121
independent parameters, so that the full numerical solution over the PDF distribution can
be carried out. The assumption underlying this strategy is that the effects of intra-patch
heterogeneity of key parameters are more important than the detailed representation of
processes for an individual, "averaged", surface type.
An alternative strategy to account. for intra-patch het.erogeneity is to keep the complete
set of ESEM equations and include PDF integration only for a few critical variables and
only over some of the non-linear terms entering the equations rather than the full coupled
set. {From section 3 it is evident that two of the variables which ent.er most non-linear
terms are the soil moisture content relative t.o saturation and the temperature of canopy
and soil. In addition, it can be expect.ed that, because of variations in microenvironmental
conditions, climatic forcing, terrain structure and soil properties, these variables may have
a relatively large range of values within the scale of an AM grid box.
Therefore, a statistical-dynamical model of surface processes could begin with the
assumption that surface temperature and soil water content are distributed according to
analytical PDFs. If these PDFs have the properties given by Eqs. (61) and (62), by
applying the operator
(65)
to the various terms of the equations of an ESEM (where X in Eq. (65) represents the
term of the equation), prognostic equations can be obtained for the grid-point averaged
values of soil water content relative to saturation and the surface temperatures which
account for the effects of non-linearities associated with these variables.
Such a model, limited to the hydrologic processes of evaporation, infiltration and
runoff, was developed by Enthekaby and Eagleson (1989) who assume that the soil moisture relative to saturation within a grid box, s, follows a two-parameter gamma PDF:
A, Ct, s ~ 0
(66)
where Ct is inversely proportional to the coefficient of variation of the distribution. The
appropriate choice of the distribution function should allow analyt.ical integration of the
non-linear terms over the full, or sub-regions of, the space domain, and is thus partially
dictated by the form of the equations. As an example, the non-linearities present in the
soil water equation are in the form of power functions (see Eq. 27). Power functions are
also the non-linearities associated with temperature, both in t.he infrared emission term
and in the evaporation term since, as pointed out, the temperature dependency of the
saturation vapor pressure can be accurately approximated with polynomial expressions
as a function of temperature. Thus, a suitable choice of PDF which would allow analytical and computationally efficient integration of the operator FpdJ would be a linear or
polynomial function.
independent parameters, so that the full numerical solution over the PDF distribution can
be carried out. The assumption underlying this strategy is that the effects of intra-patch
heterogeneity of key parameters are more important than the detailed representation of
processes for an individual, "averaged", surface type.
An alternative strategy to account. for intra-patch het.erogeneity is to keep the complete
set of ESEM equations and include PDF integration only for a few critical variables and
only over some of the non-linear terms entering the equations rather than the full coupled
set. {From section 3 it is evident that two of the variables which ent.er most non-linear
terms are the soil moisture content relative t.o saturation and the temperature of canopy
and soil. In addition, it can be expect.ed that, because of variations in microenvironmental
conditions, climatic forcing, terrain structure and soil properties, these variables may have
a relatively large range of values within the scale of an AM grid box.
Therefore, a statistical-dynamical model of surface processes could begin with the
assumption that surface temperature and soil water content are distributed according to
analytical PDFs. If these PDFs have the properties given by Eqs. (61) and (62), by
applying the operator
(65)
to the various terms of the equations of an ESEM (where X in Eq. (65) represents the
term of the equation), prognostic equations can be obtained for the grid-point averaged
values of soil water content relative to saturation and the surface temperatures which
account for the effects of non-linearities associated with these variables.
Such a model, limited to the hydrologic processes of evaporation, infiltration and
runoff, was developed by Enthekaby and Eagleson (1989) who assume that the soil moisture relative to saturation within a grid box, s, follows a two-parameter gamma PDF:
A, Ct, s ~ 0
(66)
where Ct is inversely proportional to the coefficient of variation of the distribution. The
appropriate choice of the distribution function should allow analyt.ical integration of the
non-linear terms over the full, or sub-regions of, the space domain, and is thus partially
dictated by the form of the equations. As an example, the non-linearities present in the
soil water equation are in the form of power functions (see Eq. 27). Power functions are
also the non-linearities associated with temperature, both in t.he infrared emission term
and in the evaporation term since, as pointed out, the temperature dependency of the
saturation vapor pressure can be accurately approximated with polynomial expressions
as a function of temperature. Thus, a suitable choice of PDF which would allow analytical and computationally efficient integration of the operator FpdJ would be a linear or
polynomial function.