 introduced an approach to include temperature and soil water heterogeneitywithin complex land surface schemes. This paper has presented a hierarchy of validation and sensitivity experiments using a stand-alone version of the model described by G97.
The model was first run in point mode using observed fields and compared with actual observations. Three datasets have been used for this purpose, the Cabauw, HAPEX, and ARME datasets. In all cases, and especially at Cabauw, the model reproduced observed sensible heat flux, latent heat flux, and surface temperatures reasonably well. A limited number of sensitivity experiments to various parameter values was also presented.
As a second testing step, the model was driven by the climatology of SGD, which is characteristic of a region of a few hundred kilometers size in the central-eastern United States, and results were compared with those of BATS. Finally, the SGD experiment setup was used to test various aspect of the temperature and soil water heterogeneity representation. The two main findings of this series of experiments were that 1) temperature heterogeneity mostly affected the cold season surface energy and water cycles, primarily by modifying the cycle of snow formation and melting; and 2) soil water content heterogeneity significantly affected the surface hydrologic cycle, in particular where nonlinear processes were dominant. An attempt to verify the heterogeneity formulation has been presented by comparing results from heterogeneous runs with aggregated results from point-mode runs. Although differences are still present between results from heterogeneous and corresponding aggregated point-mode runs, these experiments have shown that our formulation considerably improves the simulation of the surface water and energy budgets in the presence of heterogeneity compared to point-mode runs that employ gridbox-averaged forcing.
It should be pointed out that, rigorously, the conclusions from the sensitivity analysis are limited to the PDF chosen, although they may provide useful indications about the sensitivity that could be found using other PDFs. Also, the model sensitivity is representative of a sensitivity in real conditions to the extent that the chosen PDF is representative of observed distributions.
A key aspect of our heterogeneity formulation consists of how to evaluate the PDF parameter values. Our symmetric PDF only requires two independent parameters and the sensitivity experiments presented in this article indicate a relatively low sensitivity to the height ratio γpdf, at least for temperature and soil water content. This leaves the half-width αpdf as the most critical parameter in the proposed approach. It should be noted that, because of the simplified treatment of heterogeneity proposed in this approach, a very accurate estimation of PDF parameters is not warranted, and realistic estimates that would allow the representation of first-order effects may suffice for most applications. In this paper, an example was given of how to estimate parameters of temperature PDFs based on subgrid-scale topographical information and on the assumption of a 6.5 K km−1 lapse rate. Admittedly, this assumption, which is widely used in meteorology for vertical temperature interpolation (e.g., in the calculation of sea level pressure) may not account for effects of downdrafts, frontal advection, solar radiation, and surface fluxes, but it is a plausible first-order guess. More generally, temperature may actually be a relatively easy quantity for the PDF parameter estimation because satellite data could be used for this purpose, which could even provide a global coverage.
The issue is more complicated for soil water content. That soil moisture shows a high degree of spatial variability has been known and studied for a long time (e.g., Kalma and Sivapalan 1995); however, high-density distributed direct observations are not, and likely will not be soon, available. As a result, Entekhabi and Eagleson (1989) rely on a“reasonable operational choice” of soil moisture PDF parameters for use in GCMs. Perhaps the most promising approach to PDF parameter estimation for soil moisture is that suggested by Famiglietti and Wood (1994), who estimate parameter values for their Gamma soil moisture distribution from topography and soil information based on the topographic-soil index of Beven and Kirkby (1979). Ad hoc simple formulations of runoff and precipitation effects within the parameter estimates could also be included based on intuitive reasoning.
Further ways to estimate PDF parameters could make use of ensembles of point-mode simulations. It may also be possible to devise simple prognostic equations for the PDF parameters. For example, the effect of diffusion or fractional precipitation area in modifying the soil water PDF could be included in such equations. The primary objective of the next phase of model development will be to devise procedures to estimate PDF parameter values for the model distributed variables.
The analysis presented in this paper was limited to stand-alone experiments and mostly served to provide a basic assessment of the importance of the heterogeneous temperature and soil water terms. Much more testing is necessary to fully evaluate the present approach and its importance for atmospheric model simulations. As a next step in model evaluation, and as a preliminary step toward its use in a global model, the author plans to implement the present heterogeneous representation of surface processes within a regional climate model to study its effects on surface climatology and atmospheric circulations.
I would like to thank R. Dickinson and D. Pollard for making available portions of BATS and LSX, respectively, and for valuable suggestions and comments throughout the completion of this work. I also thank J. F. Mahfouf and P. Viterbo for making available the HAPEX, Cabauw, and ARME datasets, P. Martin and M. Verstraete for useful discussion during the early stages of this work, and M. R. Marinucci for help in producing the figures for this paper. Finally, I thank three anonymous reviewers for their comments on this work. Part of this work was completed while I was a scientific visitor of the Institute for Remote Sensing Applications of the Joint Research Centre, Ispra, Italy.
Corresponding author address: Dr. Filippo Giorgi, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307-3000.
* The National Center for Atmospheric Research is sponsored by the National Science Foundation.