RAINFALL AND RUNOFF FACTOR FOR EROSION ESTIMATES · RAINFALL AND RUNOFF FACTOR FOR EROSION ESTIMATES — PRAIRIE REGION J. M. Wigham and W. J. Stolte Civil EngineeringDepartment,

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 UnitedStates for predicting soil loss from cultivated areas. The equation consists of a number of terms, one of which is therainfall factor. This factor is determined from the product of the kinetic energy of rainfall and the maximum 30-minrainfall intensity. The studies by Wischmeier et al. showed that this factor was the one of many rainfall-related variablestested 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 prairieprovinces now is sufficient both in number of stations and length of record for the statistics of kinetic energies and rainfallfactors to be determined. A study was initiated to determine the two quantities and their frequency of occurrence, foreach recorded storm during summer periods and for all stations in Manitoba, Saskatchewan and Alberta. The seasonalrainfall factors were determined for all stations for which sufficient raingauge data were available. The log-normaldistribution, fitted by the method of moments, produced the most consistent fit to the seasonal data. The standarddeviations of the rainfall factor tended to be high at all stations because one or more high intensity storms usually greatlyinfluenced 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-normaldistribution, provide the information for calculationof seasonal rainfall factor for any location and for any return period. Estimates of soil erosion rates then can be madeusing the Universal Soil Loss Equation.

INTRODUCTION

Soil removal from land surfaces bywater action occurs slowly under naturalconditions. The erosion rate may, andusually does, increase dramatically whenman's activities cause removal or dis

turbance of protective vegetal covers ormulches. Some farming practices, for example, are conducive to the developmentof accelerated erosion. In fact, erosionrates in excess of rates of soil regenerationhave been estimated (Coote 1983) for theprairie region. The cumulative soil removal that results from such rates ofmovement can mean substantial lossesin productivity of agricultural areas. Increased technological inputs can offsetsoil losses to some degree but at increasedcost. Some estimates of the costs resultingfrom soil erosion have been made (PFRA1982) and show the seriousness of theproblem.

Because of the severe economic impactof erosion, it is important to be able topredict soil loss from cultivated areas andto be able to evaluate changes due tochanges in farming practices. The universal soil loss equation developed by Wischmeier and Smith (1965) has been used formany years in the United States for thispurpose. Some use has been made of it ina Canadian context to predict soil erosiondue to rainfall; however, data on some of

the terms of the equation are not available.The universal soil loss equation may be

written as

MRKLSCP (1)

where M is a conversion constant, A is theseasonal soil loss per unit area, R is therainfall and runoff factor, K is the soilerodibility factor or soil loss rate per unitarea, L is the slope length factor, S is theslope steepness factor, C is the cover andmanagement factor, and P is the supportpractice factor. The soil loss, A, is expressed as a weight per unit area for theduration for which the rainfall is determined; the appropriate SI units for Ashould be Newtons per square metre perseason.

The equation can be used to predict soilloss from an area, due to rill and inter-rillerosion from rainfall runoff, by evaluatingeach of the terms on the right-hand side ofthe equation using methods described byWischmeier and Smith (1978). The slopelength and slope steepness factors can bedetermined from equations when topographic data for an area are available. Thecover and management factor and the support practice factor can be determinedfrom knowledge of the farming practicesemployed and through the use of chartsand tables. Knowledge of soil character

CANADIAN AGRICULTURAL ENGINEERING, VOL. 28, NO. 2, SUMMER 1986

istics for the area permits evaluation of thesoil erodibility factor, again using charts.

The value of the conversion constant,M, depends on the units used to determineR, K and A. It is 0.129 if R and A are inmetric units but with all other terms in theequation determined by the English unitproceduresof Wischmeierand Smith. Useof this conversion constant means allterms, other than R and A, have the samevalue regardless of the units used.

The studies by Wischmeier and Smith(1965) showed that the rainfall factor wasthe one of many rainfall related variablestested that correlated best with measuredsoil loss quantities. It is the summationover a season of the erosion index unitsdivided by 100 where each unit is theproduct of the kinetic energy of rainfalland the maximum 30-min rainfall rate foreach storm. The rainfall factor can be obtained in the United States from mapsshowing contours of the mean annual value. These maps were developed throughcalculation of point values of the rainfallfactor using long-period, rainfall ratedata.

The rainfall rate data base for the prairieprovinces now is sufficient both in number of stations and length of record (22 yr)for the statistics of kinetic energies andrainfall factors to be determined. A studywas initiated to determine the two quan-

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RETURN PERIOD IN YEARS

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Figure 1. Return period of seasonal rainfall factors for Regina Saskatchewan log-normal IIdistribution.

TABLE I. SEASONAL RAINFALL FACTORS

Seasonal rainfall factors

Length of Standard

record Mean deviationStation name (yr) (MJ-mm)(ha-h) (MJ-mm)(ha-h)

Beaverlodge 22 401 408Brandon 22 792 539Calgary 22 321 290Dauphin 22 847 436Edmonton 21 398 244Lethbridge 22 302 287Moose Jaw 22 488 345Prince Albert 22 501 333Regina 22 634 459Saskatoon 22 379 233Winnipeg 22 1259 666

tities, and their frequency of occurrence,on a storm and seasonal basis. The objective was to determine the statistical characteristics of these terms for ease in extrapolation to return periods beyond theperiod of record and to provide the datafor the production of maps of the meanand coefficient of variation of the rainfallfactor.

PROCEDURES

The original equation for the storm kinetic energy used by Wischmeier andSmith (1978) was given in English units.When conversions to the SI system areperformed the equation becomes

£, = 2 [(11.93 + 8.73logI()/„)/] (2)

where ET is the total storm kinetic energyin Nm/m2; /n is a uniform rainfall rate inmm/h within the storm; / is the rainfalldepth in mm, resulting from /n; and n isthe number of periods in a storm duringwhich the rainfall rate was uniform. Theequation for the storm rainfall factor, /?, is

R = EtWIOO (3)

where R is the storm rainfall factor in(MJ/ha) •(mm/h) (Foster et al. 1981); ETis as defined previously; and 730 is themaximum 30-min rainfall intensity inmm/h. Limitations applying to theseequations include a maximum of 76.2mm/h for /n and 63.5 mm/h for /30(Wischmeier and Smith 1978).

The procedure used by Wischmeier andSmith (1965) for finding the total kinetic

energy, ET, was to search the storm masscurve for the total precipitation occurringat a selected rainfall intensity, 7n, and tomultiply it by the kinetic energy calculated for that value of/n. This was donefor all values of 7n, and the total kineticenergy of the storm, £T, was the sum ofall the incremental kinetic energies.

The rainfall data used in this study werehourly precipitations as these are theshortest duration data commonly available for the prairies in digital form. Stormkinetic energy was determined by calculating incremental kinetic energies fromEq. 2 for each hour of the storm and summing for the duration of the storm. Calculation of the kinetic energy in this manneris basically no different from the procedure used by Wischmeier and Smith(1978) except that periods of equal intensity are limited to integer multiples ofhours because of the discrete nature of the

data.

The accuracy with which storm erosionand rainfall factors could be determined

using hourly data was evaluated, however, by comparison with the same factorscalculated using 15-min duration rainfalldata. Short-period data were obtained forSaskatoon for some 50 storms recorded at

seven tipping-bucket gauge locations.Data for major storms at 12 other stationsin Manitoba, Saskatchewan and Albertawere also used. Storm kinetic energiesusing 15-min and hourly data were calculated for each storm and a ratio of the twovalues was determined. The ratios were,with one exception, larger than 1 andranged from 1.02 to 1.79 for very smallkinetic energies and from 1.01 to 1.09 forthe larger, dominant storms. A correctionfactor equation, to convert storm kineticenergies calculated using hourly data tothat which would be calculated usingshort-period data, was developed andused for all stations.

A storm was defined as all hourly precipitations separated by less than an arbitrarily chosen 10 h. Examination of theresults in comparison with the resultsusing other separation periods showedthat the 10-h period was acceptable.

The erosion index is the product of thestorm kinetic energy and the maximum30-min rainfall intensity for the storm.Values for the maximum 30-min rainfallintensity, 730, were obtained for each dayof record, from Atmospheric Environment Service records. These data were

merged with the appropriate hourly datarecords to allow calculation of storm

rainfall factors. When a storm began andended on the same day, the maximum 730for that day was assumed to apply to the

72 CANADIAN AGRICULTURAL ENGINEERING, VOL. 28, NO. 2, SUMMER 1986

MEANS

SEASONAL RAINFALL EROSIVITY FACTORPRAIRIE PROVINCES

Figure 2. Contour plot of mean seasonal rainfall factors for the Prairie Region.

storm; when the storm began and endedon different days, the maximum 730 duringthese days was assumed to apply to thestorm. Perusal of some of the recordedhourly precipitations and 730 data indicatedthese assumptions were reasonable.

The computer program yielded a datafile giving for each storm the beginningand ending month, day, and hour of thestorm, the rainfall depth during the storm,and the storm kinetic energy and rainfallfactor. Another data file produced by theprogram detailed the missing hourly precipitation data.

The influence of missing data on thecalculated storm and seasonal erosion indexes and rainfall factors was of someconcern. Missing data included missingstorms, missing hourly precipitations andmissing30-min intensities. The amount ofmissing storm and hourly data is a mea

sureof the reliabilityof the equipment andof the care taken in obtaining the data andmaintaining the equipment. Given thatboth equipment and procedures have improved over the years, the problem ofmissing data is less now than in the past.Examination of the missing hourly datafile confirmed this in that the number ofhours for which there obviously were nodata was small relative to the total numberof rainfall hours.

The maximum 30-min intensity rainfallwas used to calculate the erosion indexand rainfall factor for every storm and,therefore, missing intensity data wereof much more importance than missinghourly precipitations. Accordingly, aregression equation relating recordedmaximum 30-min intensity to the corresponding maximum recorded hourly precipitation amount during a storm was

CANADIAN AGRICULTURAL ENGINEERING, VOL. 28, NO. 2, SUMMER 1986

developed for every station. Missing intensity data for given storms were evaluated by using the appropriate regressionequation together with recorded maximum hourly precipitation data for thestorm.

Seasonal rainfall factors are the sum ofall the storm rainfall factors throughoutthe season, which was defined as extending from 15 Apr. through 31 Oct. The15 Apr. beginning date was setby the endof the snowmelt period and the 31 Oct.ending date was set by the beginning ofthe snow accumulation period. Seasonalrainfall factors were only included foranalysis if records for June and July wereavailable, since these are the monthswhen the major rainstorms occur in theprairies.

The mean, standard deviation, and coefficient of skewness of the seasonal and

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COEFFICIENTS OF VRRIRTION

SEnSONFIL RniNFRLL EROSIVITY FACTORPRAIRIE PROVINCES

Figure 3. Contour plot ofcoefficients ofvariation ofrainfall factors for the Prairie Region.

storm rainfall factors weredetermined bymeans of another computer program designed to perform frequency analyses(Dumontier 1978) of data according tovarious probability distributions. Thesedistributions included the two- and three-parameter log-normal distributions asevaluated by the method of moments andthe two-parameter Pearson distributionalso evaluated by the method of moments.The log-normal II and Pearson II distributions were also fitted using the method ofmaximum likelihood (Yevjevich 1972).The intent of this analysis was to decide,on the basis of visual inspection of thefrequency plots, the probability distribution most consistently fitting the data atthe various locations.

The stations analyzed were chosenbased on considerations of period ofrecord and geographic distribution. Origi

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nally the analysis was limited to Saskatchewan stations due to the amount of datarequired but was extended to include datafrom all stations in Manitoba and Alberta,for which hourly precipitation recordswere available.

RESULTS AND DISCUSSIONAs mentioned, five combinations of

frequency distributions or fitting methodswere used on the seasonal and storm rainfall factor data. A visual assessment of thegoodness of fit of the distributions to datafrom each of 54 stations showed that thelog-normal III distribution fitted by themethod of moments or the log-normal IIfitted by maximum likelihood were fairlygood. The log-normal II distribution fittedby the method of moments was preferredhowever, on the basis of consistency andsimplicity. This finding is consistent with

the conclusions of Wischmeier and Smith(1978). An example of this distributionfitted to data for Regina is shown inFig. 1.

Magnitudes of mean, seasonal rainfallfactors are shown in Table I for selectedprairieregion stations withlongperiods ofrecord. Also shown are the standard deviation values for each station which,together with use of the log-normal IIdistribution, allow calculation of rainfallfactors of any desired return period.

The regional, spatial variability whichcan be noted by comparing tabulated values for stations in Manitoba to those forstations in Alberta is also shown on thecontour of mean seasonal rainfall factors(Fig. 2). The contours of the plot weredeveloped using data from all of theprairiestations for which hourly datawereavailable and for which at least 10 yr of

CANADIAN AGRICULTURAL ENGINEERING, VOL. 28, NO. 2, SUMMER 1986

data existed. The contouring was donewith a computer contouring package.

The mean seasonal rainfall factors are,on average, lowest in the far north, ofmoderate to low magnitude in southernAlberta and southwestern Saskatchewan,and increase dramatically in southeasternSaskatchewan and southern Manitoba.The values for central to northern Manitobaalso tend to be higher than elsewherein the total region. Rainfall factor magnitudes near the U.S. A. border are consistent with those reported by Foster et al.(1981) for the northern states just belowthe prairie region.

The coefficiency of variation of the seasonal rainfall factor, shown contoured onFig. 3, is between 0.6 and 0.7 throughoutmost of the region, regardless of the magnitude of the mean seasonal rainfall fac

tor. From this, one could conclude that itis a very stable parameter and thus fairlyreliable as calculated.

As noted earlier and as reported byWigham and Stolte (1984) the seasonalrainfall factors follow a log-normal distribution as fitted by the method of moments, at least to an acceptable degree.The log-normal distribution fitted by thismethod requires only the mean and thestandard deviation of the rainfall factors.

From these maps of the mean and coefficient of variation in the rainfall factor, it ispossible to determine the rainfall factorthat correponds to any given return periodfor 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 onFig. 3 are fairly high indicating a considerable variation in seasonal rainfall fac

tors. An even greater variation in stormrainfall factor was observedfor particularlocations. The largest single storm rainfallfactor in a season often constitutes 90% ormore of the corresponding seasonal valueat some stations. This occurs because the

kinetic energy of rainfall is a function ofterminal velocity and raindrop size, bothof which increase rapidly with rainfall intensity. Storms of high intensity are veryimportant elements, therefore, in the determination of the rainfall factor. Theprairie region is subject to considerablethunderstorm activity so the accuracy ofany calculated rainfall factor is a functionof how well the frequency of occurrenceof intense storms at a station represent theregional frequencies of occurrence.

CONCLUSIONS

The seasonal factors are lower than val

ues quoted in the literature (Wall et al.1983) for annual values for those stationsfor which comparisons are possible. Theannual values used for comparison were,however, calculated using an alternateprocedure to that originally used byWischmeier and Smith (1965, 1978). Theseasonal rainfall factors determined here

in encompass the period when agriculturalland is most susceptible to erosion andshould, therefore, provide acceptablecomparative values for calculation of soilerosion potential.

The log-normal frequency distributionfitted by the method of moments providesthe best fit to the seasonal rainfall factor

data, considering the criteria of consistency and simplicity. The data on meanseasonal rainfall factors and coefficients

of variation can be used with the log-

CANADIAN AGRICULTURAL ENGINEERING, VOL. 28, NO. 2, SUMMER 1986

normal distribution characteristics to calculate a rainfall factor for a selected returnperiod for any location on the prairies.

REFERENCESCOOTE, D. R. 1983. Soil degradation in Can

ada— an overview. Proceedings of the 8thB.C. Soil Science Workshop on soil degradation in British Columbia, pp. 6-30.

DUMONTIER, G. 1978. Flood frequencyplotting package, Unpublished report,Dept. of Civil Engineering, University ofSaskatchewan, Saskatoon, Sask.

FOSTER, G. R., D. K. MCCOOL, K. G.RENARD, and W. C. MOLDENHAUER.1981. Conversion of the Universal Soil

Loss Equation to S.I. Metric Units. J. SoilWater Cons. 36: 355-359.

PFRA 1982. Land degradation and soil conservation issues on the Canadian prairies — anoverview. PFRA, Soil and Water Conservation Branch Report.

WALL, G. J., W. T. DICKINSON, andJ. GREUEL. 1983. Rainfall erosion indices

for Canada east of the Rocky Mountains.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 CivilEngineering 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. Agricultural Handbook No. 537.

WISCHMEIER, W. H. and D. D. SMITH.1965. Predicting rainfall — erosion lossesfrom cropland east of the Rocky Mountains. U.S. Dep. Agric, Washington, D.C.Agricultural Handbook No. 282.

YEVJEVICH, V. 1972. Probability and statistics in hydrology. Water Resources Publications, Fort Collins, Colorado.

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