HYDROLOGY 469
channels: this process is termed interfl ow and is much debated
because it is so diffi cult to measure. At very high rainfall rates,
the surface infi ltration rate may be exceeded and then direct
surface run-off will occur. Direct run-off is rare from soil sur-
faces but does occur from certain impervious soil types, and
from paved areas. Much work has been done to evaluate the
relative signifi cance of these various processes and is well
documented in references (1,2,3).
Such qualitative descriptions of the run-off process are
helpful, but are limited because of the extreme complexity
and interrelationship of the various processes. Various meth-
ods have been developed to by-pass this complexity and to
give us usable relationships for hydrologic calculations.
The simplest method is a plot of historical events, showing
run-off as a function of the depth of precipitation in a given
storm. This method does not allow for any antecedent soil
moisture conditions or for the duration of a particular storm.
More complex relationships use some measure of soil
moisture defi ciency such as cumulative pan-evaporation or
the antecedent precipitation index. Storm duration and pre-
cipitation amount is also allowed for and is well illustrated
by the U.S. Weather Bureau’s charts developed for various
areas (Figure 2). It is a well to emphasize that the anteced-
ent precipitation index, although based on precipitation, is
intended to model the exponential decay of soil moisture
between storms, and is expressed byI N ( I 0 k^ M^ I M ) k ( N − M )where I 0 is the rain on the fi rst day and no more rain occurs
until day M, when I M falls. If k is the recession factor, usually
about 0.9, then I N will be the API for day N. The expres-
sion can of course have many more terms according to the
number of rain events.
Before computers were readily available such calcu-
lations were considered tedious. Now it is possible to use
more complex accounting procedures in which soil moisture
storage, evapo-transpiration, accumulated basin run-off,
percolation, etc. can all be allowed for. These procedures
are used in more complex hydrological modeling and are
proving very successful.RUN-OFF: SNOWMELTAs a fi rst step in the calculation of run-off from snow, meth-
ods must be found for calculating the rate of snowmelt. This
snowmelt can then be treated similarly to a rainfall input.
Snowmelt will also be subject to soil moisture storage effects
and evapo-transpiration.
The earliest physically-based model to snowmelt was
the degree-day method which recognized that, despite the
complexity of the process, there appeared to be a good cor-
relation between melt rates and air temperature. Such a
relationship is well illustrated by the plots of cumulative
degree-days against cumulative downstream fl ow, a rather
frustrating graph because it cannot be used as a forecast-
ing tool. This cumulative degree-day versus fl ow plot is anexcellent example of how a complex day-to-day behavior
yields a long-term behavior which appears deceptively
simple. Exponential models and unit hydrograph methods
have been used to turn the degree-day approach into a work-
able method and a number of papers are available describing
such work (Wilson,^38 Linsley^32 ). Arguments are put forward
that air temperature is a good index of energy fl ux, being
an integrated result of the complex energy exchanges at the
snow surface (Quick^33 ).
Light’s equation^31 for snowmelt is based on physical
reasoning which models the energy input entirely as a tur-
bulent heat transfer process. The equation ignores radiation
and considers only wind speed as the stirring mechanism,
air temperature at a standard height as the driving gradient
for heat fl ow and, fi nally, vapour pressure to account for
condensation–evaporation heat fl ux. It is set up for 6 hourly
computation and requires correction for the nature of the
forest cover and topography. It is interesting to compare
Light’s equation with the U.S. Crops equation^36 for clear
weather to see the magnitude of melt attributed to each term.
By far the most comprehensive studies of snowmelt have
been the combined studies by the U.S. Corps of Engineers
and the Weather Bureau (U.S. Corps of Engineers 36,37 ).
They set up three fi eld snow laboratory areas varying in
size from 4 to 21 square miles and took measurements for
periods ranging from 5 to 8 years. Their laboratory areas
were chosen to be representative of certain climatic zones.
Their investigation was extensive and comprehensive, rang-
ing from experimental evaluation of snowmelt coeffi cient
in terms of meteorological parameters, to studies of ther-
mal budgets, snow-course and precipitation data reliability,
water balances, heat and water transmission in snowpacks,
streamfl ow synthesis, atmospheric circulations, and instru-
mentation design and development.
A particularly valuable feature of their study appears to
have been the lysimeters used, one being 1300 sq.ft. in area
and the other being 600 sq.ft. (Hilderbrand and Pagenhart^30 ).
The results of these lysimeter studies have not received the
attention they deserve, considering that they give excellent
indication of storage and travel time for water in the pack. It
may be useful to focus attention on this aspect of the Corps
work because it is not easy to unearth the details from the
somewhat ponderous Snow Hydrology report. Before leav-
ing this topic it is worth mentioning that the data from the
U.S. studies is all available on microfi lm and could be valu-
able for future analysis. It is perhaps useful at this stage to
write down the Light equation and the clear weather equa-
tion from the Corps work to compare the resulting terms.
Light’s equation^31 (simple form in °F, inches of melt and
standard data heights)DU()0 001 84.. Ta⋅ 10 0 0000156. 0 00578()e6 11⋅whereU = average wind speed (m.p.h.) for 6 hr period
T = air temperature above 32°F for 6 hr periodC008_003_r03.indd 469C008_003_r03.indd 469 11/18/2005 10:29:26 AM11/18/2005 10:29:26 AM