Hydrology 3

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Institute of Hydraulic Engineering, Universität Stuttgart, Germany Pfaffenwaldring 61 * D-70550 Stuttgart

Phone: 0711/685-4679 * Fax: 0711/685-4681 * e-mail: [email protected]

Hydrology III

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1 Fundamental principles of river basin

modeling

1.1 Scope

The basis of rational water use and -management is understanding the temporal and spatial characteristics of water flow. Since transport- and transposition processes take place in the water, and large-scale input of man-made substances occurs, the description of water quality is closely related to the description of water flow. According to the actual problem different statements are required that origin from either statistical or deterministic approaches. The main tasks are:

• Computation of large-scale balances

(provides basic information about the water regime and its spatial variations applying long time means and statistical approaches.).

• Design of water management structures

(e. g. flood protection, river development, flood-control reservoirs, carryover storage) usually statistical approaches (e. g. HQ100, NQ10, MQ).

• Real-time forecasting

(e. g. inflow for storage management, flood-forecast service, storage operation) statistical and/or deterministic approach (e. g. forecast by statistical time-series models, prediction of time and height of peak flow computations.

• Design and assessment of management measures and evaluation of alternatives

(e. g. water body development, flood retention, storage operation) usually deterministic approaches (e. g. computation of retention effect of a flood-control reservoir.

• Process studies

for better comprehension of complex hydrologic processes, predominantly deterministic approach.

Another possible subdivision derives from the examination time period: • short-term minutes, hours, days (e. g. flood events)

• medium-term weeks, months (e. g. low water, storage management) • long-term years (e. g. mean water, sizing of water power plants)

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1.2 Principal methods of river basin modeling

1.2.1 Statistical methods

Statistical methods are exclusively based on the description of observed values without consideration of the underlying causes. Statistical methods are used for:

• theoretical probability distributions (distribution functions) of observed values at a certain river cross-section (e. g. extreme-value statistics of floods and low water),

• time-series analysis and -synthesis of outflow hydrographs, • regionalization,

Transformation of e. g. outflow characteristics applying regression/geostatistics based on typical features of the gaged and ungaged sites (e. g. size of drained area, inclination, geologic and morphologic features).

The meaningfulness of statistical investigations is dependent on the density of the gage network and the duration of observation. For more detailed information, see lecture “Hydrological simulation techniques“.

1.2.2 Deterministic methods

Deterministic methods investigate the correlation between cause and effect. It is essential to be able to quantify and mathematically describe the causes and the structure of the affiliated effects. These interrelations may derive from physical laws or from the analysis of short-term observations.

Runoff may be attributed to various causes, therefore several different mathematical formulations and mathematical models may be applied. All natural outflow is primarily dependent on precipitation.

• Precipitation (e. g. rainfall-runoff models, drainage basin model); (see Chapter 2). Additionally secondary effects of precipitation may be regarded as causes for surface runoff. • Volume of groundwater storage (e. g. low-water models),

• Melting of snow and glaciers (e. g. equations of snow-melt),

• Discharge of the upper courses (e. g. streamflow models); (see Chapter 7), • Operation and management (e. g. storage, discharge, withdrawal).

The results of the methods are as follows:

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• continuous outflow hydrographs (river basin model with temporal and areal resolution).

1.2.3 Combined methods

The combination of statistical and deterministic model calculations leads to combined methods. Often the natural impacts on a hydrologic system can only be described with statistics, whereas the effects of the system can be derived from physical laws.

An example of this is the computation of extreme floods for an area that features only short or no outflow measurements. Therefore an extrapolation to determine rare peak discharges is impossible. Assuming that precipitation observations of sufficient duration at one or several representative gage sites are available, extreme flow may be computed from precipitation. This is accomplished by application of a mathematical discharge model which derives from short- term discharge measurements or physical approaches. The relevant input can be obtained from the statistical distribution of precipitation. In the case of a direct dependence the probability of the effect (discharge) equals the probability of the cause (precipitation). Prerequisite for this method is that the effect for the hydrologic system and the probability distribution of the input are known. The method provides the probability distribution of the output (see Figure 1.1).

Statistical technique (distribution)

Variable propability p1

e.g. Precipitation e.g. Discharge

Deterministic model

Result

propability p₁

Figure 1.1: Combination of statistical and deterministic methods

1.3 Structure of river basin models

1.3.1 Partial models

A hydrologic river basin model generates outflow according to the relevant hydrologic processes by transforming input (precipitation, meltwater supply, evaporation) into output (discharge at the outlet cross-section of the basin). Consequently the model describes the movement and the storage of the precipitated water on the land surface, subsurface and in the stream itself by partial models. The purpose of a mathematical river basin model is therefore the spatial and temporal reproduction of waterflow within a river basin. Thereby the basin is

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broken down in the horizontal and vertical direction, however, the selection of the appropriate subdivision is always dependent on the actual problem. In principle a river basin model contains at least models to compute the soil moisture regime, groundwater and evaporation and reproduces the formation of outflow in the drainage basin, runoff concentration in the water-body system and the temporal course of discharge in the streams of the basin. The following three basic elements that contain the previously mentioned subdivisions are applied:

• precipitation drainage basin

rainfall-runoff model, drainage basin model (see Chapter 2-6), • river course

streamflow models, flood-routing (see Chapter 5 and 7), • natural or artificial storage structures (storage operation model).

The application of partial models based on physics is recommended if river basin models are applied to rarely gaged or ungaged areas, to assess human impact on the water cycle of an area or if models are coupled to evaluate water quality.

1.3.2 Drainage basin models

The drainage basin model serves the determination of discharge/streamflow caused by precipitation within the basin. The computation is related to a single cross-section of the receiving stream which can be considered the outlet cross-section for the drainage basin above it (see Figure 1.2). The actual size of the area is defined by the borders of the drainage basin.

Subsequently precipitation is only considered in liquid matter, which means as rain. The conversion of snow melting is reproduced by a suitable model.

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areal precipitation drainage basin discharge Q i_N iN(t) Q(t) time t time t d is c h a r g e Q p re ci p it a ti o n iN

Figure 1.2: Principle of a drainage basin model, determination of streamflow from areal precipitation

First of all, the model must contain a method to convert the punctual precipitation measurements at gage sites to areal precipitation.

• areal precipitation

temporal and spatial distribution of precipitation from the local data of the gage network (see lecture Hydrology I, Chapter 2.5).

The transformation of areal precipitation to streamflow takes place in two phases. • Formation of outflow

Transformation of precipitation considering evapotranspiration and the retention effect of the basin. Formation of runoff takes place at each point of the drainage basin. However, only a portion of precipitation is transformed into runoff.

• Concentration of outflow/ streamflow

Concentration of the runoff in the outlet cross-section. In this regard it is important to determine the temporal distribution of outflow.

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point precipitation model of areal precipitation model of discharge formation model of discharge concentration discharge at catchment outlet

Figure 1.3: Basic elements of a drainage basin model

Usually drainage basin models are considered basic units that are not subdivided any further. Therefore each rainfall event must be spread evenly throughout the basin (block rain). Significant variations or partial rainfall is not permitted. Since natural precipitation may only on small-scale areas be considered evenly distributed, this provides the upper limit of the size of the model basin size. If the block rain assumption does not provide sufficient precision for the model, other vertical divisions must be found. Some areas always feature a typical areal rainfall distribution (e. g. mountain rims) that can replace block rain. Subdividing the area by hydrologic characteristics (hydrotopes) is often useful and easily applicable. However, since the complexity of the model increases with the horizontal division it is useful only up to a certain degree. The size of the drainage basin is a decisive factor for the design of the model. The respective topographic and orographic characteristics must be taken into consideration. For river basins in southgerman low mountain ranges models covering an area of up to 500 km² are applied.

1.3.3 Streamflow models

Streamflow models reproduce the flow of flood waves in the streams which means instationary open channel flow. The river bed and its piedmonts constitute a retention space which holds the flood wave temporarily back. Continuous retention leads to a flattened flood wave ( wave distortion, see Chapter 7).

The streamflow models usually applied in hydrology do not compute the streamflow all the way along the stream, the results are limited to a single control section (outlet cross-section of the examined river cross-section).

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river course time t d is c h a rg e Q Z inflow _outflow Q_A Q_Z time t d is ch a r g e Q A

Figure 1.4: Principle of a streamflow model, one tributary, (floodrouting) Three tasks can be distinguished:

• Flow of a flood wave in stream without lateral inflow or withdrawal; this means one inflow Q_Z(t) and one outflow Q_A(t) and therefore identical water volume (see Figure 1.4). • Confluence of several flood waves from different streams; this means several inflows

QZ,i(t) and one outflow QA(t) (see Figure 1.5).

• Flow of a flood wave in an open channel with punctual or continuous lateral inflow from the traversed intermediate drainage basin. Combination of streamflow model in the open channel and rainfall-runoff model in the traversed drainage basin (see Figure 1.6).

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time t p r e ci p it a ti o n i N areal precipitation i _N Q_Z inflow QA outflow

intermediate catchment time t

time t d is ch a r g e Q A d is c h a rg e Q Z

Figure 1.6: Principle of a streamflow model with intermediate drainage basin

1.3.4 Complex river basin model

From the previously introduced single components, a complex river basin model including storage spaces may be established. The interfaces of the model components must be selected in a way such that the structure of the model matches the natural formation of outflow (see Figure 1.8).

• Streamflow gages, water-level gages • Confluence of tributaries

• Points of limited discharge capacity, control sites (bridges, villages, etc.) • Points that offer management possibilities (e. g. barrages)

1.4 Model approaches

The character of the individual model is selected with the previous knowledge of the hydrologic system. According to the state of knowledge about the physical laws and the extend of required data the model is selected.

• Hydraulic mathematical models

are based on physical laws (e. g. conservation of mass and energy, model with previous physical knowledge). The model is developed using detailed geometric and hydraulic measurements (e. g. channel cross-sections, bed slopes, roughness coefficients). To describe the complex spatial flow characteristics of a drainage basin hydraulic models are unsuited. The plurality of essential measurement values and the considerable amount of

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calculations limit their application in hydrologic study. Therefore, hydraulic models are applied merely to compute instationary open channel flow.

• Model concepts

are based on simplified physical concepts (e. g. continuity- or storage relations, translation). The complex physical transformation mechanisms are replaced by coarsened model assumptions. The model is defined by a number of parameters (as few as possible) that mostly are derived from only a few and not necessarily very precise geometric and/or hydraulic data or from calibrations against observed values. In this case there is no correspondence between the natural system and the model parameters, just a relation. The application of systemhydrologic models is therefore limited in the case of combined discharge-, transport- and quality analysis.

• Black-box models

contain a merely mathematical description of the transformation characteristics according to systemtheoretical methods (input-output models, models without previous knowledge). Physical principles are completely disregarded. The model is defined by empirical system parameters (see Chapter 4). After the model has been defined the parameters are calibrated against observed outflow values (observed in- and output).

Models that use previous knowledge explain the underlying physical processes, models that do not use previous knowledge only model the processes.

time t

water flow model

intersection

rainfall - runoff model

time t _{time t}

time t time t

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rainfall-runoff-model catchment model

S_i intersections _K

i gage site (discharge gage) water flow model

water flow mo including catch (lateral inflow)

Figure 1.8: Structure of a complex river basin model

Many model concepts can be described by methods of system theory. The advantage is that both approaches are based on the same mathematical foundations. Furthermore, due to the connection a direct comparison of the transformation characteristics is possible.

Another aspect for the arrangement of model approaches is the relation to the size and shape of the hydrologic system.

• Models that consider the size of the hydrologic system

Defining the model considers the areal extension of flow. The parameters are assigned to spatial-, areal- or linear gridpoints (hydraulic- and some conceptual models).

• Concentrated models, block models

A limited space is considered a hydrologic unit. Flow is therefore artificially concentrated at one point (black box- and most conceptual models).

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1.5 Analysis and synthesis

1.5.1 Calibration of the model against observed in- and output values

Models that either use very little or no previous knowledge at all (black box) obtain their transformation characteristics only from analysis of an output based on a known input. Consequently in- and output data such as precipitation-, in- and outflow hydrographs are required. However, the parameters that derive from characteristic values of the hydrologic system are subject to substantial uncertainties. Calibration against in- and outflow data on the other hand provides a means to suit the model better to the respective aim. This is also valid for hydraulic model approaches.

The analysis compares the transformation characteristics of the natural hydrologic system and the model. For the same input the respective outputs should match as closely as possible. By specific optimization the model can be suited to the hydrologic system (see Figure 1.9). Thereby either the values of the hydrographs or characteristic hydrograph values such as the moments are compared.

The data flow in the course of analysis and synthesis is displayed in Figure 1.10.

1.5.2 Synthetic streamflow hydrographs

To compute synthetic streamflow hydrographs the input values and the transformation characteristics of the model must be known (cause-transformation-effect). For drainage basin models the input is precipitation, for streamflow models it is inflow.

observed inpu t e.g. discharge, precipitation observed output natural hydrological system e.g. discharge calibration of parameters comparison

m athem atical model (parameters)

generated output

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The synthesis of streamflow hydrographs can be split up into three tasks. • Check of historic events

If only the input (precipitation or inflow) or output (outflow) of a historic event are known, the unknown values may be found by applying a model. Thus short-term observation data and gaps in observation time-series may be augmented.

• Estimation of extreme flows

For design reasons rare outflow magnitudes of small exceedence probability or probability that outflow falls below the respective value are required. Therefore the input must be connected to a corresponding statistical statement (e. g. 100-year exceedence precipitation as input for a drainage basin model). Methods to determine design precipitation are discussed in the lectures "Hydrology I, Chapter 2.6" and "Hydrologic simulation methods".

• Prediction of effects of water management projects

The effect of water management structures (e. g. storage) can only be assessed applying model calculations. The model simulation is based on historic and/or synthetic outflows.

analysis known input model, analysis of parameters known output known input model, known parameters synthesis of output synthesis

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2 Structure of drainage basin models

2.1 Formation of runoff and runoff concentration

Only a fraction of the precipitation that falls above a drainage basin eventually appears as runoff. Already through the course of the precipitation event evaporation returns a fraction of the water back to the atmosphere. The portion of precipitation that later appears as runoff (effective rainfall) infiltrates dependent on intensity and duration of the precipitation event into different stratums of the drainage basin. Usually flow is separated into three components of roughly uniform character (DIN 4049, Part 1, see Figures 2.1 and 2.2).

• Surface runoff

The portion of flow that moves into the receiving stream on the surface. • Interflow

The portion of flow that flows through the subsurface towards the receiving stream. Interflow may be further subdivided into delayed and fast interflow (unsaturated soil zone).

• Groundwater flow

The portion of flow that flows delayed towards the receiving stream from the groundwater body (saturated soil zone).

The total of surface runoff and fast interflow is termed direct runoff. Base flow is formed from groundwater flow and delayed interflow.

The formation of runoff is reproduced in the model as a two-phase process. • Separation of precipitation into two parts:

The first part, called net precipitation or effective rainfall contributes directly to the surface runoff. The other part is composed of losses to interception, evaporation, depression storage, and regional storage.

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vertex flood hydrograph Q surface runoff_Q O interflow Q_I direct outflow Q base flow

assumed drought outflow hydrograph

time t [h] o u tf lo w Q [ m 3/s

Figure 2.1: Separation of flow components of a flood wave

areal precipitation iN overall outflow Q direct outflow QD interflow Q_I effective precipitation iNe infiltration evaporation transpiration interception surface runoff QO base flow Q_B formation of outflow concentration of outflow

Figure 2.2: Separation of areal precipitation in the course of flow formation components of runoff concentration

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2.2 Formation of outflow and runoff concentration in

simple drainage basin models

Simple drainage basin models only consider two flow components (see Figure 2.3). • Direct runoff, QD

Portion of the flood wave that arises directly and quickly from a precipitation event. • Base flow, QB

Portion of the outflow that is not directly concerned with the flood event. Base flow is a constant flow that changes only slowly.

Both components are connected to their causes and are treated separately in the setup of the model. The actual precipitation event causes direct runoff, whereas base flow is dependent on the regional soil moisture and the groundwater volume and pertains to the long-term precipitation history (previous precipitation).

Simple models compute outflow by separating it into two components (see Figure 2.3). • effective rainfall or net precipitation, iNe

Precipitation that eventually appears as direct flow hydrograph at the basin outlet. • losses, iV

Combination of all components that are not included in the direct runoff (evaporation, regional storage, etc.).

areal precipitation iN effective precipitation iNE direct outflow QD overall outflow Q precipitation losses iNV base flow QB precipitation history, initial soil moisture concentration of outflow formation of outflow

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Base flow is the portion of outflow that is not directly associated with the precipitation event. In comparison to direct runoff base flow shows only small magnitudes. Its effects on the flood wave, especially on the peak flow are only marginal. Assuming the base flow hydrograph as a straight line provides sufficient precision, it may even be regarded as constant.

The separation is carried out graphically by a horizontal or slightly inclined straight line from the starting point of the flood wave (see Figure 2.4). The starting point is indicated by a recognizable increase of flow.

Subtraction of the base flow Q_B from the overall flow Q provides the direct runoff Q_D: at the beginning and at the end the direct runoff hydrograph has a value of zero.

( )

D i i B i Q t =Q t −Q t (2.1) time t [h] st a rt d is ch a rg e Q [ m 3/s ]

Figure 2.4: Separation of base flow by a a) horizontal or b) slightly inclined straight line

Computation of synthetic outflow is conducted separately for base flow and direct runoff. Adding the two components provides the overall outflow hydrograph. Considering accuracy, the base flow is only of minor importance. Assigning the mean outflow at dry-weather conditions to base flow is of sufficient precision. (For further investigations the analysis of the coaxial graphical plot is recommended, see Chapter 3.1.2).

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3 Models of runoff formation

3.1 Runoff coefficient

3.1.1 Overall runoff coefficient for single rainfall-runoff events

The overall runoff coefficient is the volumetric ratio of direct runoff to areal precipitation. It is the fraction of a precipitation event that contributes to runoff.

Ne D N N h V V h ψ = = (3.1)

ψ [-] overall runoff coefficient

V_D [m3] volume of direct runoff

V_N [m3] volume of precipitation

h_Ne [mm] overall effective depth of precipitation of the event

hN [mm] overall depth of precipitation of the event

Usually the direct runoff hydrograph Q_D(t_i) is plotted as a succession of linear interpolations between discrete values. The first (i = 0) and the last (i = k) always equals zero. Consequently the discrete integration is reduced to the trapezoidal algorithm.

( )

1 1 3600 k D D i i V t Q t − = = ∆ ⋅ ⋅

∑

(3.2) Q_D [m3/s] direct runoff 3600 [s/h] conversion factor

The volume of the observed areal precipitation is the product of the overall depth of precipitation and the size of the drainage basin.

1000

N E N

V = ⋅A ⋅h (3.3)

A_E [km2] size of the drainage basin

h_N [mm] overall depth of precipitation 1000 [m3/(km2⋅mm)] conversion factor

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3.1.2 Antecedent precipitation index and coaxial graphical plot

The overall runoff coefficient of a rainfall-runoff event is related to the duration of the precipitation event, the overall depth of precipitation, the soil moisture and the season. A measure for the initial soil moisture is the Antecedent Precipitation Index (API) h_VN.

The seasonal variation of evapotranspiration and the detention storage may be considered by a continuous array of numbers n_W assigned to each week.

(

, , ,

)

Ne N Ne N N VN w h =h −h = f h T h n (3.4) Ne N h h ψ = (3.5) h_Nv [mm] precipitation losses

hN [mm] overall depth of precipitation of a single event (e. g. design

precipitation)

h_Ne [mm] effective depth of precipitation of a single event

TN [h] duration of the precipitation event

hVN [mm] antecedent precipitation index

n_W [-] week number (of the year)

The antecedent precipitation index is based on the assumption that the soil moisture after a precipitation event decreases exponentially. The more time elapsed between precipitation events, the smaller is its impact and vice versa. The weighted daily depths of precipitation of a limited period of time preceding the actual event are taken into consideration. The individual weights α are always smaller than 1.

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t_0-n t_0-n t_0-i t_0-i time t [d] time t [d] h_VN (t₀) p re c ip it a ti o n hN [ m m ] a n te c ed e n t p r ec . in d e x hV N [ m m ]

Figure 3.1: Antecedent precipitation index

The antecedent precipitation index is computed as:

( )

0

( )

0 1 n i VN N i i h t h t ₋ = =

∑

⋅α (3.6)

t₀ [d,h] start of precipitation event,

hVN [mm] antecedent precipitation index,

h_N [mm] daily depth of precipitation,

n [-] number of days preceding the event, α [-] empirical weighting factor α < 1.

Usually the impact is limited to a time of ≈ 30 days. Preceding precipitation is not considered. Empirical investigations have suggested a weighing factor of α = 0.9.

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duration of precipitation TN [h] depth of precipitation hN [h] depth of prec. losses

hNV [mm] (calculated) week number d ep th o f p re c . lo ss es hN V [m m ] ( o b se rv ed ) a n te ce d en t p re ci p it a ti o n i n d ex hV N [m m ]

Figure 3.2: Coaxial graphical plot of precipitation losses h_{N V} for a given drainage basin, reading example

The interdependences may be found applying multiple non-linear regression or graphically by trial and error.

The equation displayed below indicates a possible non-linear regression to represent the coaxial diagram as a formula, however, here instead of the antecedent precipitation index h_VN the base specific discharge q_B is used, and instead of the week number n_W the month M is used to compute the precipitation losses h_NV.

(

)

(

)

sin 4 6 sin 4 6 N B N B N D T C q N NV E h C q D T N N h e e A B M h h h e e e A B M ⋅ ⋅ ⋅ ⋅ ⋅  π  ⋅ ⋅ ⋅_ + ⋅ _ ⋅ − __     =  π  + ⋅ ⋅ ⋅ ⋅_ + ⋅ _ ⋅ − __     (3.7)

h_NV [mm] precipitation losses (regional storage)

q_B [l/s/km2

] base specific discharge at the beginning of the event

M [-] month

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A...E [-] parameters

3.1.3 The SCS approach

According to the DVWK (1984), for the estimation of effective or net precipitation in the case of rain storm events and small drainage basins the application of the SCS approach developed by the U.S. Soil Conservation Service is recommended. This method considers effective rainfall h_Ne as a function of the depth of precipitation h_N and a curve number CN dependent on the drainage basin:

2 5080 50.8 20320 203.2 N Ne N h CN h h CN  ₋ ₊      = + − (3.8)

The CN value again is a function of the soil type, land cover, cropping practice and the antecedent moisture condition, which is dependent on the antecedent precipitation of the preceding 5 days and the season. Table 3.1 displays CN-values for various soil types and land cover/ cropping practice for antecedent moisture condition II. From Table 3.2 the current antecedent moisture condition may be taken. In case it deviates from II, the final CN-value may be determined applying Figure 3.3.

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Table 3.1: CN-Values for antecendent soil moisture condition II

Land use CN for hydrologic soil group

A B C D

Bare soil

Root crops, wine Wine (terraced) Corn, forage plants Pasture (normal) (barren) Meadow Forest (open) (medium) (dense) impervious areas 77 70 64 64 49 68 30 45 36 25 100 86 80 73 76 69 79 58 66 60 55 100 91 87 79 84 79 86 71 77 73 70 100 94 90 82 88 84 89 78 83 79 77 100 Hydrologic soil group A: Soils with great infiltration potential, even after antecedent

wetting (e. g. thick sand and gravel stratums)

Hydrologic soil group B: Soils with medium infiltration potential, thick and moderately thick stratums, fine or moderately coarse texture (e. g. moderately thick sand stratums, loess, loamy sands)

Hydrologic soil group C: Soils with low infiltration potential, sorts of fine or moderately coarse texture or with impervious layers (e. g. thin sand stratums, sandy loams)

Hydrologic soil group D: Soils with considerably low infiltration potential, clay, thin soil stratums overlying impervious layers, soils with constantly high groundwater stage.

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Table 3.2: Current antecedent moisture condition antecedent moisture

condition

accumulated depth of precipitation within the preceding 5 days in unit [mm]

vegetation period other

I II III < 30 30 - 50 > 50 < 15 15 - 30 > 30

CN for soil moisture class II

C N f o r so il m o is tu r e cl a ss I , II I

Figure 3.3: CN for antecedent moisture condition I and III cross-linked to antecedent moisture condition II

This method should only be applied for rain storm events. Experiences in the past have shown that for depth of precipitation lower than 50 mm the method underestimates effective rainfall. Modifications of equation (3.8) try to compensate this.

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3.2 Models to compute effective rainfall

3.2.1 Model requirements

The current formation of outflow is dependent on the intensity i_N, the duration t_N and the variations (e. g. breaks) of precipitation. Due to depression storage and wetting the initial losses (or after breaks) are larger than during periods of intensive precipitation.

In hydrologic practice usually two methods are applied. Runoff coefficient method

In the course of a precipitation event only a portion of precipitation is transformed into direct runoff. The runoff coefficient ψ is the ratio of effective or net precipitation iNe(t) to the

observed precipitation i_N(t).

( )

Ne N

i t = ψ ⋅i t (3.9)

In principle the runoff coefficient is a function of precipitation intensity and -duration.

( )

(

_N ,

)

f i t t

ψ = (3.10)

However, simple models consider the runoff coefficient as constant throughout the whole precipitation event (see Chapter 3.2.2).

Index approaches

Only the precipitation that is equal to or more than a certain infiltration capacity i_v ( = losses) contributes to direct runoff.

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( )

_{( )}

( )

_{( )}

( )

0 N Nv N Nv Ne N Nv i t i for i t i t i t for i t i t  − >  =  _≤ 

time t [h] time t [h] time t [h]

p r e c. in te n si ty i [ m m /h ] p r e c. in te n si ty i [ m m /h ] p r ec . i n te n si ty i [ m m /h ]

const. coeff. of discharge phi - index method loss variable with time

Figure 3.4: Models to determine effective rainfall

a) constant runoff coefficient, runoff coefficient method b) constant loss ratio, Φ-index method

c) loss ratio decreasing exponentially

overall outflow Q(t) areal precipitation i_N (t) separation of base flow formulation for effective prec. base flow Q_B (t) direct outflow Q_D (t) effective precipitation iNe (t) precipitation losses i_Nv (t)

Figure 3.5: Determination of outflow formation for simple drainage basin models, sequence and data flow

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The initial rate is always larger than at the end of a precipitation event. Usually the temporal development of losses can be represented by an exponentially decreasing function (see Figure 3.4c).

( )

(

( )

,

)

Nv N

i t = f i t t (3.12)

The simplest model features a constant loss rate, which is only a very rough approximation (see Chapter 3.2.3).

The analysis of rainfall-runoff events is carried out in established steps (see Figure 3.5). • Separation of base flow (linear course)

• Computation of overall runoff coefficient from the volumes of direct runoff and areal precipitation

• Computation of effective rainfall (runoff coefficient method or index approaches)

3.2.2 Runoff coefficient method

The runoff coefficient ψ remains constant throughout the entire course of the precipitation event (see Figure 3.4a). It corresponds to the overall of discharge (see Chapter 3.1). Consequently computation of effective rainfall is reduced to the simple formula displayed below.

( )

Ne i N i

i t = ψ ⋅i t (3.13)

i_Ne(t_i) [mm/h] effective rainfall intensity in time interval t

iN(ti) [mm/h] observed precipitation intensity in time interval t

ψ [1] overall runoff coefficient

3.2.3 Index approaches,

Φ-index

A constant loss rate in the course of a precipitation event is referred to as Φ-index.

( )

.

V

i t =const = Φ (3.14)

Φ [mm/h] constant loss rate, Φ-index

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( )

_{( )}

( )

0 N N Ne N i t for i t i t for i t  − Φ > Φ  =  _{≤ Φ} 

The Φ-index must be determined by step-by-step iterations since negative precipitation is impossible. The iteration provides a constantly increasing Φ-index; the procedure is repeated until it equals the known effective rainfall.

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4 Basis

and

methods

of

systems

hydrology

4.1 Definition of system properties

A system is a distinguished arrangement of interrelated structures (DIN 19226). Each system features an entrance where the cause (input) u_i affects the system, and an exit where the effect (output, system answer) v_i occurs (see Figure 4.1). The interrelations of these values describe the system. The system-operation establishes a definite relation between input and output.

load result

system

input output

Figure 4.1: System with several in- and output variables (input vector u_i(t) and output vector v_i(t))

The simplest case is the definite relation between one output magnitude v and one input magnitude u (see Figure 4.2), e. g. effective rainfall - direct runoff.

The mathematical relation between in- and output can be represented by

( )

{ }

( )

v t = ϕ u t (4.1)

where,

u(t) time-dependent input signal,

v(t) time-dependent output signal,

ϕ system operator.

system

output variate input variate

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The system operation may be defined by certain regularities that allow a classification of the model systems. As an example these regularities are applied to the precipitation-runoff-relation in drainage basins. The input magnitude is the areal effective rainfall expressed as intensity in unit [mm/h], the output magnitude is the outflow at the basin outlet in unit [m3/s] (see Figure 4.3). It is essential to estimate effective rainfall correctly, as the system input affects the quality of the computed unit hydrograph.

• The drainage basin is an open, dynamic system.

A system is termed dynamic if at any time t₁ the output signal v(t₁) is not merely dependent on the input signal u (t)i at the same time t1 but also from preceding input signals u(t) for t < t1.

In physical regard this feature corresponds to a temporal storage of the input magnitude which can be regarded as a system memory. Drainage basins, open channels and storage structures can be considered dynamic systems, because they temporarily store outflow and deliver it later and damped (retention). Figure 4.3a displays how a precipitation event of short duration T_N is discharged as a flood wave of much longer duration T_b.

• Theory of proportionality

Any input signal multiplied by a constant C produces an output signal multiplied by the same constant.

( )

{

C u t

}

C

_{{ }}

u t

( )

ϕ ⋅ = ⋅ϕ (4.2)

The effective rainfall displayed in Figure 4.3b is double the amount as in Figure 4.3a and produces a doubled outflow hydrograph. The duration T_b of the outflow hydrograph remains constant.

• Theory of superposition

The system answer to accumulated input signals equals the total of the single output signals.

( )

{

u t1 u t2

}

_{

u t1

( )

_}

_{

u t2

( )

_}

ϕ + = ϕ + ϕ (4.3)

• Theory of linearity

The combination of the theory of proportionality and superposition provides the theory of linearity.

( )

{

C u t1 1 C u t2 2

}

C1

_{

u t1

( )

_}

C2

_{

u t2

( )

_}

ϕ ⋅ + ⋅ = ⋅ϕ + ⋅ϕ (4.4)

• Theory of time invariance

The system operation is not time-dependent. Shifting an input signal by the time interval T results in an output signal shifted by the same time interval without changing the signal itself (see Figure 4.4c). T_b is preserved.

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(

)

{

u t T

}

v t T

(

)

ϕ − = − (4.5)

Applying the theory of time-invariance and proportionality, consecutive input signals of different intensity can separately be assigned to individual output signals (see Figure 4.4d). The theory of superposition allows to overlay the individual signals to one.

unit hydrograph of discharge principle of linearity precipitation discharge time d is ch a r g e Q D [ m 3/s ] in te n si ty iN e [ m m /h ] precipitation discharge time d is ch a r g e QD [ m 3/s ] in te n si ty iN e [ m m /h ]

Figure 4.3: a) dynamic system operation, unit streamflow hydrograph

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in te n si ty iN e [ m m /h ] in te n si ty iN e [ m m /h ]

principle of time invariance principle of superposition precipitation discharge time d is ch a r g e Q D [ m 3/s ] d is ch a rg e Q D [ m 3/s ] time precipitation discharge

Figure 4.4: a) Theory of time-invariance b) Theory of superposition

4.2 Unit hydrograph

The unit hydrograph method is a linear, time-invariant model to determine outflow.

effective precipitation i_Ne*A_e drainage basin g_E direct outflow Q_D output input system (linear, dynamic, time-invariant)

Figure 4.5: Runoff concentration model as linear, dynamic, time-invariant system

To describe the system operation it is sufficient if the output function that pertains to one constant input signal is known. Using the theory of linearity and superposition the system answer to any series of discrete input signals can be determined. It is useful to relate the characteristic output function to a constant unit input of the duration ∆t and the magnitude one (see Figure 4.3a). This function is referred to as discrete weighting function g(∆t,t_i) with

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reference interval ∆t. Note that not the intensity, but the volume of the unit input possesses the value 1 and therefore the intensity is 1/∆t. The reference time interval ∆τ is a defining feature of the weighting function g(∆t,t)i.

For a linear and time-invariant drainage basin model the weighting function g(∆t,ti) is

replaced by the unit hydrograph gE(∆t,ti) that considers the different dimensions of

precipitation and outflow and the size of the drainage basin. The unit hydrograph describes the system operation of effective rainfall to direct runoff (see Figure 4.5).

The unit hydrograph is a characteristic outflow hydrograph of a surface drainage basin that develops from constant effective rainfall of uniform distribution of 1 mm in depth and defined length (DIN 4049 Part 1).

Thereby, effective rainfall is expressed by the depth, not by the intensity of precipitation.

4.3 Analysis and synthesis of the unit hydrograph by the

black box method

The determination of the unit hydrograph for a system can be achieved directly by the analysis of the observed rainfall-runoff events. This approach ignores the physical structure of the system and applies only the system properties and is referred to as black-box.

First, a reference time interval ∆t is chosen that is valid for the discretion of all time-related data. When applying the computer program, a time interval that splits the flood hydrograph into 30-50 units is recommended. Separation of the base flow ( see Chapter 2.3) provides the system input, the direct runoff Q_D(t_i). Subsequently the overall runoff coefficient and the effective rainfall h_Ne(t_i) is computed (see Chapter.3.2).

The flood hydrograph is composed of the hydrographs of the individual precipitation intervals (see Figure 4.6). Each time interval ti provides (assuming that the unit hydrograph is given)

the direct runoff (synthesis). The overall outflow function that results from the overall inflow function may be developed by superposition of the hydrographs that pertain to the individual precipitation events. This procedure is referred to as superposition. Subsequently the equation system for a simple example enclosing n = 3 effective rainfall ordinates and m = 5 ordinates of the unit hydrograph is displayed:

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1 1 1 2 1 2 2 1 3 1 3 2 2 3 1 4 1 4 2 3 3 2 5 1 5 2 4 3 3 6 2 5 3 4 7 3 5 Q N G Q N G N G Q N G N G N G Q N G N G N G Q N G N G N G Q N G N G Q N G = ⋅ = ⋅ + ⋅ = ⋅ + ⋅ + ⋅ = ⋅ + ⋅ + ⋅ = ⋅ + ⋅ + ⋅ = ⋅ + ⋅ = ⋅ (4.6) where

Q_i = Q_D(t_i) [m3/s] direct runoff with time t_i,

N_i = h_Ne(t_i) [mm] effective rainfall in the interval between t_i-1 and t_i,

G_i = g_E(∆t,t_i) [m3/(s⋅mm)] unit hydrograph at time t_i.

The equation system may be reduced to a differential equation, the so-called discrete equation

of superposition. For a given time t_i:

( )

(

)

(

1

)

1 , i D i E k Ne i k k Q t g t t h t_{− +} = =

∑

∆ ⋅ (4.7)

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areal and effective precipitation principle of superposition time t [h] time t [h d ir ec t r u n o ff p r ec ip it a ti o n in te n si ty

Figure 4.6: Theory of superposition of the unit hydrograph method

Since the initial and terminal value of the direct runoff hydrograph and the unit hydrograph always equal zero, these times are not considered. Therefore the number of discrete values and intervals are

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1 1 o m n m o n = + − = − + (4.8)

o [1] number of discrete values of the unit hydrograph method Q_D ≠ 0,

n [1] number of effective rainfall intervals h_Ne, precipitation duration/∆t,

m [1] number of discrete values of the unit hydrograph g_E ≠ 0.

For the analysis the ordinates of the unit hydrograph must be computed from the linear equation system. For m + n - 1 equation and m unknown values the equation system is n - 1 times overdefined. The optimum solution provides computed runoff QD,ber as close to the

observed runoff Q_D,gem as possible. This is accomplished by applying the method of the lowest square error.

( )

(

)

2 1 , , 1 ! m n D ber i D gem i i Q t Q t Minimum + − = − =

∑

(4.9)

The solution of an overdefined equation system by the method of the lowest square error is available in closed form (BRONSTEIN-SEMENDJAJEV p.513-514). For a lower number of unknown values the solution may be found by trial and error. A test of plausibility derives from the definition of the unit hydrograph as the direct runoff hydrograph caused by the effective rainfall of 1 mm in depth within a time interval ∆t. Therefore

(

)

1 3.6 , 1 m E i i E t g t t A = ⋅ ∆ _⋅ _∆ ₌

∑

(4.10) A_E [km2

] size of drainage basin, 3.6 [ 2 3 mm km s h m ⋅ ⋅ ⋅ ] conversion factor, ∆t [h] time interval.

The runoff concentration within a drainage basin may only approximately be represented by a linear system. In the individual case the unit hydrograph method provides satisfactory results even though it features a certain variation when analyzing several rainfall-runoff events of same duration ∆t. The generally applicable unit hydrograph is the mean of all obtained unit hydrographs while still considering the test of plausibility.

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Oftentimes no suitable runoff- or precipitation measurements for the drainage basin are available. A number of methods exist to compute the unit hydrograph if measurements are unavailable; two of them are commonly used:

Direct application of the unit hydrograph of a similar drainage basin. Selection according to the drainage basin properties. From similar, well-observed drainage basins the dimensionless system may be taken. The reference drainage basin is selected according to size, geology, slope, soil type, characteristic values of the receiving stream and land use (Literature: DVWK-Merkblätter 1982, 1988 catalogue of system operations).

Formulation of a synthetic unit hydrograph by regionalization of the drainage basin properties: Relations between the parameters of a system operation and the properties of a drainage basin can be established. However, it is essential to examine a plurality of drainage basins and to apply weighting functions that can be described analytically (e. g. triangular hydrographs, hydrograph of constant rise and exponential recession or the gamma-function). Important parameters of the system operation are the time of rise, the peak and various recession parameters etc. The correlation of unit hydrograph and basin parameters can be described by e. g. regressions.

The Geomorphologic Unit Hydrograph (GUH) is a physically based model. It makes use of stream network characteristics to determine the probability of occurrence of individual water particles at the basin outlet (Literature: SIVAPALAN et al. 1990).

4.4 System operation and instantaneous unit hydrograph

The system operation of linear, time-invariant systems, as outlined in Chapter 4.2, can be represented by a characteristic answer to a constant input signal of duration ∆t. Other typical input signals exist that define the system by their affiliated output function. For several linear, time-invariant conceptual models (see Chapter 5) The output functions may be determined analytically. A special system input is the unit step function ε(t) (see Figure 4.7):

( )

0 0 1 0 for t t for t <  ε _{= } ≥  (4.11)

ε(t) [1] unit step function

The unit step function is equivalent to an on-switch at time t = 0. The affiliated system answer is termed s-curve h(t). For a dynamic system that features a storage effect, the system operation transforms the input to an s-shaped, lagged and dampened outflow hydrograph (see Figure 4.7). In practice, this may occur if a river or channel weir is suddenly opened. For

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most hydrologic systems the system answer has the value zero at the time t = 0 which simplifies the subsequent formulas.

( )

{ }

t h t

( )

with h t

( )

0 for t 0 ϕ ε = = ≤ (4.12) h(t) [1] system operation 0 1 u E (t) t Input u (t): unit jump E (t) 0 for t < 0 E (t) = 1 for t > 0 0 1 h (t) t Output v (t): system operation h (t) 0 for t < 0 h (t) = 0 - 1 for 0 <= t < infinity 1 for t = infinity v

Figure 4.7: unit step function ε(t) and system operation h(t)

In case the system operation h(t) is known, the output function for a constant input signal of the volume 1, duration ∆t and the intensity 1/∆t may be computed. The jump function of the intensity 1/∆t is superpositioned by a negative jump function lagged by -1/∆t (see Figure 4.8a). This provides a constant pulse of the volume 1 and duration ∆t.

( )

1

( )

1

(

)

1

(

( ) (

)

u t t t t t t t

t t t

= ⋅ε − ⋅ε − ∆ = ⋅ ε − ε − ∆

∆ ∆ ∆ (4.13)

Applying the theory of linearity and time-invariance the output function develops from the superposition of the respective system operations (see Figure 4.8a). The output flood related to a pulse of the volume 1 and duration ∆t is the known ∆t-weighting function.

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( )

_{{ }}

( )

(

)

( ) (

)

(

)

(

)

1 1 1 , _i v t u t h t h t t t t h t h t t g t t t = ϕ = ⋅ − ⋅ − ∆ ∆ ∆ = ⋅ − − ∆ = ∆ ∆ (4.14) g{∆t,t_i) [1/h] ∆t-weighting function

For an infinitesimally small time ∆t the input function becomes a theoretical function known as the needle pulse, instantaneous unit function or Dirac’s superposition pulse function and contains one unit volume of input. It corresponds to a generalized differentiation of the unit step function (see Figure 4.8b). According to the common rules of differentiation, single steps cannot be differentiated, hence the generalized differentiation for technical systems is introduced (The expression is not a numeric value and hence according to the common rules of differentiation no limit exists).

( )

lim₀ 1

(

( ) (

)

( )

t d t t t t t dt ∆ → δ = ε − ε −∆ = ε ∆ (4.15)

( )

0 0 0 0 0 for t t for t for t ≤   δ = ∞_ =  _≥  (4.16)

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pulse dt pulse magnitude 1 dt weighting function needle pulse weighting function for h (t = 0) = 0 :

Figure 4.8: a) development of the ∆t-pulse and the ∆t-weighting function g(∆t,t) b) needle pulse function δ(t) and weighting function (0,t)

The affiliated output function derives in the same way from the system operation by a limit approach.

( )

{ }

( )

lim₀ 1

(

( ) (

)

(

0,

)

t v t t h t h t g t t ∆ → = ϕ δ = − −∆ = ∆ ∆

( )

0 0, 0 0 , 0, t d g t h t for h or h t g t dt dt = + = =

∫

(4.17) g(0,t) [1/h] weighting function

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The system output that results from a needle pulse corresponds to the differentiation of the system operation h(t) and is termed the weighting function g(t) (with a reference time interval ∆t that approaches zero the expression g(0,t) appears, see Figure 4.8b).

The discrete finite difference form of the theory of superposition is transformed to the analytical integral of superposition (integral of convolution, or, in another expression also termed Duhamel integral).

Discrete superposition in finite difference form with the weighting function

g(∆t,t_i)⋅ (u(t_i) = const. Within time interval ∆t_i):

( )

(

) (

)

(

) ( )

1 1 1 1 , , i i k i k k i i k k k v t g t t u t t g t t u t t − + = − + = = ∆ ⋅ ⋅ ∆ = ∆ ⋅ ⋅ ∆

∑

(4.18)

where u(t_i) = 0 for t_i < 0.

Analytical superposition with the weighing function g(0,t):

( )

( ) (

)

(

) ( )

0 0 0, 0, t t v t g t u t t dt g t t u t dt ′ ′ ′ = ⋅ − ⋅ ′′ ′′ ′′ = − ⋅ ⋅

∫

(4.19)

where u(t) = 0 for t < 0.

Model concepts try to represent these processes in the natural system by simple mathematical models. The system is identified. The parameters are determined to approximate the model input as closely as possible to the represented process. Most model concepts assign linear, time-invariant system operation to the hydrologic system or the individual system components. The system operation h(t) or the weighting function g(0,t) may derive from the model approach (see Chapter 5). Since usually discrete in- and output data are available, the superposition must be accomplished with the differential equation and discrete values. Also the ∆t-weighting function g(∆t,ti) must be determined.

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(

)

₍

( ) ( )

₎

( )

(

)

( )

1 1 1 0 0 1 , 1 0, 0, 1 0, i i i i i i i t t t t g t t h t h t t g t dt g t dt t g t dt t − − − ∆ = ⋅ − ∆   ′ ′ ′′ ′′ = ⋅_ ⋅ − ⋅ _ ∆   ′ ′ = ⋅ ⋅ ∆

∫

(4.20) g(∆t,t_i) [1/h] ∆t-weighting function

t’,t’’ [h] auxiliary variable for the integration

4.5 Computation of the unit hydrograph from the

∆t-weighting function

the input variable u(t) on the drainage basin is the volume of the effective or net areal precipitation. The output variable v(t) is the direct runoff at the basin outlet.

( )

3.6 E Ne D A u t i t v t Q t = ⋅ = (4.21)

u(t) [m3/s] system input

v(t) [m3/s] system output

i_Ne(t) [mm/h] intensity hydrograph of the effective rainfall

AE [km

2

] size of the drainage basin

QD(t) [m

3

/s] direct runoff hydrograph 3.6 [ 2 3 mm km s h m ⋅ ⋅ ⋅ ] conversion factor

The in- and output variables, here direct runoff and effective rainfall, are affiliated by the equation of superposition.

Hydrology 3

Hydrology III

Table of contents

1

Fundamental principles of river basin

modeling

1.1 Scope

1.2 Principal methods of river basin modeling

1.2.1

Statistical methods

1.2.2

Deterministic methods

1.2.3

Combined methods

1.3 Structure of river basin models

1.3.1

Partial models

1.3.2

Drainage basin models

1.3.3

Streamflow models

1.3.4

Complex river basin model

1.4 Model approaches

1.5 Analysis and synthesis

1.5.1

Calibration of the model against observed in- and output values

1.5.2

Synthetic streamflow hydrographs

2

Structure of drainage basin models

2.1 Formation of runoff and runoff concentration

2.2 Formation of outflow and runoff concentration in

simple drainage basin models

( )

( )

( )

3

Models of runoff formation

3.1 Runoff coefficient

3.1.1

Overall runoff coefficient for single rainfall-runoff events

( )

∑

3.1.2

Antecedent precipitation index and coaxial graphical plot

(

)

( )

( )

∑

(

)

(

)

3.1.3 The SCS approach

3.2 Models to compute effective rainfall

3.2.1

Model requirements

( )

( )

( )

(

)

( )

( )

( )

( )

( )

( )

( )

(

( )

)

3.2.2 Runoff coefficient method

( )

( )

3.2.3 Index approaches,

Φ-index

( )

_{( )}

_{( )}

_{( )}

_{{ }}

_{

_}

_{

_}

_{

_}

_{

_}