#270

OF STOCHASTIC PROCESSES

Edited by

H. W. Shera

Colorado State University, U.S.A.

and

H. Kikkawa

Waseda University, Japan

WATER RESOURCES PUBLICATIONS

P.O. Box 2841

Littleton, Colorado, U.S.A. 80161

i

TAB 1.1: 01* CONTENTS

Chapter

1 Field Data Collection and Analysis for Sediment Yieldand Transport

K. Ashida, Kyoto University, Kyoto, Japan

2 Data Collection and AnalysisC. F. Nordin, Jr., U.S. Geological Survey,Denver, Colorado

3 Stochastic Aspects of Watershed Sediment YieldD. A. Woolhiser, U.S. Department of Agriculture,Fort Collins, ColoradoK. G. Renard, U.S. Department of Agriculture,Tucson, Arizona

4 An Proposal of Stochastic Evaluation for WatershedSediment Yields

A. Murota, Osaka University, Osaka, Japan

5 Bed Forms and Hydraulic Relations for Alluvial StreamsT. Kishi, Hokkaido University, Sapporo, Japan

6a Bed Forms in Alluvial Streams: Some Views on CurrentUnderstanding and Identification of UnresolvedProblems

J. F. Kennedy, The University of Iowa, Iowa City,Iowa

6b Sediment Transport in Migrating Bed FormsJ. C. Willis, U.S. Department of AgricultureSedimentation Laboratory, Oxford, MississippiJ. F. Kennedy, The University of Iowa, Iowa City,Iowa

7 Alluvial Bedforni Analysis I: Formation of AlternatingBars and Braids

T. Hayashi, Chuo University, Tokyo, JapanS. Ozaki, Central Research Institute of ElectricPower Industry, Abiko City, Japan

8 Diffusion and Dispersion of SedimentM. Hino, Tokyo Institute of Technology, Tokyo,Japan

9 Interaction Between Turbulent Fluid and SuspendedSediments

S. Fukuoka, Tokyo Institute of Technology,Tokyo, Japan

Chapter 3

STOCHASTIC ASPECTS OF WATERSHED SEDIMENT YIELD

D. A. Woolhiser, Research Hydraulic Engineer, USDA, Science andEducation Administration, Colorado State University, Engineering

Foothills Campus, Fort Collins, Colorado

and

K. 6. Renard, Research Hydraulic Engineer, USDA, Science andEducation Administration, Southwest Watershed Research Center,

442 E. 7th St., Tucson, Arizona

3.1 Abstract 3-13.2 Introduction 3-23.3 Background 3_23.4 Deterministic Estimates of Sediment Yield .... 3-53.5 Stochastic Description of Sediment Yield 3-9

Definition and notation 3-9Sediment yield from a plot 3-10Sediment yield from a small watershed(field) 3-13Sediment yield from a large watershed 3-14

3.6 Application of Stochastic Sediment Yield Models. . 3-163.7 Discussion and Conclusions 3-173.8 Figures 3_2l3.9 References 3_26

3'. 2 INTRODUCTION

Sediment yield is defined as "the total sediment outflowfrom a watershed or drainage basin, measurable at a point ofreference and a specified period of time" (ASCE Task Committee,1970). It is the portion of the gross erosion within a watershed(sum of all erosion from land and channels) that is not depositedbefore being transported from the watershed.

If we consider a sequence of measurements of sediment yield(annual sediment yield, for example), sediment yield can be considered a random variable and can be described by a probabilitydistribution function. The processes of soil particle detachment,entrainment, transport, and deposition, which are involved insediment yield, can be described as stochastic processes (processoccurring in time in a manner controlled by probabilistic laws).The purpose of this paper is (1) to briefly review deterministicmodels that are often included as components of stochastic sediment yield descriptions; (2) to develop some general definitionsand notation describing the sediment yield process; (3) to consider analytical and Monte Carlo methods of obtaining the distribution function of sediment yield in T years; and (4) finally,to examine some examples of application of stochastic sedimentyield models.

3.3 BACKGROUND

The sediment yield process may be divided into two categories - - the upland phase and the lowland phase (Bennett, 1974).Sediment detachment and transport processes predominate in theupland phase and depend on individual rainfall (or snowmelt)events. Sheet and gully erosion, landslides, and constructionsites are included in this category. Predominantly silt and clayparticle sizes are produced by sheet erosion. Erosion productsof a larger size may be more commonly produced by gully erosionand landslides, but frequently they will be transported onlyshort distances before they are deposited. The magnitude oferosion from the upland phase depends on factors like slopesteepness and shape, cover, soil type, land use, and rainfallcharacteristics.

In the lowland phase, sediment transport and depositionprocesses predominate, and the bed load transport capacity becomes a significant factor because channel flow is normallycapable of transporting all the fine material available. Sediment transport rates are determined by hydraulic variables, likethe flow depth and velocity, channel slope, water temperature,and physical properties of the sediments. The effect of individual precipitation events is highly damped.

The substantial differences between the temporal patternsof sediment yield in the upland phase and the lowland phase havebeen documented (McGuinness, Harrold, and Edwards, 1971). Monthly sediment yield from 0.4- to 1-ha (1- to 2-acre) watershedscorrelated well with rainfall characteristics and vegetative

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cover factors, whereas monthly suspended sediment transport froma 15,500 km2 (6,000 mi?) watershed was correlated with river flowand was unrelated to precipitation pattern. Much of the sedimenteroded from upland watersheds is deposited where it may be erodedlater by lowland erosion processes.

The seasonal variation in the upland sediment yield process will depend on seasonal variations in precipitation, cover,soil erodibility, and infiltration characteristics. These periodicities must be considered in constructing mathematical modelsof erosion processes and may be important when sediment is considered as a pollutant. They will be less important in problemsof sediment deposition in reservoirs. Sediment yield from uplandsheet/rill erosion sources is generally greater than that fromother sources (Leopold et al. 1966, Glymph 1951). Piest et al.(1975) reported that only about 20% of the total sediment yieldfrom a 30-ha (74-ac) watershed in the loessial soils of Missouriresulted from gully erosion in an area with visible gully development.

Because sediment yield responds quite readily to man's activities, various types of nonstationary processes may be presentin sediment yield data. For example, if the percentage of theland area in row crops is increasing or decreasing, it would bereflected in the sediment yield from upland areas. Constructionof reservoirs or sediment basins may result in a sudden decreasein sediment yield immediately downstream. The reduction may bedue partly to reduced runoff, and partly to the release of waterrelatively free of sediment from the reservoir. Urban development or surface mining may cause large increases in erosion rates,followed by a transient period as vegetation becomes reestablished and new surface drainage nets develop. Land treatment measures are basic elements of watershed projects developed underthe U.S. Watershed Protection and Flood Prevention Act of 1954(P.L. 566) and will have a significant effect in reducing sediment yields from upland areas (ASCE Task Committee, 1969).

Because erosion changes the land forms, which in turn affecterosion rates, clearly in the geological time scale, sedimentyield can never be a stationary process. However, stationaritycan frequently be assumed for engineering projects for times,like 50 years, assuming there are no major changes in the landuse during this period.

Many components of the sediment yield process are bestdescribed by their probabilistic structure because randomnessis the very essence of the individual process. The most obvious source of randomness in sediment yield from upland areas isprecipitation. Indeed, most investigators have considered precipitation as the only source of randomness and have coupledstochastic rainfall models with deterministic runoff and erosionmodels to develop simulation models of sediment yield. In thelowland phase, there have been many investigations of the relationship between river discharge and sediment discharge. Manyof these studies showed a relationship of the form:

3-3

n + r. (3-DG = aQ" + e

where Gs 1s sediment discharge, Qis water discharge annare oarameters, and E is a random component. The distributionfunction of c and its dependency structure has not been systematica lv examined, and undoubtedly should include provisions toaccounl\or the seasonal hysteresis of Gs +Q, observed in manydaU Other sources of randomness in the sediment ylelc| Processinclude fluctuations in rainfall drop size, wind effects, tillage operations, vegetative cover, soil infiltration rates ande9rod?bimy! gilly head-cutting and bank-caving, landslides andvarious random processes (both natural and man "duced) thattrigger potential instabilities in channels and valley floors.Fire insect infestations, and earth movements provide additional sources of random variability with relatively low fre-qUeRCC °oCfCtheseCsiochastic processes have a complex depend-encv structure. For example, the probability of a forest fire 1vear after one ha occurred is much lower than it would be 30years lTer!. Asimilar dependence structure is evident ininsect infestations that kill perennial vegetation. .

s r lated to the expected sediment yield the trap efficiencyif the reservoirs, and the sediment size distribution (ASCE Task?ommfttee 1969) No provision is made for the random variabill-tvTf sediment yield. Recent studies using Bayesian decisiontheorv have shown that information on the distribution function1 cln^pnt vield is required to determine the worth of addition-of sediment yieia is requneu u« « . . . . rp<;prvoiral data in the Bayesian sense (Jacobi, 1971) and that reservoir

desiqns based only on the expected sediment yield, are often1optimal Smith! Davis, Fogel, 1977). These findings sug-oested that all design procedures requiring estimates of sedimentvield should be examined with respect to their sensitivity to the"natural variability of sediment yield and to uncertainties in thenaramptprs describinq this variability.^"^ though the stochastic nature of the sediment yiel process

Sdb ;::tl!»l resource projects and the deve op-

HcalSSstat st p?ation'of stochastic process models.UnLrrintive statistics, the sample is of primary interest; in

i m^i n»t1st1cs the sample is important only for the in-Zll i t ]n;'concerning the population it representsAnalytical statistics requires experimental desion, including

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the construction of formal statistical models and statisticalinference in analysis and interpretation. The application ofstochastic processes requires construction of formal (mathematical) models describing the chance mechanisms involved in thephenomena. The stochastic models usually include deterministiccomponents developed from fundamental physical considerations,and the parameters may have some physical interpretations. Mostresearch on sediment yield is included in the first and secondcategories; stochastic models are a recent development.

3.4 DETERMINISTIC ESTIMATES OF SEDIMENT YIELD

Because the specific needs for sediment yield predictionsare so varied, no general model can meet these needs without aconsiderable loss of efficiency; i.e., the model would be toocomplicated to afford the required estimates without considerable inputs of prototype data and computer costs. Onstad,Mutchler, and Bowie (1977) discussed these needs and categorizedthe models on the basis of length of model event time, area tobe simulated, and sediment sources to be considered.

A deterministic model of upland erosion, based on physicalprinciples of hydrology, hydraulics, sediment transport, anderosion mechanics was presented by Foster and Meyer (1972). Themodel is based upon the continuity equation for mass transportunder steady-state conditions

If •Dr +D1 <3"2>where G = sediment load (weight/unit time/unit width), x = distance downslope, Dp = detachment (or deposition) rate of rillerosion (weight/unit time/unit area), and Di = delivery rate ofparticles detached by interrill erosion to rill flow (weight/unit time/unit area).

They also included an approximate expression to account forthe interaction of detachment (or deposition) and sediment load:

Dr r

cr cr

where Dcr = the detachment capacity of rill flow, and Tcr = thetransport capacity of rill flow.

Dcr« TCr. and Dj are functions of rainfall and runoff characteristics, soil properties and vegetative cover.

Equations (3-2) and (3-3) can be combined by eliminatingDr.

ff =̂(Tcr-G)+D. (3-4)The rill transport capacity during erosion (Tcr > G) is not the

3-5

svame as during deposition (Tcr < G) because of changes in thesurface geometry.

The delivery rate of particles detached by interrill erosion, Di , depends on rainfall energy and on the transport capacity of interrill flow, which in turn depends on rainfall intensity, rainfall excess, and the ril1-interrill geometry.

The detachment capacity of rill erosion, DCr> and the rilltransport capacity, Tcr, are similar functions of the bed shear,t. An approximation that may be used when the shear is large ascompared with a critical shear, t , is

Dcr = Cct3/2 (3-5)

and

Tcr •CtT»« (3-6)

substituting these expressions into equation (3-4), we obtain

~ = Cd {Ctl3/? ' G) "h D1 (3"7)ax

where C.= ^

For steady state conditions, x is a function of rainfall excessrate, slope, and distance along the slope, x, and closed formsolutions can be obtained. Foster and Meyer (1975) wrote equation (3-7) in a dimensionless form and presented solutions forplane and simple concave and complex slopes. They showed thatif the sediment yield from a plane slope for a constant rainfallexcess for a given time was set at 1.0, the sediment yields forthe convex and concave slopes (with the same total relief) were1.12 and 0.494, respectively.

Recently, Lane and Shirley (1978) used the equations ofrill and interrill erosion and transport with the kinematic flowequations

andVt +U -R <3-8>q = Kh"1 (3-9)

where h = flow deptht = time

q = runoff rate per unit widthx = distance

R = rainfall excess rateK = slope-resistance coefficient andm = dimensionless exponent

to obtain analytic solutions for the general case of sedimentconcentration in overland flow for the rising, equilibrium, andrecession hydrograph. This runoff-erosion model was tested

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using data from a small watershed in Arizona. Simulation resultswith optimal model parameters seemed to give good fits to observed runoff and sediment concentration data (Figure 1).

More important than the agreement between observed and simulated data was the ability of such a model to illustrate sedimentdischarge rate on a plane (Figure 2). The heavy lines in thisfigure are the boundaries of different solution domains projectedto the surface from the x-t plane. Seven regions are involved insuch a simulation model involving domains of flow establishment,established flow, pre-recession, and recession. The 1.3 ha watershed used in the simulation was modeled as a plane with 194-mlength, 47-m width, and a total relief of 7.8-m. Experiments arenow being planned to verify the model parameters using a rainfallsimulator.

As Onstad et al. (1977) stated:"Erosion and sediment yield models of this type haveappeared only recently because it soon became apparentthat empirical models were being modified and used beyond their original objectives and scope. This approachhas been accelerated with the recent interest and re

quirements of national water quality management. Thisdoes not mean that empirical parametric approaches arenot useful. On the contrary, demands for immediateanswers to many water quality problems necessitates useand modification of existing techniques to accommodatethe vast amounts of data that have been accumulated byresearch and action agencies at all levels of government. However, elements of time-variant deterministicmodels, based on theoretical approaches, will graduallyreplace many empirical functions as they become available and are verified and accepted."Although the processes controlling transport and deposition

of sediment are complex, the deterministic components describingthese processes in stochastic sediment yield models have beensimple ones.

Murota and Hashino (1969) used a bed load formula of theBrown type:

dm 1(2.- 1) gdV p ' 3 m

(3-10)

where u* is the shear velocity, dm is the average sediment size,g is the acceleration of gravity, o/p is the specific weight ofthe sediment and K and n are experimental constants.

Renard and Lane (1975) used the Laursen (1958) transportrelation: 7 ^___

c-Z[p$ Sli-U f(^)l (3-11)*• Tc ••

where c is the mean concentration of total sediment in percent,

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p = bed material fraction of diameter d, y = depth of flow infeet, t0' = boundary shear stress associated with sediment diameter, t0 = tractive force at the stream bed, tc = criticaltractive force, y = specific weight of water, p = density ofwater and w = fall velocity of sediment.

In both of the above examples the sediment transport rateis a unique function of the water discharge and sediment sizeand the sediment yield was obtained by numerical integration.It is further assumed that there is no limitation on the amountof sediment available for transport.

Renard and Lane (1975) suggested that the mean and standarddeviation of the sediment size should be treated as random variables in ephemeral streams because of substantial variations observed in these statistics.

Statistical equations have also been widely used for predicting sediment yield. Generally, such models use equation(s)relating sediment yield to one or more watershed and climaticfactors. Such models by their nature require relatively largequantities of data on watershed parameters and on sediment discharge, and thus require considerable time and expense to collect adequate data. These models are widely used for problemsrequiring sediment yields averaged over long periods. Theirwidest use is for larger watersheds, including those used forwater supply. Such problems do not generally necessitate detailedlocations of sediment sources because the main interests are

storage reservoirs, delta formations, or channel capacity. Inaddition to regression equations, the flow-duration, sediment-rating curve procedure (ASCE, 1975) can provide adequate sediment yield estimates for a particular watershed, but the resultsare difficult to extrapolate to other watersheds, particularlyungaged areas .

Notable examples of regression-type sediment yield modelsin the U.S. are Flaxman (1972), Anderson (1976), Branson andOwen (1970), Tatum (1963), Hindall (1976), and Herb and York(1976).

Dendy and Bolton (1976) derived sediment yield equationsfrom reservoir deposition data for about 800 reservoirs in theUnited States, representing watershed areas from 2.6- to 60,000-km2 and runoff ranging from near zero to about 125 cm/year. Inareas where runoff is less than 2-in (5-cm), they derived theequation:

S= 1280 Q0-46 (1.43 - 0.26 log A) (3-12)

and for other areas (runoff greater than 2-in)

S = 1958 e-°-055Q (1.43 _ 0.26 log A) (3-13)

where S = sediment yield (t/mi2/yr)and Q = runoff (in)

A = watershed area (mi2).The coefficient of determination for these two equations is 0.75.

3-8

Although it may be unwise to use these equations for any specific locations because of widely varying local factors, the equations express a general relationship for sediment yields on aregional basis.

Because of the dependence between the variates related tosediment yield, multivariate analysis can be helpful. Wallisand Anderson (1965) utilized multivariate techniques includingprinciple component analysis, and principle components regres-.sion to develop a simplified prediction equation for sedimentyield. They pointed out that one of the limitations of statistical approaches is that they cannot be used where changes willoccur due to man's activities and suggested that a better methodwould be to develop a probabilistic model of watershed performance for computer simulation.

3.5 STOCHASTIC DESCRIPTION OF SEDIMENT YIELD

Consider the generalized watershed shown in Figure 3 withprocesses distributed in time and space. Such a system is bounded on the bottom by rock, by an imaginary side surface, s, andby the imaginary top surface, A. Input to such a system consistsof the precipitation flux to the surface A, denoted as the stochastic process £i(x, y, t). Rainfall excess, f,2(*> y» t) is theamount of precipitation greater than that which infiltrates soiland results in the surface runoff, ci(t), at the watershed outlet. Besides the runoff and sediment yield, ;2(t), outputs fromthe system include the evapotranspiration, f^tx, y, t), and porous media flow through surface S, n(x, y, z, t).

Definition and Notation

Suppose that we have accurate instantaneous measurements oftotal sediment transport at a stream draining a watershed witharea A. We shall denote this instantaneous rate of transport as{^U). teT), where <;2(t) 1S tne total sediment transport rate(MT~i) and T = {t >, o}. From physical considerations, we knowthat c2U) is dependent (among other things) on the stream discharge, Ci(t), and the precipitation rate, ?i(x,y,t). The sediment-yield process, Y2(t), can then be expressed as:

Y2(t) =/t C2(s)ds (3-14)0

It is evident that Y2(t+At) >, Y2(t), therefore, the sedimentyield process represents a stochastic process of non-decreasingsample functions. The distribution function of sediment yield,frequently of interest in design problems, is

Ft(y) - P{Y2(t) * y) (3-15)

Let T(y) be the minimum time required for the accumulated sediment yield to equal or exceed the amount y, or

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T(y) = infft.Yp(t) >. y) (3-16)

which is often called the first-passage time. The distributionfunction for the first-passage time is

Gy(t) = P{T(y) .< t} (3-17)

Now P(T(y) > tl = P[Y?(t) <;. y)

Therefore,

Gy(t) =1- Ft(y) " (3-18)Equations (3-14) through (3-18) are general and can be ap

plied to any situation. To develop useful stochastic descriptions of sediment yield, however, we must consider the structureof the other physical processes controlling sediment yield. Aconvenient way to do so is to consider the variables and parameters, discussed by Foster and Meyer (1975), and to consider howthey might vary for plots, small watersheds (field size), andlarge watersheds. This is essentially the same approach utilizedby Woolhiser and Blinco (1972).

Sediment Yield from a Plot

Case I in Table I corresponds to sediment yield from a plotand is by far the simplest case. We will first consider some ofthe approaches that might be used for this case, and then consider possible extensions to more difficult cases. Considersediment yield from the standard fallow plot, used by Wischmeierand Smith (1960 and 1965), in developing the Universal Soil LossEquation. It is 22.1-m (72.6-ft) long, has a 9 percent slope,and is tilled in the direction of maximum slope. From Table Iwe see that the interrill particle delivery rate, Dj, can be assumed to be spatially constant, but, because it is dependent onrainfall intensity, it will vary with time.

The transport capacity and detachment capacity of rills,TCr and Dcr> vary in both space and time. A characteristic runoff response time (which might be the steady state storage onthe surface divided by the rainfall excess rate) might be a fewminutes.

We will be concerned with the four dependent stochasticprocesses Ci(t), c2(t), ci(t), and c2(t), which represent therainfall rate, rainfall excess rate, runoff rate at the lowerboundary, and sediment transport rate at the lower boundary, respectively. Schematic sample functions of these processes areshown in Figure 4. In the following, our notation will conformas closely as possible to Woolhiser and Todorovic (1974), although we have introduced an additional process £2(t). Now, therainfall excess process, f.2(t), is primarily subordinate to therainfall process but also is related to runoff because infiltration can take place during periods of no rainfall when water is

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still present on the surface. The sediment yield process issubordinate to the other three processes, in a manner describedby Foster and Meyer (1972 and 1975) (discussed earlier). If weassume that the rainfall process, Ci(t), 1s given, and the parameters in an infiltration equation and an erosion equation areknown, it is possible to derive the runoff and sediment yieldprocesses by numerically solving these deterministic equations.This simulation approach was utilized with models of varyingcomplexity by Negev (1967), Foster and Meyer (1975), Holton,Yen, and Comer (1972), and Smith (1977) and could be utilized indeveloping distribution functions for sediment yield from aplot. Fleming (1975) presented a good general discussion of thesimulation approach.

Another approach would be to write the Foster and Meyersteady-state sediment transport equation as a function of discharge and rainfall intensity. If short erosion records wereavailable or if the parameters could be estimated a priori, itwould be possible to integrate with respect to time and to estimate sediment yield for storm periods or any arbitrary longerperiod.

As an example of the approach, the steady state equationfor sediment yield as a function of runoff rate and rainfallrate for a plot of length, L , and slope, S,

BL h{t)l B L^l(t)^ G S.C2(t)-Cl ^(Od-d-^^d-expt-l^^y})^^ (3-19!

8g'where d ={£)** CT Sf*:

y = specific weight of waterg = acceleration of gravityf = Darcy-Weisbach friction factor

CT = coefficient in the relationship TcQ = C* t3/2, wheret is the bed shear

B = the coefficient in the relationship D^B Ci (t)2and

Cp = the coefficient in the relationship Dc = Cp t3/2.Equation (3-19) is a dimensional version of Foster and Meyers'(1975) equations 11 and 12 and involves several simplifyingassumptions. By writing the sediment transport rate as a function of time, it is assumed that unsteady, free surface flowover a plot occurs as a series of steady states. It does provide a direct linkage between the sediment yield process, therunoff rate, and the rainfall rate. Obviously, it is only validwhen runoff Ci(t) > o.

A rainfall event is defined as any continuous period ofrainfall SiU) > o. A runoff event is any continuous period ofrunoff, and a sediment yield event is associated with each runoff event. Associated with the itn rainfall event is the timeof ending, Ri. Similarly, Ti refers to the time of ending ofthe itn runoff and sediment yield event.

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0

.004 r

003

.002

001

— orr.envro nuNorr

-- fll !EU RUNOPF

?0 40

TiME I MINUTES)

80

O8SERVE0 CONCENTRATION

FITTED CONCENTRATION

R* • 0 98

20 40

TIME (MINUTES)60 80

Fig. 3-1. Example of observed and fitted data, Watershed 63.101,Event 1, 7/14/73 (Lane and Shirley, 1978).

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TIME

(MIN

)

Figure

3-2.

Three

dimensional

view

ofsediment

discharge

rate

ona

plane

(Land

and

Shirley,

1978)

WATERSHEDBOUNDARY

RAINFALL EXCESS

EVAPOTRANSPIRATION

£0) RUNOFF

C2(») SEDIMENT

:ig. 3-3. Watershed-stochastic processes (modified fromWoolhiser and Blinco, 1975).

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• RAINFALL { (l)

RAINFALL EXCESS ^(l)

RUNOFF C,l»)

SEOIMENT TRANSPORT C2*»J

Fig. 3-4. Sample functions of the processes £,(t), c..(t),and c2(t).

WATERSHEOBOUNOARY

IELO BOUNOARY

CONTOURZ:CONSTANT

Fig. 3-5. Definition sketch - small watershed,

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distance duwnslope (m)

2 s 10

5 10 20 30 40

DISTANCE DOWNSLOPE (FT.)

Fig. 3-6. Rill losses for plot R-3, determined from microreliefdata, and interrill losses, computed as differencebetween total R-3 erosion in Fig. 3-1 and rill losses.Adapted from Meyer, Foster, and Romkens (1975).

2.0

1.0

0.5-

0.2

OISTANCE 0OWNSLOPE (m)4 10

0.3

4 10 30 60 80

OISTANCE DOWNSLOPE (FT.)

Fig. 3-7. Erosion of three mechanically shaped slopes from 5inches of intense simulated rain. Crofton silt loam

(deep loess soil) 18-1/3 ft wide, 9 percent averagesteepness (range of concave and convex 5 to 15 percent). (Adapted from Young and Mutchler, p. 169.)

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3.9 REFERENCES

ASCE, 1969. Chapter V, Sediment control methods, introductionand watershed area. J. of Hydr. Div. , ASCE 95(HY2):649-678.

ASCE, 1970. Chapter IV, Sediment sources and sediment yields.J. of Hydr. Div., ASCE 96(HY6):1283-1330

Ainer. Soc. of Civ. Eng., 1975. Sedimentation Engineering. VitoVanoni , ed. 745 p.

Anderson, H. W., 1976. Reservoir sedimentation associated withcatchment attributes, landslide potential, geologic faults, andsoil characteristics. Proc. 3rd Federal Inter-Agency Sedimentation Conf. p. 1.35-1.46

Bennett, J. P., 1974. Concepts of mathematical modeling ofsediment yield. Water Resources Research, 10(3):485-492.

Branson, F. A., and Owen, J. B., 1970. Plant cover, runoff, andsediment yield relationships on Mancos shale in Western Colorado,Water Resour. Res. 6(3):783-790.

Dendy, F. E. , and Bolton, G. C, 1976. Sediment yield-runoff-drainage area relationships in the United States. J. Soil andWater Conserv. 31(6):264-266.

Dragoun, F. J., 1962. Rainfall energy as related to sedimentyield. J. of Geoph. Res,. 67(4) :1495-1502.

Duckstein, L., Szidarovszky, F. F., and Yakowitz, S., 1977.Bayes design of a reservoir under random sediment yield. WaterResour. Res., 13(4)713-719.

Flaxman, E.M., 1972. Predicting sediment yield in westernUnited States. J. of Hydr. Div. , ASCE 98(12):2073-2085.

Fleming, G. , 1975. Sediment erosion-transport-deposition simulation: State of the Art. J_n Present and Prospective Technologyfor Predicting Sediment Yields and Sources, USDA-ARS-S-40.p. 274-285.

Foster, G. R., and Meyer, L. D., 1972. A closed-form soil erosion equation for upland areas, in Sedimentation Symposium toHonor Professor Hans Albert Einstein, edited by H. W. Shen. pp.12-1-12-19, Colorado State University, Fort Collins.

Foster, G. R. , and Meyer, L. D., 1975. Mathematical simulationof upland erosion by fundamental erosion mechanics. Xn Presentand Prospective Technology for Predicting Sediment Yields andSources. USDA-ARS-S-40. p. 190-207.

Glymph, L. M. , 1951. Relation of sedimentation to acceleratederosion in the Missouri River Basin. USDA-SCS-TP-102.

Herb, W. J., and Yorke, T. H., 1976. Storm-period variablesaffecting sediment transport from urban construction areas.Proc. 3rd Federal Inter-Agency Sedimentation Conf. p. 1.181-1.192.

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Hindall, S. M. , 1976. Prediction of sediment yields in Wisconsinstreams. Proc. 3rd Federal Inter-Agency Sedimentation Conf. p.1.205-1.218.

Holtan, H. N., Yen, C. L. , and Comer, G. H. , 1975. Potentialsof USDAHL models for sediment-yield predictions. j_n Present andProspective Technology for Predicting Sediment Yields and Sources,USDA-ARS-S-40. p. 214-219.

Jacobi, S. , 1971. Economic worth of sediment load data in astatistical decision framework. PhD Dissertation, ColoradoState University, Fort Collins.

Krumbein, W. E., 1968. Statistical models of sedimentology.Sedimentology (Amsterdam) 10(l):7-23.

Lane, L. J., and Shirley, E. D., 1978. Mathematical simulationof erosion on upland areas. Manuscript submitted to WaterResources Research.

Laursen, E. M., 1958. The total sediment load of streams.J. of Hydr. Div. , ASCE 84(HY1):1530.1-1530.36.

Leopold, L. B., Emmett, W. N. , and Myrick, R. M., 1966. Channeland hillslope processes in a semiarid area in New Mexico. USGSProf. Paper 352-G.

McGuinness, J. L. , Harrold, L. L., and Edwards, W. M., 1971.Relation of rainfall energy and streamflow to sediment yieldfrom small and large watersheds. J. Soil and Water Conserv.26(6):233-235.

Meyer, L. D., Foster, G. R., and Romkens, M. J. M., 1975. Sourceof soil eroded by water from upland slopes. ]m Present and Prospective Technology for Predicting Sediment Yields and Sources.USDA-ARS-S-40. p. 177-189.

Murota, A., and Hashino, M. , 1969. Simulation of river bed variation in mountainous basin. Proc. 13th Congress IAHR, Kyoto,Japan, 5(1):245-248.

Onstad, C. A., Mutchler, C. K. , and Bowie, A. J., 1977. Predicting sediment yields. Proc. Nat'1 Symp. on Soil Erosion andSedimentation by Water. ASAE Publication 4-77, p. 43-58.

Piest, R. F. , Kramer, L. A., and Heinemann, H. G. , 1975. Sediment movement from loessial watersheds, hi Present and Prospective Technology for Predicting Sediment Yields and Sources.USDA-ARS-S-40, p. 130-141.

Renard, K. G. , and Lane, L. J., 1975. Sediment yield as relatedto a stochastic model of ephemeral runoff, hn Present and Prospective Technology for Predicting Sediment Yields and Sources.USDA-ARS-S-40. p. 253-263.

Renard, K. G., and Laursen, E. M., 1975. Dynamic behavior modelof ephemeral stream. J. of Hydr. Div., ASCE, 101(HY5): 511-528.

3-27

o

Smith, J. H., Davis, D. R., and Fogel, M., 1977. Determinationof sediment yield by transferring rainfall data. Water Resources Bulletin 13(3):529-541.

Smith, J., Fogel, M., and Duckstein, L., 1974. Uncertainty insediment yield from a semi-arid watershed. Proceedings 18thAnnual Meeting of the Arizona Academy of Sciences, Flagstaff,Arizona, April, pp. 258-268.

Smith, R. E. , 1977. Field test of a distributed watershed erosion/sedimentation model, jji Soil Erosion: Prediction andControl. Special Publication No. 21. Soil Cons. Soc. ofAmerica, p. 201-209.

Tatum, F. E. , 1963. A new method of estimating debris-storage .requirements for debris basins. USDA Misc. Pub. 970, p. 886-898.

Wallis, J. R., and Anderson, H. W., 1965. An application ofmultivariate analysis to sediment network design. IASH Pub. 67,pp. 357-378.

Williams, J. R. , 1975. Sediment-yield predictions with universalequation using runoff energy factor. Jjn Present and ProspectiveTechnology for Predicting Sediment Yields and Sources. USDA-ARS-S-40, p. 244-252.

Wischmeier, W. H. , 1959. A rainfall erosion index for a universal soil-loss equation. Soil Science Society of America Proceedings 23(3):246-249.

Wischmeier, W. H. , and Smith, D. D., 1960. A universal soil-lossequation to guide conservation farm planning. 7th InternationalCongress on Soil Science Transactions 1:418-425.

Wischmeier, W. H., and Smith, D. D., 1965. Predicting rainfall-erosion losses from cropland east of the Rocky Mountains. U.S.Department of Agriculture, Agriculture Handbook No. 282, 48 pp.

Wolman, M. Gordon, and Miller, John P., 1960. Magnitude andfrequency of forces in geomorphic processes. J. of Geo!. 68(1):54-74.

Woolhiser, D. A., and Blinco, P. H., 1975. Watershed sedimentyield - a stochastic approach. Jji Present and Prospective Technology for Predicting Sediment Yields and Sources. USDA-ARS-S-40, p. 264-273.

Woolhiser, D. A., and Todorovic, P., 1974. A stochastic modelof sediment yield for ephemeral streams. Proc. USDA-IASPSSymposium on Statistical Hydrology. U.S. Department of Agriculture Misc. Pub. 1275. pp. 232-246.

Young, R. A., and Mutchler, C. K., 1969. Soil movement on irregular slopes. Water Resources Research 5(1):184-189.

3 -28

The total sediment yield for the vth event can be writtenas

fT,Yv =|" C:.(s)ds (3-20)

where c2(t) can be related to the rainfall and runoff processes

C2(t) =C, CittJ^fcjdJ.E^t)] (3-21)

onnr?,-ilC^WA/,l(t,J is the relationshiP given in brackets inequa cion IJ- IyJ.

Previous investigators have examined various relationshipsbetween Y and various functions of the rainfall and runoff processes. Dragoun (1962) for example, found the relationships

Yy =a + bEn ; R? = 0.613

Yv =a+b(Q +qp); R? =0.811 (3"22>where Ev is the kinetic energy of the rainfall event, Q is thevolume of storm runoff and qp is the peak rate of storm runoff.Wischmeier (1959) found that'the sediment yield per event ishighly correlated with the product of the total rainfall kineticenergy and maximum 30-min intensity for storms greater than 0 5in (12.7 mm). Woolhiser and Blinco (1975) added a random variable representing the error term to the regression relationship,i.e.,

Yv =K(Vv "m> +S 0-23)where m is acoefficient, and Ev is an independent normally distributed random variable with zero mean, and variance, o* Theytreated the sediment yield per event as a function of the sum oftwo independent random variables, the product of rainfall kinetic energy, E , and the maximum 30-min intensity, I , and theerror term, e . v

While Foster and Meyer did simulate sediment transport andcompared predicted erosion with observed sediment losses, apparently no one has simulated a large series of erosion events froma fallow plot to examine the form of the distribution functionof sediment yield per event. Such a distribution would dependupon soil erosion parameters, the length and slope of the plotand the rainfall and runoff characteristics.

Sediment Yield from a Small Watershed (Field)

A more complicated situation is Case II in Table I Thiscorresponds to a watershed about a few hectares in size. The

3-12

rainfall erosion potential, expressed as Di, is uniform over thearea, as are the cover and management practices. This case differs from Case I because the slopes are not uniform, and therecan be significant convergence of rills, so that the runoffleaves the watershed in a concentrated flow.

Woolhiser and Blinco (1975) constructed a wery simple modelutilizing isochrones to reduce the problem to one dimension andintroduced p(r), the probability that a soil particle erodedalong an isochrone during an event will reach the mouth of thewatershed during that event. Because the time-varying sedimenttransport was not considered, it seems unnecessary to introducea translation time. Contour lines and elevation could be usedmore easily. Also, if an erosion model similar to that discussed earlier is used, the requirement for p(r) disappears. Consider the schematic drawing of the very small watershed in Fig-

dAure 5. The length of the contour is -g^-, where A is the surface

area. Let sv (z) be the average weight of soil per unit areaeroded (or deposited) during the vtn event.

s..(z) °t±'z

v • I.1 fLz [Dr(z,s) + D.(z,s)]vds (3-24)

where Dr(z,s) + D-j(z.s) are the rill and interrill erosion forthe vtn event, expressed as functions of the distance, s, alongthe ztn contour, and L2 is the contour length. The total erosion during the vtn event is then

Y =f2"1 s(z) ^ dz (3-25)v v dz

Jzowhere z0 is the elevation of the mouth of the watershed, and zis the maximum watershed elevation.

The key to utilizing this relationship is in evaluating thefunction sv(z). It is a random function, depending on the rainfall intensity patterns, and infiltration characteristics of thevth event. The only advantage in using this approach over astraightforward simulation would be if certain regularities canbe found in the form of sv(z). If such regularities exist, itmay be possible to relate them to various geomorphic parametersand rainfall and runoff characteristics. This could lead tosubstantial savings of computer time.

One method of evaluating sv(z) would be to perform simulations with observed rainfall inputs, using a deterministic erosion model based on physical principles. An alternative wouldbe to make microrelief measurements after each of a series ofnatural or simulated rainstorms.

Meyer, Foster, and Romkens (1975) presented data showingerosion losses at a cross section versus distance downslope forsimulated rainfall on plots. Some of these data, for a simulated rainstorm of 6.35 cm/hr for 2-hr on a 3.65- x 10.66-m plotare shown in Figure 6. The total erosion loss at each cross-

3-13

section (which would be equivalent to sv(z)L(z)) is subdividedinto rill and interrill erosion. Therefore, the sums of each ofthese functions divided by the width of the plot would be equivalent to sv (z). Both Dj(z,s) and Dr(z,s) would be much moreerratic for a small watershed than for a plot because the lengthof slope and slope characteristics above each point on a contourmay vary considerably. Convergence of flow into rills may alsoresult in a "breakdown" and very rapid erosion at certain partsof the watershed (Meyer, Foster, and Romkens (1975)).

The effect of concavity or convexity of the profile ons0(z) is demonstrated by data presented by Young and Mutchler(1969) (Figure 7). The rapid decrease of average depth of erosion with slope length for the concave slope suggests that botherosion and deposition are occurring at the base of the slope,but that erosion is still greater than deposition. Quite likely,this function would become negative (net deposition) near themouth of the watershed for many cases.

Sediment Yield from a Large Watershed

If we consider the characteristics of upland erosion fromlarge watersheds (watersheds having spatial variability of precipitation, soils, vegetation, and land use), as shown in TableI, we see that all of the factors affecting sediment yield havesignificant spatial variability, and that the characteristicrunoff response time may vary from hours to weeks. Streamflowand sediment transport at the mouth of the watershed may occurcontinuously, but surface runoff and erosion on the upland areaswill be intermittent. Much of the sediment eroded from an upland area during a storm may be deposited in the stream channelsystem or flood plain where it may subsequently be eroded by increasing discharges when the entire watershed is contributing tostreamflow (but upland erosion may be small).

The stochastic model developed by Murota and Hashino (1969),is an example of an approach that develops a deterministic relationship between sediment transport and daily rainfall, and thendevelops distribution functions for total sediment yield in ann-day period. Murota and Hashino assumed a simple linear relation between daily rainfall and runoff so that the runoff response to rainfall of X, on the yth wet dav ]n a period oflength, n, can be given by

Q (t) = X h(t) (3-26)

where h(t) is the impulse response function. Then, knowing therelationship between runoff and stage, they could calculate thesediment yield due to the hydraulic forces on the channel byutilizing a sediment discharge formula like that of Brown. Upland erosion was ignored. By assuming independence betweenevents, they could then calculate a threshold amount of rainfall,w, below which no sediment would move for an isolated event.Also, given a sequence of n rainfall events in an N day period,

3-14

they could calculate the runoff and sediment yield. It is notquite clear from their paper if they used Monte Carlo simulationor numerical integration and summation to obtain distributionsof sediment yield for nine periods within a calendar year.

Renard and Lane (1975) and Renard and Laursen (1975) usedsimulation techniques to estimate distribution functions of sediment yield from semiarid watersheds. Basically, their modelconsisted of a stochastic runoff model and a deterministic equation to calculate sediment yield given a runoff hydrograph.Because flow from the watersheds is intermittent, they could develop an event-oriented model specified by eight parameters thatwere estimated from experimental data. Two variables were usedto describe the runoff season - - the starting date and the number of runoff events. The temporal distribution was describedby two variables - - the time of day and the interval betweenevents. Each runoff event was described by the runoff volumeand the peak discharge. Hydrographs were assumed to be triangular in shape and sediment transport rates were computed at threeor five points on the hydrograph, using Manning's equation tocalculate depth and Laursen's (1958) equation to calculate instantaneous sediment transport rates. No explicit provision wasmade for upland erosion.

Smith, Fogel, and Duckstein (1974) used two empirical expressions relating runoff to rainfall characteristics, and sediment yield to both rainfall and runoff characteristics in developing a stochastic sediment yield model for a semiarid watershed,The relationship between runoff volume, Qv, and effective rainfall per event, Xv, is given by the Soil Conservation Serviceformula

«v= iirrsr (3-27)V

where S is a watershed parameter. The peak discharge per event,qv, is related to the runoff volume and the rainfall duration bythe equation

484 AQ

% •(a,D +a;) (3"28)A. L

where Dv is the duration of the v event, A 1s the drainagearea (mi2), and aj and a2 are coefficients assumed constant fora given drainage basin. The 484 coefficient is specific for theEnglish system of units as are the other terms.

The sediment yield per event is given by the Modified Universal Soil Loss Equation (Williams, 1975).

Yy =95 (Qvqv)'56 KCPLS (3-29)where Y = sediment yield in tons

qu = peak flow rate in cfsQv = runoff volume in acre-ft

v

3-15

K = soil erodibility factorC = cropping - management factorP = erosion control practice factorLS = slope length and gradient factors.

Thus the sediment yield per event is a function of the random variable Xv and Dv. They assumed that Xv and Dv were distributed as a bivariate gamma distribution. The total seasonalsediment yield could be calculated by assuming that the numberof events was Poissonian.

3.6 APPLICATION OF STOCHASTIC SEDIMENT YIELD MODELS

There are \/ery few instances in the literature in whichstochastic sediment yield models have been applied to practicalproblems, even in the form of examples. Jacobi (1971) utilizedan elementary sediment yield model in evaluating the economicworth of sediment yield data in a statistical decision framework,He assumed that the annual sediment yield has a log-normal distribution with mean \i and variance o2. He further assumed thatthe annuaj series is independent. Because the estimates of uand a2, X and s2, respectively, are not known with certainty, heassume^ that the conditional joint distribution of u and a2,given X, s2 and the number of years of record, n was given by

(ft)Tthe following:

f(u,o2|n,X,s2) * exp { - n(u-X)

(sCT

a

alt

n-1

2 ,.exp{-

• ns?9a2

•}

r(^)c2(3-30)

A goal function, G(Q |n,a2), was then introduced. This

goal function is a penalty function and indicates the excesscost of a project due to over or underdesign of sediment storage

QJj . The Bayes risk is the expectation of the goal functionwith respect to f(u,a2).

The optimal design is to choose the alternative sedimentstorage Q£ that minimizes Bayes risk:

f

R(0J min

naltGtQ^ln.o2) f(,i,a?)du da2 (3-31)

Thus the storage Q| is the Bayes solution. If the true valuesof the parameters (pt, at2) were known, the information would

give Q , the alternative that gives the minimum variable risk

G(Q mi n

,alt[G(QasU|V"a in (3-32)

3-16

The opportunity loss (OL) is:

0L(Q*|ut,ot2) -G(Q*|Vot2) -G(Q^Mt,at2) (3-33)The opportunity loss represents extra costs because the decisionwas made without perfect knowledge.

The Expected Opportunity Loss (EOL) can then be defined as

EOL =(((G(Q*|u,o2) -G(Q>,o2)}-f(u,a2)du do2 (3-34)

where Qg is the design alternative that minimizes the goal function for each particular set \i, a2.

The EOL is the expected value of perfect information andthe decrease in EOL as more information is obtained is a measureof the reduction of uncertainty.

More information can be obtained by (1) use of all existingprimary data; (2) postpone project to obtain additional data;(3) use of regression models to augment the primary data set.

As an example, Jacobi (1971) evaluated the worth of additional data in the design of sediment storage for Cochiti Dam inNew Mexico. He found empirically that the expected opportunityloss was inversely proportional to record length, n, and thatthe expected marginal worth of 1 year's data was inversely proportional to n2.

Duckstein, Szidarovszky, and Yakowitz (1977) demonstratedthe conjunctive use of event-based simulation with a Bayesianapproach to decision making to make maximum use of a very limited amount of data available to estimate sediment yield from asemiarid watershed. In an example, they demonstrated that although the estimated mean sediment yield for the Charleston DamSite in Southern Arizona is 103.3 ac-ft, the optimal designvalue (in the sense that it minimizes the Bayes Risk) is 143.6ac-ft.

3.7' DISCUSSION AND CONCLUSIONS

In this section we should like to focus attention on three

questions: (1) Under what circumstances is it desirable tohave an estimate of the distribution function of seasonal orannual sediment yield rather than only an estimate of its mean?(2) What advantages, if any, does the stochastic approach haveover the more traditional approaches such as regression analysisand multivariate analysis? (3) Is there a place for analyticalsolutions to simplified models as compared to Monte Carlo simulation with more physically realistic models?

To answer the first question, we must reconsider the theoretical and applied problems in which estimates of sedimentyield and its distribution are relevant. From the standpoint ofunderstanding geological processes such as denudation, geomor-phologists are interested in the relative importance of extremeor "catastrophic" events and more frequent events of smaller

3-17

magnitude (Wolman and Miller, 1960). Knowledge of the mean sediment yield rate is not sufficient for this purpose. Appliedproblems include the determination of sediment storage capacityof a reservoir, the estimation of maintenance costs for removingsediment from various hydraulic structures, evaluation of theprobability of severe erosion during a period of construction orrevegetation, transport of chemicals attached to sediment, anddesign of sediment traps and debris basins.

The fact that in most cases the sediment capacity of a reservoir is set equal to the expected sediment yield for the design life should not lead us to believe that we could not usemore detailed information. It is simply a commentary on therudimentary state of design procedures. As Jacobi (1971) pointsout, and Duckstein, Szidarovszky, and Yakowitz (1977) demonstrated by an example, the expected sediment yield is not the optimal value and to estimate the optimal value one must have anestimate of the distribution of sediment yield. For large reservoirs with a long time horizon, the exact form of the distribution of annual sediment yield is unimportant, because by thecentral limit theorem, the sum of n annual sediment yields isasymptotically normal. For smaller structures or for shorterdesign periods, this approximation is not valid, and knowledgeof the distribution for annual or seasonal periods would be useful.

The essential difference between the stochastic approachand the statistical methods described earlier is that the stochastic approach explicitly includes the time sequence of sedimentyield, and therefore is a more general formulation. For example,once a stochastic model has been formulated and the parametersidentified one can readily consider functionals of the processsuch as distributions of the first passage time, distribution ofT year sediment yields, distribution of the largest occurrencein T years, etc. Stochastic models utilizing simulation can bemore physically realistic than statistical models, and can beused (with caution) to examine effects of system changes. Thereis still a definite need for statistical models in application,but as a field for research it appears to have limited value.

From the earlier sections of this paper it should be readily apparent that in order to obtain mathematically tractablemodels, we must greatly simplify the physical processes involved.The question naturally arises - why should one consider analytical models? -First, there may be some cases where the simplifications clearly do not affect the accuracy of the interpretationof model results. In these cases it would be foolish to use

simulation if analytical approaches are available. However,where there are some questions regarding the effects of simplification, one frequently must carry out simulations to comparethe alternative models. In these circumstances the analyticmodel would be worthwhile only if it provided some insight whichcould not be obtained from a simulation model of comparable accuracy or if it provides a means of generalizing or regionalizing which was not available from the simulation model. It is

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Table I. Characteristics of Sediment Yield Models

Sediment Yield Case I Case II Case III

Variables Plot Small watershed Large watershed

D. D(t) D(t) D(x,y,t)

Delivery rateof particlesdetached byinterrill ero

sion uniform uniform spatially varied

Tcr T(x,t) T(x,t) T(x,y,t)Transportcapacity varies varies spatiallyof rills downslope downslope varied

Dcr D(x,t) D(x,t) D(x,y,t)Detachment

capacity varies varies spatiallyof rills downslope downslope varied

Characteristicrunoff response min min to hrs hr to weeks

3-20