• Abu-Hamdeh, N. H., and R. C. Reeder, 2000: Soil thermal conductivity: Effects of density, moisture, salt concentration, and organic matter. Soil Sci. Soc. Amer. J., 64, 12851290.

    • Search Google Scholar
    • Export Citation
  • ASHRAE, 2009: Material properties. 2009 ASHRAE Handbook: Fundamentals, American Society of Heating, Refrigerating and Air-Conditioning Engineers, 26.1–26.22.

    • Search Google Scholar
    • Export Citation
  • Au, S. K., and J. L. Beck, 2001: Estimation of small failure probabilities in high dimensions by subset simulation. Probab. Eng. Mech., 16, 263277.

    • Search Google Scholar
    • Export Citation
  • Au, S. K., and J. L. Beck, 2003: Subset simulation and its application to seismic risk based on dynamic analysis. J. Eng. Mech., 129, 901917.

    • Search Google Scholar
    • Export Citation
  • Au, S. K., J. Ching, and J. L. Beck, 2007a: Application of subset simulation methods to reliability benchmark problems. Struct. Saf., 29, 183193.

    • Search Google Scholar
    • Export Citation
  • Au, S. K., Z. H. Wang, and S. M. Lo, 2007b: Compartment fire risk analysis by advanced Monte Carlo simulation. Eng. Struct., 29, 23812390.

    • Search Google Scholar
    • Export Citation
  • Bou-Zeid, E., J. Overney, B. D. Rogers, and M. B. Parlange, 2009: The effects of building representation and clustering in large-eddy simulations of flows in urban canopies. Bound.-Layer Meteor., 132, 415436.

    • Search Google Scholar
    • Export Citation
  • Brutsaert, W., 2005: Hydrology: An Introduction. Cambridge Press, 605 pp.

  • Campbell, G. S., C. Calissendorff, and J. H. Williams, 1991: Probe for measuring soil specific heat using a heat-pulse method. Soil Sci. Soc. Amer. J., 55, 291293.

    • Search Google Scholar
    • Export Citation
  • Chen, F., and Coauthors, 2011: The integrated WRF/Urban modeling system: Development, evaluation, and applications to urban environmental problems. Int. J. Climatol., 31, 273288.

    • Search Google Scholar
    • Export Citation
  • Coceal, O., T. G. Thomas, I. P. Castro, and S. E. Belcher, 2006: Mean flow and turbulence statistics over groups of urban-like cubical obstacles. Bound.-Layer Meteor., 121, 491519.

    • Search Google Scholar
    • Export Citation
  • Defraeye, T., B. Blocken, and J. Carmeliet, 2010: CFD analysis of convective heat transfer at the surfaces of a cube immersed in a turbulent boundary layer. Int. J. Heat Mass Transfer, 53, 297308.

    • Search Google Scholar
    • Export Citation
  • Fernando, H. J. S., 2010: Fluid dynamics of urban atmospheres in complex terrain. Annu. Rev. Fluid Mech., 42, 365389.

  • Grimmond, C. S. B., and T. R. Oke, 1999: Aerodynamic properties of urban areas derived, from analysis of surface form. J. Appl. Meteor., 38, 12621292.

    • Search Google Scholar
    • Export Citation
  • Grimmond, C. S. B., and Coauthors, 2010: The International Urban Energy Balance Models Comparison project: First results from phase 1. J. Appl. Meteor. Climatol., 49, 12681292.

    • Search Google Scholar
    • Export Citation
  • Grimmond, C. S. B., and Coauthors, 2011: Initial results from phase 2 of the International Urban Energy Balance Model Comparison. Int. J. Climatol., 31, 244272.

    • Search Google Scholar
    • Export Citation
  • Hamdi, R., and G. Schayes, 2007: Validation of Martilli’s urban boundary layer scheme with measurements from two mid-latitude European cities. Atmos. Chem. Phys., 7, 45134526.

    • Search Google Scholar
    • Export Citation
  • Harman, I. N., M. J. Best, and S. E. Belcher, 2004: Radiative exchange in an urban street canyon. Bound.-Layer Meteor., 110, 301316.

  • Hastings, W. K., 1970: Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97109.

  • Johnson, G. T., T. R. Oke, T. J. Lyons, D. G. Steyn, I. D. Watson, and J. A. Voogt, 1991: Simulation of surface urban heat islands under ‘ideal’ conditions at night part 1: Theory and tests against field data. Bound.-Layer Meteor., 56, 275294.

    • Search Google Scholar
    • Export Citation
  • Kusaka, H., H. Kondo, Y. Kikegawa, and F. Kimura, 2001: A simple single-layer urban canopy model for atmospheric models: Comparison with multi-layer and slab models. Bound.-Layer Meteor., 101, 329358.

    • Search Google Scholar
    • Export Citation
  • Lemonsu, A., S. Belair, and J. Mailhot, 2009: The new Canadian urban modelling system: Evaluation for two cases from the Joint Urban 2003 Oklahoma City Experiment. Bound.-Layer Meteor., 133, 4770.

    • Search Google Scholar
    • Export Citation
  • Liu, Y., J. N. Chen, W. Q. He, Q. Y. Tong, and W. F. Li, 2010: Application of an uncertainty analysis approach to strategic environmental assessment for urban planning. Environ. Sci. Technol., 44, 31363141.

    • Search Google Scholar
    • Export Citation
  • Loridan, T., and Coauthors, 2010: Trade-offs and responsiveness of the single-layer urban canopy parametrization in WRF: An offline evaluation using the MOSCEM optimization algorithm and field observations. Quart. J. Roy. Meteor. Soc., 136, 9971019.

    • Search Google Scholar
    • Export Citation
  • Martilli, A., A. Clappier, and M. W. Rotach, 2002: An urban surface exchange parameterisation for mesoscale models. Bound.-Layer Meteor., 104, 261304.

    • Search Google Scholar
    • Export Citation
  • Mascart, P., J. Noilhan, and H. Giordani, 1995: A modified parameterization of flux-profile relationship in the surface layer using different roughness values for heat and momentum. Bound.-Layer Meteor., 72, 331344.

    • Search Google Scholar
    • Export Citation
  • Masson, V., 2000: A physically-based scheme for the urban energy budget in atmospheric models. Bound.-Layer Meteor., 94, 357397.

  • Metropolis, N., A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, 1953: Equations of state calculations by fast computing machines. J. Chem. Phys., 21, 10871092.

    • Search Google Scholar
    • Export Citation
  • Molders, N., and G. Kramm, 2009: Permafrost modeling in weather forecasts and climate projections. New Permafrost and Glacier Research, M. I. Kruger and H. P. Stern, Eds., Nova Science, 51–88.

    • Search Google Scholar
    • Export Citation
  • Nadeau, D. F., and Coauthors, 2009: Estimation of urban sensible heat flux using a dense wireless network of observations. Environ. Fluid Mech., 9, 635653.

    • Search Google Scholar
    • Export Citation
  • Niceno, B., A. D. T. Dronkers, and K. Hanjalic, 2002: Turbulent heat transfer from a multi-layered wall-mounted cube matrix: A large eddy simulation. Int. J. Heat Fluid Flow, 23, 173185.

    • Search Google Scholar
    • Export Citation
  • Nunez, M., and T. R. Oke, 1977: The energy balance of an urban canyon. J. Appl. Meteor., 16, 1119.

  • Oke, T. R., 1982: The energetic basis of the urban heat island. Quart. J. Roy. Meteor. Soc., 108, 124.

  • Oleson, K. W., G. B. Bonan, and J. Feddema, 2010: Effects of white roofs on urban temperature in a global climate model. Geophys. Res. Lett., 37, L03701, doi:10.1029/2009GL042194.

    • Search Google Scholar
    • Export Citation
  • Panofsky, H. A., and G. W. Brier, 1958: Some Applications of Statistics to Meteorology. The Pennsylvania State University, 224 pp.

  • Refsgaard, J. C., J. P. van der Sluijs, A. L. Hojberg, and P. A. Vanrolleghem, 2007: Uncertainty in the environmental modelling process—A framework and guidance. Environ. Modell. Software, 22, 15431556.

    • Search Google Scholar
    • Export Citation
  • Roberts, C., and G. Casella, 1999: Monte Carlo Statistical Methods. Springer, 680 pp.

  • Sailor, D. J., 2011: A review of methods for estimating anthropogenic heat and moisture emissions in the urban environment. Int. J. Climatol., 31, 189199.

    • Search Google Scholar
    • Export Citation
  • Sailor, D. J., and H. L. Fan, 2002: Modeling the diurnal variability of effective albedo for cities. Atmos. Environ., 36, 713725.

  • Skamarock, W. C., J. B. Klemp, J. Dudhia, D. O. Gill, D. M. Barker, W. Wang, and J. G. Powers, 2005: A description of the Advanced Research WRF version 2. NCAR Tech. Note TN-468+STR, 88 pp.

    • Search Google Scholar
    • Export Citation
  • Thunnissen, D. P., S. K. Au, and E. R. Swenka, 2007a: Uncertainty quantification in the preliminary design of a spacecraft attitude control system. AIAA J. Aerospace Comput. Info. Commun., 4, 902917.

    • Search Google Scholar
    • Export Citation
  • Thunnissen, D. P., S. K. Au, and G. T. Tsuyuki, 2007b: Uncertainty quantification in estimating critical spacecraft component temperatures. J. Thermophys. Heat Transfer, 21, 422430.

    • Search Google Scholar
    • Export Citation
  • Wang, Z. H., E. Bou-Zeid, and J. A. Smith, 2011: A spatially-analytical scheme for surface temperatures and conductive heat fluxes in urban canopy models. Bound.-Layer Meteor., 138, 171193.

    • Search Google Scholar
    • Export Citation
  • Zio, E., and N. Pedroni, 2009: Estimation of the functional failure probability of a thermal–hydraulic passive system by subset simulation. Nucl. Eng. Des., 239, 580599.

    • Search Google Scholar
    • Export Citation
  • View in gallery

    Schematic of the resistance network of energy transport in the single-layer urban canopy model developed and used in this study (T is temperature; H is the sensible heat; G is the conductive heat flux; za is the reference height in the atmospheric layer; zT is the representative height in the street canyon; subscripts a, R, W, G, “can,” and i denote the atmosphere, roof, wall, ground, canyon, and building interior, respectively; numeric subscripts denote numbering of heterogamous subsurfaces).

  • View in gallery

    Schematic diagram of subset simulation procedure depicting (a) level-0 (initial phase) direct Monte Carlo simulation, (b) determination of the first conditional level F1 given conditional exceedance probability p0, (c) populating conditional samples in the first conditional level by Markov chain Monte Carlo procedure, and (d) forwarding algorithm to subsequent conditional levels.

  • View in gallery

    Histogram of conditional samples at different conditional levels for (a) a parameter with high sensitivity (e.g., heat capacity of roof CR) and (b) a parameter with low sensitivity (e.g., roof emissivity εR) with peak sensible heat Hu (Ce = 0.6) as the monitored output.

  • View in gallery

    Atmospheric forcing from SNOP on 20 Mar 2010 (a clear day) as input to the UCM: (a) atmospheric temperature and wind speed; (b) downwelling radiation.

  • View in gallery

    Estimates of exceedance probability vs maximum sensible heat over the urban area Hu, with atmospheric forcing conditions on 20 Mar 2010 (clear day).

  • View in gallery

    Estimates of exceedance probability vs maximum Hcan, HR, LEu, and Rn with the atmospheric forcing conditions on 20 Mar 2010 (clear day).

  • View in gallery

    Estimates of exceedance probability vs maximum conductive heat fluxes through roof (right curve) and wall (left curve) with the atmospheric forcing conditions on 20 Mar 2010 (clear day).

  • View in gallery

    Estimates of PSI for monitored critical (maximum) diurnal fluxes of (a) Hu, (b) Hcan, (c) HR, (d) LEu, (e) GR,i, (f) GW,i, and (g) Rn under both clear-sky conditions (20 Mar 2010) and cloudy conditions (1 Jul 2010), all with the default Ce = 0.6.

  • View in gallery

    Estimates of exceedance probability vs maximum surface temperatures of roof (right curve) and wall (left curve), with atmospheric forcing conditions on 20 Mar 2010 (clear day).

  • View in gallery

    Estimates of exceedance probability vs maximum TG_imp and TG_veg with the atmospheric forcing of 20 Mar 2010 (clear day) and 1 Jul 2010 (cloudy day).

  • View in gallery

    Illustration of the impact of different regions of normalized building height h on TG_imp with a distinct radiative trapping feature, using atmospheric forcing on 20 Mar 2010 (clear day).

  • View in gallery

    Estimates of PSI for monitored maximum (a) TR, (b) TW, (c) TG_imp, and (d) TG_veg under both clear-sky (20 Mar 2010) and cloudy conditions (1 Jul 2010), all with the default Ce = 0.6.

  • View in gallery

    Coefficient of variation of exceedance probability estimates; c.o.v. is a normalized measure of dispersion of probability distributions.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 14 14 14
PDF Downloads 13 13 13

Analyzing the Sensitivity of WRF’s Single-Layer Urban Canopy Model to Parameter Uncertainty Using Advanced Monte Carlo Simulation

View More View Less
  • 1 Department of Civil and Environmental Engineering, Princeton University, Princeton, New Jersey
  • | 2 Department of Building and Construction, City University of Hong Kong, Kowloon, Hong Kong
  • | 3 Department of Civil and Environmental Engineering, Princeton University, Princeton, New Jersey
Full access

Abstract

Single-layer physically based urban canopy models (UCM) have gained popularity for modeling urban–atmosphere interactions, especially the energy transport component. For a UCM to capture the physics of conductive, radiative, and turbulent advective transport of energy, it is important to provide it with an accurate parameter space, including both mesoscale meteorological forcing and microscale surface inputs. While field measurement of all input parameters to a UCM is rarely possible, understanding the model sensitivity to individual parameters is essential to determine the relative importance of parameter uncertainty for model performance. In this paper, an advanced Monte Carlo approach—namely, subset simulation—is used to quantify the impact of the uncertainty of surface input parameters on the output of an offline modified version of the Weather Research and Forecasting (WRF)-UCM. On the basis of the conditional sampling technique, the importance of surface parameters is determined in terms of their impact on critical model responses. It is found that model outputs (both critical energy fluxes and surface temperatures) are highly sensitive to uncertainties in urban geometry, whereas variations in emissivities and building interior temperatures are relatively insignificant. In addition, the sensitivity of the model to input surface parameters is also shown to be very weakly dependent on meteorological parameters. The statistical quantification of the model’s sensitivity to input parameters has practical implications, such as surface parameter calibrations in UCM and guidance for urban heat island mitigation strategies.

Corresponding author address: Zhi-Hua Wang, Dept. of Civil and Environmental Engineering, Princeton University, Princeton, NJ 08544.E-mail: zhihuaw@princeton.edu

Abstract

Single-layer physically based urban canopy models (UCM) have gained popularity for modeling urban–atmosphere interactions, especially the energy transport component. For a UCM to capture the physics of conductive, radiative, and turbulent advective transport of energy, it is important to provide it with an accurate parameter space, including both mesoscale meteorological forcing and microscale surface inputs. While field measurement of all input parameters to a UCM is rarely possible, understanding the model sensitivity to individual parameters is essential to determine the relative importance of parameter uncertainty for model performance. In this paper, an advanced Monte Carlo approach—namely, subset simulation—is used to quantify the impact of the uncertainty of surface input parameters on the output of an offline modified version of the Weather Research and Forecasting (WRF)-UCM. On the basis of the conditional sampling technique, the importance of surface parameters is determined in terms of their impact on critical model responses. It is found that model outputs (both critical energy fluxes and surface temperatures) are highly sensitive to uncertainties in urban geometry, whereas variations in emissivities and building interior temperatures are relatively insignificant. In addition, the sensitivity of the model to input surface parameters is also shown to be very weakly dependent on meteorological parameters. The statistical quantification of the model’s sensitivity to input parameters has practical implications, such as surface parameter calibrations in UCM and guidance for urban heat island mitigation strategies.

Corresponding author address: Zhi-Hua Wang, Dept. of Civil and Environmental Engineering, Princeton University, Princeton, NJ 08544.E-mail: zhihuaw@princeton.edu

1. Introduction

The interaction between the atmosphere and urban areas involves complex physics emanating from the wide range of scales and processes, such as the complexity of built terrains, the variability of the flow and turbulence fields, and the strong heterogeneity of the urban environment. Classical approaches for coupling urban surfaces with climate models represent urban areas as “flat” surfaces with high roughness length and modified surface properties (e.g., Nadeau et al. 2009). At local scales, building-averaged models have been developed to study the urban surface energy budgets, the “canyon” model (Nunez and Oke 1977; Johnson et al. 1991) being a prominent example. The gap between the mesoscale atmospheric models and microscale (building, block, and city scales) land surface models remained wide until the emergence of the physically based urban canopy model (UCM) approach (e.g., Masson 2000; Kusaka et al. 2001; Martilli et al. 2002). These UCM schemes attempted to incorporate the detailed physics of turbulent transport, radiative trapping, and conduction in solid media while keeping the simple geometry of an urban-canyon representation. The urban canopy scheme has been adopted in the widely used Weather Research and Forecasting (WRF) model (Skamarock et al. 2005; Chen et al. 2011) and tested extensively against field measurements (e.g., Hamdi and Schayes 2007; Lemonsu et al. 2009; Grimmond et al. 2010, 2011). The appeal of this modeling approach is that it is computationally efficient as compared with numerical thermofluid models that resolve all the processes involved, such as direct numerical simulations (Coceal et al. 2006) or large-eddy simulations (Bou-Zeid et al. 2009) of urban areas, flow and heat transport simulations over cube matrices (Niceno et al. 2002), or Reynolds-averaged numerical simulations of similar problems (Defraeye et al. 2010). This reduced computational cost allows the coupling of a UCM to mesoscale meteorological models such as WRF.

The UCM is generally suitable for modeling urban–atmosphere energy exchanges, but its performance depends largely on the accuracy of the input parameters. Among the two groups of input parameters (see Kusaka et al. 2001) of a UCM, the atmospheric forcing—that is, temperature, pressure, humidity, wind speed, and solar radiation—can be readily measured in the atmospheric surface layer or they could be provided by the atmospheric component of a coupled model (normally at the first grid node above the surface in a mesoscale model). In contrast, the surface parameters of the UCM—including thermal properties of walls, roofs, and ground soils, (normalized) dimensions of canyon geometry, and internal building temperatures—are rarely measured at urban scales. Although thermal properties can be selected for common engineered (concrete, asphalt, gravel, etc.) or natural (bare soils, grass, trees, etc.) materials and calibrated for a particular case study, these results are usually only limited in applicability. The uncertainty in the surface parameters of a UCM for particular studies is therefore ubiquitous and is the norm rather than the exception. This uncertainty can reduce the quality of the model output, although these models realistically capture most of the physical processes occurring in urban areas. Grimmond et al. (2010, 2011), for example, in an intercomparison study featuring numerous UCMs and several bulk schemes, did not observe any significant improvement in the performance when sophisticated models were compared with much simpler models.

Modeling of uncertainties through error propagation analysis, stochastic methods, and similar approaches is common in weather forecasting (Panofsky and Brier 1958; Molders and Kramm 2009) and environmental studies (Refsgaard et al. 2007; Liu et al. 2010). To evaluate the sensitivity of the turbulent energy exchange between urban areas and the atmosphere using a UCM, the conventional (and the most direct) approach will be setting up a “control” case with a set of base parameters. The sensitivity of the model prediction to an individual parameter is then investigated by changing the values of that particular parameter while keeping the remaining parameters fixed. This approach, when the number of uncertain parameters gets large, becomes computationally expensive and statistically problematic. By fixing the parameter space but allowing one parameter to change at a time, the resulting statistical correlations between the uncertain parameters can be an artifact of the base scenario choices. Loridan et al. (2010) recently applied a more advanced statistical procedure to assess the skill of the offline WRF-UCM. By calibrating the parameters for the UCM to reproduce targeted net radiation and turbulent heat budgets, the study showed that the parameterization scheme is highly sensitive to roof properties.

In this paper, we use an advanced Monte Carlo simulation tool, subset simulation (Au and Beck 2001), to perform a sensitivity analysis for the uncertainties inherent in the surface parameters in the offline WRF-UCM. Subset simulation was originally developed to solve dynamic problems involving input stochastic processes and later found its applications in a broad range of engineering problems such as seismic risk analysis (Au and Beck 2003), aerospace engineering (Thunnissen et al. 2007a,b), fire risk analysis (Au et al. 2007b), and nuclear engineering (Zio and Pedroni 2009). This method is computationally efficient as compared with the direct Monte Carlo method, particularly in investigating small probability events. Model sensitivity and characterization of individual surface parameters of the UCM are evaluated using the conditional samples generated in the subset simulation.

2. Urban canopy model

In this paper, we use a physically based single-layer urban canopy model, which was first introduced by Masson (2000) to include urban energy budgets in atmospheric models. The basic framework has been expanded by a number of researchers (e.g., Kusaka et al. 2001; Martilli et al. 2002), and its application has been extended to a wide range of urban scales from street canyons to cities. We have implemented the UCM used in WRF offline and developed it considerably. The main developments in the model we use here as compared with WRF-UCM are the implementation of a spatially analytical scheme for heat diffusion in solid media and the generalization of the model to be able to include multiple surface types for the ground, walls, and roofs (Wang et al. 2011). A schematic of the resistance network of turbulent energy transport in the UCM is shown in Fig. 1. The geometric configuration is adopted from the WRF urban canopy scheme (Kusaka et al. 2001), in which building arrays are represented as one-dimensional (1D) infinite street canyons with equal height on both sides. Subscripts R, W, G, i, and “can” denote roofs, walls, ground, indoor, and canyon, respectively. As underlined above, the partitioning of urban surfaces presented in Fig. 1 is different from WRF-UCM since our model takes into account the surface heterogeneity of urban terrain. Urban surface types in Fig. 1 consist of impervious/green roofs, brick/glass walls, and asphalt/concrete/vegetated ground surfaces, although the model is general and more types can be included. The subsurfaces are assumed to be uniformly distributed in space such that radiative shading and trapping are the same for all subsurfaces. Building physics such as air conditioning and ventilation systems can also be resolved in our UCM framework, but this level of detail is outside the scope of this paper and will not be discussed here. In this paper, the canyon ground is divided into impervious (paved) and vegetated fractions and the roof and the wall are kept as uniform surfaces.

Fig. 1.
Fig. 1.

Schematic of the resistance network of energy transport in the single-layer urban canopy model developed and used in this study (T is temperature; H is the sensible heat; G is the conductive heat flux; za is the reference height in the atmospheric layer; zT is the representative height in the street canyon; subscripts a, R, W, G, “can,” and i denote the atmosphere, roof, wall, ground, canyon, and building interior, respectively; numeric subscripts denote numbering of heterogamous subsurfaces).

Citation: Journal of Applied Meteorology and Climatology 50, 9; 10.1175/2011JAMC2685.1

Lists of parameters of surface and meteorological input are shown in Tables 1 and 2, respectively, with symbols following Kusaka et al. (2001). In this paper, we focus on uncertainties in surface parameters, which constitute the response system of urban surfaces to meteorological forcing.

Table 1.

Input surface parameters used in urban canopy model; symbols follow Kusaka et al. (2001).

Table 1.
Table 2.

Input meteorological parameters used in urban canopy model; symbols follow Kusaka et al. (2001).

Table 2.

The energy balance equation solved by the UCM is given by
e1
where Rn = S + LSL is the net radiation with S and L denoting shortwave and longwave radiative budgets, respectively; downward and upward arrows denote the downwelling and upwelling components, respectively; QF is the anthropogenic heat flux; and H, LE, and G are the sensible, latent, and conductive heat fluxes, respectively. Because of limited data available in the literature for measurement and quantification of the anthropogenic heat, in this paper we ignore the QF component [as seen in Eq. (1), its inclusion would be relatively similar to an increase in a local forcing term in urban canopies]. For a comprehensive discussion on the estimate of the anthropogenic heat and moisture emission in the urban environment, readers are referred to the recent review by Sailor (2011). In a similar way for the advection component, the major effect is implicitly included by direct measurements of the air temperature as input to UCM, except for a small advective effect that may be present inside the street canyon. During a diurnal cycle, all incoming solar radiation is treated as direct radiation that is partially blocked from reaching all canyon surfaces (walls and ground surfaces) by the shading effect of the buildings. The UCM computes surface temperatures and turbulent fluxes averaged over the sunlit and the shaded fractions. Heat conduction through building surfaces (roofs and walls) is determined using a constant interior building temperature subject to uncertainty within the human comfort range. An adiabatic boundary is assumed for heat conduction through ground surfaces at a sufficiently large depth. The UCM is driven by atmospheric forcing including air temperature, pressure, humidity, wind speed, and downwelling shortwave/longwave radiation.

a. Effective surface properties for radiative trapping

Introducing heterogeneous subsurfaces increases the dimensionality of the numerical model. Thus we cannot assume uniform temperature distribution on the walls, roofs, or ground surfaces (see Fig. 1). It is desirable for the UCM to be able to predict the heterogeneous temperature distributions on different subsurfaces, but there are instances (e.g., radiative exchange computations) in which representative thermal properties are needed for the “entire” urban surface to keep the model complexity reasonable, and the “lumped” properties have to be derived on the basis of the energy balance.

For radiative trapping in street canyons, computation of the shortwave and longwave heat fluxes involves albedos and emissivities of all the participating subsurfaces. This is straightforward for the direct incident solar radiation, which only involves the albedo of the receiving subsurface, and corresponding values can be used. For reflection of shortwave radiation, the computations are complicated by the interaction between different surfaces through view factors. By assuming that all subsurfaces are homogeneously distributed, we can use the average albedo to compute the shortwave radiation reflected off a surface, and absorption is then computed for each subsurface using its individual absorptivity.

For the longwave radiation, the procedure is slightly more complicated. The thermal properties for receiving surfaces are readily available, but those of the emitting surfaces have to be determined in a lumped sense, that is, from summation of the longwave radiative fluxes from each emitting subsurface. Here we present a unified approach for determining the representative (effective) thermal parameters of an emitting surface consisting of heterogeneous subsurfaces (e.g., ground consisting of roads and lawns). A generic form of reflected longwave radiation can be written as (Kusaka et al. 2001; Harman et al. 2004)
e2
where Ωij is the incoming radiation at surface i emitted from surface j (indices i and j represent wall, roof, and sky), Fj→i is the view factor, ε is the emissivity, and is the blackbody irradiance emitted from j. Note that in Eq. (2) Kirchhoff’s relation is assumed (absorptivity = emissivity) for longwave radiation. If the emitting surface j is heterogeneous (consisting of N types of subsurface), then Eq. (2) can be written as
e3
where fk is the fraction of each subsurface k. In the above equation, we assume all N subsurfaces are uniformly distributed in space, such that . We further assume the temperature of the emitting surface can be represented by an effective value , such that . It is clear that with this simplification, the emissivity of the heterogeneous emitting surface can be represented by a single value as
eq1
which is the area average of the subsurface emissivities.
The effective temperature of the emitting surface can be computed using the energy balance of the turbulent heat exchange between the surface j (be it wall or ground) and the canyon as
e4
where A is the surface area; ρa and cp are the density and specific heat of air, respectively; and RES is the aerodynamic resistance adopted from Masson (2000). Within the canyon, the aerodynamic resistance is the same for ground surfaces and for walls:
e5
The terms Ucan and Wcan are the horizontal and vertical wind speeds along walls, respectively:
e6
e7
where Ua is the wind speed at height za (the height of the first atmospheric model layer), z0,town is the effective roughness length of the entire urban area (Masson 2000), Δz = zazR, and Cd is the drag coefficient computed using the stability coefficient of Mascart et al. (1995). Given that the turbulent transport efficiency and its influence on RES are modulated by the mean stability and flow in the canyon, rather than by the stability over each subsurface, we can assume , and Eq. (4) can be simplified to
e8

It is clear that the effective temperature of the emitting surface is the area average of all of the subsurface temperatures. Replacing Tcan with Ta in Eq. (4), we can obtain the same result for the effective roof temperature. We note that this average temperature is only used in the computation of longwave radiative exchanges and is an ad hoc quantity to simplify notations when the lumped contribution is considered from individual subsurfaces. Calculation of individual subsurface temperatures is treated with a spatially analytical scheme that is different from the convectional discrete layer method (see Wang et al. 2011). With these effective thermal parameters, the parameterization of the heat fluxes—in particular, the radiative budgets with canyon trapping presented by Masson (2000) and Kusaka et al. (2001)—can be readily extended to heterogeneous surfaces.

b. Parameterization of latent heat flux

In the current offline UCM, only evaporation from vegetated surfaces is considered and the water-holding capacity of engineered materials is ignored. The latent heat flux arising from a vegetated surface is parameterized using the actual evaporation given by
e9
where Lυ is the heat of evaporation, Ee is the equilibrium evaporation rate, and Ce = αeβe, with αe being the amplifying constant (on the order of 1.20–1.30) due to large-scale advection and βe being a reduction factor that reflects moisture availability. Depending on environmental conditions, Ce can vary from 0 (dry) to 1.26 (fully saturated) (Brutsaert 2005). The equilibrium evaporation is calculated using the resistance method as
e10
where qa is the specific humidity of air, and is the saturated specific humidity computed from the surface temperature of the vegetated surfaces using the Clapeyron equation. The total sensible and latent heat fluxes over the urban area are then given by
e11
e12
where r and w are the nondimensional roof and canyon widths, respectively.

3. Subset simulation

In the context of urban environmental study, the capability of assessing critical responses to the anthropogenic stressors is of paramount practical importance—for example, extreme high urban center temperatures in summers, “hotspots” of surface sensible heat seen from the atmosphere, local concentration of pollutants, and high frequency/intensity of local precipitation. Subset simulation is an adaptive stochastic simulation procedure that is particularly efficient in capturing short-tail probabilities (but is very well adapted to long-tail probabilities as well) that are associated with critical events (Au and Beck 2001; Au et al. 2007a). It stems from the idea that a small exceedance probability can be expressed as a product of larger conditional exceedance probabilities for some intermediate exceedance events, thereby converting a rare-event simulation problem into a sequence of more frequent ones. Here the exceedance probability P(Y > y) is defined as the probability of a critical response Y (in our case, e.g., surface temperature) exceeding a threshold value y.

During a simulation, conditional samples are generated from specially designed Markov chains so that they populate each intermediate exceedance region. The procedure illustrated in the schematic in Fig. 2, is the following: At the initial phase (level 0), the choice of uncertainty parameters follows the prescribed probability distribution function (PDF), the same as in the direct Monte Carlo method (Fig. 2a). At the end of the initial stage, the first conditional level, defined as F1 at which P(Y > y1), is determined in terms of a given conditional probability p0 (Fig. 2b). Conditional samples in the first level are then generated using the Markov chain Monte Carlo (MCMC) procedure (Fig. 2c) on the basis of the parameter values that caused exceedance of y1 in the initial-stage simulations. The subsequent conditional levels are determined as exceedance events Fi at which , respectively, and the algorithm continues until simulations reach the final target (rare) exceedance region (Fig. 2d). Subset simulation is robust—in particular, for uncertain parameter space with high dimensionality, that is, for problems with a large number of uncertain parameters.

Fig. 2.
Fig. 2.

Schematic diagram of subset simulation procedure depicting (a) level-0 (initial phase) direct Monte Carlo simulation, (b) determination of the first conditional level F1 given conditional exceedance probability p0, (c) populating conditional samples in the first conditional level by Markov chain Monte Carlo procedure, and (d) forwarding algorithm to subsequent conditional levels.

Citation: Journal of Applied Meteorology and Climatology 50, 9; 10.1175/2011JAMC2685.1

The efficient generation of conditional samples is highly nontrivial but is pivotal to the success of subset simulation. It is made possible through the machinery of a class of powerful Metropolis algorithms (Metropolis et al. 1953; Hastings 1970; Roberts and Casella 1999) on which the MCMC procedure is based. In MCMC, successive samples are generated from a specially designed Markov chain whose limiting stationary distribution tends to the target PDF as the length of the Markov chain increases. An essential aspect of the implementation of MCMC is the choice of the “proposal distribution,” which governs the generation of the next sample from the current one and consequently the efficiency of the algorithm. For application robustness, it is desirable to automate the choice of proposal distribution, at the expense of giving up possible gains in efficiency. For this purpose, it is found from previous experience that a normal distribution or a uniform distribution centered at the current sample gives reasonable accuracy (Au and Beck 2003; Au et al. 2007b; Molders and Kramm 2009). These two classes of PDFs are therefore used in this study to enhance the statistical efficiency of subset simulations: normal distributions for surface thermal properties and uniform distributions for morphological surface parameters.

As a practical example, we run a typical subset simulation with the critical sensible heat flux Hu monitored as the model response, using a set of uncertain parameters in Table 3 (to be discussed in detail in section 4). In this study, we use four conditional levels and a conditional probability of p0 = 0.1: this means that, at each level, simulations yielding the highest 10% of the monitored output values are considered to exceed the intermediate threshold. In Fig. 3, the threshold values of the sensible heat Hu that correspond to the exceedance probability of 10−1, 10−2, and 10−3 at conditional levels 1, 2, and 3 are 183, 228, and 259 W m−2, respectively (these values are reflected in Fig. 5 with Ce = 0.6 and will be discussed in the following section). Typical histograms of conditional samples of uncertainty parameters, at different exceedance probability levels extracted from a typical simulation, are plotted in Fig. 3. The distribution of the conditional samples at different levels is used to determine the sensitivity of the model to each individual uncertain parameter. As shown in Fig. 3a, the distribution of the conditional sample for a typical parameter with high sensitivity, experiences a significant deviation from the unconditional distribution (i.e., the predefined PDF, plotted as the dashed line). This indicates that, at higher exceedance probabilities, the parameter was skewed in one direction from its unconditional distribution and hence contributed to the exceedance. On the other hand, the histogram of an insensitive parameter (Fig. 3b) at high exceedance probability levels exhibits insignificant deviation from the unconditional one; that is, there is no significant relation between the distribution of the parameter and the exceedance rate.

Table 3.

Statistics of uncertain parameters.

Table 3.
Fig. 3.
Fig. 3.

Histogram of conditional samples at different conditional levels for (a) a parameter with high sensitivity (e.g., heat capacity of roof CR) and (b) a parameter with low sensitivity (e.g., roof emissivity εR) with peak sensible heat Hu (Ce = 0.6) as the monitored output.

Citation: Journal of Applied Meteorology and Climatology 50, 9; 10.1175/2011JAMC2685.1

4. Sensitivity study of surface parameters

In this section, we apply subset simulation to model the uncertainties in surface parameters of the UCM. The meteorological forcing of the UCM is prescribed using measurements of a typical clear day (20 March 2010) from a standard meteorological station of the Sensor Network Over Princeton (SNOP) project [for more information, see Wang et al. (2011) and online at http://efm.princeton.edu/SNOP]. The inputs of atmospheric temperature, wind speed, and downwelling radiation are plotted in Fig. 4. To enhance the statistical significance, in total we have run more than 1500 simulations (30 independent runs/simulations per monitored output), each having 1850 realizations of the set of 24 uncertain parameters (as discussed below; subset simulation samples all the 24 uncertain parameters once on the basis of the prescribed initial distributions during one realization).

Fig. 4.
Fig. 4.

Atmospheric forcing from SNOP on 20 Mar 2010 (a clear day) as input to the UCM: (a) atmospheric temperature and wind speed; (b) downwelling radiation.

Citation: Journal of Applied Meteorology and Climatology 50, 9; 10.1175/2011JAMC2685.1

a. Parameter uncertainty

Both groups of inputs (cf. Tables 1 and 2) to the UCM, meteorological and surface parameters, are associated with uncertainties, and their characterization is of fundamental importance. The meteorological forcing is subject to chaotic atmospheric dynamics and cloud formation processes with extremely high uncertainties. Thus the forcing uncertainties can be more appropriately modeled as a stochastic process. In this paper, however, we focus on the uncertainties in surface parameters, which are (scalar valued) random variables with their PDFs chosen to lie within a physically realistic range.

The statistics of surface parameter uncertainties are listed in Table 3, where subscripts “imp” and “veg” denote impervious and vegetated ground surfaces, respectively. Sources for the mean (representative) values and the range (maximum and minimum values) of thermal properties of engineered materials include albedos from Sailor and Fan (2002), emissivities from the University of California, Santa Barbara, emissivity library (online at http://www.icess.ucsb.edu/modis/EMIS/html/em.html), and thermal conductivities and heat capacities from the American Society of Heating, Refrigerating and Air-Conditioning Engineers (ASHRAE) handbook (ASHRAE 2009). Thermal properties for a vegetated surface are chosen so that the albedo and the emissivity are representative of grass (Brutsaert 2005), and the thermal conductivity and the heat capacity are those of sublayer soils (Abu-Hamdeh and Reeder 2000; Campbell et al. 1991). All thermal properties are associated with a normal distribution for which the standard deviation is 25% of the mean values, except for emissivities, for which a smaller variance is used on the basis of both physical (not to exceed practical limits) and statistical (a larger standard deviation contaminates the normal distribution of emissivity) considerations. Uniform PDFs are assigned to normalized dimensional parameters (the normalized building height h, r, and fveg), roughness lengths, and thickness of roofs and walls, weighing all possible values equally in the range listed in Table 3. The choice of the normalized dimensions for real urban areas is based on Grimmond and Oke (1999), with conversion from building-block representation of urban areas to a 1D infinite canyon. Note that the range of normalized building heights is wide, from 0.2 (town houses in suburban areas) to 3.0 (skyscrapers in megacities). Moreover, we associate the uncertainties of roughness lengths of heat with those of momentum through zh = zm/10, for both roofs and canyons [see Nadeau et al. (2009) for a discussion of the relation of the two parameters for urban areas].

b. Conditional samples

Using the prescribed forcing, we run the UCM using subset simulation. One realization of the statistics of surface parameters in Table 3 yields a prediction of diurnal variation of heat fluxes (H, LE, and G) and surface temperatures. We record the critical (peak) values of heat fluxes or temperatures during the diurnal cycle, which are then used, one at a time, as monitored responses to define exceedance probabilities. In all subsequent simulations, we use four levels, a conditional probability of p0 = 0.1, and 500 samples per level. The statistical distributions of the set of input parameters associated with these simulations are then compared with the original imposed distribution (see Fig. 3). Here we define a quantitative index, the “percentage sensitivity index” (PSI, expressed in percentage), to measure the relative sensitivity of each uncertain parameter:
e13
where j = 1, 2, … , Nlevel is the index of conditional sampling level, with Nlevel being the number of levels of exceedance probability (in this case, Nlevel = 4), E[X] is the expected (statistical mean) value of the unconditional distribution (as in Table 3) of the uncertain parameter of interest X, and E[X|Y > yj] is the expected value of X at conditional level j. Note that the magnitude of the PSI indicates the sensitivity (deviation of conditional means). In addition, the sign of PSI indicates the sign of correlation between the monitored output and the uncertain input parameter; for example, a negative PSI implies that increasing the uncertain input value results in a decrease of the monitored model output, and vice versa.

Before proceeding to examine the results in detail, we need to emphasize here that this sensitivity study is on the UCM and its physical parameterization and application in meteorological models. The results discussed below represent the real physics of urban areas only to the extent that the model is faithful to these physics.

c. Sensitivity of heat budgets

We monitored the critical values of nine heat budgets including four total sensible heat fluxes over the urban area Hu with Ce = 0.0, 0.3, 0.6, and 1.2 [cf. Eq. (9)]; the sensible heat from the canyon Hcan; the sensible heat from the roof HR; the total latent heat over the urban area LEu; the indoor conductive heat fluxes through the roof GR,i and the wall GW,i; and the net radiative flux over the urban area Rn. Note that Ce = 0 implies that evaporation is suppressed over the vegetated surface. For monitored heat flux other than Hu, the evaporation from the vegetated surface is active with a moderate coefficient Ce = 0.6 by default, unless otherwise specified.

Plots of exceedance probabilities versus the selected peak/critical heat budgets during a diurnal cycle, each averaged over 30 simulations, are shown in Figs. 57. Figure 5 shows that with the given uncertainty in the surface parameter space, there is a significant variation of the peak diurnal sensible heat fluxes, ranging from 50 to 200 W m−2 (with 10% exceedance probability). This is the range that the simulation can produce when the parameters are varied within the limits we defined. Similar results are observed for other critical turbulent fluxes (peaks of diurnal Hcan, HR, and LEu, GR,i, GW,i, and Rn). This illustrates the importance of uncertainty in the surface parameter space: significant errors in model prediction could arise from the inaccurate determination of surface parameters. Figure 5 also shows that if the vegetated surface is dry (Ce = 0.0), the critical sensible heat flux, over the entire urban area during a diurnal cycle, is higher than when the vegetated surfaces are evaporating. This is obvious because when evaporation is suppressed over the vegetated surface, more energy is available for release as sensible heat into the atmosphere from the underlying urban area. On the other hand, with evaporative vegetated surfaces, the critical Hu decreases with increasing Ce, but the decrease is not large. We note, however, that the latent heat is the least adequately parameterized scheme [as noted by Grimmond et al. (2010)] in the current UCMs. The parameterization scheme for latent heat has relatively weak dependence on the surface temperature through the saturated humidity .

Fig. 5.
Fig. 5.

Estimates of exceedance probability vs maximum sensible heat over the urban area Hu, with atmospheric forcing conditions on 20 Mar 2010 (clear day).

Citation: Journal of Applied Meteorology and Climatology 50, 9; 10.1175/2011JAMC2685.1

Fig. 6.
Fig. 6.

Estimates of exceedance probability vs maximum Hcan, HR, LEu, and Rn with the atmospheric forcing conditions on 20 Mar 2010 (clear day).

Citation: Journal of Applied Meteorology and Climatology 50, 9; 10.1175/2011JAMC2685.1

Fig. 7.
Fig. 7.

Estimates of exceedance probability vs maximum conductive heat fluxes through roof (right curve) and wall (left curve) with the atmospheric forcing conditions on 20 Mar 2010 (clear day).

Citation: Journal of Applied Meteorology and Climatology 50, 9; 10.1175/2011JAMC2685.1

Estimates of the PSI for the monitored turbulent fluxes are shown in Table 4. To facilitate visualization of these results, a bar graph for PSI (with Ce = 0.6 for Hu) estimates is also shown in Fig. 8. Note that although the uncertain parameters are independently generated from their proposal PDF their counterparts in the conditional samples are not independent because of the acceptance/rejection process. This explains the small (usually <5% and insignificant) PSI in Table 4 between physically uncorrelated input/output parameters—for example, dependence of HR on aW—as statistical artifact. For all the heat flux outputs, the canyon dimensions (i.e., normalized building height h and roof width r) are of general importance in modulating the fluxes, with PSI being >20% in most cases. This implies that the geometric configuration of street canyons plays an essential role in determining the urban–atmosphere energy exchange. Moreover, critical sensible heat fluxes also have strong correlations with the roughness lengths (zm,R for the roof and the total sensible heat flux, and zm,can for canyon sensible heat flux) with PSI around 50%, and with the thickness of roofs and walls with PSI around 30%. For evaporation from the urban area, the critical latent heat is mainly controlled by the presence of vegetated surfaces. It is not surprising that the critical latent heat is most sensitive to the fraction of vegetated surface fveg. Otherwise, the dimensions of the canyon (h and r = 1 − w) determine the local turbulence intensity inside the canyon and therefore have high impact on the model since they control the efficiency of turbulent transport of heat from the street canyon. One interesting observation from Table 4 is that an increase of h is associated with increases in both HR and Hcan (with positive PSI) but leads to a decrease in Hu (negative PSI). This is an intriguing result that is probably due to the complex interactions of the model parameters and their effects on the output. For example, an increase in r (with positive PSI for Hu) leads to a decrease in HR but an increase in Hcan. This could partially compensate the effect of reduction in Hcan and HR due to decrease in h in maximizing Hu. The negative correlation between h and Hu is in agreement with Loridan et al. (2010), although they did not investigate the correlation for the components HR and Hcan.

Table 4.

Estimates of PSI for monitored critical (maximum) diurnal fluxes with the atmospheric forcing conditions of 20 Mar 2010 (clear day).

Table 4.
Fig. 8.
Fig. 8.

Estimates of PSI for monitored critical (maximum) diurnal fluxes of (a) Hu, (b) Hcan, (c) HR, (d) LEu, (e) GR,i, (f) GW,i, and (g) Rn under both clear-sky conditions (20 Mar 2010) and cloudy conditions (1 Jul 2010), all with the default Ce = 0.6.

Citation: Journal of Applied Meteorology and Climatology 50, 9; 10.1175/2011JAMC2685.1

Now consider the thermal parameters. For total sensible heat over urban areas, the roof properties control the maximum energy transport, with aR, kR, and CR having PSI values greater than 15% (as compared with those of the walls and paved and vegetated ground surfaces with PSI < 5%). The only exception is the heat capacity of the wall CW (PSI of ~ 10%–20%). This is also clear from Fig. 6, where for the entire range of exceedance probability, the sensible heat from the roof is always greater than the one from the canyon. The generic conductive fluxes GR,i and GW,i, which are representative of the heating/cooling loads required in these buildings, are modulated by the thermal properties of roof and wall, respectively. Sensible heat Hcan is mainly controlled by kW. The same set of parameters controls HR and GR,i, but with the opposite signs in the PSI of kR indicating the competition between the conduction and convection processes over the roof layer. It is noteworthy that all of the heat fluxes, whether from the canyon or the roof, are relatively insensitive to the thermal properties of the ground surface, be it paved (impervious) or natural (vegetated).

Emissivities, with the given uncertainty statistics, have minimal significance for the model output (all PSI < 0.5%). For most engineered and natural materials, the mean emissivity over a range of wavelength is ~ 0.9–0.95 (cf. Table 3). With the upper and lower limits of 1.0 and 0.8, respectively, the absolute maximum magnitude of PSI that can be achieved by variation of emissivity is 11%. Numerical experiments (not presented here) show that for emissivity to achieve comparable range of PSI of other uncertain parameters, the conditional samples at higher levels (2 or 3) will be deviated and distributed in the range 0 < ε < 0.5. This is clearly not a realistic range for real materials. In a similar way, maintaining the temperature inside the building in the comfort range [20°, 28°C] results in the uncertainty in TB being insignificant for model predictions (PSI < 2%).

A closer examination of the PSI of uncertain parameters for critical Hu responses apparently suggests that the critical sensible heat exchange between the urban canopy and the atmosphere is primarily modulated by the buildings rather than by the paved or vegetated ground surfaces. It is conceivable that buildings affect the turbulent (sensible) energy transport arising from urban areas in the following ways:

  1. Use of engineered materials (nearly impervious)—in particular, in the roof—yields excessive heating of the surface and transport of energy back to the atmosphere primarily as sensible heat (Fernando 2010),

  2. radiative trapping inside the street canyon by building arrays reduces the effective albedo of the urban surface,

  3. heating of the atmosphere due to the energy use inside buildings to maintain the comfort range of internal temperature (either by indoor heating and conduction/leakage to the outside or by the rejection of the heat gained through the building envelope through heat pumps used for cooling), and

  4. increase in roughness length due to the presence of buildings, with enhanced vertical energy transport.

The importance of the building parameters to UCM predictions is also demonstrated in the PSI for maximum net radiation (Fig. 8). Although critical Rn is highly sensitive to albedos of the roof and wall; its sensitivity to thermal properties of ground surfaces (whether vegetated or impervious) is insignificant.

The parameter sensitivity is similar for vegetated surfaces having different evaporative power, ranging from Ce = 0 to 1.2 with critical Hu monitored. One interesting difference between these cases is the role of vegetated surface. The PSI of fveg is positive if evaporation from the vegetated surface is suppressed (Ce = 0), but is negative otherwise. This reflects the fact that as the vegetated surface evaporation increases, turbulent sensible heat necessarily decreases. Therefore, an increase in the fraction of vegetated surface reduces the total sensible heat exchange between urban areas and the atmosphere (negatively correlated). If the vegetated surface does not evaporate, however, then it has a higher contribution to sensible heat (positively correlated) than does a paved ground surface because of its low heat storage capacity.

d. Sensitivity of surface temperatures

Estimates of exceedance probability versus critical roof, wall, and ground surface temperatures are plotted in Figs. 9 and 10. Note that with meteorological forcing of 20 March 2010 (clear day) there is a log concavity in the exceedance probability versus maximum ground surface temperature for both the impervious and the vegetated surfaces. The log concavity found in TG_veg is also responsible for the similar pattern observed in Fig. 6 for LEu. The existence of a log concavity in exceedance probability suggests that the model response of ground surface temperature is dictated by the radiative trapping inside the canyon, which exhibits distinct features under different regions of normalized building height h. The higher ground surface temperature range corresponds to smaller values of h, whereas the lower ground surface temperature range corresponds to larger values. The log concavity demarks the switching between these two influence regions, as illustrated in Fig. 11. As verification, Fig. 10 also shows that for a cloudy day (on 1 July 2010), where radiative trapping effect is not as significant as that in a clear day, the log concavity of surface temperature response disappears as expected.

Fig. 9.
Fig. 9.

Estimates of exceedance probability vs maximum surface temperatures of roof (right curve) and wall (left curve), with atmospheric forcing conditions on 20 Mar 2010 (clear day).

Citation: Journal of Applied Meteorology and Climatology 50, 9; 10.1175/2011JAMC2685.1

Fig. 10.
Fig. 10.

Estimates of exceedance probability vs maximum TG_imp and TG_veg with the atmospheric forcing of 20 Mar 2010 (clear day) and 1 Jul 2010 (cloudy day).

Citation: Journal of Applied Meteorology and Climatology 50, 9; 10.1175/2011JAMC2685.1

Fig. 11.
Fig. 11.

Illustration of the impact of different regions of normalized building height h on TG_imp with a distinct radiative trapping feature, using atmospheric forcing on 20 Mar 2010 (clear day).

Citation: Journal of Applied Meteorology and Climatology 50, 9; 10.1175/2011JAMC2685.1

PSI estimates for the critical temperature of different surfaces are listed in Table 5 and Fig. 12. As expected, critical temperatures are dictated by the thermal properties of the corresponding surfaces. The influence of emissivities and interior building temperatures, given the physical range of variation, is insignificant in comparison with other thermal properties. Again, canyon dimensions—in particular, the canyon height h—strongly modulate surface temperatures. Roughness lengths of the roof and the canyon influence the surface temperature of roof and canyon surfaces, respectively. The surface temperature of the building enclosure (roof and wall) is also determined by the thickness of the enclosure (dR and dW). The moderate sensitivity of critical ground surface temperatures to dW, on the other hand, is likely to be the result of the complex interactions of model parameters or to be due to multiple radiative reflection inside the street canyon.

Table 5.

Estimates of PSI for monitored critical (maximum) diurnal surface temperatures with the atmospheric forcing conditions of 20 Mar 2010 (clear day).

Table 5.
Fig. 12.
Fig. 12.

Estimates of PSI for monitored maximum (a) TR, (b) TW, (c) TG_imp, and (d) TG_veg under both clear-sky (20 Mar 2010) and cloudy conditions (1 Jul 2010), all with the default Ce = 0.6.

Citation: Journal of Applied Meteorology and Climatology 50, 9; 10.1175/2011JAMC2685.1

e. Effect of weather conditions

To investigate the effect of different weather conditions on the sensitivity study, we ran a second set of simulations driven by meteorological forcing on 1 July 2010 (characterized by the presence of large cloud-cover fraction and high air temperatures). PSI estimates for monitored critical responses of heat flux and surface temperature are listed in Tables 6 and 7 , respectively. In comparing Tables 4 and 5 with Tables 6 and 7, it is found that the sensitivity of surface parameters, for both the critical heat flux and the critical surface temperature response of the model, is generally independent of the meteorological forcing. Plots of PSI bars in Figs. 8 and 12 exhibit very similar trends under either clear or cloudy weather conditions. One prominent difference is the PSI of the three important canyon dimensional parameters—h, r, and fveg—when maximum Hcan is monitored. Both h and r have positive PSI for the Hcan response in the diurnal variation of a clear day (20 March), whereas their PSI was negative for the cloudy day. This indicates that the extremely high sensible heat flux arising from the canyon is likely to happen in areas with high building density (larger h and r values) during a clear day. In contrast, for a cloudy day (negative PSI), the model prediction of extreme Hcan response is likely to happen in suburban areas. Also note that the response of Hcan is highly sensitive to fveg during a cloudy day but is relatively insensitive during a clear day, showing that the vegetated surface plays an important role in regulating Hcan when it is cloudy. In addition, it is noteworthy that the response of critical total sensible heat has higher sensitivity to the thicknesses dW and dR during a clear day.

Table 6.

Estimates of PSI for monitored critical (maximum) diurnal fluxes with the atmospheric forcing conditions of 1 Jul 2010 (cloudy day).

Table 6.
Table 7.

Estimates of PSI for monitored critical (maximum) diurnal surface temperatures with the atmospheric forcing conditions of 1 Jul 2010 (cloudy day).

Table 7.

f. Statistical error

Subset simulation is much more numerically efficient than classic Monte Carlo simulation (MCS). In this section, we investigate the statistical error by computing the coefficient of variation (c.o.v.; c.o.v. is equal to the standard deviation/mean and is a normalized measure of the dispersion of the PDF) of exceedance probability estimates using 30 independent simulations. The results are plotted in Fig. 13. The number of samples used for the estimate of exceedance probability at different levels—namely, P(F) = 10−1, 10−2, 10−3, and 10−4—are NT = 500, 950, 1400, and 1850, respectively. For comparison, the c.o.v. of exceedance probability estimate produced by direct MCS is given by δ = [(1 − PF)/(PFNT)]1/2, which is also plotted in Fig. 13. We see that the c.o.v of direct MCS grows drastically with decreasing exceedance probability, indicating that the statistical error for small exceedance probability is high. In contrast, the c.o.v. of subset simulation increases much more slowly as the exceedance probability decreases. It is clear that, using the same number of samples, subset simulation yields much lower statistical error for exceedance probability estimation, as compared with direct MCS.

Fig. 13.
Fig. 13.

Coefficient of variation of exceedance probability estimates; c.o.v. is a normalized measure of dispersion of probability distributions.

Citation: Journal of Applied Meteorology and Climatology 50, 9; 10.1175/2011JAMC2685.1

g. Practical implication of parameter sensitivity

A direct consequence of anthropogenic stressors on urban areas, as manifested by the turbulent energy exchange and urban surface temperatures, is the so-called urban heat island (UHI) effect (Oke 1982). One important implication of this sensitivity study is related to the determination of strategies to mitigate urban heat island intensity. Oleson et al. (2010) investigated the effects of white roofs on UHI mitigation. They found that the annual mean heat island decreased by 33%, averaged over all urban areas, and pointed out that “changing roof albedo should have the largest impact on near-surface urban climate.” Their conclusion is confirmed by our study in the sense that the energetics of urban areas are indeed dominated by the presence of buildings (in particular, roofs) rather than impervious pavements. Options also include application of increasingly popular designs of “green” roofs with evaporative potential. In practice, however, increasing the albedo using white roofs can be done with relative ease as compared with other options (e.g., green roofs), because it only involves changing the material skin property.

The results of the sensitivity study can also be extended to provide guidelines for parametric studies and calibration of surface parameter involving UCM. Because field measurements of all surface parameters are rarely available for a particular application of UCM, knowing the model sensitivity relative to parameter uncertainties can greatly reduce the effort in parameter calibration procedures to yield better model predictions. A similar research work was recently conducted by Loridan et al. (2010) to assess the skill of a single-layer UCM in numerical weather prediction models, using a systematic statistical procedure. Their analyses started with a default set of urban surface parameters. In the process of optimizing model prediction of urban surface energetics, sensitivity of these uncertain urban surface parameters was also evaluated. It is therefore a “local” sensitivity analysis, in the sense that uncertain parameters were fine tuned to yield the optimal performance of the UCM as compared with measurement at specific sites, and the permissible variation of a specific parameter largely depends on tuning of the rest of the parameter space. The sensitivity analysis in this study is “global,” in the sense that the uncertain parameter space covers the entire range of physically possible values, weighted by density distribution functions and is not limited to any specific urban morphology or climate. In addition, the statistical sampling of any given uncertainty parameter in subset simulation is independent of the rest of the parameter space. It is noteworthy, however, that our broad conclusions are in agreement with Loridan et al. (2010); for example, both studies indicate that roof properties are significantly more important that canyon properties.

5. Concluding remarks

Subset simulation, an advanced Monte Carlo procedure, is used to quantify statistically the sensitivity of surface parameter uncertainty in a modified offline version of WRF-UCM. To evaluate the sensitivity of individual parameters, we devise a percentage sensitivity index that measures the deviation of means of conditional samples from the means of the predefined distribution. Results show that critical heat exchange between urban areas and the atmosphere is largely dictated by the presence of buildings and their thermal properties, whereas impervious pavement or vegetated ground has a relatively lower impact. Model output of both critical heat fluxes and surface temperature is highly sensitive to the uncertainties in urban geometry, characterized by the normalized building height and roof width. Uncertainties in thermal parameters and thickness of building enclosures (roofs and walls) largely modulate the model output from the corresponding surfaces. Variations in roughness lengths of roof and canyon also have a significant effect on the transport of energy and surface temperatures in urban areas. In contrast, surface emissivities and building interior temperatures, given the physically realistic range of variation, exhibit minimal influence on the UCM predictions. In general, the meteorological forcing in a UCM, which depends on weather conditions, has a relatively low impact on the characterization of parameter uncertainties. The results indicate that the sensitivity of the model to building morphology (h and r) is different on cloudy or clear days, however. It is also noteworthy that the anthropogenic heat, not explicitly included in this study, has an important impact on the urban surface energy balance. Its inclusion is similar to a local source term inside urban canopies and may be subject to variation of other canopy parameters. It is recommended that, for the result of this study to be applied to any specific site, it is preferable to evaluate the local anthropogenic heat whenever conditions permit. There are many foreseeable practical applications of this sensitivity study, for example, guidance to parametric studies involving UCMs and improving UHI mitigation strategies.

Acknowledgments

This work is supported by the High Meadows Sustainability Fund of Princeton University, the Mid-Infrared Technology for Health and the Environment (MIRTHE) NSF center at Princeton University, and the NSF under Grant CBET-1058027. We are also grateful to Sensorscope, LLC (http://www.sensorscope.ch/), the manufacturers of the Sensor Network over Princeton stations, for their assistance, especially in data management. The third author (S. K. Au) is supported by the Hong Kong Research Grant Council through General Research Fund 9041484 (CityU 110109).

REFERENCES

  • Abu-Hamdeh, N. H., and R. C. Reeder, 2000: Soil thermal conductivity: Effects of density, moisture, salt concentration, and organic matter. Soil Sci. Soc. Amer. J., 64, 12851290.

    • Search Google Scholar
    • Export Citation
  • ASHRAE, 2009: Material properties. 2009 ASHRAE Handbook: Fundamentals, American Society of Heating, Refrigerating and Air-Conditioning Engineers, 26.1–26.22.

    • Search Google Scholar
    • Export Citation
  • Au, S. K., and J. L. Beck, 2001: Estimation of small failure probabilities in high dimensions by subset simulation. Probab. Eng. Mech., 16, 263277.

    • Search Google Scholar
    • Export Citation
  • Au, S. K., and J. L. Beck, 2003: Subset simulation and its application to seismic risk based on dynamic analysis. J. Eng. Mech., 129, 901917.

    • Search Google Scholar
    • Export Citation
  • Au, S. K., J. Ching, and J. L. Beck, 2007a: Application of subset simulation methods to reliability benchmark problems. Struct. Saf., 29, 183193.

    • Search Google Scholar
    • Export Citation
  • Au, S. K., Z. H. Wang, and S. M. Lo, 2007b: Compartment fire risk analysis by advanced Monte Carlo simulation. Eng. Struct., 29, 23812390.

    • Search Google Scholar
    • Export Citation
  • Bou-Zeid, E., J. Overney, B. D. Rogers, and M. B. Parlange, 2009: The effects of building representation and clustering in large-eddy simulations of flows in urban canopies. Bound.-Layer Meteor., 132, 415436.

    • Search Google Scholar
    • Export Citation
  • Brutsaert, W., 2005: Hydrology: An Introduction. Cambridge Press, 605 pp.

  • Campbell, G. S., C. Calissendorff, and J. H. Williams, 1991: Probe for measuring soil specific heat using a heat-pulse method. Soil Sci. Soc. Amer. J., 55, 291293.

    • Search Google Scholar
    • Export Citation
  • Chen, F., and Coauthors, 2011: The integrated WRF/Urban modeling system: Development, evaluation, and applications to urban environmental problems. Int. J. Climatol., 31, 273288.

    • Search Google Scholar
    • Export Citation
  • Coceal, O., T. G. Thomas, I. P. Castro, and S. E. Belcher, 2006: Mean flow and turbulence statistics over groups of urban-like cubical obstacles. Bound.-Layer Meteor., 121, 491519.

    • Search Google Scholar
    • Export Citation
  • Defraeye, T., B. Blocken, and J. Carmeliet, 2010: CFD analysis of convective heat transfer at the surfaces of a cube immersed in a turbulent boundary layer. Int. J. Heat Mass Transfer, 53, 297308.

    • Search Google Scholar
    • Export Citation
  • Fernando, H. J. S., 2010: Fluid dynamics of urban atmospheres in complex terrain. Annu. Rev. Fluid Mech., 42, 365389.

  • Grimmond, C. S. B., and T. R. Oke, 1999: Aerodynamic properties of urban areas derived, from analysis of surface form. J. Appl. Meteor., 38, 12621292.

    • Search Google Scholar
    • Export Citation
  • Grimmond, C. S. B., and Coauthors, 2010: The International Urban Energy Balance Models Comparison project: First results from phase 1. J. Appl. Meteor. Climatol., 49, 12681292.

    • Search Google Scholar
    • Export Citation
  • Grimmond, C. S. B., and Coauthors, 2011: Initial results from phase 2 of the International Urban Energy Balance Model Comparison. Int. J. Climatol., 31, 244272.

    • Search Google Scholar
    • Export Citation
  • Hamdi, R., and G. Schayes, 2007: Validation of Martilli’s urban boundary layer scheme with measurements from two mid-latitude European cities. Atmos. Chem. Phys., 7, 45134526.

    • Search Google Scholar
    • Export Citation
  • Harman, I. N., M. J. Best, and S. E. Belcher, 2004: Radiative exchange in an urban street canyon. Bound.-Layer Meteor., 110, 301316.

  • Hastings, W. K., 1970: Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97109.

  • Johnson, G. T., T. R. Oke, T. J. Lyons, D. G. Steyn, I. D. Watson, and J. A. Voogt, 1991: Simulation of surface urban heat islands under ‘ideal’ conditions at night part 1: Theory and tests against field data. Bound.-Layer Meteor., 56, 275294.

    • Search Google Scholar
    • Export Citation
  • Kusaka, H., H. Kondo, Y. Kikegawa, and F. Kimura, 2001: A simple single-layer urban canopy model for atmospheric models: Comparison with multi-layer and slab models. Bound.-Layer Meteor., 101, 329358.

    • Search Google Scholar
    • Export Citation
  • Lemonsu, A., S. Belair, and J. Mailhot, 2009: The new Canadian urban modelling system: Evaluation for two cases from the Joint Urban 2003 Oklahoma City Experiment. Bound.-Layer Meteor., 133, 4770.

    • Search Google Scholar
    • Export Citation
  • Liu, Y., J. N. Chen, W. Q. He, Q. Y. Tong, and W. F. Li, 2010: Application of an uncertainty analysis approach to strategic environmental assessment for urban planning. Environ. Sci. Technol., 44, 31363141.

    • Search Google Scholar
    • Export Citation
  • Loridan, T., and Coauthors, 2010: Trade-offs and responsiveness of the single-layer urban canopy parametrization in WRF: An offline evaluation using the MOSCEM optimization algorithm and field observations. Quart. J. Roy. Meteor. Soc., 136, 9971019.

    • Search Google Scholar
    • Export Citation
  • Martilli, A., A. Clappier, and M. W. Rotach, 2002: An urban surface exchange parameterisation for mesoscale models. Bound.-Layer Meteor., 104, 261304.

    • Search Google Scholar
    • Export Citation
  • Mascart, P., J. Noilhan, and H. Giordani, 1995: A modified parameterization of flux-profile relationship in the surface layer using different roughness values for heat and momentum. Bound.-Layer Meteor., 72, 331344.

    • Search Google Scholar
    • Export Citation
  • Masson, V., 2000: A physically-based scheme for the urban energy budget in atmospheric models. Bound.-Layer Meteor., 94, 357397.

  • Metropolis, N., A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, 1953: Equations of state calculations by fast computing machines. J. Chem. Phys., 21, 10871092.

    • Search Google Scholar
    • Export Citation
  • Molders, N., and G. Kramm, 2009: Permafrost modeling in weather forecasts and climate projections. New Permafrost and Glacier Research, M. I. Kruger and H. P. Stern, Eds., Nova Science, 51–88.

    • Search Google Scholar
    • Export Citation
  • Nadeau, D. F., and Coauthors, 2009: Estimation of urban sensible heat flux using a dense wireless network of observations. Environ. Fluid Mech., 9, 635653.

    • Search Google Scholar
    • Export Citation
  • Niceno, B., A. D. T. Dronkers, and K. Hanjalic, 2002: Turbulent heat transfer from a multi-layered wall-mounted cube matrix: A large eddy simulation. Int. J. Heat Fluid Flow, 23, 173185.

    • Search Google Scholar
    • Export Citation
  • Nunez, M., and T. R. Oke, 1977: The energy balance of an urban canyon. J. Appl. Meteor., 16, 1119.

  • Oke, T. R., 1982: The energetic basis of the urban heat island. Quart. J. Roy. Meteor. Soc., 108, 124.

  • Oleson, K. W., G. B. Bonan, and J. Feddema, 2010: Effects of white roofs on urban temperature in a global climate model. Geophys. Res. Lett., 37, L03701, doi:10.1029/2009GL042194.

    • Search Google Scholar
    • Export Citation
  • Panofsky, H. A., and G. W. Brier, 1958: Some Applications of Statistics to Meteorology. The Pennsylvania State University, 224 pp.

  • Refsgaard, J. C., J. P. van der Sluijs, A. L. Hojberg, and P. A. Vanrolleghem, 2007: Uncertainty in the environmental modelling process—A framework and guidance. Environ. Modell. Software, 22, 15431556.

    • Search Google Scholar
    • Export Citation
  • Roberts, C., and G. Casella, 1999: Monte Carlo Statistical Methods. Springer, 680 pp.

  • Sailor, D. J., 2011: A review of methods for estimating anthropogenic heat and moisture emissions in the urban environment. Int. J. Climatol., 31, 189199.

    • Search Google Scholar
    • Export Citation
  • Sailor, D. J., and H. L. Fan, 2002: Modeling the diurnal variability of effective albedo for cities. Atmos. Environ., 36, 713725.

  • Skamarock, W. C., J. B. Klemp, J. Dudhia, D. O. Gill, D. M. Barker, W. Wang, and J. G. Powers, 2005: A description of the Advanced Research WRF version 2. NCAR Tech. Note TN-468+STR, 88 pp.

    • Search Google Scholar
    • Export Citation
  • Thunnissen, D. P., S. K. Au, and E. R. Swenka, 2007a: Uncertainty quantification in the preliminary design of a spacecraft attitude control system. AIAA J. Aerospace Comput. Info. Commun., 4, 902917.

    • Search Google Scholar
    • Export Citation
  • Thunnissen, D. P., S. K. Au, and G. T. Tsuyuki, 2007b: Uncertainty quantification in estimating critical spacecraft component temperatures. J. Thermophys. Heat Transfer, 21, 422430.

    • Search Google Scholar
    • Export Citation
  • Wang, Z. H., E. Bou-Zeid, and J. A. Smith, 2011: A spatially-analytical scheme for surface temperatures and conductive heat fluxes in urban canopy models. Bound.-Layer Meteor., 138, 171193.

    • Search Google Scholar
    • Export Citation
  • Zio, E., and N. Pedroni, 2009: Estimation of the functional failure probability of a thermal–hydraulic passive system by subset simulation. Nucl. Eng. Des., 239, 580599.

    • Search Google Scholar
    • Export Citation
Save