HYDROLOGY 481
whereBM = Daily melt in an elevation band
TM = Mean daily temperature
TX = Maximum daily temperature
TN = Minimum daily temperature
PTM = Point melt factor (Assumed value
2.1 mm/°C/Day)k 1 , k 2 are constants (assumed value, k 1 = 8, k 2 = 10°C).
A modifi ed formulation is used for open areas, where
additional weighting is given to the maximum temperature
and this equation is also used for glaciated regionsBM TXTN
kTN PTM
2∗⎛
⎝⎜⎞
⎠⎟∗ (50)A negative melt budget is maintained during ripening peri-
ods and following cold periods.
The soil moisture budget When designing a watershed
model, there is considerable computational advantage if all
the non-linear watershed behavior can be concentrated in
one section of the model. This advantage is even greater if
the non-linearities are handled completely in each time step,
rather than being distributed over the many time steps of the
routing procedure.
In the UBC watershed model, non-linearities have been
confi ned to the soil moisture budget section of the model.
The soil moisture budget section subdivides the total rain
and snowmelt inputs into fast, medium, slow and very slow
components of run-off. This subdivision of the total run-off
depends on the present status of each section of the soil mois-
ture and groundwater components, and so the sub division
process is non-linear. The degree of non-linearity is in the
hands of the model designer and his concept of the various
hydrologic processes. For example, in the UBC model, the
non-linearities are greatest at the soil surface, where the sub-
division between fast and medium run-off is determined by
soil moisture defi cit conditions. In contrast, the deep ground-
water zone has no non-linear behavior within itself; it simply
accepts what fi nally arrives. This gradation of non-linear
behavior, which is maximum at the surface and decreases
with depth, is probably realistic in mountain conditions,
but might require redesign for some soil conditions in fl at
terrain.
Once the fl ow has been subdivided into components
of run-off, each component can be routed separately to the
watershed outfl ow point. Some models might assume that
water can be exchanged between the various run-off com-
ponents as the water migrates towards the stream system.
Although it is probably quite realistic to make such an
assumption, it also introduces considerable complexity and
extra data handling. More complex versions of Sugawara’s^51
tank model could be designed to operate in this fashion.
The routing process itself can also be linear or non-
linear. If a linear routing procedure is used, then each fl ow
increment can be routed independently of any other fl ow, andthe routing procedure is computationally simple. Non-linear
routing, on the other hand, can become quite complex.
The simplicity of linear routing procedures is attractive
to the model designer, but such simplicity is only of value
if it is also a realistic representation of actual run-off pro-
cesses. In the following section, some brief consideration
will be given to the physical nature of the various run-off
processes and to their mathematical representation.
Two major types of routing procedure are widely referred
to in the literature: unit hydrograph routing, Nash^17 and kine-
matic routing, Hayami.^29 More complex procedures, usually
referred to as hydraulic routing, are not usually used within
watershed models, but are occasionally used in some more
detailed channel routing procedures. The unit hydrograph
is defi ned as a linear process Sherman,^16 yet, some work-
ers make changes to hydrograph shape for extreme rain-
fall events. This non-linearing of the unit hydrograph can
be avoided by recognising that total watershed response is
made up of several different unit hydrograph responses, as
shown schematically in the fi gure. Soil moisture budgeting
switches varying amounts of water to the various run-off
components, so that the composite hydrograph can be more
peaked or less peaked according to the severity of the storm.
Kinematic routing can be linear or non-linear according to
the assumed functional form between incoming volume and
fl ow rate. In channels, this relationship is usually expressed
as a velocity-depth or a discharge-area relationship.THE UNIT HYDROGRAPHFor the unit hydrograph to be truly linear, then, as Nash^23 has
pointed out, storage and outfl ow must be linearly related.
A cascade of storages will then yield various possible hydro-
graph shapes. This simple model assumes a uniform instan-
taneous input of run-off volume over an δ A in time δ t. A real
watershed receives a non-uniform input of run-off volume
over an area, A, in a fi nite time period, t. Strictly speaking,
a watershed should be subdivided into regions where uniform
input is a reasonable approximation. With such a subdivision,
the fi nal outfl ow hydrograph would become not only a func-
tion of watershed shape and varying response characteristics
of sub-areas, but also of rain and snowmelt distribution across
the watershed. For example, there should be a recognizable
difference between rainfall response and snowmelt response.
Snowmelt, which is highly temperature dependent, is great-
est at low elevations and least at the top of the watershed.
Rain, on the other hand, undergoes orographic enhancement,
so that it is greatest at the highest parts of the watershed and
least in the valley portions. This orographic effect is always
present to some extent, but, as a general rule, the warmer the
air mass, the less is the orographic enhancement. In addition
to this complete reversal of elevation dependent distribution
of run-off, there is also a considerable contrast in the intensi-
ties of water input from rain and snow events. Snowmelt, even
under quite extreme snowmelt conditions, rarely exceeds 80
mm of water equivalent per day, and some 75% of this melt
occurs during the daylight period. Rainfall daily volumesC008_003_r03.indd 481C008_003_r03.indd 481 11/18/2005 10:29:30 AM11/18/2005 10:29:30 AM