## Introduction

Much interest has been generated in the use of remotely sensed variables to estimate surface heat fluxes. Urban areas, both in terms of surface materials and morphometry, provide a contrast to the wholly or partially vegetated systems that thus far have been the primary focus of study. In this study, observed surface sensible heat fluxes for a light industrial area within the city of Vancouver, British Columbia, Canada, are compared with modeled values based on bulk transfer or microscale variability approaches. The bulk transfer approach represents the surface as a single entity and employs bulk transfer equations applicable at the local scale (the bulk response) (e.g., Kustas et al. 1989). The microscale variability approach represents the surface as multiple entities, which then are aggregated within the local scale area (e.g., Sun and Mahrt 1995a).

The site chosen consists almost entirely of bluff-body elements with very little vegetation cover. The bluff-body elements differ less in their composition from the plane surface upon which they are built than is normally the case for vegetation upon a soil surface. The general simplicity of the bluff elements (buildings) allows them to be represented as solid rectangular objects. Thus, the surface configuration can be simplified to a six-component system based upon geometric configuration and materials: roofs, four walls of given orientations, and ground-level surfaces. This approach makes the system more complex than the two-component system that may be applied to vegetation (e.g., Norman et al. 1995; Sun and Mahrt 1995a). The increase in surface components is offset by the ability to obtain detailed temperature observations representative of these surfaces, however.

The main objective of this study is to investigate bulk transfer and microscale temperature variability approaches for estimating sensible heat fluxes from an urban surface and compare the results with measured fluxes and results from other studies. In the implementation of both of these methods, we incorporate the flux source area model of Schmid (1994, 1997) to define appropriate source areas on the urban surface for extracting surface temperatures from the remotely sensed imagery. We also implement methods intended to provide more representative urban surface temperatures that explicitly include vertical surfaces within the flux source areas, and we assess the spatial variability of these temperatures and their effect upon modeled heat fluxes. We combine these procedures to produce estimates of the parameter *kB*^{−1} (defined below) over a simple, relatively unvegetated urban surface.

This paper reviews the modeling approaches in section 2 and describes the study site, observations, and methods of temperature representation in section 3. The results from the bulk surface modeling and microscale variability approach are contained in sections 4b and c, respectively. All variables used in this paper are described in the appendix.

## Modeling approaches

### The bulk surface response

*r*

_{b}is defined (Verma 1989):

*r*

_{am}is the aerodynamic resistance for momentum,

*r*

_{ah}is the aerodynamic resistance for heat,

*k*is von Kármán’s constant (0.4),

*u*∗ is the friction velocity, and Ψ

_{H}is the stability correction for heat. The sensor height is

*z*

_{s}. The level at which the wind speed extrapolates to zero, via the logarithmic wind profile, is at

*z*

_{d}+

*z*

_{0m}(the zero-plane displacement length plus the roughness length for momentum). Similarly, the aerodynamic surface temperature

*T*

_{0}is the temperature extrapolated down to a surface that is at the height

*z*

_{d}+

*z*

_{0h}(roughness length for heat). These surfaces are depicted graphically in Fig. 1. The local-scale surface sensible heat flux

*Q*

_{H}then can be calculated from the bulk transfer equation:

*Q*

_{H}

*C*

_{a}

*T*

_{0}

*T*

_{a}

*r*

_{ah}

*T*

_{0}

*T*

_{a}

*C*

_{H}

*U,*

*C*

_{a}is the volumetric heat capacity of air,

*T*

_{a}is the air temperature,

*C*

_{H}is the exchange coefficient for heat, and

*U*is the wind speed. To implement (2), both

*T*

_{0}and

*z*

_{0h}are needed, which are difficult to measure (Troufleau et al. 1997). However, it is possible to measure a radiometric surface temperature

*T*

_{R}and to replace

*T*

_{0}in (2) (Stewart et al. 1994) so that

*Q*

_{H}

*C*

_{a}

*T*

_{R}

*T*

_{a}

*r*

_{h}

*r*

_{ah}is replaced with

*r*

_{h}(Stewart et al. 1994):

*r*

_{h}

*r*

_{ah}

*r*

_{r}

*r*

_{r}is the radiometric excess resistance (Fig. 2).

*kB*

^{−1}(Owen and Thomson 1963), where

*B*

^{−1}is a dimensionless parameter. The aerodynamic definition is

*r*

_{h}is calculated as

*r*

_{h}=

*r*

_{am}+

*r*

_{T}(Stewart et al. 1994), where

*r*

_{T}=

*r*

_{b}+

*r*

_{r}represents the resistance between

*z*

_{0m}and the surface (Fig. 2). Furthermore, the stability corrections for momentum and heat (Ψ

_{M}and Ψ

_{H}), which should be included in

*r*

_{b}[Verma 1989, Eq. (13)], rarely are incorporated. Thus, (5) is redefined as

*R*and

*T*make explicit the dependence of the results on the use of radiometric temperatures and the method for calculating the resistance [e.g., see also Troufleau et al. (1997)].

*kB*

^{−1}, which rely to a greater or lesser extent on measured fluxes. These methods may be categorized broadly as (a) those that use Reynolds and sometimes also Prandtl numbers and (b) those that use additional information about the surface, for example, the anisothermal roughness length for heat [Brutsaert and Sugita 1996, Eq. (18), hereinafter referred to as BS96]. An example of the first approach is the Brutsaert [1982, Eq. (5.29)] equation for bluff-rough situations:

*z*

_{0h,B82}

*z*

_{0m}

^{0.25}

_{∗}

*z*

_{0m}

*u*∗/

*ν*is the roughness Reynolds number, with a kinematic molecular viscosity

*ν*of 1.461 × 10

^{−5}m s

^{−1}.

*z*

_{0h}=

*z*

_{0hR,T}for four periods of growth of a mixed pasture and conclude that “

*z*

_{0h}=

*z*

_{0hR,T}is not quite acceptable” for the entire stage of vegetation growth. They obtain the lowest correlations in the earliest and latest stages. In their approach,

*z*

_{0h}was determined from Brutsaert’s (1979) theoretical model. Although they provide a linear regression relation, it does not have general utility. BS96 have developed an equation based on Brutsaert’s (1979) model that expresses the roughness length for heat in terms of canopy structure, turbulence, and temperature. This approach takes into account the anisothermal heating, which is expressed as a function of the isothermal roughness length

*z*

_{0h,I}with

*z*

_{h}is the height of the roughness elements, and

*r*

_{2}= [

*a*− (

*a*

^{2}+ 4

*C*

_{2})

^{0.5}]/2. The anisothermal scalar roughness length is then expressed as

*ρ*is air density,

*c*

_{p}is the specific heat capacity of air,

*T*

_{g}is the potential surface temperature near the ground surface,

*T*

_{h}is the potential surface temperature near

*z*

_{h}(here taken to be the roof surface temperature),

*b*is an extinction coefficient that ranges between 1 (isothermal) and 7 (steep temperature gradient) (BS96), and

*a*is an extinction coefficient in the exponential shear stress profile that is about 1 for a sparse canopy and 3 for a dense canopy (BS96). Here the complete aspect ratio

*λ*

_{c}is used in coefficient

*C*

_{2}:

*λ*

_{c}is the ratio of the complete surface area (the three-dimensional areas of vegetation and buildings) to the plan area, calculated using the method of Voogt and Oke (1997). The transfer coefficient Ct

_{f}is given by

_{f}

*C*

_{L}

^{−m}

_{∗}

^{−n}

*C*

_{L},

*m,*and

*n*are parameters that may depend on the shape and orientation of the roughness elements, and intensity of the turbulence which, following Verhoef et al. (1997), were set at 0.25, 0.25, and 0.36, respectively; and a Prandtl number Pr of 0.7 at 293 K (Verhoef et al. 1997) was used.

*w*in (9) is a weighting parameter, which ensures that there is not undue variability in

*z*

_{0h}from the measurement strategy of the surface temperature. Practical measurement considerations usually dictate that

*T*

_{0}as required by (2) is replaced by a remotely sensed radiative temperature

*T*

_{R}. If the surface is aerodynamically smooth and flat,

*T*

_{0}may equal

*T*

_{R}. However, when the surface consists of an array of variously oriented surface elements, exposed to a wide array of irradiances and wind speeds, as in the urban environment,

*T*

_{0}will not equal

*T*

_{R}(Huband and Monteith 1986; Troufleau et al. 1997). In addition, significant measurement problems with

*T*

_{R}are incurred because of anisotropy of the emitted thermal radiation from rough surfaces when viewed by a narrow field-of-view instrument (Voogt and Oke 1997). According to BS96, the scalar roughness for an anisothermal surface (9) will remain the same as its isothermal value (8) when

To estimate a representative radiative temperature of the urban surface for the bulk surface representation, we first use the flux source area model of Schmid (1994, 1997) to define the area of the surface, which influences the tower-based measurement of *Q*_{H}. Then, within the source area, airborne and ground-based radiometric temperatures are combined in a geographic information system (GIS) framework to estimate a representative radiative source area temperature that combines temperatures of both horizontal and vertical surfaces. Further details are provided in section 3e. Following the specification of the surface temperature within the source area, we then can use the measured values of *Q*_{H} to back-calculate *kB*^{−1}.

*C*

_{H}is replaced with

*C*

_{HR}. Variation in

*C*

_{HR}can be explained by

*T*∗ is a temperature scale, and

*H*is the kinematic heat flux. Based on the analysis of a number of tower- and aircraft-derived datasets of sensible heat flux and surface radiative temperature, Sun and Mahrt (1995b) found

*a*

_{SM}= 0.11, and

*a*

_{SM}= 0.12 when only the aircraft data are used. The differences are suggested to be due in part to different averaging methods. This result suggests that

*C*

_{HR}can be derived independently of the roughness length for heat. The single empirically determined curve suggested by (14) includes both the height dependence of the temperature difference Δ

*T*and the effect of variable vegetation (surface) cover on Δ

*T.*However, it is suggested that these dependencies do not strongly affect the coefficient

*a*

_{SM}. The roughness of the surface influences

*u*∗ but exerts only a secondary influence on

*C*

_{HR}, which can be expressed in terms of the drag coefficient

*C*

_{D},

*a*

_{SM}=

*C*

^{0.5}

_{D}

*z*

_{0hR,T}

*c*

_{1}

*c*

_{2}

*T*

*T*

*c*

_{1}and −0.37 for

*c*

_{2}.

### Microscale surface variations

*C*

_{HR}can be calculated by assigning the roughness length for heat equal to that of momentum in (2). This equivalence avoids the simultaneous tuning of the relation for aerodynamic temperature and thermal roughness length (Sun and Mahrt 1995a). The surface (within the precalculated source area in this case) is divided into

*N*different surface types, each with fractional area coverage

*f*

_{i}such that

*T*

_{R,i}) by

*ĝ*

_{i}is the normalized heat conductance (the inverse of resistance) for the

*i*th surface type, and

*g*

_{i}is the conductance for surface type

*i*such that

*ĝ*

_{i}

*f*

_{i}in (18) is a weighting function subject to

*f*

_{r}), north wall (

*f*

_{n}), east wall (

*f*

_{e}), south wall (

*f*

_{s}), west wall (

*f*

_{w}), sunlit street (

*f*

_{su}), and shaded street (

*f*

_{sh}). If vegetation is introduced, the fractions become more numerous. This approach can be simplified to consider the fraction that is sunlit (

*F*

_{sun}) or shaded (

*F*

_{sha}), subject to the constraint

*F*

_{sun}+

*F*

_{sha}= 1. The mean radiative temperature for the surface is

## Study description

### Study area

The study area is a light industrial (LI) area (approximately 0.65 km^{2}) within the city of Vancouver. The LI area is characterized by one–three-story buildings, most with flat roofs, and a notable lack of vegetation (plan area cover <5%). The buildings are arranged in city blocks, which have an east–west/north–south orientation, with the major block axis oriented east–west. Alleys run east–west between each street. Several blocks have buildings that share a common east/west wall, thereby reducing the exposed wall area. The average complete-to-plan area ratio (*A*_{c}/*A*_{p}; also referred to as the complete aspect ratio *λ*_{c}) is 1.4. The complete area *A*_{c} is the total area of the urban surface, which includes all horizontal and vertical surfaces, and *A*_{p} is the area of plan (horizontal) surfaces.

A database of building plan area was created by digitizing building outlines from aerial photographs and planning maps. The database includes 733 buildings. Building heights were estimated from the number of floors. The vector database was converted to a raster image with a 1 m × 1 m resolution, and separate images of building roofs and walls (assumed to be of a north, south, east, or west orientation only) were created. For the analyses here, a reduced subset of the study area centered about the flux observation tower is considered. This area (which includes 517 buildings) extends 400 m west of the tower and 300 m to the north, south, and east, reflecting the predominant wind direction during the day, which varies from southwest to northwest. The average building height in this area is 6.9 m, with a standard deviation of 2.5 m. The distribution of building heights in the entire study area is presented in Voogt and Oke (1997).

### Flux observations

Measurements were conducted in the framework of the surface energy balance. Instruments were mounted on a 28.5-m tower. Ancillary measurements of temperature, humidity, wind speed, and direction were made from a 9-m tower in close proximity. Net allwave radiation was measured using a Radiation and Energy Balance Systems, Inc., (REBS) Q*6 net pyrradiometer, and the turbulent sensible and latent heat fluxes were measured directly using the eddy correlation approach. The fast-response instruments, mounted less than 0.15 m apart, consisted of a Campbell Scientific, Inc., (CSI) one-dimensional sonic anemometer and fine-wire thermocouple system (CA27) to measure the fluctuations of vertical wind velocity and temperature and a CSI krypton hygrometer (KH20) to measure those of absolute humidity. The vertical wind velocity, air temperature, and humidity fluctuations were sampled at 5 Hz. Covariances were determined for 15-min periods. Flux corrections were made for oxygen absorption and air density (Webb et al. 1980; Tanner and Greene 1989; Tanner et al. 1993). No corrections were made for frequency response or spatial separation of the sensors. Fluxes and ambient meteorological conditions were measured for a period of 15 days, 11–25 August 1992, inclusive.

### Flux source areas

A flux source area (or flux footprint) is the effective area on the ground that influences the measurement of a scalar flux. The model used here is the flux source area model (FSAM) described by Schmid (1994, 1997). This model calculates a source weight distribution function for a horizontal plane at the level of the zero-plane displacement height. This function defines the relative contribution of ground-level point sources to a flux measurement made at a specified height and integrates to unity (Schmid and Lloyd 1999). The interpretation of this function is that a flux measured by a sensor can be represented as the weighted average of the point sources that make up the surface and that an appropriate averaging is defined by the convolution of the source area function with the surface source strength distribution (e.g., here temperature), by using the source area function as a linear, low-pass filter.

The source area model requires as input *z*_{s}, *z*_{0m}, *z*_{d}, *L* (Obukhov length), and the acrosswind turbulence near the surface *σ*_{υ}/*u*∗ (Schmid 1994), where *σ*_{υ} is standard deviation of crosswind velocity. Dimensions of the source area are dependent upon (in order of sensitivity):*z*_{s}/*z*_{0m} (increases with height), stability [decreases as (*z*_{s} − *z*_{d})/*L* decreases], and *σ*_{υ}/*u*∗ (increases for increased crosswind turbulence) (Schmid and Lloyd 1999). The model assumes the surface to be flat and impermeable and does not take into account canopy geometry or within-canopy flows. Rough surfaces are handled simply through the general specification of the roughness length and zero-plane displacement (Schmid and Lloyd 1999). The radiative effects of vertical surfaces are included in the analysis by calculating an area-weighted urban surface temperature within the source area from off-nadir and vehicle-traverse temperature data (see section 3e).

FSAM source weight functions were calculated for the hourly observed conditions at the site for times that bracket the time of the remote-sensing overflight. This procedure yields two source areas for each of the three overflights for which surface temperature data are acquired. Typical input conditions for the majority of midday and late-afternoon times tested were *z*_{0m} = 0.26 m, *z*_{d} = 3.1, *σ*_{υ}/*u*∗ = 2.0–3.0 and (*z*_{s} − *z*_{d})/*L* = −0.68 to −0.95, which yield source areas ranging from 18 × 10^{3} to 32 × 10^{3} m^{2}.

An example of a source weight function overlain on the composite thermal imagery is shown in Fig. 3, in which the source weight function values take on darker tones for low values and are more transparent for higher values. These tones represent the three-dimensional shape of the source weight function, which exhibits an increasing value upwind of the sensor along the mean-wind direction to a maximum and then decreases further upwind, with a symmetrical Gaussian shape in the crosswind direction.

### Input variables

The variables used in calculations were derived in the following manner.

Roughness parameters were calculated using Raupach’s (1992, 1994, 1995) method, as reported in Grimmond and Oke (1999a). The average values are

*z*_{0m}≈ 0.26 m and*z*_{d}≈ 3.1 m.Friction velocities, with stability corrections, were derived from tower measurements of

*U*(log-law corrected to 30 m),*T*_{a}, and*Q*_{H}with an iterative solution as reported in Grimmond and Cleugh (1994).Stability corrections for momentum and heat are the Paulson (1970) stability functions. For Ψ

_{M}, when*L*is less than 0, the Högström (1988)-modified Dyer (1974) equation was used, and when*L*is greater than 0, the van Ulden and Holtslag (1985) equation was used. For Ψ_{H}, the Högström (1988)-modified Dyer (1974) equations were used.

### Surface temperature

Radiative surface temperature data were collected using a combination of airborne and ground-based instrumentation to ensure adequate sampling of the major urban surface-component temperatures. A thermal scanner (AGEMA 880 long wave band), temporarily mounted in a helicopter, was used to obtain thermal images of the study area from both nadir and 45° off-nadir viewing angles in each of four viewing directions orthogonal to the main street orientation. Flights were made three times during 15 August 1992 (denoted as year/yearday 92/228) and four times in the period 1500 Pacific Daylight Time (PDT) 24 August 1992 (92/237)–0500 PDT 25 August 1992. Only daytime imagery is used in this study.

Imagery was corrected for atmospheric effects using locally launched radiosondes. Coincident ground-based sampling of select surfaces was performed as a check on the accuracy of the imagery. A summary of the flight details is provided in Table 1. Further details of the system configuration, flights, and correction procedures are provided in Voogt and Oke (1998a).

Corrections for surface emissivity were applied following image compositing (see below). In the absence of high-resolution data on surface type and radiative properties, a uniform surface emissivity of 0.95 was assumed (Arnfield 1982). The hemispheric incoming sky radiance in the scanner wave band was estimated using the quadrature method of Lacis and Oinas (1991), with sky radiance modeled at 5 cm^{−1} intervals using LOWTRAN-7 (low-resolution transmittance model and code, Kneizys et al. 1988) and the observed vertical profiles of temperature and humidity.

Building-wall temperatures in the study area were sampled using an array of infrared radiometers mounted on a vehicle (Voogt and Oke 1998b) that traversed all streets and alleyways in the study area. Corrections for surface emissivity were applied to the vehicle-traverse data using the methods outlined in Voogt and Oke (1998b). These take into account the effect of sensor position and the temperatures of the canyon facets. In addition, a set of fixed infrared sensors was used to monitor continuously the individual facets of select buildings near the flux observation tower.

Individual nadir images cover a limited area, so an image composite of the LI area was made. This procedure consists of identifying numerous (generally between 15 and 50) control points in both the nadir imagery and building footprint database, followed by resampling the imagery onto a common 1 m × 1 m template. A visual inspection of the remapped images was used to assess the suitability of the procedure. The final nadir image composite for flights 1–3 consists of 10–20 images. Off-nadir image composites were also made for each of the four view directions made. The image composite for flight 9 required only six images because of the much higher aircraft altitude.

#### Surface temperature representations for urban areas

The composite thermal images provide high-resolution, spatially extensive surface temperature data. However, the images are limited in that each composite is associated with a particular view direction, which misses or undersamples portions of the “complete” urban surface (the full ground–atmosphere interface). Ideally, temperature information is required for all surfaces that make up the atmosphere–ground interface. Voogt and Oke (1997) generated estimates of the complete surface temperature *T*_{c} by combining frequency distributions of temperature obtained from airborne and ground-based observing systems, which were used to represent horizontal and vertical (wall) surfaces, respectively. Single estimates for the entire study area were calculated. To estimate *T*_{c} for a flux source area, a new method was developed (Voogt 2000) using the remotely sensed data and a surface database (GIS). This method can be used to derive representations of surface temperature that take into account the three-dimensional structure of the buildings. Each of the representations combines nadir thermal imagery with temperatures of building walls derived from the vehicle traverses. Off-nadir imagery can also be used to estimate the building wall temperatures; these two methods show generally good agreement, with some differences observed because of the relative sensor position (Voogt and Oke 1997).

##### Area-weighted thermal images

Area-weighted thermal images were constructed in which individual pixels represent the area-weighted temperature *T*_{aw} for both the horizontal and vertical surfaces associated with that pixel. The image was created by combining the building database, wall temperatures derived from vehicle traverses, and the composite nadir thermal imagery. Images of area-weighted temperature show cooler pixels that represent building edges, because those pixels represent not only the horizontal area of exposed roof but also the area of the wall below. Pixels that form corners of buildings are the most highly affected. Pixels that do not include a wall component are unaffected. Variations in the general method can be incorporated; for example, we have tested area-weighted temperature images that add only the projected windward wall areas of the buildings to the plan imagery.

##### Complete surface temperature of source areas

A complete surface temperature for a given spatial domain is the area-weighted temperature for an area that is sufficiently large to include the dominant surface structure (e.g., a building or block length). It therefore requires (a) the complete area within the domain and (b) temperatures of all surfaces within the domain. Here, the spatial domain used is that of the calculated flux source area.

*T*

_{c}is based upon prior calculation of an area-weighted thermal image. The area-weighted temperature image is multiplied by the image of total pixel area to recover the correct pixel-by-pixel temperature–area product. The sum of these pixels over the selected domain is divided by the total area of the domain to estimate

*T*

_{c}. An unweighted spatial average of the area-weighted temperature image is not equivalent to

*T*

_{c}. Notationally, we have

*A*

_{i}and

*T*

_{i}represent the area and temperature of surface elements

*i*within a pixel containing

*n*surface elements, and the outer summations are performed over the spatial domain defined by pixels

*j*= l,

*N,*which cover the flux source area. The numerator of the inner bracketed term in (23) represents the pixel area-weighted temperature image. The source weight distribution function can be incorporated in (23) by multiplying both the numerator and denominator of the inner bracketed term by the appropriate source weight for each pixel.

The final temperature representation used (*T*_{cb}) calculates a complete surface temperature for each building in the study area (rather than the flux source area itself) and applies the results to the plan area of each building. The remaining (plan) area temperature is specified by the nadir thermal composite image. This method provides an alternate temperature representation that may be convolved with the source weight distribution function to estimate a temperature within the source area.

## Results

### Surface heat flux observations

The synoptic conditions on both study days (92/228 and 92/237) were characterized by generally clear, sunny skies; warm temperatures; and a sea-breeze circulation—typical of the conditions for the full 15-day period during which energy balance measurements were made (Fig. 4). Lower-than-average air temperatures on day 237 can be explained by the passage of a cold front that day, which subsequently stalled and developed into a stationary front over central British Columbia. Air temperature on day 228 followed average conditions very closely until 0600. For the next 4 h (0600–1000), temperatures were warmer than average, reverting back to average values at 1100 with the onset of the sea breeze.

At this site, the sensible (*Q*_{H}) and storage heat fluxes (Δ*Q*_{S}) are the most significant output fluxes. As expected, the latent heat flux *Q*_{E} is very small. Daytime *β* values (*Q*_{H}/*Q*_{E}) were approximately 2.5. Grimmond and Oke (1999b) present a fuller discussion of the fluxes at this site relative to other urban areas.

In terms of conditions on the specific study days, the high sensible heat flux on day 228 at 0900 was the largest measured at that time during the study period. For much of the rest of day 228, however, *Q*_{H} was depressed relative to the ensemble. The surface temperatures (east and south walls especially) were warmer than average, as was the internal wall surface temperature (one facet was measured from an IR thermometer positioned inside the building). This result is consistent with the higher Δ*Q*_{S} flux observed that day. There were no synoptic disturbances that can explain the variability that was observed; the day was characterized by clear conditions of increasing pressure following the passage of a surface low and cold front two days earlier. This kind of hour-to-hour and day-to-day variability in fluxes is evident in energy balance data collected from a wide array of urban sites and is the subject of further investigation (see discussion in Grimmond and Oke 1999b). The sensible heat fluxes on day 237 (Fig. 4) were much more typical of the average conditions during the measurement period.

### Results from the bulk approach

#### Surface temperatures from different methods

The bulk approach adopted in this paper is based upon a remotely measured surface temperature. In the analysis here, directional radiometric surface temperatures are modified to incorporate corrections for (a) atmospheric effects, (b) bulk surface emissivity effects, and (c) anisotropy (directional variations in observed emitted radiation caused by microscale patterns of temperature induced by the three-dimensional surface structure). Corrections for (a) and (b) are standard practice [e.g., see techniques and discussion by Byrnes and Schott (1986); Wan and Dozier (1989); Desjardins et al. (1990); Prata (1994)]. The term “bulk” as applied to surface emissivity denotes the application of a single average emissivity rather than a pixel-by-pixel varying surface emissivity.

Because of the rough nature of the urban surface and the combination of viewing angles for a remote sensor, a large range of directional radiometric surface temperature measurements are possible (see section 3e). Figure 5 illustrates the range of temperature values calculated for the source areas using various definitions of the “surface” temperature. Two general categories of temperature estimates exist: 1) direct surface estimates from remote sensors (nadir and off-nadir temperature estimates), and 2) area-weighted temperatures (those which include information on the temperature of the building walls). Anisotropy is represented in Fig. 5 by the variability of directional radiometric temperatures (i.e., those that include atmospheric and emissivity corrections but do not take into account the surface structure) for the five view directions. All temperatures have been weighted by the source area weighting function applicable at the time for the tower-based measurements. These temperatures show a range of up to 5°C near midday (Fig. 5) at the scale of the flux source area. This range highlights the importance of surface structure and the effective anisotropy of surface thermal emissions.

Problems associated with anisotropy are addressed by estimating a radiometric temperature that includes the effects of vertical and horizontal surfaces (i.e., *T*_{aw}, *T*_{c}, or *T*_{cb}). The effect of including vertical surfaces into the source area temperatures yields a decrease in temperature; decreases are from 0.5° to 1°C when area-weighted temperatures are compared with nadir values, 2°–3.5°C when *T*_{cb} is used, and 2°–5°C when *T*_{c} is used.

##### Spatial variability

In this section, the spatial variability of the bulk surface temperature representation (temperature calculated for the domain of a flux source area) is assessed. Flux source area position depends upon the wind direction, and the source area size is a function of measurement height, thermal stability, and crosswind turbulence intensity and the surface structure (Schmid 1994).

To evaluate the effect of source area position upon the source area–averaged temperature, two tests were performed. In the first, a flux source distribution function (and flux source area) is calculated for a given set of conditions (here we use those for 1400 PDT, 92/228). This source area is rotated about the tower to simulate the source area position for different wind directions (but similar measurement height, stability, crosswind turbulence, and general surface structure). The source weight distribution function then is convolved with the image of *T*_{aw}, and the resultant source area temperature is recorded. These results are shown as empty circles in Fig. 6a. The frequency distribution of *T*_{aw} (from individual pixel values) for the entire study area (solid line) as well as within the calculated flux source area (dashed line) is also plotted for comparison. A similar test is performed by using source areas calculated for each flight on 92/228 and by calculating *T*_{c} for each source area position (Fig. 6b). Results for flight 9 are also shown in Fig. 6b; these results display the average nadir temperature from the source area, because wall surface temperatures were available for only a single building on that day.

The second test uses the calculated source weight distribution function as a filter. This filter is applied to the entire image, and the resultant image is analyzed. This process is analogous to a low-pass filtering operation, except the weights for the filter array in this case are not equal but rather are specified from the source weight distribution function (see also Schmid and Lloyd 1999). Temperatures from the resultant image are extracted (edge effects are masked) and are plotted as the frequency distribution represented by shaded bars in Fig. 6a. To speed the filtering operation and to accommodate the filter size, the image and filter resolution is degraded to 5 m × 5 m.

The results of these two tests show spatial variability of up to 5°C near midday based upon source area position. This variability is reduced later in the day when the overall range of surface temperatures narrows and the source area increases in size. The rotation test shows a consistent pattern related to the temperature structure of the ground. Temperature minima are observed when the source area fraction of shaded area increases from either the height and/or spacing of the buildings (Fig. 6c). Local maxima in temperature are observed for directions in which there are shorter buildings with large plan-to-complete area ratios and/or for which street areas dominate. Some of the source area temperature variability could be traced to individual buildings with extreme temperature values such as a very hot roof or a roof with a low surface emissivity, not accounted for in the correction procedure. These tests demonstrate that using the source area concept is an important step in specifying the most appropriate surface temperature for use in the bulk heat transfer equation.

Source area size is coupled to atmospheric and surface conditions. Thus, direct observation of the variability of source area temperature for changing source area dimensions is not strictly possible without thermal imagery taken under each of the conditions to be tested. In the absence of this coupled information, a test of the variability of *T*_{c} within various domain sizes, which have as their starting point different components of the urban surface, was performed. For simplicity, square domains ranging from 3 m × 3 m to 251 m × 251 m were used with equal weightings applied to all points. The domain midpoints were selected to be middle of a street intersection, middle of a roof, midpoint of an east–west alleyway, and the flux observation tower. In accordance with previous results (Schmid and Oke 1992), the majority of variability is at small scales, in this case well within the source area size dimension (Fig. 7). Notably, most of the temperature variation is confined to length scales less than 60 m. Variability in *T*_{c} at different spatial scales for this study area is similar to that of directional temperature measurements (e.g., *T*_{nadir}). This similarity results from the fact that the dominant control on the variability is the surface structure, and the breakpoints at which major temperature changes occur (building, lot, and block scales) remain the same. The effect of including building wall temperatures does not alter the range of observed surface temperatures, nor does it add a new spatial scale of temperature variability.

#### Derived coefficients

Calculation of *kB*^{−1} was performed using three methods.

Method 1 (

*kB*^{−1}|_{R,T}). Eq. (3) is used with*r*_{h}=*r*_{am}+*r*_{T}, and an explicit*r*_{b}in (1) is neglected. This approach is the most commonly used method (Stewart et al. 1994) that provides a direct back-calculated value.Method 2 (

*kB*^{−1}|_{B82}). Here*z*_{0hB82}is determined from (7) proposed by Brutsaert (1982) for bluff-rough surfaces.Method 3 (

*kB*^{−1}|_{I},*kB*^{−1}|_{A}). Here*z*_{0h,I}is determined for the isothermal case using (8), and*z*_{0h,A}for the anisothermal case is determined using (9).

When method 1 is used, the values of *kB*^{−1}|_{R,T} range from 10 to 27 (Fig. 8a) and clearly depend on the surface temperatures used. In general, the values calculated with nadir and off-nadir surface temperatures are higher than those that use the complete temperature values. A diurnal pattern is evident in the *kB*^{−1}|_{R,T} values calculated, similar to that observed elsewhere [for a review see Verhoef et al. (1997)]. Higher values occur in the afternoon, generally rising later in the day (Fig. 8a). The smaller *z*_{0hR,T} (larger *kB*^{−1}|_{R,T}) values are associated with the largest (*T*_{R} − *T*_{a}) values but not the largest *Q*_{H}.

For the nadir thermal imagery (without consideration of wall temperatures), *kB*^{−1}|_{R,T} values are mostly in the range 13–27. The use of *T*_{c} reduces *kB*^{−1}|_{R,T} by 3–6. Values derived from off-nadir directional temperatures span a wide range in accordance with the variability of surface temperature as seen in the various view directions. The use of the off-nadir temperature in the direction of the most shaded facet provides *kB*^{−1}|_{R,T} values that are approximately the same as the *T*_{c}-based estimates. This result is in accordance with the similarity between the surface temperatures obtained using these two methods (Voogt and Oke 1997). This use of off-nadir temperatures to estimate the overall surface temperature is similar to the oblique measurements recommended by BS96. The variation in the values of *kB*^{−1}|_{R,T} using *T*_{c} or nadir temperatures when the source areas are rotated about the tower position is 4–5 (Fig. 8a), with the value from the actual wind direction being within ±1 standard deviation in all cases except at 17 h, the time period with the smallest standard deviation. This result suggests that the range of values that would be obtained from the general area around the tower would be within the other sampling error (<1.5).

The *kB*^{−1}|_{R,T} values reported here for the light industrial site are large relative to those determined for natural or agricultural surfaces. Most reported values of *kB*^{−1}, derived for vegetated surfaces, range from 1 to 10. A frequently cited value for natural surfaces is 2 (see, e.g., Stewart et al. 1994; Verhoef et al. 1997). Methods 2 and 3 provide an independent assessment of the appropriate size of *kB*^{−1}. Using Method 2, *kB*^{−1}|_{B82} is simply a function of Re*, so the data plot on to the bluff rough curve from Brutsart (1982) (Fig. 8b). For comparison purposes, the permeable rough curve from Stewart et al. (1994) also is included. The *kB*^{−1}|_{B82} values of 25–27 obtained from (8) are at the high end of those obtained from Method 1 (Fig. 8b). However, they are more reasonable than those from the permeable rough curve. Using Method 3 (BS96), it is possible to begin to address the issue of the effect of the combination of surface and sensor angle geometry, which changes as the solar zenith angle varies through the day. As the sun rises, the solar radiation penetrates deeper into the canopy (urban canyon), and the surface temperature increases. In the urban environment, because of the bluff-body nature of the roughness elements, there is a strong directional dependence (Fig. 4 and Fig. 5) in this heating. However, in this industrial/commercial area with very little vegetation, we do not see the situation that others have observed in which the bottom surface temperature exceeds the temperatures at the top of the canopy (Fig. 9a). This disparity is because the upper layer, in this case roofs of buildings, is not transpiring. Therefore, it continues to heat up through the day (Fig. 9a). In addition, although the top layers of the canopy are where most of the turbulent exchange takes place (BS96), in this environment the vertical components of the canopy (walls) also are important. The temperature difference between the roof (*T*_{h}) and the ground (*T*_{g}) decreases through the day (Fig. 9a), whereas *T*_{R} − *T*_{a} has more of a parabolic diurnal trend, with the maximum in the middle of the day. Throughout the day, *T*_{R} − *T*_{h} is negative, indicating that the roof temperature is much larger than the composite surface temperature (irrespective of method used to determine *T*_{R}). The size of the difference, however, decreases through the day. The difference between *T*_{R} and *T*_{g} is positive in the morning, but, by the late afternoon, with most methods it also is negative (Fig. 9a).

When *z*_{0hR,T} is calculated from (3) and plotted against solar elevation (Fig. 8d), we find the following:

With use of

*T*_{nadir}, the afternoon data show a similar pattern to BS96 (their Fig. 3) with*z*_{0hR,T}decreasing with increasing solar elevation. However, the morning value of*z*_{0hR,T}(at a zenith angle of 51°) is the largest (based on BS96 it would be expected to be of a similar size to that obtained late in the afternoon).The off-nadir west-facing sensor (with a view angle of 45°) shows the least variability with solar elevation. This result suggests that back-calculated values should be the least biased by solar elevation.

The off-nadir north-facing sensor is the only view angle that shows an increase in

*z*_{0hR,T}in the afternoon with increasing solar elevation; all others show a decrease.

We can use the BS96 method as expressed in (8) to determine the isothermal *z*_{0h,I} value, which should be independent of solar elevation and instrument geometry used to determine *T*_{R}. To use this method, a value needs to be assigned to *a* (all other terms are assigned values as outlined in section 2a). The present environment is extremely sparse in terms of vegetation but not in terms of roughness elements. The surface, however, is not densely covered by roughness elements. Based on this fact, *a* values of 1, 1.5, and 2 were assigned (from sparse to more dense canopy), which results in *z*_{0h,I} values of 10^{−5}, 10^{−7}, and 10^{−9} m and *kB*^{−1}|_{I} values of 11, 16, and 21, respectively (Fig. 8c). These values fall in the general range of the *kB*^{−1}|_{R,T} values (Fig. 8c).

It then is possible to determine the isothermal surface temperature (*T*_{I}) that would produce the observed *Q*_{H} in (2) with *z*_{0h,I} used in (1). Figure 9b shows the relation between *T*_{I} and *T*_{R} stratified by the *a* value assigned and by the method used to determine *T*_{R}. The ideal method to determine *T*_{R} is the one most consistent with *T*_{I}. This reasoning also indicates which *z*_{0h,I} value is the most appropriate. Based on the mean differences and the mean absolute differences (Table 2), it is likely that the most appropriate value for *a* is 2, and the *T*_{R} method that most closely follows *T*_{I} is the off-nadir sensor that is west facing. If we relax the assessment to the mean difference *T*_{aw} all have mean values that are, on average, less than 1°C different (but with std dev that are larger than 2.5°C).

The time period when the west-facing off-nadir radiometric surface temperature is most different from *T*_{I} is at 10 h (Fig. 9b). Therefore, it is likely that the *kB*^{−1} value at 10 h (Fig. 8a) probably is too low and the appropriate value for this environment is closer to that at 14 and 17 h; for *kB*^{−1}|_{R,T} this value is 24, and from *kB*^{−1}|_{I} it is 21. These values correspond to *z*_{0hR,T} of 7.5 × 10^{−11} and *z*_{0h,I} of 4.2 × 10^{−10} m, respectively.

With use of (12) to determine the weighting factor *w,* the BS96 anisothermal *z*_{0h,A} can be determined from (9). The results, shown in Fig. 8c, have a range similar to the *kB*^{−1}|_{R,T} values.

The bulk radiative heat exchange coefficient *C*_{HR} ranges in size from 1.43 × 10^{−3} to 3.38 × 10^{−3} (Fig. 10a). When compared with *C*_{HR} as in (14), the coefficient *a*_{SM} is calculated as 0.111. This value compares favorably with the value Sun and Mahrt (1995b) derived for their data. However, it is important to note that the data reported here are for a much smaller range of conditions than those considered by Sun and Mahrt (1995b). Thus, the full extent of variability in this environment is not documented.

The radiometric roughness lengths for heat (*z*_{0hR,T}) are extremely small, ranging from 10^{−4} to 10^{−12} m; those *z*_{0h,I} predicted by (8) are on the order of 10^{−9} m (Fig. 8d). This result suggests that similiarity theory is predicting physically unrealistic values to compensate for the inadequacy of the stability dependence of the exchange coefficient or aerodynamic resistance [as documented previously by Sun and Mahrt (1995b)]. These small values have been found also by others, for example, Sugita and Brutsaert (1990) and Malhi (1996). The values determined here are likely to be close to the extreme because of the lack of vegetation at the site.

When the relation between *z*_{0hRT} and −Δ*T*/*T*∗ is analyzed, after Sun and Mahrt (1995b) [(12)], the coefficients are found to be *c*_{1} = 3.22 and *c*_{2} = −0.40 (Figure 10b). These values are similar to those derived by Sun and Mahrt (1995b), showing consistency, once again, with other environments.

##### Modeled *Q*_{H}

It clearly is inappropriate to use the back-calculated *kB*^{−1}|_{R,T} values to evaluate the ability to calculate *Q*_{H}, and insufficient data exist to evaluate independently the performance of the *Q*_{H} model. Thus, our approach here is to investigate the implications of assigning particular *kB*^{−1}|_{R,T} values on modeled *Q*_{H} [using (3) and *r*_{T} from (6)]. This approach is of more general relevance, because those wishing to undertake this kind of modeling in urban environments in situations where measured *Q*_{H} is not available will likely select just one, or a limited range, of *kB*^{−1}|_{R,T} values. Our approach is to assign a range of *kB*^{−1}|_{R,T} values (2–60) and then to determine the average absolute difference between the observed and the modeled fluxes *Q*_{H} for the closest hour of observations. Note that all of the measurements will have errors that would result in different values. These errors are not considered here; others already have addressed this issue (see, e.g., Verhoef et al. 1997).

First, the effect of the method to determine bulk surface temperatures is considered (Fig. 11a). Day 237 only has nadir temperatures, thus for all other methods only the average of the three flights on day 228 is presented. The lowest overall ^{−2}) were obtained using the west-facing off-nadir method source area temperature, with a *kB*^{−1} of 21 (Fig. 11a). The second-lowest minimum *kB*^{−1}|_{R,T} of 23–26, ^{−2}. The minimum *kB*^{−1}|_{R,T} of 17 (for *T*_{c}) and 25 (off-nadir north facing). The choice of appropriate *kB*^{−1}|_{R,T} value, that is, the one that minimizes the overall difference between measured and modeled heat fluxes, is sensitive to the off-nadir view direction of the surface. This result is consistent with the findings of BS96. Generally, the nadir and off-nadir results have a lower overall difference with measured data than those modeled using a pixel-by-pixel approach (*T*_{cb}, *T*_{c}). The largest difference is obtained using *T*_{c} (20.1 W m^{−2}).

If we ignore the method to determine *T*_{R} and just average the results of all methods together and examine variations through the course of the day, interesting patterns emerge (Fig. 11b). As noted earlier, based on the discussion of Fig. 8a, a diurnal trend in *kB*^{−1}|_{R,T} is evident. The difference *kB*^{−1}|_{R,T} of 13 for the first flight, and at 22–23 for flights 2 and 3. The afternoon data on day 237 are very similar, although not identical, to those on day 228. The smaller *T*_{R} data. If we consider results for just one temperature method, namely, the nadir method *T*_{R}, the minimum ^{−2} (Fig. 11c). Thus, the difference in results between times (Fig. 11b) is greater than that between methods (Fig. 11a). This result is supported when we consider all of the methods but only on day 228. It also is notable, at the later period (228/17), that the range of *kB*^{−1}|_{R,T} values with a ^{−2} becomes larger. This result implies that the effect of selecting a slightly different *kB*^{−1}|_{R,T} value is less for any of the temperature methods at this time.

Figure 11d summarizes the results across all times and methods and illustrates the relative error in *Q*_{H} for a given *kB*^{−1}|_{R,T} is 21–23. However, the error variation around this range (Fig. 11d) is small;*kB*^{−1}|_{R,T} values from 19 to 24 all yield an error less than 20%. This result indicates that, if the general fluxes are of interest rather than specific values for a given time, the approach is fairly robust and insensitive to the exact *kB*^{−1}|_{R,T} value selected. These values correspond well with those predicted using the BS96 *kB*^{−1}|_{I} with *a* equal to 2 (Fig. 8c, 20–22).

We can consider the implications of the exact area that is used to determine *T*_{R}. For the four flights, the area for which the remotely sensed data were collected is large enough to sample a series of different directions (source areas) at 10° intervals around the tower (228/10: 35 directions, 228/14: 36, 228/17: 36, and 237/16: 25). The calculated source area is placed over the nadir thermal composite image, and the temperatures are weighted by the source weight distribution function to determine a representative temperature for the flux source area. This temperature is used with the observed *Q*_{H} and *T*_{a}, which are assumed to represent reasonable local-scale estimates. Differences between measured and modeled fluxes ^{−2} at *kB*^{−1}|_{R,T} of 11–12 for 228/10, at *kB*^{−1}|_{R,T} of 17 for 228/14, at *kB*^{−1}|_{R,T} of 19–24 for 228/17, and at a *kB*^{−1}|_{R,T} of 21–24 for 237/16, when measured *Q*_{H} was 125 W m^{−2}. If, for each of the individual directions, we consider the minimum difference (<10 W m^{−2}) between measured and modeled values, this difference occurs with a narrow range of *kB*^{−1}|_{R,T} values [(*T*_{c}) 9–11, 17–18, 22–26, and (*T*_{nadir}) 22–25, for the four times, respectively]. Therefore, we can conclude that, for this study site under the conditions analyzed, the exact area that is sampled is not that critical; that is, the results spatially are very consistent.

Last, if *z*_{0h,I} is used to model *Q*_{H} (using all *T*_{R} assignment methods or using the off-nadir west-facing *T*_{R} data), as expected, the *a* is 2 and are greatest when *a* is 1 (Table 3). There is a reduction in *n* is only 3). This result suggests that using a BS96 *z*_{0h,I} value would generate reasonable results, but assigning the appropriate *a* value will have an effect on the size of the errors. To assign the appropriate parameter ahead of time requires additional studies of different urban areas.

### Microscale variability

An alternative approach to model *Q*_{H}, proposed by Sun and Mahrt (1995a), uses temperature gradients and conductances for surfaces within the domain of interest. The relative simplicity of the study area and assessment of the temperature distribution allows a generalization of the surface into a limited number of components, which then can be used to assess this method. The surface fractions considered are plan surfaces of roof, sunlit street and shaded street, and each of the wall directions.

#### Surface temperatures

The urban surface is heterogeneous at small scales, with large variations in surface temperature occurring over short distances (Figs. 3 and 12). Frequency distributions of image temperature are multimodal in character. Peaks within the distribution are distinguished most strongly in the morning (Voogt and Oke 1998a) and can be related to surface type. Figure 12 illustrates the frequency distribution of plan surface temperatures for flight 1. Despite the wide range in surface temperature, the distributions suggest an underlying reduction of the surface to a few main components. The uppermost peak of the surface temperature distribution is related to roof temperature, the middle peak to open sunlit streets, and the lowest peak to fully shaded surfaces. The component temperature distributions are broadened somewhat by misregistration of pixels because of the simplified building outlines used, errors in compositing the thermal imagery, and very low emissivity surfaces. In general, the surface temperature in the shaded areas is only slightly warmer than the air temperature, especially prior to solar noon, whereas the sunlit areas are much warmer. Thus, the heat flux probably is dominated by the sunlit area.

Separation of the shaded and sunlit street surface temperatures was accomplished using a limiting temperature for shaded surfaces. This choice was made over a more independent calculation of shaded areas using the building database and solar geometry because of the simple three-dimensional structure represented by the building database. The disadvantage of the method is misclassification of shaded surfaces. This problem is most apparent following solar noon when hot surfaces with large heat capacities are now shaded yet exhibit relatively high surface temperatures in comparison with surfaces that have been shaded throughout the day, for example, east-facing wall temperatures (Fig. 4) and street temperatures (Fig. 12) in the afternoon. Despite this problem, temperature bounds for shaded surfaces, extracted from inspection of temperature distributions, yield qualitatively correct shading patterns when combined with the masks of roof and street surfaces. Misregistration of pixels because of low surface emissivity is minimized by first masking off building roofs, which are much more likely to exhibit low surface emissivity values.

The fraction of area that is sunlit (*F*_{sun}) and shaded (*F*_{sha}) varies through time (Table 4) and also depends on the source area size and position. This variation is caused by the change in both the vertical and the horizontal areas that are sunlit. In terms of the vertical components, clearly, east-facing walls are sunlit first and west-facing walls are the last to be sunlit. The shadows that are cast on the low-lying horizontal surfaces, primarily roads, change as the sun’s position changes through the day. However, the roof area remains primarily sunlit throughout the daylight hours. The fractional significance of this is dependent on the height of the walls. In this area, in which there are large numbers of warehouse-type buildings, the wall area is less significant than if the site was a downtown area with taller buildings. The ratio of wall to complete area in the study site is 0.3, the ratio of roof to complete surface area is 0.3, and the remaining 0.4 is streets (open ground). The fractional roof area is also dependent upon the source area dimensions and position. Figure 13 shows some variability of fractional roof area as source area size increases, even at the scale of the calculated source areas for day 228. In addition, the calculated fraction can be sensitive to the center position of the source area, as indicated by the difference in the calculated fractions for different starting points that have the same source area.

#### Variation in conductances/resistances because of surface characteristics

Using the methods of Sun and Mahrt (1995a), the effect of the surface characteristics is considered (see section 2b), with the surface divided into sunlit (*F*_{sun}) and shaded (*F*_{sha}) fractions. The fractions vary with time (Tables 4 and 5), with *F*_{sun} increasing from 0.45 to 0.57 through the day. Five methods (reference numbers given in square brackets) were used to determine *C*_{HR}: [1] (1) with *z*_{0h} = *z*_{0m}, [2] (1) with *z*_{0h} = 0.1(*z*_{0m}), [3] Malhi [1996, (30)] with *z*_{0h} = *z*_{0m}, [4] Malhi [1996, (30)] with *z*_{0h} = 0.1(*z*_{0m}), and [5] (1) with *z*_{0h} = *z*_{0h,I} (with *a* = 2). The *z*_{0h} = *z*_{0m} assumption follows Sun and Mahrt’s assumption; the *z*_{0h} = 0.1(*z*_{0m}) is a common a priori assumption for rough environments, based on Brutsaert (1982). Case 5 is based on the results presented in section 4b (2).

The component conductances and the aerodynamic surface temperatures *T*_{0}, determined from (2), are given in Table 5 for each time period and case. We summarize the results as follows.

(a) In cases 1–4, *T*_{0} generally is similar in size to *T*_{sha};the differences between *T*_{0} and *T*_{sun} are larger (10–23 K). The maximum *T*_{0}–*T*_{a} differences are less than 4 K (case 2).

(b) In cases 1–4, *T*_{0} is much smaller than for case 5.

(c) In case 5, the *T*_{0}–*T*_{a} differences are 13–18 K, and *T*_{0}–*T*_{sha} differences are 11–19 K, with the larger *T*_{0}–*T*_{a} differences associated with larger *T*_{0}–*T*_{sha} differences.

(d) In some situations the conductances become negative. In cases 1–4, these situations occur when *T*_{0} is less than *T*_{sha}. However, in case 5, this situation occurs when *T*_{0} is greater than *T*_{sun} (228/10).

(e) Case 2 produces no negative conductances.

(f) In cases 1–4, the shade conductance *g*_{sha} is larger than the sunlit conductance *g*_{sun}. The opposite is true for case 5.

The results of this work clearly fall into two groups:those with the larger *z*_{0h} values (cases 1–4), and case 5. In the first instance, the results are very different from Sun and Mahrt’s (1995a) study of a sunlit black spruce tree canopy and shaded ground at 1300 LST, in which *T*_{0} was very similar to the treetop temperature. On the other hand, the case-5 results are similar both in terms of the relative temperature differences (large *T*_{sha} − *T*_{0} and small *T*_{sun} − *T*_{0}) and the ratio of the two conductances (*g*_{sun}/*g*_{sha}), which for their study was on the order of 16. Here we can look also at two other time periods, in addition to the middle of the day. In the late afternoon, the ratio of the conductances has dropped to about 2. In the morning, the method predicts a negative *g*_{sha} value that clearly leads to a negative ratio, although the absolute magnitude is similar to that of the middle of the day.

*H*

*F*

_{sun}

*g*

_{sun}

*T*

_{sun}

*T*

_{a}

*F*

_{sha}

*g*

_{sha}

*T*

_{sha}

*T*

_{a}

*H*and the effect of the size of the temperature gradients (Δ

*T*

_{sun}=

*T*

_{sun}−

*T*

_{a}and Δ

*T*

_{sha}=

*T*

_{sha}−

*T*

_{a}) on

*g*

_{sun}and

*g*

_{sha}are investigated (Fig. 14). Only positive conductance values (physically reasonable) are shown. From Fig. 14, it is clear that, for conditions around

*g*

_{sha}of 0.01–1 s m

^{−1}, it is possible to have physically unreasonable

*g*

_{sun}values with small changes in ambient conditions. These results suggest that there are wide ranges of conditions under which this method will fail, not just limited to those we encountered in this study. If Δ

*T*

_{sha}becomes negative, then both

*g*

_{sun}and

*g*

_{sha}increase in size.

As noted earlier (section 4a), the *Q*_{H} values were low on day 228. If the entire measurement period ensemble *Q*_{H} values are used in the calculations instead (which also influences the *C*_{H} values used) with all other inputs held constant, *g*_{sun}/*g*_{sha} becomes slightly larger in the afternoon. However, *g*_{sun} values remain negative for cases 1, 3, and 4, and *g*_{sha} is negative for case 5.

From this kind of analysis, it clearly is important to have a reasonable estimate of *z*_{0h}. The isothermal value from BS96 seems to provide a good first approximation. However, the results do show that there are problems with the method at certain times of the day. These problems may be related to the definition of the fractions of the surface. For example, the case-5 negative value could have been avoided if the sunlit fraction had been defined as only the roof (Tables 3 and 4).

## Conclusions

Bulk and microscale methods of estimating surface sensible heat flux have been examined for an urban area almost devoid of vegetation and characterized by rectangular buildings that act as bluff-rough elements. The explicit recognition of temperatures of vertical surfaces often undersampled by nadir remote sensors is included in this analysis. This inclusion has the effect of reducing the surface temperature by 2–7 K during the daytime observation period.

For this environment, a reasonable estimate for *kB*^{−1}|_{R,T} appears to be about 20–27, which is larger than those observed over vegetated and agricultural surfaces and suggests extremely small *z*_{0h} values. This range represents the results obtained by three independent methods. The values determined for the bluff-rough curve (Brutsaert 1982) provide the largest values. At the lower end are those values predicted by the isothermal method of Brutsaert and Sugita (1996). Intermediate values (21–23) are obtained from the back-calculation method. From the analysis presented in Fig. 11d, the implications for the likely size of error in *Q*_{H} can be determined. From this analysis, we conclude that, for this kind of urban environment with little vegetation, an appropriate *kB*^{−1} value could be obtained by using the Brutsaert (1982) method as expressed in (7) or the isothermal method of Brutsaert and Sugita (1996) as expressed in (8). However, the assessment of the appropriate *a* value for (8) (Brutsaert and Sugita 1996) in this environment requires further investigation.

The exact *kB*^{−1}|_{R,T} value varies, depending on the method used to determine the surface temperatures. The off-nadir west-facing temperature data provided the lowest error in modeled *Q*_{H} in this study. Generally, the nadir and off-nadir results have a lower overall difference with measured data than those modeled using a pixel-by-pixel approach (*T*_{aw}, *T*_{cb}, *T*_{c}). There is a very small difference between methods to determine wall temperatures.

Diurnal variations in *kB*^{−1}|_{R,T} are evident. However, the suggestion by Brutsaert and Sugita (1996) that these variations may be due in part to solar elevation also appears to be a reasonable explanation. Solar geometry plays an important role in creating urban local- and microscale climates. This is particularly the case at this light industrial site because of the large size of the roughness elements, the orderly location of the elements relative to the solar path, the large building surface area with large thermal inertia, and low evaporative ability. Consequently, this site provides an opportunity to consider the BS96 suggestion that the variability in *z*_{0h} may be due to the interaction of solar and surface geometry. In this study, the combined role of geometry is considered in three different ways, as follows.

*Bulk surface temperature:*A series of different combinations of solar and surface geometries are obtained through use of different bulk surface temperature representations. Representations include a range of directly observable radiometric temperatures from nadir and off-nadir view angles as well as calculated composite temperatures that incorporate all surfaces.*Partitioning of the surface by position:*The surface is divided, and temperatures are determined for the ground level and*z*_{h}level. In an urban area (without trees), the*z*_{h}level is the roof. This representation neglects the building walls. In a vegetated area, a temperature determined for*z*_{h}will inherently involve some sort of integrated response of the top layers of the canopy, given the porous nature of vegetative canopies and the view angle of the sensor. The BS96 anisothermal model partitions the surface in this way.*Partitioning of the surface by radiative condition:*In this case, all aspects of the surface area are taken into account but relative position is not. Thus, the sunlit fraction constitutes roof, some ground, and some wall areas. The actual areas and fractions vary through time and have different thermal histories, which, because of the thermal inertia, can be significant. This type of partitioning is used in our application of the Sun and Mahrt microscale variability method.

The coefficients proposed for relations with −*T*∗/Δ*T* by Sun and Mahrt (1995b) from a range of vegetated surfaces are shown to agree well with our results for a dry urban environment. This agreement suggests that these empirical relations may hold across a wider range of conditions.

In this environment, when the surface is subdivided into sunlit and shaded fractions of the surface, the estimates of subcomponent conductances do not produce meaningful results throughout the day. We attribute this to the fact that, during the later portion of the day, there are surfaces (both horizontal and vertical) that are no longer sunlit but are still very warm. Thus, the anisothermal behavior suggested by BS96 and identified here in the bulk values may explain also the inability to model the microscale conductances throughout the day. Clearly, this result has implications in other environments where strong thermal inertia may be evident. We also conclude that the initial assumption of the size of *z*_{0h} is important for this method. In this environment with large bluff bodies, the initial assumption of Sun and Mahrt (1995a) is not appropriate.

Our results show that first-order estimates of heat fluxes can be estimated using this approach, but careful attention needs to be directed to the method by which surface temperatures are defined and determined and to the diurnal variability of model parameters, notably *kB*^{−1}|_{R,T}. Clearly more research in more complex urban environments is needed.

## Acknowledgments

We thank our many field assistants, Dr. T. R. Oke for his advice on many aspects of this work, and Dr. H. P. Schmid for assistance with FSAM. We also thank the three anonymous reviewers for their useful comments and suggestions on an earlier version of the manuscript. This research was funded by NSERC (Voogt) and a USDA Forest Service cooperative grant (Grimmond).

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## APPENDIX

### List of Symbols

Note: all temperatures *T* are potential temperatures.

*A*_{c}complete surface area*A*_{i}area of surface type*i**A*_{p}plan (horizontal) surface area*A*_{roof}roof area*A*_{st(sha)}shaded street area*A*_{w}wall area*a*extinction coefficient in the exponential shear stress profile*a*_{SM}coefficient for*C*_{HR}relationship from Sun and Mahrt (1995b)*B*^{−1}dimensionless parameter =*r*_{b}*u*∗ = ln(*z*_{0m}/*z*_{0h})/*k**B*^{−1}|_{R,T}dimensionless parametedimensionless parameter =*r*_{T}*u*∗ = ln(*z*_{0m}/*z*_{0hR,T})/*k**b*extinction coefficient for the vertical canopy temperature profile*C*_{a}volumetric heat capacity of air*C*_{D}bulk drag coefficient for momentum*C*_{H}bulk exchange coefficient for heat*C*_{HR}bulk radiometric exchange coefficient for heat*C*_{L}fitting parameter for Ct_{f}*c*_{1}coefficient for use with (16)*c*_{2}coefficient for use with (16)*c*_{p}specific heat capacity*F*_{sha}total shaded fraction of the surface*F*_{sun}total sunlit fraction of the surface*f*_{i}fractional area of coverage by surface type*i**g*bulk surface heat conductance*g*_{i}surface heat conductance for surface type*i**ĝ*_{i}normalized surface heat conductance for surface type*i**g*_{sha}surface heat conductance for shaded surfaces*g*_{sun}surface heat conductance for sunlit surfaces*H*kinematic heat flux*k*von Kármán constant*kB*^{−1}the natural logarithm of the ratio of roughness length of heat to momentum*kB*^{−1}|_{B82}*kB*^{−1}derived using Brutsaert (1982) [Eq. (7)]*kB*^{−1}|_{R,T}*kB*^{−1}derived using*z*_{0hR,T}(back-calculated value) [Eq. (3)]*kB*^{−1}|_{I}*kB*^{−1}derived using*z*_{0h,I}(BS96 isothermal case) [Eq. (8)]*kB*^{−1}|_{A}*kB*^{−1}derived using*z*_{0h,A}(BS96 anisothermal case) [Eq. (9)]*L*Obukhov length*m*fitting parameter in Ct_{f}*N*number count*n*fitting parameter in Ct_{f}Pr Prandtl number

*Q** net allwave radiative flux*Q*_{E}latent heat flux*Q*_{H}sensible heat fluxaverage absolute difference between observed and modeled sensible heat fluxes Δ

*Q*_{S}storage heat fluxRe* roughness Reynolds number

*r*_{2}coefficient for BS96*r*_{ah}aerodynamic resistance to heat transfer between the height*z*_{d}+*z*_{0h}and*z*_{s}*r*_{am}aerodynamic resistance for momentum*r*_{b}bulk aerodynamic excess resistance*r*_{h}aerodynamic resistance to heat*r*_{r}excess resistance from different heat source and momentum sink locations*r*_{T}excess resistance between*z*_{0m}and the surface*T*_{R}*T*_{0}aerodynamic surface temperature*T*_{a}air temperature*T*_{aw}area-weighted radiometric surface temperature for an image pixel*T*_{c}complete (or area weighted) radiometric surface temperature for a specified spatial domain*T*_{cb}complete (radiometric) temperature for an individual building*T*_{g}surface temperature near the ground*T*_{h}surface temperature near*z*_{h}*T*_{i}temperature of surface element*i*within a pixel*T*_{I}isothermal surface temperature*T*_{nadir}nadir (directional) radiometric temperature*T*_{R}radiometric surface temperature*T*_{sun}radiometric sunlit surface temperature*T*_{sha}radiometric shaded surface temperature*T*∗ temperature scale = −*H*/*u*∗*U*mean horizontal wind speed*u*∗ friction velocity*w*weighting fraction for surface temperature*z*_{0h}roughness length for heat*z*_{0h,B82}roughness length for heat, calculated using Brutsaert (1982) method [Eq. (7)]*z*_{0h,A}anisothermal scalar roughness length*z*_{0h,I}isothermal roughness length for heat*z*_{0hR,T}radiometric roughness length for heat, calculated using*r*_{T}*z*_{0m}roughness length for momentum*z*_{d}zero-plane displacement length*z*_{h}height of the roughness elements*z*_{s}sensor or measurement height*β*Bowen ratio*λ*_{c}complete aspect ratio =*A*_{c}/*A*_{p}*ν*kinematic molecular viscosity*ρ*density of air*σ*standard deviation*σ*_{ν}standard deviation of the crosswind velocityΨ

_{H}stability correction for heatΨ

_{M}stability correction for momentum

Summary of helicopter flights over the study area (year is 1992)

Differences between isothermal temperature and surface temperature (determined by a range of methods) (K). The values are the average (and standard deviation *σ*) of absolute differences (|*T*_{I} − *T*_{R}*T*_{I} − *T*_{R}).

Differences between measured and modeled *Q*_{H} (W m^{−2}) using *z*_{0}_{h, I} for all *T*_{R} methods and for off-nadir west *T*_{R}.

Component surface temperatures and the fraction of the flux source area that they represent for 1992 yearday 228

Conductances determined for sunlit and shaded fractions of the surface (see section 2b). Methods used to determine *C _{HR}* are [1] (1) with

*z*

_{0h}=

*z*

_{0m}, [2] (1) with

*z*

_{0h}= 0.1(

*z*

_{0m}), [3] Malhi [1996, (30)] with

*z*

_{0h}=

*z*

_{0m}, [4] Malhi [1996, (30)] with

*z*

_{0h}= 0.1(

*z*

_{0m}), and [5] (1) with

*z*

_{0h}=

*z*

_{0h,I}(with

*a*= 2). Assumptions are surface partitioned into seven fractions: roof (

*ƒ*), north wall (

_{r}*ƒ*), east wall (

_{n}*ƒ*), south wall (

_{e}*ƒ*), west wall (

_{s}*ƒ*), sunlit street (

_{w}*ƒ*), and shaded street (

_{su}*ƒ*

_{sh}