RAINFALL AND RUNOFF FACTOR FOR EROSION ESTIMATES —
PRAIRIE REGION
J. M. Wigham and W. J. Stolte
Civil Engineering Department, University of Saskatchewan, Saskatoon, Saskatchewan S7N 0W0
Received 4 November 1985, accepted 7 February 1986
Wigham, J. M. and W. J. Stolte.
1986. Rainfall and runoff factor for erosion estimates—Prairie Region. Can.
Agric. Eng. 28: 71-75.
The universal soil loss equation, developed by Wischmeier and Smith, has been used for many years in the United
States for predicting soil loss from cultivated areas. The equation consists of a number of terms, one of which is the
rainfall factor. This factor is determined from the product of the kinetic energy of rainfall and the maximum 30-min
rainfall intensity. The studies by Wischmeier et al. showed that this factor was the one of many rainfall-related variables
tested that related best to the soil loss quantities. The kinetic energy of rainfalls and the rainfall factor are of interest,
therefore, in evaluating the potential soil erosion rate for a given location. The rainfall rate data base for the prairie
provinces now is sufficient both in number of stations and length of record for the statistics of kinetic energies and rainfall
factors to be determined. A study was initiated to determine the two quantities and their frequency of occurrence, for
each recorded storm during summer periods and for all stations in Manitoba, Saskatchewan and Alberta. The seasonal
rainfall factors were determined for all stations for which sufficient raingauge data were available. The log-normal
distribution, fitted by the method of moments, produced the most consistent fit to the seasonal data. The standard
deviations of the rainfall factor tended to be high at all stations because one or more high intensity storms usually greatly
influenced the seasonal value. Contour plots of mean seasonal rainfall factor and coefficient of variation were produced.
These plots, together with statistical relationships for the log-normal distribution, provide the information for calculation
of seasonal rainfall factor for any location and for any return period. Estimates of soil erosion rates then can be made
using the Universal Soil Loss Equation.
INTRODUCTION
Soil removal from land surfaces by
water action occurs slowly under natural
conditions. The erosion rate may, and
the terms of the equation are not available.
The universal soil loss equation may be
The value of the conversion constant,
written as
M, depends on the units used to determine
(1)
R, K and A. It is 0.129 if R and A are in
where M is a conversion constant, A is the
seasonal soil loss per unit area, R is the
equation determined by the English unit
proceduresof Wischmeierand Smith. Use
usually does, increase dramatically when
MRKLSCP
metric units but with all other terms in the
man's activities cause removal or dis
turbance of protective vegetal covers or
mulches. Some farming practices, for ex
ample, are conducive to the development
of accelerated erosion. In fact, erosion
rates in excess of rates of soil regeneration
have been estimated (Coote 1983) for the
prairie region. The cumulative soil re
moval that results from such rates of
movement can mean substantial losses
istics for the area permits evaluation of the
soil erodibility factor, again using charts.
rainfall and runoff factor, K is the soil
of this conversion constant means all
erodibility factor or soil loss rate per unit
area, L is the slope length factor, S is the
slope steepness factor, C is the cover and
management factor, and P is the support
practice factor. The soil loss, A, is ex
terms, other than R and A, have the same
pressed as a weight per unit area for the
tested that correlated best with measured
value regardless of the units used.
The studies by Wischmeier and Smith
(1965) showed that the rainfall factor was
the one of many rainfall related variables
in productivity of agricultural areas. In
creased technological inputs can offset
soil losses to some degree but at increased
duration for which the rainfall is deter
soil loss quantities. It is the summation
mined; the appropriate SI units for A
over a season of the erosion index units
should be Newtons per square metre per
cost. Some estimates of the costs resulting
season.
divided by 100 where each unit is the
product of the kinetic energy of rainfall
from soil erosion have been made (PFRA
The equation can be used to predict soil
1982) and show the seriousness of the
problem.
Because of the severe economic impact
of erosion, it is important to be able to
loss from an area, due to rill and inter-rill
erosion from rainfall runoff, by evaluating
each of the terms on the right-hand side of
the equation using methods described by
predict soil loss from cultivated areas and
Wischmeier and Smith (1978). The slope
to be able to evaluate changes due to
length and slope steepness factors can be
determined from equations when topo
graphic data for an area are available. The
cover and management factor and the sup
port practice factor can be determined
from knowledge of the farming practices
employed and through the use of charts
and tables. Knowledge of soil character
changes in farming practices. The univer
sal soil loss equation developed by Wisch
meier and Smith (1965) has been used for
many years in the United States for this
purpose. Some use has been made of it in
a Canadian context to predict soil erosion
due to rainfall; however, data on some of
CANADIAN AGRICULTURAL ENGINEERING, VOL. 28, NO. 2, SUMMER 1986
and the maximum 30-min rainfall rate for
each storm. The rainfall factor can be ob
tained in the United States from maps
showing contours of the mean annual val
ue. These maps were developed through
calculation of point values of the rainfall
factor using long-period, rainfall rate
data.
The rainfall rate data base for the prairie
provinces now is sufficient both in num
ber of stations and length of record (22 yr)
for the statistics of kinetic energies and
rainfall factors to be determined. A study
was initiated to determine the two quan-
71
3000
energy, ET, was to search the storm mass
curve for the total precipitation occurring
at a selected rainfall intensity, 7n, and to
•
/
2000
multiply it by the kinetic energy calcu
lated for that value of/n. This was done
for all values of 7n, and the total kinetic
q:
o
\o
energy of the storm, £T, was the sum of
all the incremental kinetic energies.
The rainfall data used in this study were
hourly precipitations as these are the
shortest duration data commonly avail
able for the prairies in digital form. Storm
kinetic energy was determined by calcu
lating incremental kinetic energies from
Eq. 2 for each hour of the storm and sum
ming for the duration of the storm. Calcu
lation of the kinetic energy in this manner
is basically no different from the pro
cedure used by Wischmeier and Smith
(1978) except that periods of equal in
tensity are limited to integer multiples of
1000
<s%
800
•
/%
jS*
-J
<
600
<
a:
y^
400
<
z
•
o
<
UJ
in
#
200
<s
hours because of the discrete nature of the
data.
100
1.01
I.I
1.25
RETURN
2
PERIOD
5
10
25
50
100
Figure 1. Return period of seasonal rainfall factors for Regina Saskatchewan log-normal II
distribution.
TABLE I. SEASONAL RAINFALL FACTORS
Length of
seven tipping-bucket gauge locations.
Data for major storms at 12 other stations
Standard
in Manitoba, Saskatchewan and Alberta
record
Mean
deviation
Station name
(yr)
(MJ-mm)(ha-h)
(MJ-mm)(ha-h)
Beaverlodge
22
401
408
Brandon
22
792
539
Calgary
Dauphin
22
321
290
22
847
436
Edmonton
21
398
244
Lethbridge
22
302
287
Moose Jaw
22
488
345
Prince Albert
22
501
333
Regina
22
634
459
Saskatoon
22
379
233
Winnipeg
22
1259
666
tive was to determine the statistical char
acteristics of these terms for ease in ex
trapolation to return periods beyond the
period of record and to provide the data
for the production of maps of the mean
and coefficient of variation of the rainfall
factor.
lated for each storm and a ratio of the two
with one exception, larger than 1 and
ranged from 1.02 to 1.79 for very small
kinetic energies and from 1.01 to 1.09 for
the larger, dominant storms. A correction
factor equation, to convert storm kinetic
energies calculated using hourly data to
that which would be calculated using
short-period data, was developed and
where ET is the total storm kinetic energy
in Nm/m2; /n is a uniform rainfall rate in
mm/h within the storm; / is the rainfall
depth in mm, resulting from /n; and n is
the number of periods in a storm during
which the rainfall rate was uniform. The
equation for the storm rainfall factor, /?, is
R = EtWIOO
The original equation for the storm kin
etic energy used by Wischmeier and
Smith (1978) was given in English units.
When conversions to the SI system are
performed the equation becomes
£, = 2 [(11.93 + 8.73logI()/„)/]
72
(2)
were also used. Storm kinetic energies
using 15-min and hourly data were calcu
values was determined. The ratios were,
(3)
where R is the storm rainfall factor in
PROCEDURES
using hourly data was evaluated, how
ever, by comparison with the same factors
calculated using 15-min duration rainfall
data. Short-period data were obtained for
Saskatoon for some 50 storms recorded at
Seasonal rainfall factors
tities, and their frequency of occurrence,
on a storm and seasonal basis. The objec
The accuracy with which storm erosion
and rainfall factors could be determined
IN YEARS
used for all stations.
A storm was defined as all hourly pre
cipitations separated by less than an arbi
trarily chosen 10 h. Examination of the
results in comparison with the results
using other separation periods showed
that the 10-h period was acceptable.
The erosion index is the product of the
storm kinetic energy and the maximum
30-min rainfall intensity for the storm.
(MJ/ha) •(mm/h) (Foster et al. 1981); ET
is as defined previously; and 730 is the
maximum 30-min rainfall intensity in
mm/h. Limitations applying to these
intensity, 730, were obtained for each day
of record, from Atmospheric Environ
equations include a maximum of 76.2
merged with the appropriate hourly data
Values for the maximum 30-min rainfall
ment Service records. These data were
mm/h for /n and 63.5 mm/h for /30
records to allow calculation of storm
(Wischmeier and Smith 1978).
rainfall factors. When a storm began and
ended on the same day, the maximum 730
for that day was assumed to apply to the
The procedure used by Wischmeier and
Smith (1965) for finding the total kinetic
CANADIAN AGRICULTURAL ENGINEERING, VOL. 28, NO. 2, SUMMER 1986
MEANS
SEASONAL RAINFALL EROSIVITY FACTOR
PRAIRIE PROVINCES
Figure 2. Contour plot of mean seasonal rainfall factors for the Prairie Region.
sureof the reliabilityof the equipment and
on different days, the maximum 730 during of the care taken in obtaining the data and
storm; when the storm began and ended
these days was assumed to apply to the maintaining the equipment. Given that
storm. Perusal of some of the recorded both equipment and procedures have im
hourly precipitations and 730 data indi proved over the years, the problem of
catedthese assumptions were reasonable. missing data is less now than in the past.
The computer program yielded a data Examination of the missing hourly data
file giving for each storm the beginning file confirmed this in that the number of
and ending month, day, and hour of the hours for which there obviously were no
developed for every station. Missing in
tensity data for given storms were evalu
ated by using the appropriate regression
equation together with recorded max
imum hourly precipitation data for the
storm.
Seasonal rainfall factors are the sum of
all the storm rainfall factors throughout
the season, which was defined as ex
tending from 15 Apr. through 31 Oct. The
15 Apr. beginning date was set by the end
storm, the rainfall depth during the storm,
and the storm kinetic energy and rainfall
factor. Another data file produced by the
data was small relative to the total number
The maximum 30-min intensity rainfall
of the snowmelt period and the 31 Oct.
program detailed the missing hourly pre
was used to calculate the erosion index
ending date was set by the beginning of
cipitation data.
The influence of missing data on the
and rainfall factor for every storm and,
therefore, missing intensity data were
of much more importance than missing
the snow accumulation period. Seasonal
rainfall factors were only included for
hourly precipitations. Accordingly, a
regression equation relating recorded
maximum 30-min intensity to the corre
available, since these are the months
when the major rainstorms occur in the
prairies.
sponding maximum recorded hourly pre
cipitation amount during a storm was
The mean, standard deviation, and co
efficient of skewness of the seasonal and
calculated storm and seasonal erosion in
dexes and rainfall factors was of some
concern. Missing data included missing
storms, missing hourly precipitations and
missing30-min intensities. The amount of
missing storm and hourly data is a mea
of rainfall hours.
CANADIAN AGRICULTURAL ENGINEERING, VOL. 28, NO. 2, SUMMER 1986
analysis if records for June and July were
73
COEFFICIENTS OF VRRIRTION
SEnSONFIL RniNFRLL EROSIVITY FACTOR
PRAIRIE PROVINCES
Figure 3. Contour plot ofcoefficients ofvariation ofrainfall factors for the Prairie Region.
storm rainfall factors were determined by
means of another computer program de
signed to perform frequency analyses
(Dumontier 1978) of data according to
nally the analysis was limited to Saskatch
the conclusions of Wischmeier and Smith
ewan stations due to the amount of data
(1978). An example of this distribution
fitted to data for Regina is shown in
required but was extended to include data
various probability distributions. These
from all stations in Manitoba and Alberta,
for which hourly precipitation records
distributions included the two- and three-
were available.
parameter log-normal distributions as
evaluated by the method of moments and
the two-parameter Pearson distribution
also evaluated by the method of moments.
The log-normal II and Pearson II distribu
tions were also fitted using the method of
maximum likelihood (Yevjevich 1972).
The intent of this analysis was to decide,
on the basis of visual inspection of the
frequency plots, the probability distribu
tion most consistently fitting the data at
the various locations.
The stations analyzed were chosen
based on considerations of period of
record and geographic distribution. Origi
74
Fig. 1.
Magnitudes of mean, seasonal rainfall
factors are shown in Table I for selected
prairieregion stations withlongperiods of
RESULTS AND DISCUSSION
As mentioned, five combinations of
frequency distributions or fitting methods
record. Also shown are the standard devi
ation values for each station which,
together with use of the log-normal II
were used on the seasonal and storm rain
distribution, allow calculation of rainfall
fall factor data. A visual assessment of the
goodness of fit of the distributions to data
from each of 54 stations showed that the
factors of any desired return period.
The regional, spatial variability which
log-normal III distribution fitted by the
method of moments or the log-normal II
fitted by maximum likelihood were fairly
can be noted by comparing tabulated val
ues for stations in Manitoba to those for
stations in Alberta is also shown on the
contour of mean seasonal rainfall factors
good. The log-normal II distribution fitted (Fig. 2). The contours of the plot were
by the method of moments was preferred developed using data from all of the
however, on the basis of consistency and prairiestations for which hourly data were
simplicity. This finding is consistent with available and for which at least 10 yr of
CANADIAN AGRICULTURAL ENGINEERING, VOL. 28, NO. 2, SUMMER 1986
data existed. The contouring was done
with a computer contouring package.
The mean seasonal rainfall factors are,
on average, lowest in the far north, of
moderate to low magnitude in southern
Alberta and southwestern Saskatchewan,
tors. An even greater variation in storm
normal distribution characteristics to cal
rainfall factor was observed for particular culate a rainfall factor for a selected return
locations. The largest single storm rainfall period for any location on the prairies.
factor in a season often constitutes 90% or
more of the corresponding seasonal value
at some stations. This occurs because the
REFERENCES
itoba also tend to be higher than elsewhere
in the total region. Rainfall factor mag
kinetic energy of rainfall is a function of
terminal velocity and raindrop size, both
of which increase rapidly with rainfall in
tensity. Storms of high intensity are very
important elements, therefore, in the de
nitudes near the U.S. A. border are consis
termination of the rainfall factor. The
COOTE, D. R. 1983. Soil degradation in Can
ada— an overview. Proceedings of the 8th
B.C. Soil Science Workshop on soil deg
radation in British Columbia, pp. 6-30.
DUMONTIER, G. 1978. Flood frequency
plotting package, Unpublished report,
Dept. of Civil Engineering, University of
tent with those reported by Foster et al.
(1981) for the northern states just below
the prairie region.
The coefficiency of variation of the sea
prairie region is subject to considerable
thunderstorm activity so the accuracy of
Saskatchewan, Saskatoon, Sask.
FOSTER, G. R., D. K. MCCOOL, K. G.
any calculated rainfall factor is a function
RENARD, and W. C. MOLDENHAUER.
and increase dramatically in southeastern
Saskatchewan and southern Manitoba.
The values for central to northern Man
sonal rainfall factor, shown contoured on
Fig. 3, is between 0.6 and 0.7 throughout
most of the region, regardless of the mag
nitude of the mean seasonal rainfall fac
tor. From this, one could conclude that it
is a very stable parameter and thus fairly
of how well the frequency of occurrence
of intense storms at a station represent the
regional frequencies of occurrence.
CONCLUSIONS
The seasonal factors are lower than val
1981. Conversion of the Universal Soil
Loss Equation to S.I. Metric Units. J. Soil
Water Cons. 36: 355-359.
PFRA 1982. Land degradation and soil conser
vation issues on the Canadian prairies — an
overview. PFRA, Soil and Water Conser
vation Branch Report.
ues quoted in the literature (Wall et al.
WALL, G. J., W. T. DICKINSON, and
reliable as calculated.
1983) for annual values for those stations
J. GREUEL. 1983. Rainfall erosion indices
As noted earlier and as reported by
Wigham and Stolte (1984) the seasonal
rainfall factors follow a log-normal distri
bution as fitted by the method of mo
ments, at least to an acceptable degree.
The log-normal distribution fitted by this
method requires only the mean and the
for which comparisons are possible. The
annual values used for comparison were,
however, calculated using an alternate
procedure to that originally used by
for Canada east of the Rocky Mountains.
standard deviation of the rainfall factors.
From these maps of the mean and coeffi
cient of variation in the rainfall factor, it is
possible to determine the rainfall factor
that correponds to any given return period
for any location in the prairie provinces.
The coefficient of variation is the stan
dard deviation of the-rainfall factor di
vided by the mean. The values shown on
Fig. 3 are fairly high indicating a consid
erable variation in seasonal rainfall fac
Wischmeier and Smith (1965, 1978). The
seasonal rainfall factors determined here
in encompass the period when agricultural
land is most susceptible to erosion and
should, therefore, provide acceptable
comparative values for calculation of soil
erosion potential.
The log-normal frequency distribution
fitted by the method of moments provides
Can. J. Soil Sci. 63: 271-280.
WIGHAM, J. M. and W. J. STOLTE. 1984.
Statistics of rainfall factors for the Prairie
Region. Proceedings, Can. Soc. for Civil
Engineering Annual Conference, Halifax,
N.S. Vol. II, pp. 609-618.
WISCHMEIER, W. H. and D. D. Smith.
1978. Predicting rainfall erosion losses —
a guide to conservation planning. U.S.
Dep. Agric, Washington, D.C. Agricul
tural Handbook No. 537.
WISCHMEIER, W. H. and D. D. SMITH.
data, considering the criteria of consis
tency and simplicity. The data on mean
1965. Predicting rainfall — erosion losses
from cropland east of the Rocky Moun
tains. U.S. Dep. Agric, Washington, D.C.
Agricultural Handbook No. 282.
YEVJEVICH, V. 1972. Probability and statis
seasonal rainfall factors and coefficients
tics in hydrology. Water Resources Publi
of variation can be used with the log-
cations, Fort Collins, Colorado.
the best fit to the seasonal rainfall factor
CANADIAN AGRICULTURAL ENGINEERING, VOL. 28, NO. 2, SUMMER 1986
75