Geoscience Reference
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by the availability of GIS and remote-sensing data rather than any real advances in the understanding of
how to represent hydrological processes. They can perhaps be considered most appropriately with the
class of semi-distributed models, and we return to them in Chapter 6.
2.5 Distributed Process Description Based Models
It will be realised from the discussion in Section 2.4 that there is considerable subjectivity associated
with the definition of an ESMA-type model, even if the hydrologist is attempting, as far as possible, to
reflect her or his perceptual model of how catchments work. Hence the wide range of ESMA models
available. Another early (in the digital computer era) response to this subjectivity was the attempt to
produce models based directly on equations describing all the surface and subsurface flow processes in
the catchment. A blueprint for such a model was described in the article by Freeze and Harlan, 1969,
reproduced by Loague, 2010).
In their seminal paper, Freeze and Harlan wrote down the equations for different surface and subsurface
flow processes and showed how they could be linked by means of common boundary conditions into a
single modelling framework. Their analysis is still the basis of the most advanced distributed rainfall-
runoff modelling systems today. The equations are all nonlinear partial differential equations, that is to
say differential equations that involve more than one space or time dimension (see Box 2.3). For the type
of flow domains and boundary conditions of interest in rainfall-runoff modelling, such equations can
normally only be solved by approximate numerical methods which replace the differential terms in the
equations by an approximate discretisation in time and space (see Chapter 5). However, the descriptive
equations that they used for each process required, in all cases, certain simplifying assumptions. Thus,
for subsurface flow, it is assumed that both saturated and unsaturated flows can be described by Darcy's
law (that flow velocity is proportional to a hydraulic conductivity and a gradient of total potential, see
Box 5.1), while for surface flows it was assumed that the flow could be treated as a one-dimensional,
cross-section averaged flow either downslope over the surface or along a reach of the channel network
in a catchment (leading to the St Venant equations, see Box 5.6).
Freeze andHarlan (1969) discuss the input requirements and boundary conditions for these equations in
some detail. Meteorological data are required to define rainfall inputs and evapotranspiration losses. Their
model definition input
includes necessary assumptions on the extent of the flow domain, assumptions
about prescribed potential or prescribed flow boundaries (especially zero flow at impermeable or divide
boundaries) and the way in which the flow domain is divided to create space and time increments for
solving the process equations.
Flow parameter inputs
are then required for each element of the solution
grid, allowing consideration of heterogeneity of catchment characteristics to be taken into account.
This type of distributed model allows the prediction of local hydrological responses for points within
the catchment. The first applications of this type of model were made for hypothetical catchments and
hillslopes by Freeze (1972). The calculations required the largest computers available at the time (Al
Freeze was then working at the IBM Thomas J. Watson Research Centre at Yorktown Heights), and
even then only a limited flow domain and coarse mesh of points could be solved. The first application
to a field site of this type of model was published by Stephenson and Freeze (1974), who attempted to
model a single hillslope at the Reynolds Creek catchment in Idaho (see Section 5.3). The results were
not particularly successful but, as they pointed out, it was a complex hillslope underlain by fractured
basalts with complex flow pathways, They had limited knowledge of the inputs and initial conditions
for the simulation and computing constraints limited the number of simulations they could actually try.
Arising from these difficulties, they were also the amongst the first to discuss the difficulties of validating
hydrological models.
Distributed models of this type have the possibility of defining parameter values for every element in
the solution mesh. There may, even with continuing limitations imposed by computational constraints, be
