Hydrology 3

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UNIVERSITT STUTTGARTINSTITUTE OF HYDRAULIC ENGINEERING

CHAIR OF HYDROLOGY AND GEOHYDROLOGY

Prof. Dr. rer. nat. Dr.-Ing. Andrs Brdossy

Institute of Hydraulic Engineering, Universitt Stuttgart, Germany

Pfaffenwaldring 61 * D-70550 Stuttgart

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

Hydrology III

October 2003


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Prof. Dr. rer. nat. Dr.-Ing. Andrs Brdossy Hydrology III - I

Table of contents

1 FUNDAMENTAL PRINCIPLES OF RIVER BASIN MODELING .......................... 1

1.1 Scope ................................................................................................................................................ 1

1.2 Principal methods of river basin modeling ............. ............. ............. ............. ............. .............. ...... 21.2.1 Statistical methods .....................................................................................................................21.2.2 Deterministic methods ............................................................................................................... 21.2.3 Combined methods.................... .............. ............. ............. ............. ............. .............. ............. .... 3

1.3 Structure of river basin models ............. ............. ............. .............. ............. ............. ............. ........... 31.3.1 Partial models ............................................................................................................................31.3.2 Drainage basin models............................................................................................................... 41.3.3 Streamflow models................ ............. ............. .............. ............. ............. ............. ............. ......... 61.3.4 Complex river basin model...................... ............. ............. ............. ............. .............. ............. .... 8

1.4 Model approaches............................................................................................................................8

1.5 Analysis and synthesis ................................................................................................................... 111.5.1 Calibration of the model against observed in- and output values.......... ............. ............. ........... 111.5.2 Synthetic streamflow hydrographs.. ............. ............. ............. .............. ............. ............. ........... 11

2 STRUCTURE OF DRAINAGE BASIN MODELS................................................ 13

2.1 Formation of runoff and runoff concentration..... .............. ............. ............. ............. ............. ....... 13

2.2 Formation of outflow and runoff concentration in simple drainage basin models... ............. ....... 15

2.3 Base flow........................................................................................................................................ 16

3 MODELS OF RUNOFF FORMATION................................................................ 17

3.1 Runoff coefficient............. ............. ............. .............. ............. ............. ............. ............. .............. .... 173.1.1 Overall runoff coefficient for single rainfall-runoff events ............ ............. ............. .............. .... 173.1.2 Antecedent precipitation index and coaxial graphical plot... ............. .............. ............. ............. 183.1.3 The SCS approach.............. ............. .............. ............. ............. ............. ............. .............. ............. .. 21

3.2 Models to compute effective rainfall ............ ............. ............. ............. ............. .............. ............. .. 243.2.1 Model requirements ................................................................................................................. 243.2.2 Runoff coefficient method............ ............. ............. .............. ............. ............. ............. ............. ....... 263.2.3 Index approaches, -index............................................................................................................. 26

4 BASIS AND METHODS OF SYSTEMS HYDROLOGY ..................................... 28

4.1 Definition of system properties ............ .............. ............. ............. ............. ............. .............. ............. .. 28

4.2 Unit hydrograph ............................................................................................................................ 31

4.3 Analysis and synthesis of the unit hydrograph by the black box method................. ............. ....... 32


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Prof. Dr. rer. nat. Dr.-Ing. Andrs Brdossy Hydrology III - II

4.4 System operation and instantaneous unit hydrograph........... ............. ............. .............. ............. .. 36

4.5 Computation of the unit hydrograph from the t-weighting function .........................................41

5 CONCEPTS OF HYDROLOGIC MODELS ........................................................ 43

5.1 Translation and retention............. ............. .............. ............. ............. ............. ............. .............. .... 43

5.2 Translation models.................... ............. ............. ............. .............. ............. ............. ............. ......... 435.2.1 Linear translation, linear channel... ............. ............. ............. .............. ............. ............. ........... 435.2.2 Time of concentration ............. ............. .............. ............. ............. ............. ............. .............. .... 455.2.3 Floodplan method ....................................................................................................................475.2.4 Time-area diagram...................................................................................................................51

5.3 Reservoir routing models.......... ............. ............. ............. .............. ............. ............. ............. ......... 555.3.1 Linear reservoir................. ............. ............. ............. ............. .............. ............. ............. ........... 555.3.2 Non-linear, exponential reservoir ............ ............. ............. ............. ............. .............. ............. .. 585.3.3 Linear reservoir cascade...... ............. ............. ............. .............. ............. ............. ............. ......... 58

5.4 Parameter estimation for simple conceptual models..................... ............. ............. ............. ......... 615.4.1 Moment method for linear model concepts ............. ............. ............. .............. ............. ............. 615.4.2 Storage-outflow relation of single reservoir models ............. ............. .............. ............. ............. 655.4.3 Outflow recession curve of the linear, single reservoir.............. ............. ............. ............. ......... 67

6 COMBINATION OF MODEL CONCEPTS IN DRAINAGE BASIN MODELS..... 70

6.1 One-component models for direct runoff ............. .............. ............. ............. ............. ............. ....... 706.1.1 Clark model ............................................................................................................................. 706.1.2 Two-reservoir-model (Singh's model)................... ............. ............. ............. .............. ............. .. 716.1.3 Influence of precipitation on the model concept............... ............. ............. ............. .............. .... 71

6.2 Multi-component models, parallel reservoir cascades ............. ............. .............. ............. ............. 71

7 FLOOD ROUTING MODELS ............................................................................. 74

7.1 Flood routing......... ............. ............. .............. ............. ............. ............. ............. .............. ............. .. 74

7.2 Simple flood forecasting methods... .............. ............. ............. ............. ............. .............. ............. .. 777.2.1 Gage relation curve............. ............. ............. ............. .............. ............. ............. ............. ......... 777.2.2 Travel time curve .....................................................................................................................797.2.3 Prediction of discharge changes .............. ............. ............. ............. ............. .............. ............. .. 80

7.3 Hydraulic approaches to instationary flow............. ............. ............. ............. ............. .............. .... 82Continuity equation............. ............. ............. ............. .............. ............. ............. ............. ............. ....... 82

7.4 Hydrologic flood routing concepts....... ............. ............. ............. .............. ............. ............. ........... 887.4.1 Basic principles of hydrologic flood routing ............. ............. .............. ............. ............. ........... 887.4.2 Muskingum-model ............ ............. ............. ............. ............. .............. ............. ............. ........... 887.4.3 Kalinin-Miljukov method, basic principles............. ............. ............. .............. ............. ............. 937.4.4 Kalinin-Miljukov method, linear reservoir cascade................ .............. ............. ............. ......... 103

8 PHYSICALLY BASED HYDROLOGICAL MODELS........................................ 106


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Prof. Dr. rer. nat. Dr.-Ing. Andrs Brdossy Hydrology III - III

8.1 Fundamental principles ............................................................................................................... 106

8.2 Spatial model extension ............................................................................................................... 1078.2.1 One-dimensional models....... ............. ............. .............. ............. ............. ............. ............. ..... 1078.2.2 Two-dimensional models ....................................................................................................... 1088.2.3 Three-dimensional models ..................................................................................................... 109

8.3 Temporal and spatial model resolution... ............. .............. ............. ............. ............. ............. ..... 1108.3.1 Temporal resolution ............ ............. ............. ............. .............. ............. ............. ............. ....... 1108.3.2 Spatial resolution ................................................................................................................... 111

8.4 Modeling of single processes......... ............. .............. ............. ............. ............. ............. .............. .. 1138.4.1 Infiltration.................................................................................................................................... 1138.4.2 Evaporation.......... ............. ............. ............. ............. ............. .............. ............. ............. ......... 114

8.5 Model parameters........................................................................................................................ 1158.5.1 Parameter estimation............. ............. ............. .............. ............. ............. ............. ............. ..... 1158.5.2 Parameter variability....... ............. ............. ............. ............. ............. .............. ............. ........... 116

8.5.2.1 Temporal variability.......... ............. ............. .............. ............. ............. ............. ............. ..... 116


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Prof. Dr. rer. nat. Dr.-Ing. Andrs Brdossy Hydrology III - 1

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) statisticaland/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 studiesfor 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 certainriver 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:

simple outflow characteristics (e. g. time and magnitude of peak flow),


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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 combinedmethods. 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 dependencethe 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 basinrainfall-runoff model, drainage basin model (see Chapter 2-6),

river coursestreamflow 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

iN

iN(t)

Q(t)

time t

time t

dischargeQ

pr

ecipitationiN

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 precipitationtemporal 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 outflowTransformation 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/ streamflowConcentration 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 km 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 cross-section (outlet cross-

section of the examined river section).


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river coursetime tdischargeQ

Z

inflowoutflow

QAQZ

time tdischargeQ

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 oneinflow Q

Z(t) and one outflow

A(t)Q and therefore identical water volume (see Figure 1.4).

Confluence of several flood waves from different streams; this means several inflowsQ

Z,i(t) and one outflow

A(t)Q (see Figure 1.5).

Flow of a flood wave in an open channel with punctual or continuous lateral inflow fromthe 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).

Figure 1.5: Principle of a streamflow model with several tributaries (flood forecasting)


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time t

precipitationi

N

areal precipitation i N

QZinflow

QAoutflow

intermediate catchmenttime t

time t

dischargeQ

A

dischargeQZ

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 ttime t

Figure 1.7: Interfaces of a river basin model


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rainfall-runoff-model

catchment model

Si intersections Ki gage site (discharge gage)

water flow

model

water flow m

including catc

(lateral inflo

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

input

e.g. discharge,

precipitation

observed

output

natural hydrological

systeme.g. discharge

calibration of

parameterscomparison

mathematical model

(parameters)generated

output

Figure 1.9: Calibration of a mathematical hydrologic model


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The synthesis of streamflow hydrographs can be split up into three tasks.

Check of historic eventsIf 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 flowsFor 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 projectsThe 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

inputmodel,

known parameters

synthesis

of output

synthesis

Figure 1.10: Data flow in the course of analysis and 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 runoffThe 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 flowThe 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.

Distribution of the effective rainfall into the three components of flow.


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flood increase flood decrease

vertex

flood

hydrograph

Qsurface runoff

QOinterflow

QI

direct outflow

Q

base flow

assumed drought outflow hydrograph

time t [h]

outflowQ[m3/s]

Figure 2.1: Separation of flow components of a flood wave

areal

precipitation

iN

overall

outflow

Q

direct

outflow

QD

interflow

QI

effective

precipitation

iNe

infiltration

evaporation

transpiration

interception

surface

runoff

QO

base flow

QB

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 outflowformation of outflow

Figure 2.3: Flow formation and runoff concentration in simple drainage basin models


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2.3 Base flow

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 QB

from the overall flow Q provides the direct runoffQD: at the

beginning and at the end the direct runoff hydrograph has a value of zero.

( ) ( ) ( )D i i B iQ t Q t Q t = (2.1)

time t [h]

start

dischargeQ[m3/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-weatherconditions 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.

NeD

N N

hV

V h = = (3.1)

[-] overall runoff coefficient

VD

[m3] volume of direct runoff

VN

[m3] volume of precipitation

hNe

[mm] overall effective depth of precipitation of the event

hN

[mm] overall depth of precipitation of the event

Usually the direct runoff hydrograph QD(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

3600k

D D i

i

V t Q t

=

= (3.2)

QD

[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.

1000N E NV A h= (3.3)

AE

[km2] size of the drainage basin

hN

[mm] overall depth of precipitation

1000 [m3/(km

2mm)] 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) hVN.The seasonal variation of evapotranspiration and the detention storage may be considered by

a continuous array of numbers nW

assigned to each week.

( ), , ,Ne N Ne N N VN wh h h f h T h n= = (3.4)

Ne

N

h

h = (3.5)

hNv [mm] precipitation lossesh

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

precipitation)

hNe

[mm] effective depth of precipitation of a single event

TN

[h] duration of the precipitation event

hVN

[mm] antecedent precipitation index

nW

[-] 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 are always smaller than 1.


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t0-n

t0-n t0-i

t0-i

time t [d]

time t [d]

hVN (t0)

precipitationh

N[mm]

antece

dentprec.indexh

VN

[mm]

Figure 3.1: Antecedent precipitation index

The antecedent precipitation index is computed as:

( ) ( )0 01

ni

VN N ii

h t h t == (3.6)

t0

[d,h] start of precipitation event,

hVN

[mm] antecedent precipitation index,

hN

[mm] daily depth of precipitation,

n [-] number of days preceding the event,

[-] empirical weighting factor < 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= 0.9.The precipitation losses h

NVare displayed in coaxial graphical form (see Figure 3.2).


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duration of precipitation TN [h] depth of precipitation hN [h]

depth of prec. losses

hNV [mm] (calculated)

week number

depthofprec.losses

hNV

[mm](observed)

antecedentpr

ecipitation

indexh

VN

[m

m]

Figure 3.2: Coaxial graphical plot of precipitation losses hNV

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 hVN

the base specific discharge qB

is used, and instead of the week number nW

the monthMis used

to compute the precipitation losses hNV

.

( )

( )

sin 46

sin 46

NB

N NB

D TC qN

NV

E h D TC q

N N

h e e A B M

h

h h e e e A B M

+ = + +

(3.7)

hNV

[mm] precipitation losses (regional storage)

qB

[l/s/km2] base specific discharge at the beginning of the event

M [-] month

TN

[h] duration of precipitation


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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 thecase 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 Neh as a function of the depth of precipitation Nh and a curve number CN

dependent on the drainage basin:

25080

50.8

20320203.2

N

Ne

N

hCN

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

CNforsoilmoistureclassI,III

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 iN, the duration tN 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 is the ratio of effective or net precipitation iNe

(t) to the

observed precipitation iN(t).

( ) ( )Ne Ni t i t = (3.9)

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

( )( ),Nf i t t = (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 iv( = losses)

contributes to direct runoff.


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( )( ) ( ) ( )

( ) ( )0N 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]prec.intensityi[mm/h]

prec.intensityi[mm/h]

prec.intensityi[mm/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, -index method

c) loss ratio decreasing exponentially

overall outflow

Q(t)

areal

precipitation

iN (t)

separation of

base flow

formulation for

effective prec.

base flow

QB (t)

direct outflow

QD (t)

effective

precipitation

iNe (t)

precipitation

losses

iNv (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 Ni 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 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 ii t i t = (3.13)

iNe

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

iN(t

i) [mm/h] observed precipitation intensity in time interval t

[1] overall runoff coefficient

3.2.3 Index approaches, -index

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

( ) .Vi t const = = (3.14)

[mm/h] constant loss rate, -index

The portion of precipitation that exceeds the -index is the effective rainfall.


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( )( ) ( )

( )0N N

Ne

N

i t for i t i t

for i t

> =

The -index must be determined by step-by-step iterations since negative precipitation isimpossible. The iteration provides a constantly increasing -index; the procedure is repeateduntil 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) uiaffects the system, and an exit where the effect

(output, system answer) vioccurs (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 ui(t) and output

vector vi(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 u t = (4.1)

where,

u(t) time-dependent input signal,

v(t) time-dependent output signal,

system operator.

system

output variateinput variate

Figure 4.2: System with one input magnitude u(t) and one output magnitude v(t)


33/120




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 signali

u (t) 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 shortduration T

Nis discharged as a flood wave of much longer duration T

b.

Theory of proportionalityAny input signal multiplied by a constant Cproduces an output signal multiplied by the same

constant.

( ){ } ( ){ }C u t C u t = (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 Tb

of the outflow hydrograph remains

constant.

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

( ) ( ){ } ( ){ } ( ){ }1 2 1 2u t u t u t u t + = + (4.3)

Theory of linearityThe combination of the theory of proportionality and superposition provides the theory of

linearity.

( ) ( ){ } ( ){ } ( ){ }1 1 2 2 1 1 2 2C u t C u t C u t C u t + = + (4.4)

Theory of time invarianceThe 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). Tbis preserved.


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( ){ } ( )u t T v t T = (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

dischargeQ

D[m3/s]

intensityiN

e[mm/h]

precipitation

discharge

time

dischargeQ

D[m3/s]

intensityiN

e[mm/h]

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

b) theory of proportionality


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intensityiN

e[mm/h]

inte

nsityiN

e[mm/h]principle of time invariance principle of superposition

precipitation

discharge

time

d

ischargeQ

D[m3/s]

d

ischargeQ

D[m3/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

iNe*Ae

drainage basin

gE

direct outflow

QD

outputinput 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 tand the magnitude

one (see Figure 4.3a). This function is referred to as discrete weighting function g(t,ti) with


36/120




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 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) isreplaced by the unit hydrograph g

E(t,t

i) 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 tis chosen that is valid for the discretion of all time-relateddata. 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 QD(t

i). Subsequently the overall runoff coefficient and the

effective rainfall hNe

(ti) 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 tiprovides (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

Qi= Q

D(t

i) [m

3/s] direct runoff with time t

i,

Ni= h

Ne(t

i) [mm] effective rainfall in the interval between t

i-1and t

i,

Gi= g

E(t,t

i) [m

3/(smm)] 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 ti:

( ) ( ) ( )11

,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 superpositiontime t [h]

time t [

d

irectrunoff

precipitationintensity

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 QD 0,

n [1] number of effective rainfall intervals hNe, 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 runoffQD,gem

as possible. This is accomplished by applying the method of the lowest

square error.

( ) ( )( )21

, ,

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

iE

tg t t

A =

= (4.10)

AE

[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-Merkbltter 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(t) (see Figure 4.7):

( )

0 0

1 0

for t

t for t


41/120




most hydrologic systems the system answer has the value zero at the time t= 0 which

simplifies the subsequent formulas.

( ){ } ( ) ( ) 0 0t h t with h t for t = = (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


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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

= =

= =

(4.14)

g{t,ti) [1/h] t-weighting function

For an infinitesimally small time tthe input function becomes a theoretical function known

as the needle pulse, instantaneous unit function or Diracs 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).

( ) ( ) ( )( ) ( )0

1lim

t

dt t t t

t dt = =

(4.15)

( )

0 0

0

0 0

for t

t for t

for t

= =

(4.16)

(t) [1/h] needle pulse function


43/120




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 (t) and weighting function (0,t)

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

approach.

( ) ( ){ } ( ) ( )( ) ( )0

1lim 0,

tv t t h t h t g t

t = = =

( ) ( ) ( ) ( ) ( )0

0, 0 0 , 0,

td

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 functiong(t) (with a reference time interval

tthat 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,ti) (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(ti) = 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 themodel 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,

10, 0,

10,

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,ti) [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

ENe

D

Au t i t

v t Q t

=

=(4.21)

u(t) [m3

/s] system inputv(t) [m3/s] system output

iNe

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

AE

[km2] 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.


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( ) ( ) ( )11

,i

i k i k

k

v t g t t u t t +=

=

( ) ( ) ( )11 , 3.6

iE

D i k Ne i kk

A

Q t g t t i t t +== (4.22)

By definition of the unit hydrograph (according to DIN 4049 Part 1), the input is not the

intensity of effective rainfall iNe

(ti) but the effective depth of precipitation h

Ne(t

i) that pertains

to the respective interval. For reasons of clarification it can be termed hNe

(t,ti).

( ) ( )Ne i Ne ih t i t t = (4.23)

This formula and the t-weighting function provides the unit hydrograph that considers the

size of the drainage basin and the finite difference dimensions.

( ) ( )

( ) ( ) ( )11

, , 13.6

,

EE i i

i

D i E k Ne i k

k

A mmg t t g t t

t

Q t g t t h t +=

=

= (4.24)

QD(t

i) [m

3/s] direct runoff at time t

i

hNe

(ti) [mm] effective rainfall within the time interval ti - 1 to ti

gE(t,t

i) [

3m

s mm] unit hydrograph of the drainage basin at time t

i


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5 Concepts of hydrologic models

5.1 Translation and retentionThe complex processes that occur in the course of runoff concentration and instationary open

channel flow are in hydrologic practice replaced by simple model concepts.

The two main groups are:

translation and

retention.

The application of an individual model is usually not satisfactory in terms of accuracy. Only

the combination of translation and retention yields results that are close enough to the natural

conditions (see Chapter 6).

5.2 Translation models

5.2.1 Linear translation, linear channel

Translation is a shifting in time. The shape of the input signal u(t) reappears lagged by thetranslation time T

tas the output signal v(t). If the translation time T

tis a constant and thus not

dependent on the magnitude and the temporal occurrence of the input signal (time-invariant),

the system operation is called a linear translation (see Figure 5.1).

( ) ( )tv t u t T = (5.1)

u(t) [div.] time-related input signal

v(t) [div.] time-related output signal

Tt [h] translation time


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u, v u (t) v(t) = u(t-Tt)

dt dttime t

Tt dt

u1 v1=u1

Figure 5.1: Input and output of a linear translation system

In the finite difference form Ttshould be an integer multiple of the time interval t.

( ) ( )i i t t v t u t T with T m t = = (5.2)

The linear channel is a flow model where the flow velocity v is constant. The effects of

instationary channel flow (flow retention v(t) and local hydraulic conditions (cross-section

A(x), roughness kSt(x)) are ignored.

( ) ( ) ( ) ( )( ), , , , .Stv A x k x h x h t t v const = = (5.3)

v [m/s] flow velocity

A [m2] cross-section

kSt

[m1/3/s] coefficient of roughness

x [m,km] stream coordinate

h [m] depth of water

t [h] time

The time of flow Ttis the time required for a water particle to move a certain distance l

F(DlN

4049 Part 1).

For a linear channel the time of flow equals a constant translation time.

1

3.6

Ft

lT

v= (5.4)

Tt

[h] time of flow

lF

[km] flow distance, channel length

v [m/s] constant flow velocity


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3.6 [km s

m h

] conversion factor

The system operation of linear channels is a linear translation. Therefore it is a linear, time-

invariant model system where the weighting function g(0,t) is known.

( ) ( )0, tg t t T = (5.5)

g(0,t) [1/h] weighting function of the linear translation,

(t) [1/h] needle pulse function (see Chapter 4.4).

The outflow QA(t) of a linear channel is the inflow Q

Z(t) lagged by the time of flow T

t.

( ) ( )A Z tQ t Q t T = (5.6)

QZ

[m3/s] inflow of a linear channel,

QA

[m3/s] outflow of a linear channel.

The linear channel is only a very rough approximation of the instationary outflow since it

does not consider retention. Hence it may only be applied as a component of a combined

model.

5.2.2 Time of concentration

The water particles that fall as precipitation in a drainage basin require different time to reach

the basin outlet dependent on their position in the watershed. The time of concentration Tc is

defined as the travel time of a water particle from the hydraulically most remote point in the

basin to the outflow location, or, in other words, the time until the whole basin contributes to

the outflow. From the mean flood flow velocity Specht suggested the rough estimation below.

1 1

2 3C F F

h hT l to l

km km

= (5.7)

vm

[m/s] means flow velocity from 0.6 to 0.8 m/s

Tc [h] concentration time,l

F[km] length of the main feeder.

For mountainous regions, Kirpich established an empirical formula identical to the equation

of the U.S.-Soil Conservation Service. A uniform procedure is applied to determine the mean

inclination IF

of the longitudinal section of the drainage basin. A straight line represents the

slope of the basin in a way that the two regionsA1andA

2are of equal size (see Figure 5.2).


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time of concentration according to Kirpich (coefficients not dimensionless)

0.77

0.06625 FC

F

lT

I

=

310

FF

F

hI

l=

(5.8)

time of concentration according to the U.S.-Soil Conservation Service (coefficients not

dimensionless)

0.3833

0.868 FC

F

lT

h

=

(5.9)

Tc

[h] time of concentration,

hF

[m] relevant difference in elevation,

lF

[km] relevant path of flow,

IF

[1] mean inclination of the main channel or the longitudinal section of

the basin

length in cross-section

drainage basin boundary

outlet

drainage basin

boundary

elevation

at outlet

elevation

outlet

elevation

at basin

boundary

Figure 5.2: Time of concentration, determination of mean inclination by equalizing the

regions in the longitudinal section of the drainage basin (section that provides

the longest extension lmax

of the basin measured from the outlet)


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5.2.3 Floodplan method

The floodplan method is a simple runoff concentration model based on linear translation. The

water network of a drainage basin consists of linear channels of equal flow velocity. The

drainage basin is approximated by a rectangular plain. The mean inclination IF and the time ofconcentration cT are determined according to Chapter 5.2.2. Precipitation is considered

evenly distributed (block precipitation, see Figure 5.3).

The drainage basin dewaters with a constant flow velocity vm., consequently a travel time T

t

and an area Atcan be assigned to a point with the distance x from the outlet (A

i= A

i(T

i), see

Figure 5.3). Assuming the precipitation being a jump function, as time elapses, progressively

distant areasAtcontribute to the outflow (see Figure 5.4). In other words, up to a certain time

tonly the portion of precipitation that fell within the areaA with a corresponding travel time

t = Ttcontributes to the outflow. As soon as the time of concentration T

cis exceeded (t > T

c)

the whole basinAE dewaters and a constant outflow rate is acquired (see Figure 5.4).

On the rising limb (tTc) the area A

t(t) that contributes to the outflow is constantly increasing

with time.

( )0E C

Ct

E C

tA for t T

TA t

A for t T

= >

(5.10)

At(t) [km

2] area contributing to outflow,

AE

[km2] size of the drainage basin.

effective precipitation

drainage basin

Figure 5.3: Floodplan-model, areaAt(T

t) that contributes to outflow at time T

t


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time t [h] time t [h]

precipitation outflow

eff.prec

.iN

e[mm/h]

directru

noff[m3/s]

contributingarea

At(t

)[km

2]

normalizedcontributing

ar

ea

Figure 5.4: System operation of the floodplan for precipitation in form of a jump function,

temporal course of the area contributing to outflow At(t), system operation

function

Since the floodplan method is based on linear translation it is a linear, time-invariant system.

The function of the ratio of the contributing area to the size of the drainage basin At(t)/A

E

represents the system operation function h(t) of the floodplan method (see Figure 5.4).

( )( )

0

1

Ct

C

E

C

tfor t TA t

Th tA

for t T

= = >

(5.11)

h(t) [1] system operation function of the floodplan method

From the system operation function the unit hydrograph of the floodplan method can be

determined (see Chapter 4.4). However, it is only representative if the concentration time Tcis

a integer multiple of the time interval t.

( ) ( ) ( )

10

0, 0 0

Hydrology 3

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