Basin sediment yield modelling using hydrological variables

C. A. Onstad and A. J. Bowie

Abstract. Estimates of sediment yields from basins and prediction of sediment sources are needed increasingly throughout the world for use in designing practices to conserve soil and control pollution. Sediment yield models incorporating hydrological and hydraulic flow proper t ies are useful in this respect. Models with these characteristics have been designed to predic t sediment yields from single storms to monthly and annual yields. Models of this type can a l so be used to predict sediment sources within basins. Three models for predicting yields from single storms are compared. One model provides estimates of the magnitude of the sediment sources on the basin, which give planners an index of areas where conservation measures are most critically needed.

Modélisation de la production en sédiments des bassins versants en utilisant des variables hydrologiques

Résumé. Les calculs de rendement en sédiments et la reconnaissance des sources de sédiments sont du plus en plus nécessaires dans le monde pour décider des pratiques à appliquer en conserva-tion du sol et pour le contrôle de la pollution. Des modèles de rendements en sédiments , incluant les propriétés d'écoulement hydrique et hydraulique, sont utiles à cet égard. Des modè les de ce type ont été créés pour la prévision à l'échelle de l'averse individuelle, du mois ou d e l'année. De mêmes modèles peuvent être appliqués à la prévision des sources de sédiments. T ro i s modèles de prévision des rendements en sédiments pour des orages individuels sont comparés ici . Un modèle fournit des estimations sur les sources de sédiments et leur importance, ce qui peut permett re aux décideurs de reconnaître les lieux où il convient en priorité d'appliquer une conservation des sols.

INTRODUCTION

Estimates of sediment yield are needed for reservoir design, conservation practice design, and as a basis for predicting off-site damages. Accurate estimates of sediment yield are also needed to determine the effects of management changes in basins such as removing residue for use as an energy source and cropping pattern changes resulting from increased emphasis on agricultural productivity. Prediction needs range from sediment yield estimates for single storms and annually for both large and small basins to sediment source locations ranging in size from single fields to many square kilometres.

In this paper, we describe three models and their capabilities for predicting short and long-term sediment yields. Also described are the capabilities of one model to predict sediment sources within a basin.

SINGLE-STORM MODELS

The models presented here use the universal soil loss equation (USLE), developed by Wischmeier and Smith (1965), as a base:

A =RKLSCP (1) 191

192 C. A. Onstad and A. J. Bowie

where

A = computed soil loss per unit area, R = rainfall energy factor, K = soil erodibility factor, L = slope length factor, S = slope steepness factor, C = cropping-management factor, P = erosion control practice factor.

This equation is based on several thousand plot-years of record and is used worldwide for estimating soil erosion from agricultural lands. It was developed for use on small cropped fields, but is also widely used in conjunction with a sediment delivery ratio to estimate basin sediment yields. The USLE with sediment delivery ratio (USLE-SDR) is one of the models tested. Sediment delivery ratios are commonly represented graphi-cally with basin area as the independent variable. Renfro (1975) presented such a graph based on several different data sources.

The second model was developed by Williams (1975a) and described in detail by Williams and Berndt (1976b). Their version of the equation is

Y = a(QqpfKCPLS (2)

where

Y = sediment yield from an individual storm [tonnes], Q = storm runoff volume [m3j], qp = peak runoff rate [m

3/s], a,(3= model parameters.

Equation (2) is essentially the same as equation (1) except that the rainfall factor has been replaced by a runoff factor. Williams and Berndt (1976b) contend that this substitution eliminates the need for delivery ratios. The values for Q and qv are deter-mined from runoff data or from hydrology simulation.

The third model treats erosion in two phases, detachment and transport. The model has been described by Onstad and Foster (1975), Frère et al (1975), and Onstad et al (1976b). The total soil detachment on a slope with several segments is represented by

V Wf(KCPS)j , „ E = ZJ}LW

A(^S-^t) (3) /

where

E = total detached soil [kg/m], Xj = length of slope segment; [m], Wj = energy factor.

The constant 31 is a conversion factor. The energy factor is expressed as follows:

W = aRst + 0.22(1 -a)QqlJ3 (4)

where

a = coefficient (0 < a < 1), Rst= storm rainfall factor [EI units of USLE], Q = runoff volume [m3], (7p = peak runoff rate [m3/h].

Basin sediment yield modelling using hydrological variables 193

The numerical constant, 0.22, was established from plot data (Foster et al, 1973). The coefficient, a, indicates the relative importance of rainfall energy compared with runoff energy for detaching soil. As indicated in equation (3), each slope segment may have a unique set of parameters.

The transport capacity is represented by

WAySCP),- , < Tj = J?—!>x}-* (5)

31

where

Txj= transport capacity at position Xj [kg/m], y = transportability.

The value for y is assumed to be the same as for K, but reflects the transportability of material from upslope segments across the segment being evaluated.

Sediment yield is calculated to the bottom of each slope segment by comparing the detachment and transport capacity. If transport exceeds detachment, soil is eroded from the segment, if detachment exceeds transport, soil is deposited on the segment. Total basin sediment yield is the sum of the yields of all slopes at the channel. For small basins channel erosion or deposition is not considered.

TESTING SINGLE-STORM MODELS

The data used for testing the single-storm models were obtained from the US Depart-ment of Agriculture, Agricultural Research Service experimental watersheds W-l and W-2 near Treynor, Iowa (Saxton et al, 1971). Detailed sediment and hydrological data are available for each of these small (30.2 and 33.5 ha), single-cropped basins shown in Figs.l and 2, including data for 110 storms.

The basins were divided into a series of slopes by constructing flow lines (lines intersecting contour lines perpendicularly) on topographic maps of the basins as shown in Figs.l and 2. The areas between adjacent flow lines constitute the series of complex slopes for which sediment yield was computed with the OF (Onstad—Foster, 1975) model. To evaluate the parameter, a, in equation (4), we used an optimization pro-cedure for half of the storms available. The optimization was a least squares procedure, minimizing the sum of the squared deviations, SD

SD = 2 ^ - r / ) 2 (6) i = l

where F(- is the estimated sediment yield and Yt, the measured yield. For the Treynor basins the coefficient a for the OF model is 0.10. For Williams'

model, the values of a and j3 are 7.24 and 0.84, respectively (Williams, 1975c). When using the USLE, the optimum value for the sediment delivery ratio is 0.80.

In Table 1, SD from equation (6) is used as an objective function to compare the results for these three models on these basins. The second column indicates the per-formance for 53 predicted storms. The OF model performs best, although the Williams model is almost as good. When the worst 10 per cent of the predictions are eliminated, SD decreases about 70 per cent for the OF model, about 75 per cent for the Williams model and about 63 per cent for the USLE-SDR. The model ranking does not change.

Because the energy terms for the three models are associated with runoff, rainfall or their combination, differences in degree of fit may be associated with runoff amounts.

194 C. A. Onstad and A. J . Bowie

No. I I6 OOO Feet

No. II7

_ — ~ w »*»»

-— i i s o —

— ...

Basin sediment yield modelling using hydrological variables 195

FIGURE 2. Treynor, Iowa 2.

ment delivery ratio usually is selected based on area (Renfro, 1975) and applies to all erosion sources within the basin. The USLE predicts only sheet-rill erosion, so any other erosion sources such as gully and channel should be included before the sediment yield is calculated.

Onstad et al. (1976a) modified the USLE to estimate annual sediment yields. This algorithm considers sheet-rill erosion for a basin on a distributed basis and utilizes a sediment-routing function to calculate sediment yield at the outlet. The sheet-rill erosion equation is identical to the USLE [equation (1)] except that the energy factor (R) is replaced by

196 C. A. Onstad and A. J. Bowie

TABLE 1. Comparison of different models for predicting single storm sediment yields for two basins at Treynor, Iowa

Model

Summation of squared deviations (SD)

53 storms Best 48 storms

16 storms 0.3 t /ha or less

6 storms 8 t/ha or more

Onstad—Foster Williams USLE-SDR

256.4 378.4

1695.3

78.3 98.0

633.6

33.1 0.5

672.1

155.6 297.1 576.4

1

2 / / /

3

/

/

5 s '

6

/ 7

^-CH

V

CHANNELS

OUTLET

FIGURE 3. Hypothetical delineation.

W = b R + c Q(A{sll2Yl3 (?)

where

b,c = constants, Af = increment area as a percentage of total area, s = increment slope steepness.

Equation (7) is used on a basin that is represented by a series of rectangular incremental units, as shown in Fig.3. The term, Acs1*2, is related to peak flow rates within each unit. The constants, b and c, are evaluated from plot or small basin data. The USLE factors are determined in the customary manner for each incremented area. The results are a set of values, each representing the amount of sheet-rill erosion from an individual area.

Erosion is routed from each increment to the basin outlet, utilizing a technique that incorporates channel length and slope. The average 'channel' slope for each increment is calculated by dividing the relief from the increment centre to the basin outlet by its corresponding 'channel' length, the direct distance to the outlet. These 'channels' are also shown in Fig.3.

The routing function for each increment is as follows:

r,- = V -dTt (8)

Basin sediment yield modelling using hydrological variables 197

LEGEND -—-* Watershed Boundary

' Sub-watershed Bounda Perennial Streams

— ^ Intermittent Streams (ÎO) Gaging Station

0 Rasn Gage Locations

•coia in mi l t i

FIGURE 4. Pigeon Roost basin, Marshall County, Mississippi.

where

Yj = sediment contribution at the outlet from increment /, Yoj = total erosion from increment /, d = routing coefficient,

Tt ^CLfVCSf385, (9)

CI , ='channel' length, CSj = 'channel' slope.

The total sediment yield for the basin is the summation of Yfs. The parameter 7/ represents the time of concentration from the increment i to the outlet. The exponents in equation (9) are those used by Kirpich (1940). Williams (1975b) used a similar con-cept for sediment routing that involved travel times calculated from flood routing instead of Tj and a consideration of mean sediment particle size.

TESTING ANNUAL MODELS

The USLE modified with a sediment delivery ratio and the incremental model pre-sented by Onstad et al. (1976a) were tested on five watersheds (10, 12, 32, 34, 35) in northern Mississippi (Fig.4). These basins range from 2240 to 30 320 ha. The data available were annual measured sediment yield for 14 years at the outlets and an annual estimate of gully erosion. Topographic maps and soils and land use information also

198 C. A. Onstad and A. J Bowie 4>

s.s ë

c

5 £ 3 - a

•a

fl> N

e !«

vid

a-a O •5

T3 O «

O

w -C C "£r a> , ,

_H T3 ^ r-~ en \ o oo o o ^ ^ r ^ o o r i ^ f i r i t ^

o

o « — ON oo •^ ^

o o o o o

r-H T-H r-H

o n̂ (N ci te . s 1—< m m t-H m •£*

Basin sediment yield modelling using hydrological variables 199

TABLE 3. Comparisons of two different models for predicting annual sediment yields from five basins in Mississippi

Model

Onstad et al. (1976a) USLE-SDR

Summation of squared deviations (SD)

Ali 35 storms

411.5 427.9

Best 31 storms

271.9 191.0

10 storms 6 t/ha or less

114.0 136.2

8 storms 13 t/ha or greater

177.7 230.0

were available. Topographic data were used to calculate S and L. The C-factor was determined from cropping history and the ^-factor from a soils map. For all cases the P-factor was assumed to be 1. The USLE parameters are shown in Table 2.

Using the parameters in Table 2 with the rainfall factor, R, for each year and basin, we calculated total sheet-rill erosion. Annual sediment yield was estimated by summing the sheet-rill erosion and the gully erosion and multiplying by a sediment delivery ratio. Implicit in this technique is the assumption that sediment yield from channel sources is equal to the unmeasured bed load. The estimate of the sediment delivery ratio shown in Table 2 for each basin was obtained from Renfro (1975). This procedure resulted in poor predictions of sediment yields.

We then decided to split randomly the total record of 70 storms and use a least squares technique to determine the optimum sediment delivery ratio for all five basins together. The optimum ratio was calculated to be 0.215, or nearly 60 per cent larger than the average using the curve values shown in Table 2. The sediment delivery ratio for each basin optimized individually ranged from an underestimate of about 19 per cent to an overestimate of about 32 per cent of the overall optimum (Table 2).

Next, the optimum sediment delivery ratio of 0.215 was used together with sheet-rill and gully erosion estimates to predict the remaining 35 storms. This resulted in predictions with a standard error of about 1.80 t/ha.

Differences between the optimized sediment delivery ratios and those taken from the curve as shown in Table 2 are probably due in part to the omission of channel ero-sion losses. Partially compensating for this exclusion is the unmeasurable sediment yield termed bed load. Estimates of bed load using Colby's (1964) mean velocity formula range to more than 30 per cent of the total measured sediment yield for these basins. Mutchler and Bowie (1976) reported that average channel losses over 1 5 years on watersheds 4 and 5 (Fig.4) were about 37 and 18 per cent, respectively, of the measured sediment yield. These figures tend to support the assumption that unmeasured bed load compensates for omitting the channel losses in this analysis.

A similar procedure was used to test the annual model developed by Onstad et al. (1976a). First each basin was subdivided into 259-ha units corresponding to sections on a topographic map. For each unit, sheet-rill erosion was estimated from soils, land use, and topographic data. This was accomplished by evaluating b and c of equation (7) using data from two small basins in the same area (Mutchler and Bowie, 1976).

Next, the routing coefficient, d, of equation (8) was evaluated using a least squares technique on half of the storms. The optimum d for the five basins was 2 .01. When d was optimized individually for each basin values ranged from 1.85 to 2.20 as shown in Table 2. Using the optimum value for d, the remaining 35 storms were predicted with a standard error of 1.61 t/ha. This compared with predictions made by Williams and Berndt (1976b) on basins of about the same size where their standard error was about 0.92 t/ha.

In Table 3, predictions from the USLE with fitted SDR are compared with those from the Onstad et al. (1976a) model with fitted routing coefficient, d. For the smaller

200 C. A. Onstad and A. J. Bowie

TABLE 4. Sources of sediment on Treynor, Iowa Watershed W-l

Tube

L K C J B D I H A F E G

Sediment yield contributions

Cumulative [%]

18.3 34.4 46.1 57.7 65.6 72.8 79.3 84.4 88.7 92.9 97.0

100.00

Indidivual [%]

18.3 16.1 11.7 11.6

7.9 7.2 6.5 5.1 4.3 4.2 4.1 3.0

Area

Cumulative [%]

20.9 32.7 42.1 51.4 63.5 67.9 74.5 81.8 88.3 91.4 96.7

100.00

Individual [%]

20.9 11.8

9.4 9.3

12.1 4.4 6.6 7.3 6.5 3.1 5.3 3.3

Nonunifor-mity ratio for sedi-ment yields

0.88 1.36 1.24 1.25 0.65 1.64 0.98 0.70 0.66 1.35 0.77 0.91

storms, the sum of squared deviations for the Onstad et al. (1976a) model is about 84 per cent of that for the USLE and for the larger storms the percentage is about 77, indicating good predictions at both extremes. However, when the four storms predicted most poorly are omitted, the USLE-SDR performs better, indicating that the error in the USLE-SDR prediction is contained in just a few events.

Each of the two models has features that are desirable depending on the modelling objective. The parameters for the USLE are widely known. However, for a relatively large basin problems can be encountered when evaluating average slope length and steepness. Williams and Berndt (1976a) evaluated several techniques for determining these parameters and concluded that accurate results can be obtained with 'short cut' techniques. Problems can also arise when evaluating weighted C and K factors for large basins. Finally, the selection of the sediment delivery ratio is critical. Good results can be obtained if data are available to optimize the value of the sediment delivery ratio.

The Onstad et al. (1976a) model is used when sediment sources are required. Again, the USLE parameters are used, but in smaller units so that weighting problems are not so difficult. However, every unit requires a complete set of variables. Again, the value for the routing coefficient, d, should be fitted statistically.

SEDIMENT SOURCES

The determination of sediment sources is important for implementing on-site erosion controls efficiently and economically (Moldenhauer and Onstad, 1975). Although data regarding sediment sources are lacking, Table 4 indicates the type of information that can be predicted using the OF model. The tube designation for Treynor W-l refers to streamtube outlines shown in Fig.l. In Table 4, the tubes are ranked in order of decreasing individual sediment contribution. For example, tubes L and K produced 18.3 and 16.1 per cent of the total sediment, respectively. The corresponding area percentages for tubes L and K are 20.9 and 11.8 per cent, respectively. Although tube L produced the most sediment it also has the largest area. In fact, the area percentage is larger than the sediment yield percentage, indicating that the sediment yield from tube L is less than would be expected if the sediment contribution were uniformly dis-tributed over the basin. The ratio of the estimated sediment yield contribution from the model to that expected if the sediment were uniformly distributed is called the nonuniformity ratio (NR). If NR > 1, the predicted sediment contribution for the tube is greater than would be expected if the contributions were uniformly distributed.

Basin sediment yield modelling using hydrological variables 201

TABLE 5. Sources of sediment on Pigeon Roost Watershed W-10

Section

3 2

35 36

1 11 12 10

Sediment yield contribution

Cumulative [%]

28.8 48.8 63.6 72.8 81.2 88.8 95.9

100.0

Individual [%]

28.8 20.0 14.8 9.2 8.4 7.6 7.1 4.1

Cumulative area [%]

12.5 25.0 37.5 50.0 62.5 75.0 87.5

100.0

Nonuniformity ratio

Sediment yield

2.30 1.60 1.18 0.74 0.67 0.61 0.57 0.33

Sheet-rill erosion

0.53 0.70 1.00 1.21 1.21 0.83 1.43 1.09

Tube D had the highest nonuniformity ratio, indicating the highest potential sediment yields. If controls were initiated on this basin the tubes with NR > 1 should receive attention first.

Table 5 shows similar information for W-10 at Pigeon Roost. The section numbers refer to those shown on Fig.4. The sections are listed in order of decreasing individual sediment yield contributions. Cumulative sediment yield contributions and areas are also shown. Note that nearly half of the predicted sediment yield contribution results from only 25 per cent of the area.

Two nonuniformity ratios can be determined for each section, one for the sheet-rill erosion and the other for the sediment yield. This is possible because sediment yield is determined from the gross erosion, which includes sheet-rill erosion, and a sediment routing function. As shown in the last two columns of Table 5, section 3 has the highest NR for predicted sediment yield and the lowest for predicted sheet-rill erosion. Thus, even though sheet-rill erosion is relatively small, a high percentage is delivered to the basin outlet, probably because section 3 is close to the outlet. If sediment con-trols were to be initiated on this basin, Table 5 indicates that sections 3, 2, and 35 should receive first attention. If erosion controls to maintain overall basin crop pro-duction were to be initiated, sections 12, 36, and 1 may require further treatment.

CONCLUDING REMARKS

Estimates of sediment yields of basins are important for a number of environmental concerns. For single storms, models that incorporate flow variables showed encouraging results. Both the Onstad-Foster (OF) 1975 model and the Williams (1975) model were better than the universal soil loss equation with sediment delivery ratio (USLE-SDR) for predicting sediment yields from single storms on basins near Treynor, Iowa. For 110 single storms, the standard error was 2.2 t/ha for the OF model and 3.3 t/ha for the Williams model. For the USLE-SDR, the standard error was larger. The OF model performed better than the other two models except for extremely small storms, where the Williams model performed best.

For annual events on large basins two models were evaluated using data from five Pigeon Roost Creek basins in northern Mississippi. One was that developed by Onstad et al. (1976a) and the other was the US LE used with an optimized sediment delivery ratio (USLE-SDR). The standard errors for the first model on 35 predicted storms was 1.6 t/ha and for the USLE-SDR, 1.8 t/ha. This compares with the Williams model tested on similarly sized basins with a standard error of 0.9 t/ha. Both the USLE-SDR and the Williams model utilize weighted parameters representing the entire basin. The Onstad et al. (1976a) model subdivides the basin and uses sets of parameters, each representing one unit of the subdivision. This permits the model to predict sediment

202 C. A. Onstad and A. J. Bowie

sources within the basin. This capability should be considered when plans are being designed and tested for implementing sediment and erosion control practice.

Acknowledgements. Contribution from the North Central Soil Conservation Research Center, North Central Region, Agricultural Research Service, USDA, Morris, Minnesota, in coopera t ion with the Minnesota Agricultural Experiment Station, Sci. Paper no.9754.

REFERENCES

Colby, B. R. (1964) Discharge of sands and mean velocity relationships in sand-bed s t reams . US Geol. Survey Professional Paper no.462-A: Washington, USA.

Foster, G. R., Meyer, L. D. and Onstad, C. A. (1973) An erosion equation derived f rom basic ero-sion principles. Paper 73-2550, presented at 1973 Winter ASAE Meeting, Chicago, Illinois, USA,

Frere, M. H., Onstad, C. A. and Holtan, H. N. (1975) ACTMO, an Agricultural Chemical Transport Model: USDA-ARS-H-3.

Kirpich, Z. P. (1940) Time of concentration of small agricultural watersheds. Civil Engr (NY) 10, no.6, p.362.

Moldenhauer, W. C. and Onstad, C. A. (1975) Achieving certain soil loss limits. J. Soil and Water Conserv. 39, no.4., 166-168 .

Mutchler, C. K. and Bowie, A. J. (1976) Effect of land use on sediment delivery ratios. Proceedings of the Third Federal Inter-Agency Sedimentation Conference, Denver, Colorado, USA, pp.1-11 to 1-21 .

Onstad, C. A., Bowie, A. J. and Mutchler, C. K. (1976a) A sediment yield model for l a rge water-sheds. Paper 76-2536, presented at 1976 Winter ASAE Meeting, Chicago, Illinois, USA.

Onstad, C. A. and Foster, G. R. (1975) Erosion modeling on a watershed. Trans. Amer. Soc. Civ. Engrs 18, 288-292.

Onstad, C. A., Piest, R. F. and Saxton, R. E. (1976b) Watershed erosion model val idat ion for Southwest Iowa. Proceedings of the Third Federal inter-Agency Sedimentation Conference, Denver, Colorado, USA, pp. 1-32 to 1-34.

Renfro, G. W. (1975) Use of erosion equations and sediment-delivery ratios for predic t ing sediment yields. In Present and Prospective Technology for Predicting Sediment Yields and Sources, pp.33-45: USDA-ARS-S-40.

Saxton, K. E., Spomer, R. G. and Kramer, L. A. (1971) Hydrology and erosion of loessial water-sheds. Proc. Amer. Soc. Civil Engrs, J. Hydraul. Div. HY-11, 1 8 3 5 - 1 8 5 1 .

Williams, J. R. (1975a) Sediment-yield prediction with universal equation using runoff energy factor. In Present and Prospective Technology for Predicting Sediment Yields and Sources, pp.244-252: USDA-ARS-S-40.

Williams, J. R. (1975b) Sediment routing for agricultural watersheds. Wat. Resour. Bull. 1 i , no.5, 965-974.

Williams, J. R. (1975c) Personal communication, June 23. Williams, J. R., Berndt, H. D. (1976a) Determining the universal soil loss equation's length-slope

factor for watersheds. Proceedings of the National Erosion Conference, Lafayette, Indiana, USA.

Williams, J. R. and Berndt, H. D. (1976b) Sediment yield prediction based on watershed hydrology. Paper No. 76-2535, presented at 1976 Winter ASAE Meeting, Chicago, Illinois, USA.

Wischmeier, W. H. and Smith, D. D. (1965) Predicting Rainfall-Erosion Losses from Cropland East of the Rocky Mountains: USDA-ARS, Agriculture Handbook no.282.