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column of base area
A
with periphery
C
,this can readily (Brutsaert, 1982) be shown
to be
p
s
∂
W
∂
t
+
1
Ag
E
−
P
=
(
qV
n
)
dC dp
(4.57)
p
t
C
where
E
and
P
are evaporation and precipitation intensity averaged over
A
,
W
is the total
water vapor content per unit surface area of the control volume,
q
is the specific humid-
ity along the vertical boundary,
V
n
is the horizontal wind component normal to the same
boundary pointing outward,
p
is the pressure, with subscripts s and t referring to the surface
and the top of the column, and the water content in the solid and liquid phases is neglected.
Equation (4.57) states that the difference between the average rate of evaporation and pre-
cipitation over a given area of the Earth's surface equals the rate of increase of water vapor
over the area plus the total flux directed away from the region.
In the past this method has been used primarily over water surfaces (for a review of
the early work, see Brutsaert, 1982). Among the first applications over land was the anal-
ysis of Benton and Estoque (1954) for the North American continent, followed by those
of Rasmusson (1971) and Magyar
et al
. (1978) for the USA. In the early studies, the data
needed were usually obtained from the operational radiosounding network, with twice-daily
observations, so that
∂
W
/∂
t
was taken as the difference over 12 h. This world-wide grid
was originally not designed for the purpose of budget calculations, but rather to observe
synoptic-scale features with time scales of a few days and length scales of the order of
10
3
km. Figure 4.14 shows a comparison between results obtainable with this method
over water using radiosonde observations every 6 h at four stations with an enclosed
area
A
=
17
×
10
4
km
2
, and values obtained using a mean-profile method by Kondo (1976).
Rasmusson (1977) has made a detailed analysis of the errors in flux divergence computa-
tions resulting from the usual limits of spatial and temporal resolution and of instrumental
accuracy in typical radiosonde observations in networks of different scales. He concluded
that with the operational network and current observational schedules, the applicability of
the method to basins with an area smaller than 25
×
10
4
km
2
islimited, and that the results
are likely to be unreliable; with such data the method can yield good results for areas of the
order of 25
×
10
4
to 10
6
km
2
,butit is best suited for areas larger than 10
6
km
2
. In recent years,
however, the situation has been changing. With the advent of improved data assimilation
schemes, inwhich observational data can be combined with model calculation output, it is
expected that the accuracy of the method may be improved considerably. As a result the
method has been receiving renewed interest in the past decade (see, for example, Brubaker
et al
., 1994; Oki
et al
., 1995; Rasmusson and Mo, 1996; Berbery
et al
., 1996; Yarosh
et al
.,
1996). The appeal of the method stems mainly from the simplicity of the budget concept.
It is the only approach that can estimate evaporation over larger areas, and it can be useful
to compare or extend the results of more local methods.
4.4.3 Soil profile water budget
With soil moisture measurements
It is also possible to consider the soil profile as a control volume, to determine its water
budget. In this case, the integral of the continuity equation (1.8) (see also Equation (8.54))
forasoil column of thickness
h
so
and unit horizontal area, without lateral inflows or outflows
