Remote measurements of spectral directional radiance have been used to estimate heat fluxes at heterogeneous land surfaces (e.g. Menenti, 2000). One active research field is the observation and modeling of sensible heat flux densities at land surfaces using remotely sensed surface temperature and albedo. The basis of this method has been classical one-dimensional resistance-type transport models in which sensible heat flux can be expressed as pa is the air density; Cp is the specific heat of air at constant pressure (=1005 J kg'1 K"1); Tair is the air temperature at a reference level; Tae0r is the surface aerodynamic temperature; and rah is the aerodynamic resistance to heat transfer and can be expressed in the near-surface layer

(Brutsaert,1982) as :

where ra is aerodynamic resistance for momentum, and re is a so-called 'excess resistance' which originally arises from the different transfer mechanisms for heat and momentum at the surface so that resistance to heat transport is higher than that to momentum transport. Transport resistances are parameterized as functions of a roughness length for momentum and a roughness length for heat transport. The 'excess resistance' may therefore be expressed in terms of kB"1 (Chamberlain, 1968):

where k is the von Karman's constant, z0m and z0h are the roughness lengths for momentum and heat transfer respectively. The roughness lengths are typically measured with moderate accuracy and the experimental error on kB"1 is large.

In classical single-source resistance-type models, Taero is derived from the extrapolation of the air temperature profile to the apparent canopy height (the displacement height + the roughness length) and may not actually exist or be measurable except for smooth surfaces (Norman et al, 1995a). For practical purposes radiometric surface temperature Trad is used in place of in such single-source heat transfer models. Radiometric surface temperature can be measured by a radiometer and is more appropriate to the application of remote sensing at various spatial scales. However, when using Trad instead of Taero in Eq.(l), an empirical adjustment must be made because Trad is not equal to Taer0, which results in an additional resistance added to the resistance term in a single-source model. The moderate accuracy of values makes it very difficult to determine the two terms of the correction separately. Therefore, when using Trad in single-source models, one can consider the 'excess resistance' in terms of kB'1 as a combination of adjustments which account for the difference between Zom and Zoh and the difference between and even though these two additional resistances are different conceptually. Most of the studies on 'excess resistance' has focused on the determination of (Table 1) and the values of (or are always related to the 'surface' temperature.

For most homogeneous 'permeable-rough' surfaces such as uniform and full canopy cover, is approximately 2 to 3 (Brusaert, 1982) and the single-source resistance methods have been applied successfully (Deardorff, 1978; Kustas, 1990). Over heterogeneous sparse canopies, however, widely varying values for kB"1 are found in literature (Kustas et al, 1989; Beljaars and Holtslag, 1991; Stewart et al, 1994) (see Table 1). This implies that the value of kB"1 cannot be approximated by a constant in case of sparse cover and it must be determined through calibration. A fixed value of kB"1 (or z0h) will introduce errors into the estimation of heat flux (Kohsiek et al., 1993, Stewart et al, 1994). Some authors related kB"1 to surface wind speed and the difference between surface temperature and air temperature (Kustas et al., 1989). It seems that regressing kB"1 with wind speed and difference of surface and air temperature does not provide a general formula for any sparse canopy. Consequently, it is difficult to develop a simple method to relate kB" to surface properties.

Recently, efforts have been made to develop dual or multi-source models to estimate sensible heat flux and evaporation from partial canopies (Choudhury and Monteith, 1988; Kustas, 1990; Lhomme et al, 1994; Norman et al, 1995) so that the empirical adjustment of resistance in single-source models can be avoided. Vegetation and the substrate (i.e. the soil), in fact, interact separately with the air in the canopy space hereby affecting the sensible and latent heat flux densities in sparsely covered canopies especially when the temperatures of the cooler vegetation and the warmer soil surface are significantly different. Consequently, resistances between vegetation (foliage) and the air in the canopy space and between the soil and the air in the canopy space have to be parameterized in dual-source models. The difficulty with this approach to account for the mechanisms determining heat transfer in the vicinity of leaves and soil is that in the canopy space there is no defined surface layer, i.e. there is no defined vertical structure and no horizontal homogeneity. We propose a different conceptual model of heat transfer in the canopy space to describe separately heat exchanges between leaves, soil and air.

In our dual-source model, component temperatures have to be known. Multi-angle and multi-channel remote sensing technology such as The Along-Track Scanning Radiometer (ATSR)-l/2 on board the European Remote Sensing Satellites (ERS)-l/2 provides an opportunity to extract component temperatures from directional measurements of existance (Menenti et al 1999). A new dual-source model is developed in our study and used to estimate sensible heat fluxes based on component temperatures for incomplete canopy cover both at field scale and at regional scale. As mentioned above, our model is different from other authors' in the resistance scheme and is simplified. It is applicable at regional scale where meteorological information near surface is not always available for each pixel.

Table I. Values of kB'1 at various (sparse) surfaces

Projector site)

Surface type



Reference author(s)

Homogeneous and full

vegetated canopy

2 or 3






Beljaars&Holtslagfl 991)







Open forest



as above





as above





as above

Owens Valley




as above

Smith Creek Valley




as above

Smoke Creek Desert




as above

La Crau




as above

Owens Valley




Kustas et al,(l 989)

HE1FE Gobi

Gobi with Shrubs



this study


Desert site




this study

Modeling heat fluxes from soil and vegetation temperatures 2. THEORY

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