PREDICTION OF FLOOD FREQUENCIES BY STOCHASTIC ... · In this study, stochastic-deterministic hydrologic model was developed for de-termining the flood magnitudes and the frequencies

PREDICTION OF FLOOD FREQUENCIES BY STOCHASTIC-DETERMINISTIC HYDROLOGIC MODEL

Assoc. Prof. Dr. Yakup DARAMA Dr. Ahmet ALPASLAN

State Hydraulic Works (DSI), Investigation and Planning Department Ankara, TURKEY

ABSTRACT

A Stochastic-Deterministic hydrologic model was developed for predicting the frequency of surface runoff of various magnitudes resulting from rainfall. Stochastic aspects of precipitation and infiltration capacity were incorporated into the unit hydrograph method in order to make more realistic prediction. The model devel-oped in this study is primarily based on the method of Brater and Sherrill (1975) who incorporated the seasonal average infiltration capacity into the unit hydrograph for predicting frequencies of floods. In order to check the accuracy of the model devel-oped in this study, it was applied to the Red Run River basin in Southeastern Michi-gan, USA where rainfall runoff records covering 40 years duration were available. Application of the model to this basin showed that the magnitudes of floods were very low in the region where the frequencies were low. This was the result of very high unrealistic infiltration capacities. Thus, surface runoff from the impermeable portion of the basin and low value of groundwater flow produced these low flood magnitudes. Although the flood magnitudes obtained from the model seemed unre-alistic for both winter and summer at low frequencies, the predicted flood magni-tudes for summer matched accurately with the observed summer flood values for higher values of frequencies. Comparison of the magnitudes and frequencies of floods predicted by the method developed in this study and the method developed by Brater and Sherrill’s (1975) indicated that the model developed in this study pre-dicts the frequencies of floods more accurately than the method of Brater and Sherrill’s (1975) for this example. Since the application of the model and its concepts is simpler than the available commercial hydrological models, it can be used for Turkish basins where rainfall and runoff records are available.

Keywords: Stochastic infiltration capacity, flood frequency, hydrological model-ling

RIVER BASIN FLOOD MANAGEMENT 699

1. INTRODUCTION

A general interest in the flow of a river can be categorized into two groups. First, is the conservation of water in the reservoirs for irrigation and power develop-ment and second, is the disposal of flood flow without loss or damage. Therefore, the prediction of the frequency of surface runoff of various magnitudes resulting from precipitation is beneficial to hydrologists, watershed managers, and design engi-neers. The main benefit of the accurate prediction of flood frequencies could be the prevention of flood damage by means of improved design of structures, such as storm sewers, channel improvement, and power development structures.

In this study, stochastic-deterministic hydrologic model was developed for de-termining the flood magnitudes and the frequencies for drainage basins. The model is based on the study conducted by Brater and Sherrill (1975). The present study differs from the study of Brater and Sherrill (1975) by incorporating stochastic fea-tures of infiltration capacity and precipitation into the unit hydrograph method. This was achieved by defining the probability distributions of infiltration and precipita-tion to incorporate the random behavior of these elements. The model was applied to the Southeastern Michigan river basin Red Run where 40 year long rainfall runoff records were available to predict floods and their frequencies resulting from various excess precipitations.

2. MODEL DEVELOPMENT

Two independent processes control peak flood discharges of a drainage basin. One process determines the rate of input to the system and the other process estab-lishes the response of the system. This concept in hydrology is known as the system concept. The first process related to the system input and the system input is precipi-tation. The second process concerns with the system response to this input. Deter-mination of the system input is quite important and complex process because system input should be considered as the excess precipitation that is minor retentions and infiltration must be subtracted from the total precipitation. Thus, it is important to define the elements of input to the system.

2.1. Retention

Before overland flow begins, or during its early stages, a small portion of the ini-tial rainfall is stored and permanently abstracted from surface runoff by interception and surface or depression storage. The interception evaporates, and the depression storage either evaporates or infiltrates after the end of rainfall. The interception is

700 INTERNATIONAL CONGRESS ON RIVER BASIN MANAGEMENT

abstracted from the beginning of rainfall, whereas depression storage accumulates only after the rain intensity exceeds the infiltration capacity of the soil. However, for the purpose of flood prediction it is convenient to combine these two abstractions. Thus, in this study, the total of these two abstractions is referred to as retention (R). Brater and Sherrill (1975) studied various basins in the USA and Southeastern Michi-gan and defined a simple relationship between the retention for whole drainage basin, and retention for impermeable and permeable portions of the drainage basins.

pApRiAiRRA += (1)

Here A is the total area of the drainage basin, Aİ and AP are the areas of the im-permeable and permeable portions of the basin, R is the retention on the total area of the basin and Rİ and RP are the retention values on impermeable and permeable portions of the basin respectively. Using Eq. 1, Brater and Sherrill (1975) showed that the difference in retention value on total area (R) and retention on permeable area (Rp) was negligible and less than the uncertainty in the determination of the value of R for urbanized area. This analysis reveals that this difference would be even smaller in the case of less urbanized watersheds. This relationship was also used in this study for determining the value of retention.

2.2. Determination of Impermeable Areas

The watershed is characterized by its area, shape, slope, soils, land use, impervi-ousness, roughness, and storage. Among these, imperviousness is one of the most important characteristics for a good prediction of flood hydrograph. Although the determination of this parameter seems to be straightforward, estimation of the per-centage of imperviousness can be subtle. Brater and Sherrill (1971), Brater and Sherrill (1975) and Bedient and Huber (1992) studies stated that imperviousness and effective impervious areas in percentage of total drainage area was one of the most important physiographic characteristics of the watershed and showed that the per-centage of impervious area (Ai/A) was influenced by the population density of the drainage basin. The extent of the effectively impervious area is of particular interest in this study, because it appears obvious that this is one of the important factors whereby urbanization influences storm runoff. It is recognized that the effectively impervious portion of the basin may be larger during wet seasons than during dry seasons. As the land surface of the basin becomes more impervious, the rate and volume of infiltration decreases and this increases the excess precipitation which produces high magnitude of flood. Brater and Sherrill (1971 and 1975) defined the percentage of impermeable area (Ai/A) as “Hydrologically significant impermeable


area”, (HSIA=100(Ai/A)), and related this parameter to basins population density varying between 500/mi2 and 10000/mi2 for 12 basins in Southeastern Michigan, USA, and obtained an equation in the following form.

dCPHSIA = (2)

In which C is the correlation coefficient, Pd is the population density in thou-sands of persons per square mile, and HSIA is in percent of total area. This relation-ship was also used in the model developed in this study for the determining the effects of impermeable area.

2.3. Infiltration

The prediction of the peak flow resulting from a specific input from rainfall re-quires knowledge of the portion of the total input which will be abstracted as infiltra-tion and surface retention. Of these abstractions, infiltration is the most important. Unlike the shapes of the unit hydrographs which depend primarily on the areas of the drainage basins and on other physical of the basins which are not related to a particular geographical region, the infiltration capacity depends to a large extent on the soil in the particular location where flood predictions are to be made. Therefore one of the principal parameters sought from hydrograph analysis is the infiltration capacity and its seasonal variation. Earlier studies (Brater et al. 1974; Brater and Sherrill,1975; Darama, 1985) also showed that infiltration capacity varied seasonally as shown in Figure 1. In this figure more than 200 infiltration values were deter-mined by hydrograph analysis of 16 basins in Southeastern Michigan.

Fig. 1: Seasonal variation of infiltration capacity of sixteen basins in Southeastern

Michigan, Brater and Sherrill (1975)


As indicated above, infiltration capacity varies from time to time because of the soil moisture content and the nature of the vegetation cover. Previous efforts (Sangal, 1970; Brater et al., 1974; Brater and Sherrill, 1975) to develop runoff models using the unit hydrograph and infiltration concepts made use of seasonal average infiltration only. Even though they recognized, they did not incorporated the influence of highly variable value of basin average infiltration on the probability of flooding in their studies.

It has been recognized that the occurrences of each component of hydrologic event (infiltration, precipitation, evaporation, etc.) have random feature. Therefore it is important to consider the stochastic features of those components when the engi-neers are dealing with the prediction flood flow and design of water structures. Es-pecially random nature of infiltration capacity and precipitation is significant for the magnitude of floods. In this study, the random nature of infiltration capacity was taken into account by incorporating both the mean and variation of infiltration for the prediction of the magnitudes and frequencies of floods. Infiltration data shown in Figure 1 were used as an example for this purpose. The infiltration data shown in this figure defined seasonally into two parts, namely winter and summer and their probability distributions were defined to incorporate the random nature of winter and summer infiltration capacities. The total number of field infiltration values shown in Figure 1 is 146 for winter, and they were transformed and statistically ana-lyzed.

Statistical analysis together with the transformation showed that the Weibull Distribution model adequately fitted to the frequency distribution of the transformed winter infiltration values (Figure 2) with 97.5% confidence limits (χ2 goodness-of-fit test). After the determination of the distribution of winter infiltration capacity, the value of infiltration capacity simulated randomly using this distribution by the result of Integral Transform Theorem (Meyer 1970). From the result of this theorem the actual winter infiltration capacity, yw, is defined by,

41.0

z11ln936.2

w ey

−

−

= (3)

The total number of field infiltration values shown in Figure 1 is around 100 for the summer. These 100 summer infiltration values were also analyzed statistically. Applying the same procedure to the summer values as in the case of winter showed that the Weibull Distribution model also adequately fitted to the frequency distribu-tion of the transformed winter infiltration values (Figure 3) with 90% confidence limits. Using the procedure for the derivation of Equation 3, the actual summer infiltration capacity, ys was determined from the Weibull distribution as,


−

=893.0

s z11ln676.41y (4)

In Equations 3 and 4 zi is the random variable uniformly distributed over [0,1]

and generated by using random number generator for i=1,…,n for computing the values of winter and summer infiltration capacity values.

In Fig. 2: Frequency distribution of the transformed winter infiltration capacity

underthe Weibull model.

Fig. 3: Frequency distribution of the transformed summer infiltration capacity

under The Weibull model

2.4. Precipitation

This section concerned with the modelling of precipitation by including the ran-dom feature of the occurrence of precipitation. To proceed with the same region for the example, the data related to the total storm amount with respect to recurrence interval that occur within 24 hour were obtained from the seasonal frequency


curves of precipitation (Figure 4) for 16 basins in Southeastern Michigan (Brater and Sherrill 1975). Since the frequency studies of the precipitation for summer and winter were performed and the resultant seasonal frequency curves were derived by previ-ous studies (i.e., Brater et al. 1974; Brater and Sherrill 1975), procedure for modelling of the summer and winter storms from these curves was straight forward. Using the seasonal frequency curves in Figure 4, the probability of occurrence of winter precipi-tation was defined by an exponential distribution. Using the result of the integral transform theorem, the simulation function for winter precipitation depth (Pw) was derived as,

+=

z1ln478.05.1Pw

(5)

Similarly, the simulation function for summer precipitation depth (Ps) was de-rived as,

+=

z1ln651.06.1Ps

(6)

In Equations 5 and 6 random variable z values were generated from the uniform distribution over an interval [0,1] by using random number generator for i=1,...,n and then the depth of precipitation for winter, Pw, and the depth of precipitation for summer, Ps, were computed for each corresponding zi value. The simulated values of winter and summer precipitation by Equations 5 and 6 were also given in Figure 4 for comparison.

Fig. 4: Final seasonal frequency curves of sixteen basins in Southeastern Michigan,

(Brater and Sherrill,1975; Darama, 1985)


Fig. 5: Typical seasonal hyetographs expressed as percent of 24-hr rainfall for

sixteen basin in Southeastern Michigan, Brater and Sherrill (1975).

It is also important to define the hyetograph depicting the time variation of precipita-tion over the basin. In connection to this, Brater and Sherrill (1975) examined 80 summer and 44 winter rains of durations from 1 to 24 hr for Southeastern Michigan to determine the patterns separately and presented the values as the percentages of the daily rainfalls of the same frequency. They defined the typical hyetograph (Fig-ure 5) depicting the distribution of rainfall intensity as percent of 24 hour rainfall for summer and winter for the drainage basins located in Southeastern Michigan.

All precipitation data presented were point precipitation obtained from the rain gages and they were converted to areal precipitation of the same frequency for areas of various sizes by using the U.S. Weather Bureau (1961) curves.

2.5. Unit Hydrograph

The unit hydrograph is one of the most important elements of rainfall runoff models for the determination of the flood frequencies. One of the most important characteris-tic of a unit hydrograph is its peak because this is the value used for predicting the peak flood flows. However, all the flow values of unit hydrograph must be known in order to compute complete flood hydrograph. Brater and Sherrill (1975) showed the relationship between the drainage basin area and the two most important charac-teristics of unit hydrograph namely its peak and period of rise. Using the results of their studies, they synthesized the unit hydrograph (dashed line in Figure 6) of the Red Run river basin whose population density was 7500 people/mi2 and drainage area is 36.5 mi2 in Southeastern Michigan. Using the rainfall and runoff records, Da-rama (1985) derived the unit hydrograph of this basin and showed that the unit hy-drograph obtained by using the rainfall runoff records and synthesized unit hydro-


graph were almost identical. This unit hydrograph was used in numerical example for predicting the frequency of floods in the hydrological model developed in this study.

Fig. 6: Synthesized and derived unit hydrographs for Red Run river basin

in Southeastern Michigan (Brater and Sherrill,1975; Darama, 1985)

2.6. Stochastic-Deterministic hydrological Model

Most hydrological systems have both stochastic and deterministic components. Deterministic components can be expressed mathematically by a set of equations, a graphical relation, or set of rules, whereas the stochastic components and parameters which can be defined by means of probability distributions. System modeling de-termines the value of some output resulting from some input. If any of the inputs are stochastic, then the output is also stochastic. The input is transformed to be output by the system during one-step of system operation. This transformation represents the deterministic component of the system.

A hydrological model was developed to simulate rainfall runoff events to de-termine the magnitudes and frequencies of flood events that are part of a stochastic-deterministic hydrological system. The model is stochastic-deterministic, because it uses stochastic input of rainfall plus infiltration and deterministic transformation of rainfall excess by unit hydrograph procedure and deterministic operations into the time variation of the basin outflow. Figure 7 conceptually illustrates the nature of the stochastic-deterministic hydrological model developed in this study.


Fig. 7: Conceptual description of the Stochastic-Deterministic Hydrologic Model

2.7. Application of the Model

The stochastic-deterministic model was applied to the Red Run River basin lo-cated in the Southeastern Michigan to predict the magnitudes and frequencies of floods resulting from precipitation and to check the accuracy of the model results with the observed floods. The model accepted the distributions of precipitation and infiltration, as an input. The value of precipitation depth was randomly obtained from its respective distribution. This point precipitation value was converted to areal precipitation and distributed on the basin area by the hyetograph given in Figure 5. The value of net precipitation on the drainage basin was computed by deducting the value of infiltration capacity that randomly selected from its respective distribution.


The volume of surface runoff from the permeable and impermeable areas were com-puted by distributing the excess rainfall on the permeable and impermeable portion of the drainage basin as described by Darama (1985). The volume of surface runoff from the total area was computed by adding surface runoff volumes from the per-meable and impermeable portion that uniformly distributed over the whole drainage basin. The unit hydrograph method was applied to compute the direct runoff hydro-graph resulting from the excess rainfall. The operations described above were re-peated over 1000 times by different values of rainfall and infiltration capacity that were randomly selected from their respected distributions.

3. DISCUSSION OF THE RESULTS OBTAINED FROM THE MODEL APPLICATION

A numerical model was developed to perform the processes and computations de-scribed above. The numerical code was applied to the Red Run, a 36.2 mi2 basin on which the population density was 7500 people/mi2. The results of the simulation were analyzed and presented in Figure 7 (solid lines) together with the observed summer floods (triangle symbols) from 14 yr of records of this drainage basin for comparison. The flood frequencies of the Red Run obtained from Brater and Sherrill’s (1975) model, in which the stochastic nature of infiltration and precipitation was neglected, were also plotted in this figure (dashed lines). For the remainder of this study the stochastic-deterministic hydrological model developed in this study will be referred to as ST-DET model and the model developed by Brater and Sherrill (1975) will be referred to as deterministic model (DET model).

Fig. 8: Comparison of the predicted flood frequencies obtained from ST-DET model

and DET model with those observed summer flood frequencies for the Red Run river basin in Southeastern Michigan.


Analysis of Figure 8 showed that the magnitudes of floods obtained from the ST-DET model are very low in the region where the frequencies are low. This was the result of very high unrealistic infiltration capacities. Thus, surface runoff from the impermeable portion of the basin and low value of groundwater flow produced these low flood magnitudes on this part of figure. Although the flood magnitudes obtained from the ST-DET model seemed unrealistic for both winter and summer at very low frequencies, the predicted flood magnitudes for summer matched accu-rately with the observed summer flood values for higher frequencies. Comparison of the magnitudes and frequencies of floods predicted by the ST-DET model and DET model documented that the ST-DET model predicts the frequencies of floods more accurately than DET model for this example.

The reason for summer flood magnitudes were higher than winter flood magni-tudes predicted by both ST-DET model and DET model was because, the time inten-sity patterns shown in Figure 5 differed considerably from winter to summer. Also very intense summer storms could cause very high flood magnitudes regardless of the rate of infiltration capacity. Since the model coupled infiltration and rainfall randomly, these high intense summer storms might be coupled with the low rate of infiltration capacity, therefore caused high flood magnitudes.

4. CONCLUSIONS

Combined modeling of the stochastic and deterministic aspects of system was achieved in this study by incorporating stochastic nature of infiltration capacity and precipitation into unit hydrograph method for estimating flood frequencies and magnitudes. The model was applied to the river basin Red Run located in Southeast-ern Michigan for prediction of floods. These results obtained from the model devel-oped in this study (ST-DET model) compared with those results obtained from the method of Brater and Sherrill (1975) (DET model) to show the effects of stochastic nature of infiltration on the magnitudes of floods.

Analysis of the ST-DET model results showed that the magnitudes of floods were very low in the region where the frequencies were low. This was the result of very high unrealistic infiltration capacities. Thus, surface runoff from the imperme-able portion of the basin and low value of groundwater flow produced these low flood magnitudes. Although the flood magnitudes obtained from the ST-DET model seemed unrealistic for both winter and summer at low frequencies, the predicted flood magnitudes for summer matched accurately with the observed summer floods for higher frequencies. This situation indicated that the ST-DET model predicts the frequencies of floods more accurately than DET model for this example. Analysis of


the results from both models also showed that the flood magnitudes at a constant frequency vary with the nature of infiltration capacity. Since the application of the model does not require very complex and complete data for a river basin and it con-ceptually is much simpler than the available commercial hydrological models, it can be applied to Turkish basins where adequate rainfall and runoff records are avail-able.

LIST OF REFERENCES

Bedient, P. B., and W. C. Hubert, 1992, Hydrology and Floodplain Analysis, 2nd ed., Addison-Wesley Publishing Company, New York, N.Y., U.S.A.

Brater, E. F., and J. D. Sherrill, 1971, “Prediction of Magnitudes and Frequencies of Floods in Michigan”, Report to Michigan Department of State Highways and U.S. Bureau of Public Roads.

Brather, E. F., S. Sangal, and J. D. Sherrill, 1974, “Seasonal Effects on Flood Synthesis”, Water Resources Research, A.G.U., v.10, no. 3.

Brather E. F., and J. D. Sherrill, 1975, “Rainfall-Runoff Relations on Urban and Rural Areas”, National Environmental Research Center, Office of Research and Devel-opment, U.S. Environmental Protection Agency, Cincinnati, Ohio, 45268, U.S.A.

Darama, Y., 1985, “Incorporating a Stochastic Infiltration Capacity Into a Unit Hydro-graph Method for Flood Prediction”, M.S. Thesis, Department of Civil and Envi-ronmental Engineering, Michigan State University, Michigan, USA.

Meyer, P.L., 1970, Introductory Probability and Statistical Applications, 2nd. ed. Addison-Wesley Publishing Co., Philippines

Sangal, S., 1970, “The Surface Runoff Process During Intense Storms”, Doctoral Disser-tation, Department of Civil Engineering, The University of Michigan, USA

U.S. Weather Bureau, 1961, Rainfall Frequency Atlas of the United States for Durations from 30 Minutes to 24 Hours and Return Period from 1 to 100 Years”, Technical Paper No. 40, Washington D.C., U.S.A.

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PREDICTION OF FLOOD FREQUENCIES BY STOCHASTIC ... · In this study, stochastic-deterministic hydrologic model was developed for de-termining the flood magnitudes and the frequencies