8 Evaporation
177
(1994). They used an empirical relationship between (Tfoliage - Tsoil) and (Trad -
TaiT) to estimate the resistance for sensible heat transport. Kustas (1990) compared
a single source model, using two different procedures (to estimate the kB-1 parameter) to correct for the difference (Trad - TaeT) with a dual-source model. Better
agreement with observations was obtained for the single source model using a
theoretical equation to estimate the kB-1 parameter.
These two examples lead to a preliminary conclusion: relatively simple models
of heat transfer at the land atmosphere interface can be sufficiently accurate if a
few key-variables can be determined with sufficient accuracy and capturing the
fundamental physics of the process, as in the case of the (Trad - TaeT) difference.
8.5 Spatial variability
The work reviewed in the preceding pages indicates that spatial variability of
evaporation, and of the land surface variables determining it, is at the same time a
problem and an asset. The methods exploiting image context (e.g Price, 1990;
Bastiaanssen, 1995) take advantage of range of variability to derive additional
equations. On the other hand, spatial variability makes it impossible to use methods such as the linear relationships, which apply to small changes in TTad around
the reference value. Even more to the point, spatial variability implies that the spatial resolution of observations has to be selected on the basis of the spatial structure of the landscape.
Kustas et al. (1990) applied the LSEB approach of Moran et al. (1989) to different agricultural fields and found that errors were of the order of 2 mmd- I , which is
significant. Better accuracy was achieved in another study in a semiarid rangeland
basin by Kustas et al. (1994 a,b). To compute sensible heat flux the factor kB-1 was
calculated with an empirical equation determined earlier with observations collected in the same basin. Spatial extrapolation was done with a scheme similar to
the one described by Menenti (1979, 1984) based on the estimate of first order
differentials relative to the values of state variables and fluxes observed at a reference site. In the study of Kustas et al. (1990) spatial extrapolation was done using
the differentials with respect to surface temperature and albedo. Changes in roughness lengths for momentum and heat transfer were neglected and the values at the
reference site kept constant over the watershed. Analysis of observations did show
that this assumption was not correct, since the empirical coefficient used to estimate the roughness length for heat transfer had to be changed significantly on each
day used for the study.
Most of the equations necessary to model exchanges of momentum and heat at
heterogeneous land surfaces are non-linear. Spatial averaging of the independent
variables in these equations has therefore a potentially large impact on the variables calculated with the non-linear equations. This is also the case when using
observations at a spatial resolution insufficient to capture the spatial structure of
the landscape. More precisely, if the spatial resolution is larger than the smaller
homogeneous element in the landscape, spatial averaging will always contribute to
177
(1994). They used an empirical relationship between (Tfoliage - Tsoil) and (Trad -
TaiT) to estimate the resistance for sensible heat transport. Kustas (1990) compared
a single source model, using two different procedures (to estimate the kB-1 parameter) to correct for the difference (Trad - TaeT) with a dual-source model. Better
agreement with observations was obtained for the single source model using a
theoretical equation to estimate the kB-1 parameter.
These two examples lead to a preliminary conclusion: relatively simple models
of heat transfer at the land atmosphere interface can be sufficiently accurate if a
few key-variables can be determined with sufficient accuracy and capturing the
fundamental physics of the process, as in the case of the (Trad - TaeT) difference.
8.5 Spatial variability
The work reviewed in the preceding pages indicates that spatial variability of
evaporation, and of the land surface variables determining it, is at the same time a
problem and an asset. The methods exploiting image context (e.g Price, 1990;
Bastiaanssen, 1995) take advantage of range of variability to derive additional
equations. On the other hand, spatial variability makes it impossible to use methods such as the linear relationships, which apply to small changes in TTad around
the reference value. Even more to the point, spatial variability implies that the spatial resolution of observations has to be selected on the basis of the spatial structure of the landscape.
Kustas et al. (1990) applied the LSEB approach of Moran et al. (1989) to different agricultural fields and found that errors were of the order of 2 mmd- I , which is
significant. Better accuracy was achieved in another study in a semiarid rangeland
basin by Kustas et al. (1994 a,b). To compute sensible heat flux the factor kB-1 was
calculated with an empirical equation determined earlier with observations collected in the same basin. Spatial extrapolation was done with a scheme similar to
the one described by Menenti (1979, 1984) based on the estimate of first order
differentials relative to the values of state variables and fluxes observed at a reference site. In the study of Kustas et al. (1990) spatial extrapolation was done using
the differentials with respect to surface temperature and albedo. Changes in roughness lengths for momentum and heat transfer were neglected and the values at the
reference site kept constant over the watershed. Analysis of observations did show
that this assumption was not correct, since the empirical coefficient used to estimate the roughness length for heat transfer had to be changed significantly on each
day used for the study.
Most of the equations necessary to model exchanges of momentum and heat at
heterogeneous land surfaces are non-linear. Spatial averaging of the independent
variables in these equations has therefore a potentially large impact on the variables calculated with the non-linear equations. This is also the case when using
observations at a spatial resolution insufficient to capture the spatial structure of
the landscape. More precisely, if the spatial resolution is larger than the smaller
homogeneous element in the landscape, spatial averaging will always contribute to