final handout (intro to hydrolgy

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Adama university, Department of civil Engineering and Architecture Introduction to hydrology (CE-3601) Tessema B; Mekdim M;Belay,Suleman 1 COURSE CONTENT: CHAPTER ONE – INTRODUCTION ...................................................................................................2 1.1 General hydrology: .................................................................................................................2 1.2 Application of Hydrology .......................................................................................................3 1.3 The Hydrologic Cycle .............................................................................................................4 1.4 The basic Hydrologic Equation ..............................................................................................5 CHAPTER TWO - RUNOFF MEASUREMENT AND RATING CURVE .........................................7 2.1 Stream Flow Measurement .....................................................................................................7 2.2 Stage-discharge /Rating Curves ............................................................................................11 2.3 Methods for Extending the Stage - Discharge curve. ...........................................................12 CHAPTER THREE - PROCESSING AND ANALYSIS OF HYDROLOGICAL DATA .................14 3.1 General ..................................................................................................................................14 3.2 Meteorological data ..............................................................................................................14 3.3 Areal Estimation ...................................................................................................................18 3.4 Hydrological Data .................................................................................................................21 CHAPTER FOUR - INTENSITY DURATION FREQUENCY PROCEDURES...............................22 4.1 Intensity-Duration relationship of a Rainfall ........................................................................22 4.2 Depth - Area - Duration (DAD) Relationship.......................................................................23 4.3 Frequency analysis of rainfall (Recurrence interval of a storm)...........................................28 4.4 Intensity - Duration - Frequency Relationship......................................................................31 CHAPTER FIVE - RAINFALL RUNOFF MODELS .........................................................................34 5.1 Introduction ...........................................................................................................................34 5.2 Factors Affecting Runoff ......................................................................................................34 5.3 The hydrograph .....................................................................................................................37 5.4 Methods Used to Estimate Runoff ........................................................................................38 CHAPTER SIX - FLOOD FREQUENCY ANALYSIS ......................................................................44 a. Introduction ...............................................................................................................................44 b. Common flood frequency distributions ....................................................................................45 c. Risk, Reliability and Safety factor: ...........................................................................................50 d. Applied Examples: ....................................................................................................................51 CHAPTER SEVEN – MISCELLANEOUS TOPICS ..........................................................................53 7.1 Introduction ...........................................................................................................................53 7.2 Precipitation ..........................................................................................................................53 7.3 Depression storage ................................................................................................................58 7.4 Interception ...........................................................................................................................59 7.5 Infiltration .............................................................................................................................60 7.6 Evaporation and transpiration ...............................................................................................64

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Page 1: Final Handout (Intro to Hydrolgy

Adama university, Department of civil Engineering and Architecture

Introduction to hydrology (CE-3601) Tessema B; Mekdim M;Belay,Suleman 1

COURSE CONTENT:

CHAPTER ONE – INTRODUCTION...................................................................................................2

1.1 General hydrology: .................................................................................................................2

1.2 Application of Hydrology .......................................................................................................3

1.3 The Hydrologic Cycle.............................................................................................................4

1.4 The basic Hydrologic Equation ..............................................................................................5

CHAPTER TWO - RUNOFF MEASUREMENT AND RATING CURVE .........................................7

2.1 Stream Flow Measurement .....................................................................................................7

2.2 Stage-discharge /Rating Curves............................................................................................11

2.3 Methods for Extending the Stage - Discharge curve. ...........................................................12

CHAPTER THREE - PROCESSING AND ANALYSIS OF HYDROLOGICAL DATA .................14

3.1 General ..................................................................................................................................14

3.2 Meteorological data ..............................................................................................................14

3.3 Areal Estimation ...................................................................................................................18

3.4 Hydrological Data.................................................................................................................21

CHAPTER FOUR - INTENSITY DURATION FREQUENCY PROCEDURES...............................22

4.1 Intensity-Duration relationship of a Rainfall ........................................................................22

4.2 Depth - Area - Duration (DAD) Relationship.......................................................................23

4.3 Frequency analysis of rainfall (Recurrence interval of a storm)...........................................28

4.4 Intensity - Duration - Frequency Relationship......................................................................31

CHAPTER FIVE - RAINFALL RUNOFF MODELS .........................................................................34

5.1 Introduction...........................................................................................................................34

5.2 Factors Affecting Runoff ......................................................................................................34

5.3 The hydrograph.....................................................................................................................37

5.4 Methods Used to Estimate Runoff........................................................................................38

CHAPTER SIX - FLOOD FREQUENCY ANALYSIS ......................................................................44

a. Introduction...............................................................................................................................44

b. Common flood frequency distributions ....................................................................................45

c. Risk, Reliability and Safety factor: ...........................................................................................50

d. Applied Examples:....................................................................................................................51

CHAPTER SEVEN – MISCELLANEOUS TOPICS ..........................................................................53

7.1 Introduction...........................................................................................................................53

7.2 Precipitation ..........................................................................................................................53

7.3 Depression storage ................................................................................................................58

7.4 Interception ...........................................................................................................................59

7.5 Infiltration .............................................................................................................................60

7.6 Evaporation and transpiration ...............................................................................................64

Page 2: Final Handout (Intro to Hydrolgy

Adama university, Department of civil Engineering and Architecture

Introduction to hydrology (CE-3601) Tessema B; Mekdim M;Belay,Suleman 2

CHAPTER ONE – INTRODUCTION

1.1 General hydrology:

Hydrology is the science that deals with the processes governing the depletion and replenishment of

the water resources of the land areas of the earth. It is concerned with the transportation of water

through the air, the precipitation occurring on the ground as rainfall or snowfall, and the flow of

water over the ground surface and through the underground strata of the earth. It is the science that

treats of the various phases of the hydrologic cycle.

Hydrology is the science, which deals with the occurrence, circulation and distribution of water upon,

over, and beneath the earth surface. Hydrology treats of the water of the earth, their occurrences,

circulation, and distribution, their chemical and physical properties, and their reaction with their

environment, including their relation to living things.

As a branch of earth science, it is concerned with the water in streams and lakes, rainfall and

snowfall, snow and ice on the land and water occurring below the earth’s surface in the pores of the

soil and rocks. Hydrology as a science has thus many components and in the broadest sense would

include the movement of water into, with in and from the atmosphere but these processes is often

considered to be with in the domain of other sciences such as meteorology, climatology and soil

science. The influence of the vegetation is obviously also with in the domain of botany.

Generally the domain of hydrology embraces the full life history of water on the earth. The concept

of water balance and the concept of a catchment are both extremely useful in the applications of

hydrological principles.

We may distinguish several areas of study in hydrology as follows

- The primary process of evaporation , transpiration by vegetation , infiltration and

percolation

- Surface runoff including flow in open channels

- Groundwater hydraulics, including flow in the unsaturated zone

Engineering hydrology includes those segments of the field pertinent to planning, design, and

operation of engineering projects for the control and use of water. It is the study of those aspects of

hydrology which are relevant to the solution of engineering problems in the control of utilization of

water and in the protection of water resources. The term however refers not to a subset of

hydrological studies only but implies as a method of study or analysis which is designed to answer in

a quantitative manner , questions arising in an engineering context but with out necessarily implying

an extension of our understanding of the process involved.

Such problems usually arise and studied under the following categories.

- Forecasting or the estimation of when some hydrological event will occur

- Frequency prediction or the estimation of how often an event will occur

Page 3: Final Handout (Intro to Hydrolgy

Adama university, Department of civil Engineering and Architecture

Introduction to hydrology (CE-3601) Tessema B; Mekdim M;Belay,Suleman 3

1.2 Application of Hydrology

Knowledge of hydrology is of basic importance for the problems that deal with any water resource

development, and the use and supply of water for any purpose whatsoever. Therefore, hydrology is of

value not only in the field of engineering but also in forestry, agriculture, and other branches of

natural science.

Hydrology is basically an applied science. To further emphasis the degree of applicability, the subject

is sometimes classified as

1. Scientific Hydrology: The study, which is concerned chiefly with academic aspects.

2. Engineering or applied Hydrology: A study concerned with applications of the

hydrological principle in engineering practice.

In general sense, engineering hydrology deals with

i. Estimation of water resources.

ii. The study of hydrologic processes such as precipitation, runoff, evapotranspiration and

their interaction and

iii. The study of problems such as floods and droughts and strategies to combat them.

This course mainly deals with an elementary treatment of engineering hydrology with an introduction

to both surface and ground water hydrology, descriptions that give a qualitative judgment and

techniques that lead to a quantitative evaluation of the hydrologic processes which are of utmost

importance to civil, agricultural and water engineering works.

Page 4: Final Handout (Intro to Hydrolgy

Adama university, Department of civil Engineering and Architecture

Introduction to hydrology (CE-3601) Tessema B; Mekdim M;Belay,Suleman 4

1.3 The Hydrologic Cycle

The hydrologic cycle is a constant movement of water above, on, and below the earth's surface. It is a

cycle that replenishes ground water supplies. It begins as water vaporizes into the atmosphere from

vegetation, soil, lakes, rivers, snowfields and oceans-a process called evapotranspiration.

As the water vapor rises it condenses to form clouds that return water to the land through

precipitation: rain, snow, or hail. Precipitation falls on the earth and either percolates into the soil or

flows across the ground. Usually it does both. When precipitation percolates into the soil it is called

infiltration; when it flows across the ground it is called surface runoff. The amount of precipitation

that infiltrates, versus the amount that flows across the surface, varies depending on factors such as

the amount of water already in the soil, soil composition, vegetation cover and degree of slope.

Surface runoff eventually reaches a stream or other surface water body where it is again evaporated

into the atmosphere. Infiltration, however, moves under the force of gravity through the soil. If soils

are dry, water is absorbed by the soil until it is thoroughly wetted. Then excess infiltration begins to

move slowly downward to the water table. Once it reaches the water table, it is called ground water.

Ground water continues to move downward and laterally through the subsurface. Eventually it

discharges through hillside springs or seeps into streams, lakes, and the ocean where it is again

evaporated to perpetuate the cycle

Page 5: Final Handout (Intro to Hydrolgy

Adama university, Department of civil Engineering and Architecture

Introduction to hydrology (CE-3601) Tessema B; Mekdim M;Belay,Suleman 5

1.4 The basic Hydrologic Equation

A fundamental assumption behind the hydrologic equation is that of the hydrologic cycle being a

closed system, i.e., there are no gains or losses of water from the cycle. However, there are many

occasions upon which the hydrologist has to deal with an open system in which can only be described

by a mass balance or water budget equation in which the difference between input I and out put, Q, is

related to the change in storage, dS, with in the time interval dt:

I – Q = dS/dt

In applying this equation care must be taken in defining the so-called control volume or region over

which the budget is applicable. For example, for an open water body, such as a lake or reservoir, the

inputs to the system consist of the inflow Qin, the precipitation, P, on the water surface, and the sub-

surface inflow, Gin, and the outputs include the outflow, Qout, the evaporation from the water surface,

E, and any sub-surface outflow, Gout. If the change in storage over the chosen time period is ∆S,

which may be positive or negative, then,

Gout

Figure 2: A simplified schematization for water budget on an open water body

For the balance of water over a control region such as a catchment the inflow, Qin, can be ignored as

the catchment receives water in form of precipitation. Transpiration by vegetation and interception,

too, can be considered as part of the water budget.

Example 1: A clear lake has a surface area of 708,000m2. For the month of March, the lake had an

inflow of 1.5 m3/s and an outflow 1.25 m

3/s. A storage change of 708,000m3 was recorded during the

month. If the total depth of rainfall recorded at the local rain gauge was 225mm for the month,

estimate the evaporation loss from the lake. State any assumptions that you make in your

calculations.

Solution: The evaporation loss may be computed rearranging the hydrologic equation given above.

That is,

E = P+ Qin-Qout -∆S

Assuming seepage to be negligible,

The precipitation, P = (225/1000X708, 000) m3 = 159,300 m

3,

Inflow, Qin = 1.5 m3/s X 86,400s/d X 31 Days /month = 4,017,600 m

3,

Qin - Qout + P – E + Gin - Gout = ∆S

Or

(Qin + P+ Gin) - (Qout + E + Gout) = ∆S

Qout

Gin

Qin

P E

Page 6: Final Handout (Intro to Hydrolgy

Adama university, Department of civil Engineering and Architecture

Introduction to hydrology (CE-3601) Tessema B; Mekdim M;Belay,Suleman 6

Outflow, Qout = -1.25 m3/s X 86,400s/d X 31 Days /month = -3,348,000 m

3,

Change in storage, ∆S = +708, 000m3.

Hence, evaporation, E = 159,300 +4,017,600 - 3,348,000 – 708, 000 = 120,900 m3, or

E = 120,900 m3 X 1000mmm/m/708,000 m

2 =171mm over the lake.

In contrast, if the control volume is a catchment or drainage area bounded by its watershed or water

divide, the inputs consist of precipitation, P, and possibly ground water inflow, Gin, and the outputs

comprise the discharge, Q, at the catchment outlet, transpiration from the vegetation growing within

the catchment and evaporation from the precipitation intercepted on the vegetal canopy held in

storage on the ground, E, and possibly groundwater outflow, Gout. The changes in storage, ∆S, to be

considered are principally those in the sub-surface unsaturated and saturated zones leading to

Example 2: During the water year 1998/99, a catchment area with the size of 2500 km2 received

1,300mm of precipitation. The average discharge at the catchment outlet was 30m3/s. Estimate the

amount of water lost due to the combined effects of evaporation, transpiration and percolation to

groundwater. Compute the volumetric runoff coefficient for the catchment in the water year.

Solution: Assuming that the changes in storage, ∆S, are negligible, the above equation for the

catchment area becomes:

E + Gout – Gin = P – Q The runoff, Q = 30m

3/s X86, 400s/d X 365 Days /year X 1000 mm/m/(2500 km

2 X (1000m/km)

2)

= 378mm.

Hence, the combined loss = 1,300 – 378 = 922mm.

The volumetric runoff coefficient, C, is the ratio of the total volume of runoff to the total volume of

rainfall during a specified time interval: in this case,

C = 378/1300 = 0.29, i.e., only 29% of the rainfall reached the catchment outlet with

in the water year.

Q =P – E +Gin –Gout - ∆S

Page 7: Final Handout (Intro to Hydrolgy

Adama university, Department of civil Engineering and Architecture

Introduction to hydrology (CE-3601) Tessema B; Mekdim M;Belay,Suleman 7

CHAPTER TWO - RUNOFF MEASUREMENT AND RATING CURVE

2.1 Stream Flow Measurement

Definition

Stream flow means the discharge flowing in a river or a stream. It represents the runoff in the river at

the given section and includes surface runoff as well as ground water.

Control section/Gauging station: - is the point selected for measuring /determining the

characteristics of the stream flow. Stream gauging includes determining the river discharge and

velocity over long period of time.

Measurement of stage, velocity and discharge Stage of a stream flow can be measured as the depth of flow in the stream channel, taken from the

bottom of the control point.

Discharge is defined as the volumetric rate of flow through a given section. Mathematically it is

given as: Q = A x V

Where,

Q = discharge, m3/sec

A = cross section of area, m2

V = average flow velocity, m/sec

Discharge can be measured in the following several ways:

1. Velocity area method

2. Weir method

3. Power plant method

4. Dilution method

5. Slope area method

6. And others.

In this lesson discussions will be limited to the first two common and basic methods alone.

1) Velocity area method

This involves estimation of discharge through direct measurement of flow velocity and area of flow.

The velocity of flow can be measured by float method or current meter.

i) Float method - float method of making a rough estimate of the flow in a channel consists of noting

the rate of movement /the advancement of a floating body. A long necked bottle partly filled with

water or a block of wood may be used as a float. A straight section of the channel about 30 meters

long with fairly uniform cross-section is usually selected. Several measurements of depth and width

are made within trial section to arrive at the average cross-sectional area. A string is stretched across

each end of the section at a right angle to the direction of flow. The float is placed in the channel, a

short distance upstream from the trial section. The time the float needs to pass from the upper to

lower section is recorded. Several trials are made to the average time of travel.

Page 8: Final Handout (Intro to Hydrolgy

Adama university, Department of civil Engineering and Architecture

Introduction to hydrology (CE-3601) Tessema B; Mekdim M;Belay,Suleman 8

Figure 15: A float method illustration

To determine the velocity of water at the surface of the channel, the length of the trial section is

divided by the average time taken by the float to cross it. Since the velocity of the float on the surface

of the water will be greater than the average velocity of the stream, it is necessary to correct the

measurement by multiplying by a constant factor which is usually assumed to be 0.85. That is,

Q = (Vsurface X 0.85) X A, where A is flow area.

ii) Current meter method-The velocity of water in a stream or river may be measured directly

with a current meter. The current meter is a small instrument containing a revolving wheel or vane

that is turned by the movement of water. It may be suspended by a cable for measurements in deep

streams or attached to a rod in shallow streams. The number of revolutions of the wheel in a given

time interval is obtained and the corresponding velocity is reckoned from a calibration table or graph

of the instrument.

For feeding channels or rivers the average flow velocity, Vaverage can be fixed using one of the

following relations.

Where,

VSurface, V0.2, V0.6 and V0.8 are velocities of flow measured at the free surface, 20%, 60% and

80% of the depth of the flow.

Current meter measurements in canals and streams are generally made at metering bridges, at

cableways or at other structures giving convenient access to the stream. The channel at the measuring

section should be straight, with a fairly regular cross section. Structures with piers in the channel are

avoided when possible.

When the mean velocity of a stream is determined with a current meter, the cross section of flow is

divided into a number of sub-areas and separate measurements are made for each sub-area. The width

of sub-areas may vary from 1m to 6m, depending on the size of the stream and precision desired. It

has been found that the average of readings taken at 0.2 and 0.8 of the flow depth below the surface is

an accurate estimate of the mean velocity in a vertical plane.

V Average = V0.6

= (V0.2 +V0.8)/2

= K VSurface

L= about 30 m V average =L/taverage

A float

Page 9: Final Handout (Intro to Hydrolgy

Adama university, Department of civil Engineering and Architecture

Introduction to hydrology (CE-3601) Tessema B; Mekdim M;Belay,Suleman 9

Figure 16: Division of a river cross-section into sub areas

Both float and current methods, however, have very limited application in the irrigation practice field.

Both methods are used in open channels.

Example 1: The following velocities were recorded in a stream with a current meter.

Depth above the bed, m 0 1 2 3 4

Velocity, m/s 0 0.5 0.7 0.8 0.8

Find the discharge per unit width of the stream near the point of measurement. Depth of the flow at

the point was 5 meters.

Solution:

Yo = 5 m.

Therefore, 0.2Yo = 1 m, V0.2 = 0.5 m/s

0.8Yo = 4 m, V0.8 = 0.8 m/s

Average mean velocity = (V0.2 +V0.8)/2 = (0.5+0.8)/2 = 0.65 m/s.

The discharge per unit width = Ax V = (5 x1) * 0.65 = 3.25 m3/s/m.

Example2: the following data were collected for a stream at a gauging station. Compute the

discharge.

Solution:

Page 10: Final Handout (Intro to Hydrolgy

Adama university, Department of civil Engineering and Architecture

Introduction to hydrology (CE-3601) Tessema B; Mekdim M;Belay,Suleman 10

2) Use of weirs

Weirs are installations used for discharge measurement. There are two kinds of weirs: sharp - crested

& broad - crested weirs. Sharp -crested weirs have knife edge overflow section from where water jet

springs out. Broad -crested weirs have either flat topped or ogee crest profile.

Discharge Formula [m3/s]

Rectangular/ trapezoidal -sharp crested weirs are used to measure large flows, while V-notch weirs

are used for small flows. But sharp-crested weirs are not widely used for rivers carrying high

sediment load as material gets collected in the stilling pool above the weir, thereby rising the velocity

of approach.

For a rectangular sharp-crested weir the discharge formula is

HLgCQ d2

3

23

2= (Without velocity of approach)

H = head of water over the weir crest, m

L = length of the weir crest, m

For a V-notch weir discharge formula is

θθ

,2

tan215

82

5

HgCQ d= is the angle of notch.

H = the head over the notch, m

For a broad -crested weir the discharge formula is

Q = 1.7CdLH3/2

Page 11: Final Handout (Intro to Hydrolgy

Adama university, Department of civil Engineering and Architecture

Introduction to hydrology (CE-3601) Tessema B; Mekdim M;Belay,Suleman 11

L and H as defined above.

Ogee weir the discharge formula is

Q = 2.2 CdLH3/2

L and H as defined above.

1.1. Coordinate method of measuring discharge from pipes

Figure 17: Coordinate method of measuring discharge from pipes

Q = discharge, m3/s

C = coefficient of contraction ~ 1.0

A = cross sectional area of pipe, m2

X = x-coordinate, m

Y = y-coordinate, m

g = gravitational acceleration, m/s2

V= flow velocity, m/s

2.2 Stage-discharge /Rating Curves

These are the curves that give the relationship between the stage of the river at a given time (gauge

height) and the corresponding discharge. Graphically it is generally represented as follows.

Figure 18: Stage discharge relations

Q = (CAXg1/2

)/ (2Y) 1/2

D

Y

X

V= (Xg 1/2

)/ (2Y) 1/2

X = Vt

Y = (gt2)/2

Gau

ge

hei

ght in

Discharge [m3/s]

Station rating curve

Page 12: Final Handout (Intro to Hydrolgy

Adama university, Department of civil Engineering and Architecture

Introduction to hydrology (CE-3601) Tessema B; Mekdim M;Belay,Suleman 12

Stage-discharge curves can be prepared for both gauged and ungauged stations. Once the stage-

discharge relationship is established, it can be used to measure the discharge during different stream

flows directly.

2.3 Methods for Extending the Stage - Discharge curve.

a. Logarithmic method:-

If the river cross- section at the site of the gauge is a uniform section (or even approximates a uniform

section) to which can be fitted either a segment of a circle or a trapezoidal or a parabola or a

rectangular section, then this method can be easily adopted. The stage discharge curve for such a

stream can be expressed by an equation of the form

Q = k (g-go) m

where

Q = discharge in m3/s

g = gauge height in meters

go = gauge height in meters corresponding to zero discharge

k and m are constants.

Taking log on both sides, we get

Log Q = log (k (g-go) m

)

= log k + log (g-go) m

= log k + m log (g-go)

This is the equation of a straight line having a slope as m and log k as the intercept on Y - axis as

shown below.

Figure 17: Logarithmic method

The gauge height corresponding to zero discharge is not known and hence a graph is plotted between

log Q and log (g-go) for different assumed values of go other than 0, till a straight line is obtained for

a certain value of go.

b. YA Method (Steven's method)

go

Gau

ge

hei

ght (g

)

Discharge, Q

Page 13: Final Handout (Intro to Hydrolgy

Adama university, Department of civil Engineering and Architecture

Introduction to hydrology (CE-3601) Tessema B; Mekdim M;Belay,Suleman 13

This method is applicable for wider and shallower streams. For such streams the wetted perimeter,

P=b+2Y ≅ b for Y<<< b (Y flow depth, b channel bottom width). Therefore, the hydraulic mean

radius R (i.e. R = A/P) is equal to A/b = (bY)/b = Y = R.

SRCAQ = (By Chezy Formula)

SRCA=

Qα YA if SC= is constant.

Figure 18: Steven's method

c. General Method

Applicable to all types of river sections, but it is necessary to know the stream cross-section at the

gauging site. The method requires plotting gauge height Vs area curve. From this curve the velocity

curves can be drawn using the relation; Mean velocity, V = Discharge/ Area on the same drawing.

Figure 19: Height: discharge, velocity area relation ship

The mean velocity is given by Manning's Formula as V= SRn

2

1

3

21 . For higher river stages, S

n2

11is

almost constant. Vα R2/3

Q Vs YA

Q

AY

0.5

ht Vs A: Concave down ward

Height Vs Q

ht Vs V: Concave up ward

Gau

ge

Hei

ght, h

t

Discharge, Velocity, Area

Page 14: Final Handout (Intro to Hydrolgy

Adama university, Department of civil Engineering and Architecture

Introduction to hydrology (CE-3601) Tessema B; Mekdim M;Belay,Suleman 14

CHAPTER THREE - PROCESSING AND ANALYSIS OF HYDROLOGICAL DATA

3.1 General

Hydrological studies require extensive analysis of meteorological, hydrological and spatial data to represent

the actual processes taking place on the environment and better estimation of quantities out of it. Precipitation

is the source of all waters which enters the land. Hydrologists need to understand how the amount, rate,

duration, and quality of precipitation are distributed in space and time in order to assess, predict, and forecast

hydrologic responses of a catchment.

Estimates of regional precipitation are critical inputs to water-balance and other types of models used in water-

resource management. Sound interpretation of the prediction of such models requires an assessment of the

uncertainty associated with their output, which in turn depends in large measure on the uncertainty of the input

values.

The uncertainties associated with a value of regional precipitation consist of:

1. Errors due to point measurement

2. Errors due to uncertainty in converting point measurement data into estimates of regional precipitation

It is therefore, necessary to first check the data for its quality, continuity and consistency before it is

considered as input. The continuity of a record may be broken with missing data due to many reasons such as

damage or fault in recording gauges during a period. The missing data can be estimated by using the data of

the neighboring stations correlating the physical, meteorological and hydrological parameters of the catchment

and gauging stations. To estimate and correlate a data for a station demands a long time series record of the

neighboring stations with reliable quality, continuity and consistency.

3.2 Meteorological data

3.2.1 Principles of Data Analysis

a) Corrections to Point Measurements

Because precipitation is the input to the land phase of the hydrologic cycle, its accurate measurement is the

essential foundation for quantitative hydrologic analysis. There are many reasons for concern about the

accuracy of precipitation data, and these reasons must be understood and accounted for in both scientific and

applied hydrological analyses.

Rain gages that project above the ground surface causes wind eddies affecting the catch of the smaller

raindrops and snowflakes. These effects are the most common causes of point precipitation-measurement.

Studies from World Meteorological Organization (WMO) indicate that deficiencies of 10% for rain and well

over 50% for snow are common in unshielded gages. The daily measured values need to be updated by

applying a correction factor K after corrections for evaporation, wetting losses, and other factors have been

applied. The following equations are recommended for U.S. standard 8-Inch gauges with and without Alter

wind shields.

Correction factor for unshielded rain gauges:

Kru = 100 exp (-4.605 + 0.062 Va

0.58

) (1.1) Correction factor for Alter wind shielded rain gauges:

Kru = 100 exp (-4.605 + 0.041 Va

0.69

) (1.2) Where: Va = Wind speed at the gage orifice in m/s (Yang et al. 1998)

Page 15: Final Handout (Intro to Hydrolgy

Adama university, Department of civil Engineering and Architecture

Introduction to hydrology (CE-3601) Tessema B; Mekdim M;Belay,Suleman 15

Errors due to splashing and evaporation usually are small and can be neglected. However, evaporation losses

can be significant in low-intensity precipitations where a considerable amount could be lost. Correction for

wetting losses can be made by adding a certain amount (in the order of 0.03 – 0.10 mm) depending on the type

precipitation.

Systematic errors often associated with recording type rain gauges due to the mechanics of operation of the

instrument can be minimized by installing a non recording type gauge adjacent to each recording gauge to

assure that at least the total precipitation is measured. Instrument errors are typically estimated as 1

– 5% of the total catch (Winter (1981)).

Although difficult to quantify and often undetected, errors in measurement and in the recording and publishing

(personal errors) of precipitation observations are common. To correct the error some subjectivity is involved

by comparing the record with stream flow records of the region.

b) Estimation of Missing Data

When undertaking an analysis of precipitation data from gauges where daily observations are made, it is often

to find days when no observations are recorded at one or more gauges. These missing days may be isolated

occurrences or extended over long periods. In order to compute precipitation totals and averages, one must

estimate the missing values.

Several approaches are used to estimate the missing values. Station Average, Normal Ratio, Inverse Distance

Weighting, and Regression methods are commonly used to fill the missing records. In Station Average

Method, the missing record is computed as the simple average of the values at the nearby gauges. Mc Cuen

(1998) recommends using this method only when the annual precipitation value at each of the neighboring

gauges differs by less than 10% from that for the gauge with missing data.

[ ]....1

321 +++= PPPM

px …… ………… ……… (1.3)

Where:

Px = The missing precipitation record

P1, P2 , …, Pm = Precipitation records at the neighboring stations

M = Number of neighboring stations.

If the annual precipitations vary considerably by more than 10 %, the missing record is estimated by the

Normal Ratio Method, by weighing the precipitation at the neighboring stations by the ratios of normal annual

precipitations.

+++=

m

mx

xN

P

N

P

N

P

N

P

M

Np ....

3

3

2

2

1

1 …… ………… ……… (1.4)

Where:

Nx = Annual-average precipitation at the gage with missing values

N1 , N2 , …, Nm = Annual average precipitation at neighboring gauges

The Inverse Distance Method weights the annual average values only by their

distances, dm, from the gauge with the missing data and so does not require

information about average annual precipitation at the gauges.

∑=

−=m

m

mbdD

1

…… ………… ……… (1.5)

The missing value is estimated as:

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m

m

m

mb

x NdP ∑=

−=1

…… ………… ……… (1.6)

The value of b can be 1 if the weights are inversely proportional to distance or 2, if the weights are

proportional to distance squared.

If relatively few values are missing at the gauge of interest, it is possible to estimate the missing value by

regression method.

c) Checking the Consistency of Point Measurements

If the conditions relevant to the recording of rain gauge station have undergone a significant change during the

period of record, inconsistency would arise in the rainfall data of that station. This inconsistency would be felt

from the time the significant change took place. Some of the common causes for inconsistency of record are:

1. Shifting of a rain gauge station to a new location

2. The neighborhood of the station may have undergoing a marked change Obstruction, etc. 4. Occurrence of observational error from a certain date both personal and instrumental

The most common method of checking for inconsistency of a record is the Double-Mass Curve analysis

(DMC). The curve is a plot on arithmetic graph paper, of cumulative precipitation collected at a gauge where

measurement conditions may have changed significantly against the average of the cumulative precipitation

for the same period of record collected at several gauges in the same region. The data is arranged in the reverse

order, i.e., the latest record as the first entry and the oldest record as the last entry in the list. A change in

proportionality between the measurements at the suspect station and those in the region is reflected in a change

in the slope of the trend of the plotted points. If a Double Mass Curve reveals a change in slope that is

significant and is due to changed measurement conditions at a particular station, the values of the earlier period

of the record should be adjusted to be consistent with latter period records before computation of areal

averages. The adjustment is done by applying a correction factor K, on the records before the slope change

given by the following relationship.

changebeforeperiodforSlope

changeafterperiodforSlopeK =

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a) Double Mass Curves for Bahir b Double Mass Curves for Adet

c) Double Mass Curves for Dangila Figure 1.1: Double Mass Analysis

The updated records are computed using equation as given below:

Pcx = PxK Where the factor K is computed by equation (g) Table 1: Slopes of the DMC and correction factor K

Precipitation records at Bahir Dar and Adet meteorological station beyond November 1998 should be updated

by applying the correction factors 1.25 and 0.75 respectively.

Average slopes

Stations Slope for period after slope

change

Slope for period

before slope change

K

Bahir Dar 1.114 0.892 1.249

Adet 0.752 1.008 0.746

Dangila 1.1986 1.1986 1.000

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3.3 Areal Estimation

Rain gauges represent only point measurements. in practice however, hydrological analysis requires

knowledge of the precipitation over an area. Several approaches have been devised for estimating areal

precipitation from point measurements. The Arithmetic mean, the Thiessen polygon and the Isohyetal method

are some the approaches.

The arithmetic mean method uses the mean of precipitation record from all gauges in a catchment. The method

is simple and give good results if the precipitation measured at the various stations in a catchment show little

variation.

In the Thiessen polygon method, the rainfall recorded at each station is given a weightage on the basis of an

area closest to the station. The average rainfall over the catchment is computed by considering the

precipitation from each gauge multiplied by the percentage of enclosed area by the Thiessen polygon. The

total average areal rainfall is the summation averages from all the stations. The Thiessen polygon method

gives more accurate estimation than the simple arithmetic mean estimation as the method introduces a

weighting factor on rational basis. Furthermore, rain gauge stations outside the catchment area can be

considered effectively by this method.

The Isohyetal method is the most accurate method of estimating areal rainfall. The method requires the

preparation of the isohyetal map of the catchment from a network of gauging stations. Areas between the

isohyets and the catchment boundary are measured. The areal rainfall is calculated from the product of the

inter-isohyetal areas and the corresponding mean rainfall between the isohyets divided by the total catchment

area.

Example 3.3: compute the mean annual precipitation for the river basin shown in fig 3.16 below by

using Arithmetic method; thiessen polygon method and Isohyets method. The location of the

various rain gauge stations and the point precipitation values are indicated in the table below.

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3.4 Hydrological Data

The availability of stream flow data is important for the model calibration process in catchment modelling.

Measured hydrograph reflects all the complexity of flow processes occurring in the catchment. It is usually

difficult to infer the nature of those processes directly from the measured hydrograph, with the exception of

some general characteristics such as mean times of response in particular events. Moreover, discharge data are

generally available at only a small number of sites in any region where different characteristics of the

catchment are lumped together.

3.4.1 Missing Data and Comparison with the Precipitation Records

The data so far collected do not indicate any missing data. The potential errors in the discharge records would

affect the ability of the model to represent the actual condition of the catchment and calibrating the model

parameters. If a model is calibrated using data that are in error, then the model parameter values will be

affected and the prediction for other periods, which depend on the calibrated parameter values, will be

affected.

Prior to using any data to a model it should be checked for consistency. In data where there is no information

about missing values check for any signs that infilling of missing data has taken place is important. A common

indication of such obvious signs is apparently constant value for several periods suggesting the data has been

filled. Hydrographs with long flat tops also often as sign of that there has been a problem with the

measurement. Outlier data could also indicate the problem.

Even though there is a danger of rejecting periods of data on the basis on these simple checks, at least some

periods of data with apparently unusual behavior need to be carefully checked or eliminated from the analysis.

The available stream flow data for this analysis generally has corresponding match with the precipitation

records in the area. The high flows correspond to the rainy seasons. In some of the years there are remarkably

high flow records, for instance in the month of august 2000 and 2001 the flow records are as high as 100 and

89 m3/s compared to normal rainy season records which is between 30 and 65 m3/s. These data might be real

or erroneous. On the other hand the values match to the days of the peak rainfall records in the area in both the

cases.

Figure 1.2: Koga stream flow record compared with the precipitation record.

However, the stream flow records of 1995 are exceptionally higher and different from flow magnitudes that

had been records for long period of time at Koga River. It is not only the magnitude which is different from the

normal flow record, but also it contradicts with the magnitude of the precipitation recorded during the year.

These records might be modeled or transferred flows. Hence, the flow records of this year are excluded from

being the part of the analysis.

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CHAPTER FOUR - INTENSITY DURATION FREQUENCY PROCEDURES

Because rainfall varies from year to year in total amounts and in its timing and pattern, each year the

total amount of rain falling at a point is the usual basic precipitation figure available for many

purposes. We require data on rainfall going back sometimes over a hundred years or more for

statistical analysis. Therefore, to make use of such data for statistical analysis, the data should

represented by three important variables such as 1) intensity 2) duration 3) Frequency.

Intensity: - Is the measure of the quantity of rain falling in a given time. It is expressed in cm/hr,

mm/hr, etc.

Duration: - is the period of time during which a particular rain is falling.

Frequency: - refers to the return period of a particular rainfall characterized by a given duration,

intensity or both, falls.

4.1 Intensity-Duration relationship of a Rainfall

An idealized curve showing the intensity variation with time is known as Time-Intensity pattern. A

hyetograph is a plot of the intensity of rainfall against the time interval. The hyetograph is derived

from the mass curve and is usually presented as a bar chart. It is very convenient way of representing

the characteristics of a storm and is particularly important in the development of design storms to

predict extreme floods. The area under a hyetograph represents the total precipitation received in the

period. The time interval used depends on the purpose; for example, in urban drainage problems

small durations are used while in flood flow computations in large catchments the intervals are of

about 6 hours.

Figure 5: Time –Intensity pattern Figure 6: Rain hyetograph

If the total accumulated precipitation is plotted against time, the curve obtained is known as the mass

curve of the storm. Thus the mass curve of rainfall is the plot of the accumulated precipitation against

time plotted in chronological order. Records of float type and weighing bucket type are of this form.

A typical mass curve of rainfall at a station during a storm is shown in the following figure 11. Mass

curves of rainfall are very useful in extracting the information on the duration and magnitude of a

storm. Also intensities at various time intervals in a storm can be obtained from the slope of the

curve. For non-recording rain gauges, mass curves are prepared from the knowledge of the

approximate beginning and end of a storm and by using the mass curves of adjacent recording gauge

station as a guide.

Rai

nfa

ll inte

nsi

ty in c

m/h

r

Time in hours ►

Rai

nfa

ll inte

nsi

ty in c

m/h

r

Time in hours ►

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Introduction to hydrology (CE-3601) Tessema B; Mekdim M;Belay,Suleman 23

Figure 7: Mass curve of rainfall

Point Rainfall: - Point rainfall, also known as station rainfall refers to the rainfall data of a station.

Depending upon the need, data can be listed as daily, weekly, monthly, seasonal or annual values for

various periods. Graphically, these data are represented as plots of magnitude Vs chronological time

in the form of a bar diagram. Such a plot, however, is not convenient for discerning a trend in the

rainfall, as there will be considerable variation in the rainfall. Values of precipitation of three or five

consecutive time intervals plotted at the mid-value of the time interval is useful in smoothing out the

variations and beginning out the trend.

4.2 Depth - Area - Duration (DAD) Relationship

Since precipitation rarely occurs uniformly over an area, variations in intensity and total depth of fall

occur from the center of the peripheries of storm. The areal characteristic of a storm of a given

duration is reflected in its depth - area relationship. Such a relation is represented by a curve shown

below. Different such curves will be obtained for different storms of different duration, as shown in

Fig 13 from which maximum observed DAD curve of a particular duration can be obtained. Different

such curves will be obtained for different storm of a given duration.

Figure 9: Depth - Area – Duration curve

A depth - area duration curve expresses graphically the relation between progressively decreasing

average depth of rainfall over a progressively increasing area from the center of the storm out ward to

its edges for a given duration of rainfall. An immediate purpose of DAD analysis of a particular

storm is to determine the largest average depth of rainfall that fell over various sizes of area during

the standard passage of time in hours or days, such as the largest average depth over 500 sp. km in 1

day.

Effect of first

storm

Effect of

2nd storm

Acc

um

ula

ted p

reci

pitat

ion, cm

Time in days

Rai

nfa

ll d

epth

in

Area in km2

DAD curve for 24 hr (day)

for the given storm

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Equation of Depth - Area –duration Curve: - in the depth area curve for a rainfall of a given

duration, the average depth decreases with the area in a exponential fashion as given by the equation:

P = P0 (e)- K An

where P = average depth in cm over an area A km2,

P0 = highest amount of rainfall in cm at the storm center,

K and n = constants for the given region.

On the basis of number of severest storms in north Gondor, Debarik and Sanja (1975) have obtained

the following values for K and n for storms of different durations:

Duration K N

1 day 0.0008526 0.6614

2 days 0.0009877 0.6306

3 days 0.001745 0.5961

Since it is very unlikely that the storm center coincides over a rain gauge station, the exact

determination of P0 is not possible. Hence in the analysis of the large area storms the highest station

rainfall is taken as the average depth over an area of 25 km2. The above equation is useful in

extrapolating an existing storm data over an area.

Maximum Depth - Area - Duration Curves: - In many hydrological studies involving estimation of

severe floods, it is necessary to have information on the maximum amount of rainfall of various

durations occurring over various sizes of areas. The development of relationship between maximum

depth - area - duration for a region is known as DAD analysis and forms an important aspect of hydro

- meteorological study.

Figure10: Maximized Depth - Area – Duration curve

The above figure shows typical DAD curves for a catchment. In this the average depth denotes the

depth averaged over the area under consideration. It may be seen that the maximum depth for the

given storm decreases with the area; for a given area the maximum depth increases with the duration.

A brief description of the analysis is given below:

(i) First, the most severe rainstorms that have occurred in the region under question are

considered.

(ii) Isohyetal maps and mass curves of the storms are prepared.

24 hrs duration

12 hrs duration

6 hrs duration

Catchment area

Rai

nfa

ll d

epth

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(iii) A depth - area curve of a given duration of the storm is prepared.

(iv) From a study of the mass curve of rainfall, various durations and maximum depth of rainfall

in these durations are noted.

(v) The maximum depth - area curve for a given duration D is prepared by assuming the area

distribution of rainfall for smaller duration to the similar to the total storm.

(vi) The procedure is then repeated for different storms and the envelope curve of maximum depth

- area for duration D is obtained.

(vii) A similar procedure for various values of D results in a family of envelope curves of

maximum depth vs. area, with duration as the third parameter. These curves are called DAD

curves.

Example: The following are the rain gauge observation during a storm. Construct:

(a) Mass curve of precipitation

(b) Hyetograph

(c) Maximum intensity- duration curve and develop a formula and

(d) Maximum depth- duration curve.

Time since commencement

of storm (min.)

Accumulated rainfall (cm)

5 0.1

10 0.2

15 0.8

20 1.5

25 1.8

30 2.0

35 2.5

40 2.7

45 2.9

50 3.1

Solution:

(a) Mass curve of precipitation: The plot of accumulated rainfall (cm) Vs time (min.) gives the

mass curve of rainfall.

Mass curve of percipitation

0

0.5

1

1.5

2

2.5

3

3.5

0 5 10 15 20 25 30 35 40 45 50 55

Time (min)

(b) Hyetograph: The intensity of rainfall at successive 5 min. interval is calculated and bar graph

of i (cm/h) Vs t (min.) is constructed; this depicts the variation of the intensity of rainfall with respect

to time and is called the "hyetograph".

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Time, t (min) Accumulated rainfall

(cm) ∆P in time ∆t = 5 min. Intensity,

i = ∆P/∆t x 60 (cm/h)

5 0.1 0.1 1.2

10 0.2 0.1 1.2

15 0.8 0.6 7.2

20 1.5 0.7 8.4

25 1.8 0.3 3.6

30 2.0 0.2 2.4

35 2.5 0.5 6.0

40 2.7 0.2 2.4

45 2.9 0.2 2.4

50 3.1 0.2 2.4

Hyetograph

1.2 1.2

7.2

8.4

3.6

2.4

6

2.4 2.4 2.4

0

1

2

3

4

5

6

7

8

9

0 5 10 15 20 25 30 35 40 45 50 55

Time (min)

Intenisty(cm/hr)

(c) Maximum depth - duration curve: By inspection of time (t) and accumulated rainfall (cm)

the maximum rainfall depths during 5, 10, 15, --------------, 50 min. duration are 0.7, 1.3, 1.6, 1.8, 2.3,

2.5, 2.7, 2.9, 3.0 and 3.1 cm respectively. The plot of the maximum rainfall depths against different

duration on a log-log paper gives the maximum depth - duration curve, which is a straight line.

Time

(min.)

Acc

rainfall

(cm)

change

in ppt

Intensity

(cm/hr) Maximum Intensity of each duration (cm/hr)

Maximum depth(cm)

Maximum intensity (cm/hr)

5min 10min 15min 20min 25min 30min 35min 40min 45min 50min

5 0.1 0.1 1.2 0.1 0.7 8.40

10 0.2 0.1 1.2 0.1 0.2 1.3 7.80

15 0.8 0.6 7.2 0.6 0.7 0.8 1.6 6.40

20 1.5 0.7 8.4 0.7 1.3 1.4 1.5 1.8 5.40

25 1.8 0.3 3.6 0.3 1 1.6 1.7 1.8 2.3 5.52

30 2 0.2 2.4 0.2 0.5 1.2 1.8

1.9 2 2.5 5.00

35 2.5 0.5 6 0.5 0.7 1 1.7

2.3 2.4 2.5 2.7 4.63

40 2.7 0.2 2.4 0.2 0.7 0.9 1.2

1.9 2.5 2.6 2.7 2.9 4.35

45 2.9 0.2 2.4 0.2 0.4 0.9 1.1

1.4 2.1 2.7 2.8 2.9 3.0 4.00

50 3.1 0.2 2.4 0.2 0.4 0.6 1.1

1.3 1.6 2.3 2.9 3.0 3.1 3.1 3.72

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Maximum Depth duration curve

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

1 10 100

Time(min)

Max depth(cm)

(d) Maximum - intensity - duration curve: Corresponding to the maximum depths obtained

in(c) above the corresponding maximum intensities can be obtained by ∆P/∆t x 60 i.e. 8.4, 7.8, 6.4,

5.4, 5.52, 5.0, 4.63, 4.35, 4.0 and 3.72 cm/h respectively. The plot of the maximum intensities against

the different durations on a log-log paper gives the maximum intensity- duration curve, which is a

straight line.

The equation for the maximum intensity - duration curve is of the form:

i = k tx

Slope of the straight-line plot,

375.000.2

75.0===

dd

x

yx

k = 17 cm/h when t = 1 min. Hence, the formula becomes,

ti

375.0

17=

Which can now be verified as t = 10 min., i = 7.2 cm/h; t = 40 min., i = 4.25 cm/h which agree with

the observed data.

Maximum Intensity duration curve

y = 16.49x-0.3616

R2 = 0.9528

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

0 5 10 15 20 25 30 35 40 45 50 55

Time(min)

Maxim

um intensity (cm/hr)

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4.3 Frequency analysis of rainfall (Recurrence interval of a storm)

By recurrence interval (R.I) of a given storm is meant the time interval during which the given storm

is likely to be equaled or exceeded. This if we say that at a given station, the maximum precipitation

of 220 cm has got a recurrence interval of 5 years, we mean that on this station, the chances of

rainfall are such that once in five years, rain is likely to be equal or exceeded 220 cm. This does not

mean that a rain equal to 220 cm or so will occur after every five years. it may occur twice in one set

of five years and may not occur at all for consecutive eight years or so; what it means is that during

100 years or so a rain will occur 20 times or so frequency analysis has the following implications.

� Frequency analysis is an aid in determining the design rainfall

� Because high rainfall is comparatively infrequent, the selection of the design rainfall can be

based on the low frequency with which these high values are permitted to be exceeded.

� The design frequency is the risk that the designer is willing to accept.

� The smaller the risk, the more costly are the drainage works and structures, and the less often

their full capacity will be reached.

� Accordingly, the design frequency should be realistic – neither too high nor too low.

Methods of calculating Recurrence Interval

i.) California method (statistical or probability method)

Y = ab

where Y = the total number of years of record;

a = the recurrence interval or frequency or period;

b = the rank of the storm (the number of times the given storm is equaled or

exceeded).

This method of frequency analysis involves ranking of existing data (records). The data can be

ranked in either ascending or descending order. For ranking in descending order, the suggested

procedures are as follows:

1. Rank the total number of data (n) in descending order according to their value (x), the highest

value first and the lowest value last.

2. Assign a serial number (r) to each value X (Xr, r=1, 2, 3 ----- n), the highest value being X1

and the lowest Xn

3. Divide the rank (r) by the total number of observation plus 1 to obtain the frequency of

exceedence.

P (X > Xr) = 1n

r

+

4. Calculate the frequency of non-exceedance (also called cumulative frequency)

P (X < Xr) = 1 – P (X>Xr) =1 – 1n

1

+

� If the ranking is ascending instead of descending, we can obtain similar relation by

interchanging P (X >Xr) and P (X ≥ Xr)

� An advantage of using the denominator n+1 instead of n is that the results for ascending or

descending ranking orders will be identical.

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Example: The maximum values of 24 hrs summer precipitations at a rain gauge station expressed in

cm from 1940-1969 are indicated below.

10.7, 9.7, 17.1, 11.2, 11.3, 17.7, 10.8, 11.6, 16.9, 9.6, 11.9, 15.4, 14.9, 12.4, 13.6, 15.2, 12.7, 12.9,

12.0, 6.4, 16.3, 12.2, 10.5, 15.7, 15.8, 9.2, 16.8, 16.8, 18.7, 17.2.

Estimate the maximum precipitation having a return period RT of

a) 5 years b) 10 years c) 20 years d) 40 years

Precipitation (cm) Ranking of the

storm (b)

Return period

Tr in years (a)

18.7

17.7

17.2

17.1

16.9

16.8

16.6

16.3

15.8

15.7

15.4

15.2

14.9

13.8

12.9

12.7

12.4

12.2

12.0

11.9

11.6

11.3

11.2

10.6

10.7

10.5

09.7

09.6

09.2

06.4

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

30

15

10

7.5

6

5

4.3

3.8

a) Tr = a = 5, 30/5 =6

i.e. the 6th record in the

table i.e. precipitation 16.8

cm

b) 10 years

n = 30

Tr = 10 years

Ppt. = 17.2mm

c) 30 years

n = 30

Tr = 30 years

Ppt. = 18.7mm

d) 40 years

n = 30

Tr = 40 years

b = n/ Tr =30/40 = 0.75 < 1

which is impossible to

determine.

Limitations of this method: - If we want to determine a storm having a return period of more than the

total number of years of which the records are available, it is not possible to find out by using the

simple probability method, i.e., n = Tr m. (In that case Tr is greater than n and therefore, m is less

than one which is not known).

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In order to make use of the above method to determine the storm of such a high return period the

available records are sometimes plotted graphically and the curve is extended back, so as to

determine the storm of the desired return period.

Figure 11: Probability analysis using a log-scale

ii. Recurrence Predictions and return periods

� The return period also called recurrence interval is the average period of time within which a

rainfall of given depth and duration will be equaled or exceeded once on an average.

� Recurrence estimates are often made in terms of return periods (T), T being the number of new

data that have to be collected, on the average, to find a certain rainfall value.

� The return period is calculated as

1T

P=

Where: T is Return Period in years

F is the frequency of exceedance

)XrX(F

1T

>= =

r

1nT

+=

Where T =return period, years

n = total number of hydrological events

r = rank of events arranged in descending order of magnitude.

� Design of any soil conservation structure requires the frequency analysis of rainfall data.

� For reliable safety, it is required to deign a structure for the maximum rainfall that has ever

occurred.

� However, it is not economical to have such a design.

� Small structures like spillway, culverts etc. may last for a period of only 10 years during which the

rainfall of highest intensity may not at all occur.

� Also temporary structures for soil conservation like check dams; vegetated waterways, bund,

terraces etc. are designed on the basis of 10 years return period.

� More permanent structures like drop spillways; chute spillways etc. are designed to handle a storm

of 25 to 50 years return period.

� Therefore, it is advisable to take the risk of periodic failure than to deign for storm of the highest

intensity that may ever occur.

� Only if there is risk of human life, the design should be for the maximum possible rainfall

intensity.

Return period, log scale

Log-s

cale

, dep

th o

f

rain

fall

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Probability of occurrence: the probability of occurrence of an event (e.g. rainfall) whose magnitude

is equal to or in excess of a specified magnitude X is denoted by P. Hence the recurrence interval or

return period (T) is defined as P

1T =

If the probability of an event occurring is P, the probability of the event not occurring in a given year

is q = (1-P)

The binomial distribution can be used to find the probability of occurrence of an event r times in n

successive years, thus

qP!r)!rn(

!nqPCP

rnrrnrrn,r

n −−

−==

Where

Pr, n = Probability of a random hydrologic event (rainfall) of a given magnitude and

exceedence probability P recurring r times in n successive years. Thus for

example

a. Probability of an event of exceedence P occurring 2 times in n successive years is

P2, n = ( ) qp

!2!2n

!n 2n2 −

b. Probability of the event not occurring at all in n successive years is

P0,n = qn = (1-P)

n

c. The probability of the event occurring at least once in n successive years (R);

R = 1-qn = 1 – (1-P)

n This probability is called risk and hence represented by R.

4.4 Intensity - Duration - Frequency Relationship

Rainfall of a place can be completely defined if the intensity, duration and frequency of the various

storms occurring on that place are known. Whenever rain occurs, its magnitude and duration is

generally known from meteorological readings. Thus, at a given station the intensities of the rains of

various durations such as 5 minutes, 10 minutes, 15 minutes, …, etc are generally known.

The intensity of storm decreases with the increase in storm duration. Further, a storm of any given

duration will have a larger intensity if its return period is large. In other words, for a storm of given

duration, storm of higher intensity in that duration are rare than storms of smaller intensity. In many

design problems related to water resources development, it is necessary to know the rainfall

intensities of different durations and different return periods. The inter-dependency between the

intensity (i cm/h), duration (D-h in hours) and return period (T years) is commonly expressed in a

general form as:

KTx

i = -------

(D + a)n

where K, x, a and n are constants for a given catchment. The variation of intensity i with duration D

and return period T is shown schematically in the following figures.

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Figure 12: Intensity - Duration - Frequency Relationships

Example: The various data were obtained for rains of various durations at a station for 30 years. The

records were analyzed and 11 worst storms of various durations have been stipulated in their

decreasing order as shown in the following table. Plot the intensity-duration curves for storms of

frequencies (a) 11, (b) 1.4, and (c) 1.0

Sr.

No.

5

mts

10 mts 15 mts 30 mts 60 mts 90 mts 120

mts

m

ranking

of storm

T =

frequency

= N/m =

11/(9)

1 2 3 4 5 6 7 8 9 10

1 0.85 1.20 1.40 1.74 2.15 2.46 2.97 1 11

2 0.76 1.04 1.18 1.55 1.92 2.38 2.63 2 5.5

3 0.73 0.93 1.11 1.36 1.70 2.14 2.34 3 3.7

4 0.72 0.88 1.03 1.22 1.45 1.81 2.12 4 2.8

5 0.66 0.84 0.97 1.18 1.40 1.65 1.83 5 2.2

6 0.62 0.80 0.92 1.10 1.33 1.50 1.64 6 1.8

7 0.51 0.78 0.90 1.05 1.25 1.40 1.55 7 1.6

8 0.45 0.68 0.82 1.01 1.20 1.36 1.51 8 1.4

9 0.36 0.52 0.67 0.95 1.14 1.34 1.46 9 1.2

10 0.28 0.51 0.62 0.83 1.11 1.27 1.41 10 1.1

11 0.21 0.39 0.50 0.79 1.09 1.23 1.34 11 1.0

Solution:

(a) Frequency of 11 years Duration (minutes) Precipitation (cm) Average Intensity (cm/h)

5 0.85 10.20 (0.85 x 60/5)

10 1.20 7.20

15 1.40 5.60

30 1.74 3.88

60 2.15 2.15

90 2.46 1.64

120 2.97 1.43

Duration in minutes

20 years return period

10 years return period

5 years return period

Inte

nsi

ty, cm

/hr

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(b)Frequency of 1.4 years

Duration (minutes) Precipitation (cm) Average Intensity (cm/h)

5 0.45 5.40

10 0.68 4.01

15 0.82 3.28

30 1.01 2.02

60 1.20 1.20

90 1.36 0.90

120 1.51 0.75

(c)Frequency of 1.0 year

Duration (minutes) Precipitation (cm) Average Intensity (cm/h)

5 0.21 2.52

10 0.39 2.34

15 0.50 2.00

30 0.79 1.58

60 1.09 1.09

90 1.23 0.82

120 1.34 0.67

Probable Maximum Precipitation (PMP): - In the design of major hydraulic structures such as

spillways in large dams, the hydrologists and hydraulic engineers would like to keep the failure

probability as low as possible i.e. virtually zero. This is because the failure of such a major structure

will cause very sever damages to life, property, economy and national moral. In the design and

analysis of such structures, the maximum possible precipitation that can be expected at a given

location is used. This stems from the recognition that there is a physical upper limit to the amount of

precipitation that can fall over a specific area in a given time.

The probable maximum precipitation (PMP) is defined as the greatest or extreme rainfall for a given

duration that is physically possible over a station or basin. From the operational point of view, PMP

can be defined as that rainfall over a basin that would produce a flood flow with virtually no risk of

being exceeded.

In another way the probable maximum precipitation (PMP) may also be defined as the maximum

depth of precipitation for a given duration that may possibly occur on a given catchment at any time

of the year within the economic life the structure. Such a precipitation would result from the possible

severest storm, that may result from the worst possible storm called Probable Maximum Storm

(PMS) and will be used in the design of (large) hydraulic structures.

The development of PMP for a given region is an important procedure and requires the knowledge of

an experienced hydro-meteorologist. Basically two approaches are used: (i) meteorological methods

and (ii) the statistical study of rainfall data.

Statistical studies indicate that PMP can be estimated as:

PMP = P + Kσ

where, P = mean of annual maximum rainfall series

K = a frequency factor which depends upon the statistical distribution of the series, number

of years of records and the return period. The value of K is usually in the neighbor hood of 15 and

σ = Standard deviation of the series

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CHAPTER FIVE - RAINFALL RUNOFF MODELS

5.1 Introduction

RUNOFF:

Definition: runoff is defined as the portion of precipitation that makes its way towards rivers, or seas,

or oceans, etc as surface and sub surface flow.

The runoff generally consists of

a) Surface runoff

b) Ground runoff

c) Direct precipitation over the streams

Runoff Process

The rain falling in the catchment/ basin is disposed off in the following ways.

i) Absorbed by plant leaves /intercepted;

ii) Evaporated and passed back to atmosphere;

iii) Runoff either surface or penetrating into the soil and changing the ground water level and then

appearing in the streams lower down in the same catchment. Surface runoff is important for the

maximum flow of the river, while ground runoff is important for the minimum (base) flow of the

river;

iv) Absorbed by the ground /or collected on the ground - detention/ making it damp or wet and

ultimately evaporated again into the atmosphere;

The difference between rainfall and runoff is termed as rainfall loss and rainfall minus losses is

rainfall producing runoff, i.e.

The total precipitation [P] on the basin can be represented by the equation

P = L +G + Q or

Q = P - L- Q

where

Q is runoff L - basin yield (discharge)

P = total precipitation G = ground water accretion

5.2 Factors Affecting Runoff

i) Characteristics of precipitation

ii) Characteristics of drainage basin

i) Characteristics of Precipitation

a) Type of precipitation

If the precipitation occurs in the form of rain, it will immediately produce a runoff (peak

flow of short duration), while if the precipitation is in the form of snow, it will produce

runoff at a slow and steady rate.

b) Rain Intensity

If the intensity of rainfall increases runoff increases rapidly.

c) Duration of rainfall

If the infiltration is less, the surface runoff will be more thus in some cases, a longer

duration rain may produce considerable runoff even when its intensity is mild.

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d) Rainfall Distribution

For small drainage basins peak flows are generally the result of intense rain falling over

small areas. On the other hand, for larger drainage basins, the peak flows are the result of

storm of lesser intensity but covering larger area.

e) Soil Moisture deficiency

The runoff depends upon the soil moisture present at the time of rainfall. If a rain occurs

after a long dry spell of time, the soil is dry and it can absorb huge amounts of water, and

thus, even intense rain may fail to produce an appreciable runoff. But on the other hand, if

the rain occurs after a long rainy season, the soil will be already wet and there may be very

less infiltration, and even smaller rains may cause peak flow and considerable stream rises,

some times disastrous floods.

f) Direction of the prevailing storm

If the direction of the storm is the same as the direction of the movement of the water in the

drainage basin, water will remain in the basin for lesser periods; hence more runoff will be

produced as compared to when the storm is moving in the direction opposite to the water

movement.

g) Other climatic Factors

Various other climatic factors such as wind, temperature, humidity, etc. affect the losses

from the drainage basin and therefore, affect the runoff. If the losses are more, runoff will be

less and vice versa.

ii) Characteristics of drainage basin

a) Shape and size of the basin

If the area of the basin is large, the total flood flow will take more time to pass the outlet;

thereby, the base of the hydrograph of the flood flow will widen out and consequently

reducing the peak flow. The shape of the drainage governs the rate at which water enters the

stream. The shape of the drainage is generally expressed by form factor and compactness

coefficient, as defined below:

L

B==

basin theoflength Axial

basin theof width Average Factor Form

The axial length (L) is the distance from the outlet to the most remote point on the basin; the

average width (B) is obtained by dividing the area (A) by the axial length.

Form factor = B/L = (A/L)/L = A/L2.

Compactness coefficient is defined as:

sin

basin theofPerimeter tcoefficien sCompactnes

batheofarea

thetoequalisareawhosecircleaofnceCircumfere=

If A is the area of the basin and re is the radius of the equivalent circle: erA2π=

Circumference = erπ2 = AA πππ 2/2 =

Compactness coefficient = )2/( AP π , P = perimeter of the basin

There are two types of catchment in general.

i) Fan shaped catchment

ii) Fern leaf shaped catchment

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Figure 20: Types of catchment

Fan shaped catchment gives quicker runoff because tributaries are nearly of the same size and

therefore time of flow is the same and is smaller, whereas in fern leaf shaped catchment, the time of

concentration is more since the discharge is distributed over a long period.

b) Elevation of the watershed

The elevation of the drainage basin governs the rainfall, its amount and its type and hence

produces enough effects on the runoff. The elevation of a watershed is a variable factor from

point to point. In order to determine the average elevation (z) of the drainage basin, a contour

map of the basin is taken, and the area lying between successive contours are measured (a).

Then z is calculated as

A

az

A

zazazaz

nn ∑=

+++=

...2211

A = area of the basin

zi = values of mean elevation between 2 successive contours

ai = areas between 2 successive contours

c) Other factors

The arrangement of the stream channels formed by nature within the basin, the type of the

soil, the type of the vegetation cover, etc are the various factors that influence the runoff.

Certain important definitions

Time of concentration (Tc) - of a drainage basin is the time required by the water to reach the outlet

from the most remote part of the drainage basin called the critical point.

Time of overland flow (TOF) TL - The excess rainfall finds its way over land to the river stream and

appears as surface runoff, but only after some delay. In other words, there exists a lag between the

time excess rainfall occurs and the time when it appears as runoff at the outlet of drainage basin.

This lag is the time for which water has flown through the basin and is known as TOF. This delay

varies throughout the basin and throughout the storm period. Thus TOF is a variable factor

depending upon: - the slope, the type of the surface of the ground, the length of the flow path, and

various other complicating factors.

Main stream

Watershed

line

Tributaries

Fan shaped

Fern shaped

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5.3 The hydrograph

The rainfall amount coming directly to the earth will be first lost in any of the three losses namely

depression storage, infiltration and interception. The rest will be represented as excess rainfall and

the water that reaches the stream channel will be called as surface runoff or direct runoff .The

surface runoff can, therefore occur only from those storms, which can contribute to excess rainfall

and are simply not dissipated in fulfilling the interception, depression storage and infiltration needs

of the basin.

The total precipitation (P) on a basin can be represented by the equation

P = L +G + Q or

Q = P - L- G

Runoff = Precipitation - Basin recharge - Ground water accumulation.

This is the fundamental equation that is used to compute runoff.

The runoff of a river stream consists of the following three components:

− Direct precipitation over the surface of the stream

− Surface runoff consisting of true surface runoff and sub surface storm flow.

− Ground water inflow, or called simply as base flow

Therefore hydrograph or runoff hydrograph is graphical representation of discharge (runoff) flowing

in a river at a given location with the passage of time. Or it represents discharge fluctuation in the

river at a given site over a given period and also indicates the peak flow which gives design of

hydraulic structure.

Fig: Typical hydrograph:

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Base flow: (also called drought flow, groundwater recession flow, low flow, and sustained or fair-

weather runoff) is the portion of stream flow that comes from "the sum of deep subsurface flow and

delayed shallow subsurface flow". It should not be confused with groundwater flow.

For the derivation of unit hydrograph, the base flow has to be separated from the total runoff

hydrograph. (i.e., from the hydrograph of the gauged stream flow). Some of the well-known base

flow separation procedures are given below,

(i) Simply by drawing a line AC tangential to both the limbs at their lower portion.

This method is very simple but is approximate and can be used only for

preliminary estimates. (Refer to the above figure)

(ii) Extending the recession curve existing prior to the occurrence of the storm up to

the Point D directly under the peak of the hydrograph and then drawing a straight

line DE, where E is a point on the hydrograph N days after the peak, and N (in

days) is given by:

N = 0.83 A0.2

Where A = area of the drainage basin, km2 and the size of the areas of

the drainage basin as a guide for values of N are given below.

5.4 Methods Used to Estimate Runoff

Runoff can be estimated by using different rainfall runoff models among which the following two

methods are the most commonly known types of models.

- Deterministic models: these are the vast majority of models which permit only one outcome from

a simulation with on e set of inputs and parameter values it can be further classified to whether

the model gives empirical (black box e.g. rational method) ; lumped conceptual (grey box) or

distributed process (white box)

- Stochastic models: these are models which allow for some randomness or uncertainness in the

possible outcomes due to uncertainty in inputs variable, boundary conditions or model

parameters. Traditionally a stochastic model can then be used for the generation of long

hypothetical sequences of events with the same statistical properties.

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Fig: classification of hydrological models according to process description.

The following are some of the most common rainfall runoff models covered in this syllabus.

1) Use of runoff coefficient

2) Use of infiltration capacity curve

3) Use of infiltration indices

4) Use of unit hydrograph theory

5) US SCS method

1) Use of Runoff coefficient

This involves drawing the hydrograph- that is a plot of discharge Vs time at any section of a river.

Using Rational Formula for Determination of Peak Flow

The basis for this formula is small impervious area, in which we may assume that if rain persists at

uniform rate for a period of at least as long as TOC; Tc, the peak runoff will be equal to the rate of

rainfall. CIAQ p =

Qp = peak flow [m3/s]

C = runoff coefficient

I = intensity of rainfall, mm/hr

A = drainage area, ha.

The C values are dependant on the topography, vegetation cover and soil characteristics of the region.

I, rainfall intensity, can be found from the intensity duration curve of the region and frequency

desired as shown below.

Figure 22: A sketch to illustrate intensity - duration relations

Inte

nsi

ty, m

m/h

r

Duration

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Estimation of Time Concentration

For plots having no channels or overland flow time /inlet time/

[ ]

[ ] 3/1

3/1111

KI

LobTo = Minutes

Lo = length of over land flow in meters,

I = rainfall intensity, cm/hr,

K = runoff coefficient,

b = a coefficient, whose value is given by

[ ] 3/1

000275.0

So

CrIb

+=

So = slope of the surface

Cr = retardness coefficient

Values of Cr

Type of surface Value of Cr

1 smooth asphaltic surface 0.007

2 Concrete pavement 0.012

3 Tar and gravel pavement 0.017

4 Closely clipped soil 0.046

5 Dense blue grass turf 0.060

Note: The above equation is applicable only when ILo < 387.

For design of hydraulic structures, the following simple formula can be used for evaluating the inlet

time or overland flow time:

385.03

885.0

=

H

LTo [Hrs]

L = the distance from the furthest [critical point] to the outlet at the hydraulic structure, such as

culvert, etc, m

H = Total fall in level from critical point to the outlet, such as culverts, m.

Time of Concentration

For a small drainage basin having flow channels in it, the time of concentration would be equal to the

largest combination of overland flow time called inlet time (To) and channel flow time, (Tf), which

exists any where in the basin. Channel flow time, Tf, is generally taken as the length of longest

channel divided by the average flow velocity in the channel at about bank full stage.

Tc = To + Tf

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Example: The total catchment area of 4000 ha of a stream up to a gauging site is subdivided into four

equal parts A, B, C, and D as shown below, that the time required for the water falling on the ground

to flow point P are

For any point on the division Line EF = 1 hr;

For any point on the division Line GH = 2 hr;

For any point on the division Line JK = 3 hr;

For any point on the division Line LM = 4 hr;

Over this catchment, a storm occurs for 4 hours, uniformly falling over the entire area. The total

rainfall is 36 mm. Determine the peak flow and the hydrograph of resulting runoff at P, if the runoff

factor is constant throughout the storm for the entire area as equal to 0.8.

Solution

Qp = CIAi

= 0.8IAi, for each area

Ai = A/4 = 1000 ha

When rain falls uniformly over the area A, for 1 hr, the hydrograph of runoff produced will be a

triangle starting from 0 hr, reaching its peak at 1 hr, and again falling to zero value at 2 hrs as shown

below.

If rain further continues beyond 1 hr, for another hour then the runoff caused by that additional 2

nd

hour rain is

Q

Time in hours 1 0

2

X

M K

Area D Area C Area B Area A

L J

H

G

E

F

Catchment

Q

Time in hours 1 0 2

X

3

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The total discharge hydrograph caused by this composite 2 hr rainfall would be obtained by addition

of the two hydrographs.

The hydrograph caused by a rainfall of 4 hrs over area A is as follows

The hydrograph caused by composite 4 hrs rainfall on area B, having time of concentration 2 hrs is as

shown below.

In a similar maner the hydrograph caused by composite 4 hrs rainfall on areas C and D , having time

of concentration 3 and 4 hrs respectively can be prepared by simply starting the hydrographs at 2 hr

and 3hr which correspondingly will end at 7 hr and 8hr on the time axis.

Total 4 hours uniform rain = 36 mm.

Rain in each hour = 9 mm.

Runoff coefficient = 0.8

Runoff caused by each hour rain = 9 mm*0.8 = 7.2 mm.

Areas, A = B = C = D = 1000 ha = 1000 x 104 m

2 = 10

7 m

2.

Area of each triangle = runoff caused by each 1 hour rain due to each part area.

Therefore, ½(2hr * X) = 7.2 mm

X = 7.2 mm/hr.

Run off = sm /3600

10*

1000

2.7 37

= 20 m3/s.

1 2 3

Q

Time in hours

0

X

Q

Time in hours 1 0 2 3 5 4

X

Time in hours

Q

1 0 2 3 5 4

X

6

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Discharge caused during 4 hr rain falling over areas A, B, C, and D

Discharge caused by Area, [m3/s] Time at

the end of

hour A B C D

Total discharge

caused by entire

area, m3/s

0 0 0 0 0 0

1 20 0 0 0 20

2 20 20 0 0 40

3 20 20 20 0 60

4 20 20 20 20 80

5 0 20 20 20 60

6 0 0 20 20 40

7 0 0 0 20 20

8 0 0 0 0 0

Hydrograph

0

20

40

60

80

60

40

20

00

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5 6 7 8Time in hr

Q, m3/s

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CHAPTER SIX - FLOOD FREQUENCY ANALYSIS

a. Introduction

Hydrologic processes such as flood are exceedingly complex natural events. They are resultants of

number of component parameters and are there fore very difficult to model analytically. For example,

the floods in the catchments depends up on the characteristics of the catchments, rainfall and

antecedent conditions, each one of them depend up on a host of constituent parameters. This makes

the estimation of the flood peaks a very complex problem leading to many different approaches. The

empirical formulae and unit-hydrographs methods presented in the previous section are some of

them. Another approach to the prediction of flood flows and also applicable to other hydrologic

processes such as rain fall etc. is the statistical method of frequency analysis.

The vales of annual maximum flood from a given catchments area for large number of successive

years constitute a hydrologic data series called the annual series. The data are then arranged in

decreasing order of magnitude and the probability p of each event being equal to or exceeded is

calculated by the plotting position formula

,

1

Where

N

mp

+=

m = order number of the event

N = total number of events in the data.

T= recurrence interval =1/P

Chow (1951) has shown that most frequency-distribution function applicable in hydrologic studies

can be expressed by the following equation known as the general equation of hydrologic frequency

analysis:

,where

KxxT σ+=

xT = value of the variate x of random hydrologic series with return period, T

x = mean of the variate.

σ = standard deviation of the variate.

K = frequency factor which depends up on return period, T

and assumed frequency distribution.

Some of the commonly used frequency distribution functions for the prediction of extreme flood

values are

i) Gumbel’s extreme-value distribution

ii) Log-Pearson Type III distribution, and

iii) Log normal distribution

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b. Common flood frequency distributions

1. Gumbel’s Method

This extreme value distribution was introduced by Gumbel (1941) and is commonly known as

Gumbel’s distribution. It is one of the most widely used probability distribution function for extreme

values in hydrologic and metrological studies for the prediction of flood peaks, maximum rain falls,

minimum rainfalls maximum wind speed, etc.

Gumbel defined a flood as the largest of 365 daily flows and the annual series of flood flows

constitute a series of larges values of flows. According to this theory of extreme events, the

probability of occurrence of an event equal to or larger than a value xo is

ye

o exXp−−−=≥ 1)( ……………………………(a)

In which y is dimensionless variable given by

577.0)(2825.1

,

2825.1

45005.0

)(

+−

=

=

−=

−=

x

x

x

xxyThus

xa

axy

σ

σα

σ

α

Where, x = mean and =xσ standard deviation of the variate x. In practice the value of X for a given

P that is required and eqn. (a) transposed as

pceT

T

Ty

py

T

p

1sin],

1ln.[ln

)]1ln(ln[

=−

−=⇒

−−−=

Now rearranging the above equation, the values of the variate x with the return period T

2825.1

)577.0(

,

−=

+=

T

xT

yK

where

Kxx σ

The above Gumbel’s equations are applicable to an infinite sample size (i.e. N=∞). But practical data

series of extreme events such as floods, maximum rainfall depths etc., all have definite lengths of

records.

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Gumbel’s Equation for Practical use

The above equation giving the value of the variate x with recurrence interval T is used as

1−+= nT Kxx σ

Where, 1−nσ = standard deviation of the sample of size N

1

)( 2

−−

=N

xx

K= frequency Factor expressed as

n

nT

s

yyK

−=

In which yT = reduced variate, a function of T and given as

]ln.[ln1−−=

TT

Ty

=ny Reduced mean, a function of sample size N and is given in table

(For N ,∞→ 577.0→ny )

ducedsn Re= Standard deviation, a function sample size N and is given in table

(For N ,∞→ 2825.1→y )

These equations are used under the following procedure to estimate the flood magnitude

corresponding to a given return period based on an annual flood series.

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Confidence limits

Since the value of variate for a given return period, xT determined by Gumbel’s methods can have

errors due to the limited data used; an estimate of the confidence limits of the estimate is desirable.

The confidence interval indicates the limits about the calculated between which the true value can be

said to lie with a specific probability based on sampling errors only.

For confidence probability c, the confidence interval of the variate xT is bounded by the values x1 and

x2 given by: eT scfxx )(2/1 ±=

Where f(c) = function of the confidence probability c determined by using the table of normal variate

as

C in percent 50 68 80 90 95 99

f ( c ) 0.674 1.00 1.282 1.645 1.96 2.58

)1.13.11(

..

2

1

KKb

Nberrorprobables n

e

++=

== −σ

K= frequency factor

1−nσ = standard deviation of the sample

N = sample size

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2. Log-Pearson Type III Distribution

In this method the variate is first transformed to logarithmic form & transformed data is analyzed. If

X is the variate of a random hydrologic series, then the series of Z variate; where, Z =log x are first

obtained. For this series, for any recurrence interval T gives:

zzT Kzz σ+=

Where, Kz = a frequency factor which is a function of recurrence interval T and the coefficient of

skew Cs,

zσ =standard deviation of the z variate sample

)1(

)( 2

−= ∑

N

zz

and Cs = coefficient of skew of variate z

3

3

))(2)(1(

)(

zNN

zzN

σ−−

−∑=

=Z Mean of the z values; N = sample size = number of year of record

The variations of Kz = f (Cs, T) is given in table. After finding ZT the corresponding value of XT is

obtained by: )log( TT zantix =

Kz value with corresponding Cs (coefficient of skewness) & T (return period) is provided in table below:

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c. Risk, Reliability and Safety factor:

The designer of hydraulic structure always faces a nagging doubt risk of failure of his structure. This

is because the estimation of hydrologic design values (such as the design flood discharge and the

river stage during the design flood) involve a natural or in built uncertainty and as such a

hydrological risk of failure. The probability of occurrence of an event ( )Txx ≥ at least once over a

period of n successive years is called the Risk, R . Thus the risk is given by

−= 1R (Probability of non-occurrence of the event )( Txx ≥ in n years)

n

n

TR

PR

−−=⇒

−−=

111

)1(1

Where, P = probability P( Txx ≥ ) = 1/T

T = Return period

The reliability Re, is defined as n

eT

RR

−=−=1

11

Safety factor

In addition to the hydrological uncertainty, as mentioned above, a water resources development

project will have many other uncertainties. These may arise out of structural, constructional,

operational and environmental causes as well as from non technological considerations such as

economic sociological and political causes. As such, any water resources development project will

have a safety factor for a given hydrological parameter M as defined below

Safety factor = (SF) m= only ionsconsiderat alhydrologic from obtained Mparameter theof valuethe

project theofdesign in the adopted Mparameter of valueactual

= hm

am

C

C

The parameter M includes such items as flood discharge magnitude, maximum river stage, reservoir

capacity and free board. The difference ( hmam CC − ) is known as safety margin.

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d. Applied Examples:

Example 1: Gumbel distribution:

For the data given below, fit the Gumbel distribution and find flood magnitude with 100 & 200 year

return period.

Solution:

The mean and standard deviation values for the above data are:

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Example 2: Pearson III distribution:

For the data given above in example 1, fit the log Pearson III curve and find flood magnitude with

100 & 200 year return period.

Solution:

The details of the computations are given in the table 2.17(a) and table 2.17(b)

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CHAPTER SEVEN – MISCELLANEOUS TOPICS

7.1 Introduction

This course was initially intended to be offered as an Engineering hydrology, where only different

engineering application of hydrology was thought to be covered, consequently some important terms

of hydrology were overlooked, which otherwise were very important to understand the fundamental

concepts and principles of basic hydrology. However, it is eventually found that the course should be

offered as introduction to hydrology and some deliberately missed topics, especially precipitation and

hydrologic losses are required to be covered a bit in detail. Hence this chapter is designed to cover

those important left topics.

When the rain starts falling, some of the most commonly known types of forms where precipitation

ends are the following:

i) Interception ii) Transpiration iii) Evaporation iv) Depression v) Infiltration vi) Runoff

The various water losses that occur in nature will be discussed below. If these losses are deducted

from the rainfall, the surface runoff can be obtained; that is: Rainfall - Losses = Runoff. Runoff has

already been discussed in some of the previous chapters. The following sections will be restricted to

some of the other important terms listed above.

7.2 Precipitation

Water is present in the atmosphere as a gas (water vapour), as a liquid (cloud droplets), and as a solid

(ice crystals). The complex process of producing precipitation may be considered to begin with the

water vapour , which results from evaporation at the earth’s surface and various chemical production

process , and has concentration of about 30gm/kg at the surface , reducing with height. The maximum

amount of water vapour that can be retained in air is an increasing function of temperature, so that

when air is cooled sufficiently and reaches the temperature at which the amount of water vapour

present is a maximum (also known as the dew point); excess water condenses as droplets or ice

crystals. The precipitation in the Ethiopia is mainly in the form of rain fall.

7.2.1 Types of Precipitation

The precipitation may be due to one of the following:

(i) Convectional precipitation (Thermal convection): This type of precipitation is in the form of

local whirling thunder storms and is typical of the tropics. The air close to the warm earth gets

heated and rises due to its low density, cools adiabatically to form a cauliflower shaped cloud,

which finally bursts into a thunder storm. When accompanied by destructive winds, they are

called ‘tornados’.

(ii) Frontal precipitation (Conflict between two air masses): When two air masses due to

contrasting temperatures and densities clash with each other, condensation and precipitation

occur at the surface of contact, Fig. 2.1. This surface of contact is called a ‘front’ or ‘frontal

surface’. If a cold air mass drives out a warm air mass’ it is called a ‘cold front’ and if a warm

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air mass replaces the retreating cold air mass, it is called a ‘warm front’. On the other hand, if

the two air masses are drawn simultaneously towards a low pressure area, the front developed

is stationary and is called a ‘stationary front’. Cold front causes intense precipitation on

comparatively small areas, while the precipitation due to warm front is less intense but is

spread over a comparatively larger area. Cold fronts move faster than warm fronts and usually

overtake them, the frontal surfaces of cold and warm air sliding against each other. This

phenomenon is called ‘occlusion’ and the resulting frontal surface is called an ‘occluded

front’.

(iii) Orographic precipitation (orographic lifting): The mechanical lifting of moist air over

mountain barriers causes heavy precipitation on the windward side (Fig. 2.2).

(iv) Cyclonic precipitation (Cyclonic): This type of precipitation is due to lifting of moist air

converging into a low pressure belt, i.e., due to pressure differences created by the unequal

heating of the earth’s surface. Here the winds blow spirally inward counterclockwise in the

northern hemisphere and clockwise in the southern hemisphere. There are two main types of

cyclones—tropical cyclone (also called hurricane or typhoon) of comparatively small

diameter of 300-1500 km causing high wind velocity and heavy precipitation, and the extra-

tropical cyclone of large diameter up to 3000 km causing wide spread frontal type

precipitation.

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7.2.2 Measurement of precipitation

Rainfall may be measured by a network of rain gauges which may either be of non-recording or

recording type.

Non- Recording Rain Gauge: The most common non-recording rain gauge type Symon’s rain

gauge (Fig. 2.3). It consists of a funnel with a circular rim of 12.7 cm diameter and a glass bottle as a

receiver. The cylindrical metal casing is fixed vertically to the masonry foundation with the level rim

30.5 cm above the ground surface. The rain falling into the funnel is collected in the receiver and is

measured in a special measuring glass graduated in mm of rainfall; when full it can measure 1.25 cm

of rain. The non-recording or the Symon’s rain gauge gives only the total depth of rainfall for the

previous 24 hours (i.e., daily rainfall) and does not give the intensity and duration of rainfall during

different time Intervals of the day. It is often desirable to protect the gauge from being damaged by

cattle and for this purpose a barbed wire fence may be erected around it.

Recording Rain Gauge: There are three types of recording rain gauges—tipping bucket gauge,

weighing gauge and float gauge.

Tipping bucket rain gauge. This consists of a cylindrical receiver 30 cm diameter with a

funnel inside (Fig. 2.4). Just below the funnel a pair of tipping buckets is pivoted such that

when one of the bucket receives a rainfall of 0.25 mm it tips and empties into a tank below,

while the other bucket takes its position and the process is repeated. The tipping of the bucket

actuates on electric circuit which causes a pen to move on a chart wrapped round a drum

which revolves by a clock mechanism. This type cannot record snow.

Weighing type rain gauge. In this type of rain-gauge, when a certain weight of rainfall is

collected in a tank, which rests on a spring-lever balance, it makes a pen to move on a chart

wrapped round a clock driven drum (Fig. 2.5). The rotation of the drum sets the time scale

while the vertical motion of the pen records the cumulative precipitation.

Float type rain gauge. In this type, as the rain is collected in a float chamber, the float moves

up which makes a pen to move on a chart wrapped round a clock driven drum (Fig. 2.6).

When the float chamber fills up, the water siphons out automatically through a siphon tube

kept in an interconnected siphon chamber. The clockwork revolves the drum once in 24

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hours. The clock mechanism needs rewinding once in a week when the chart wrapped round

the drum is also replaced. This type of gauge is used by IMD.

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7.2.3 Rain-gauge density

The following figures give a guideline as to the number of rain-gauges to be erected in a given area or

what is termed as ‘rain-gauge density’. The most commonly used rain gauge density values are

tabulated as follows:

Area Rain-gauge density

Plains 1 in 520 km2

Elevated regions 1 in 260-390 km2

Hilly and very heavy rainfall areas 1 in 130 Km2 preferably with 10% of the rain-gauge stations

equipped with the self recording type

7.2.4 Optimum rain-gauge network design

The aim of optimum rain-gauge network design is to obtain all quantitative data averages and

extremes that define the statistical distribution of the hydro meteorological elements, with sufficient

accuracy for practical purposes. When the mean areal depth of rainfall is calculated by the simple

arithmetic average, the optimum number of rain-gauge stations to be established in a given basin is

given by the equation (IS, 1968)

Where N = Optimum number of rain-gauge stations to be established in the basin

Cv = Coefficient of variation of the rainfall of the existing rain gauge stations (say, n)

P = desired degree of percentage error in the estimate of average depth of RF over the basin.

The number of additional rain-gauge stations (N–n) should be distributed in different zones (caused

by isohyets) in proportion to their areas, i.e., depending upon the spatial distribution of existing rain-

gauge stations and the variability of the rainfall over the basin.

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7.3 Depression storage

Precipitation that reaches the ground may infiltrate, flow over the surface, or become trapped in

numerous small depressions from which the only escape is evaporation or infiltration. The nature of

depressions as well as their size is largely a function of the original land form and local land-use

practices. Because of extreme variability in the nature of depressions and the paucity of sufficient

measurements no generalized relation with enough specified parameters for all cases is feasible. A

rational model can, however, be suggested

Fig: Typical Depression losses in storm periods

Figure 3.1 illustrates the disposition of a precipitation input. A study of it shows that the rate at which

depressions storage is filled rapidly declines after the initiation of a precipitation event. Ultimately,

the amount of precipitation going into depression storage will approach zero, given that there is a

large enough volume of precipitation to exceed other losses to surface storage such as infiltration and

evaporation. Ultimately, all the water stored in depressions will either evaporate or seep into the

ground.

According to Linsley volume of water stored by surface depressions at any given time can be

approximated using:

Where; V=the volume actually in storage at some time of interest

S= the maximum storage capacity of the depressions

P= the rainfall excess (gross rainfall minus evaporation, interception, and infiltration)

K= a constant equivalent to l/Sd

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7.4 Interception

This loss is due to surface vegetation that is water held by plant leaves and other physical structures

on earth. The precipitation intercepted by foliage (Plant leaves, forests and buildings) and returned to

atmosphere (by evaporation from plant leaves) without reaching the ground surface is called

interception loss. Interception loss is high in the beginning of storms and gradually decreases, the loss

is in order of 0.5 to 2 mm per shower and it is greater in case of high showers than when the rain is

continuous. The modifying effect that a forest canopy can have on rainfall intensity at the ground can

be put to practical use in watershed managements schemes. The amount of water intercepted is a

function of: (1) the storm character, (2) the species age, and density of prevailing plants and trees,

and (3) the season of the year.

Most interception loss develops during the initial storm period and the rate of interception rapidly

approaches zero thereafter. Potential storm interception losses can be estimated by using the equation

L i = S + K E t

Where: L = the volume of water intercepted (in.)

S = the interception storage that will be retained on the foliage against the forces of wind

and gravity (usually varies between 0.01 and 0.05 in.)

K= the ratio of surface area of intercepting leaves to horizontal projection of this area

E = the amount of water evaporated per hour during the precipitation period.(in.)

t = time (hr)

Fig: Typical interception losses in storm periods

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7.5 Infiltration

When water falls on a given formation, a small part of it is first of all, absorbed by the top thin layer

of soil, so as to replenish the soil moisture deficiency. Thereafter, excess water moves down ward,

where it is trapped in void and becomes ground water. This process, when by the water enters the

surface strata of the soil, and moves downward towards the water table, is known as infiltration.

Infiltration (f) often begins at a high rate (20 to 25 cm/hr) and decreases to a fairly steady state rate

(fc) as the rain continues, called the ultimate fp (= 1.25 to 2.0 cm/hr) (Fig. 3.6). The infiltration rate

(f) at any time t is given by Horton’s equation.

Where f0 = initial rate of infiltration capacity

fc = final constant rate of infiltration at saturation

k = a constant depending primarily upon soil and vegetation

e = base of the Napierian logarithm

Fc = shaded area in Fig below

t = time from beginning of the storm

The infiltration takes place at capacity rates only when the intensity of rainfall equals or exceeds fp;

i.e., f = fp when i ≥ fp; but when i < fp, f < fp and the actual infiltration rates are approximately equal

to the rainfall rates. The infiltration depends upon the intensity and duration of rainfall, weather

(temperature), soil characteristics, vegetal cover, land use, initial soil moisture content (initial

wetness), entrapped air and depth of the ground water table. The vegetal cover provides protection

against rain drop impact and helps to increase infiltration.

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Methods of Determining Infiltration

The methods of determining infiltration are:

(i) Infiltrometers

(ii) Observation in pits and ponds

(iii) Placing a catch basin below a laboratory sample

(iv) Artificial rain simulators

(v) Hydrograph analysis

The following are related important terms of infiltration.

Infiltration capacity

The ground water stored in the underground depends mainly upon the number of voids present in the

soil, which, in turn, does not depend upon the size of the soil particles but rather upon the

arrangement, shape and degree of compaction. Therefore, different soils will have different number if

voids, and hence, different capacities to absorb water. The maximum rate at which a soil in any given

condition is capable of absorbing water is called its infiltration capacity. It is generally denoted by

the letter f.

Infiltration Rate

It is evident that the rain will enter the soil at full capacity rate (f) only during the periods when the

rainfall rate exceeds the infiltration capacity. When the rainfall intensity is less than the infiltration

capacity, the prevailing infiltration rate is approximately equal to the rainfall rate. Hence, the actual

prevailing infiltration rate may be equal to or less than the infiltration capacity. This actually

prevailing rate at which the water will enter the given soil at any given time is known as Infiltration

rate.

If the rain intensity (p) exceeds the infiltration capacity (f) the difference is called the excess rain rate

(pe), This excess water is first of all, accumulated on the ground as surface detention (D) and then

flows over land into the streams.

Soil Moisture

The water below the water table is known as the ground water and the water above the water table is

known as soil moisture. The region above the water table is divided into three zones:

(i) Capillary zone;

(ii) Intermediate zone, and

(iii) Soil zone

Extending above the water table, a distance usually ranging from about 0.3 to 3m, depending

principally upon texture, is a zone called the capillary zone or capillary fringe. Throughout this

capillary zone, the moisture content is maintained practically constant by capillarity.

Extending down from the ground surface, is the soil zone which is defined as being the depth of

overburden that is penetrated by the roots of vegetation. Throughout this root zone, the moisture

content varies tremendously, ranging from a partly saturated state during and immediately following

the periods of heavy rains, to a minimum content after a spell of long continued drought.

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The region between the capillary zone and the soil zone is called the intermediate zone. Throughout

this zone, except during the period of ground water accretion from rainfall, the amount of water

contained within any given space is nearly constant throughout the year. In some cases, the height of

capillary fringe (or capillary zone) may be more, and may extend up to the soil zone. In that case,

there will be no intermediate zone.

Infiltration capacity curve

Infiltration capacity curve is the graphical representation of as how the infiltration capacity varies

with time during and a little after rain.

Infiltration capacity is generally very high at the beginning of a rainstorm that occurs after a long dry

period. During the rainstorm, the infiltration capacity is considerably reduced due to the various

factors, such as, surface detention, soil moisture, compaction due to rain, washing of fines, etc. After

a certain period of time (of the order of 1 to 3 hours) the infiltration capacity tends to become

constant.

Sometimes, the value of f is to be used for computing the surface runoff due to a given storm from a

given drainage basin; in that case, the value of f is so calculated as to include interception and

depression storage in itself, and it is general practice to include interception and depression storage in

the infiltration itself.

If two values of f at two known times are known and fc is also known, the straight line van be drawn

through these two points, and the slope of the line measured; and then equating it to –1/k log10 e, k

can be determined, and hence, an equation for the Infiltration capacity (I.C.) can be written easily.

Infiltration Indices

In hydrological calculations involving floods, it if found convenient to use a constant value of

infiltration rate of the duration of the storm. The average infiltration rate is called infiltration index.

Two types of indices are in common use: these are

� φφφφ - Index: The φ index is the average rainfall above which the rainfall value is equal to the

runoff volume; the φ index is derived from the rainfall hyetograph with the edge of the resulting

runoff volume. The initial loss is also considered as infiltration. The φ value is found by treating it

as a constant infiltration capacity. If the rainfall intensity is less than φ, then the infiltration rate is

equal to the rainfall intensity; however, if the rainfall intensity is larger than φ the difference

between rainfall and infiltration in an interval of time represent the runoff volume.

The amount of rainfall in excess of the index is called rainfall excess, The φ index thus accounts of

the total abstraction and enables runoff magnitudes to be estimated for a given rainfall hyetograph.

� W - Index: In an attempt to refine the φ index the initial losses are separated from the total

abstractions and an average value of infiltration rate called the W- index is defined as:

W = P - R - Ia

te

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Where; P = total storm precipitation (cm)

R = total storm runoff (cm)

Ia = initial losses (cm)

te = duration of the rainfall excess i.e. the total time in which the rainfall intensity is

greater than W (in hours) and

W = average rate of infiltration (cm/h)

Since In values are difficult to obtain, the accurate estimation of the W index obtained under very wet

soil conditions, representing the constant minimum rate of infiltration of the catchment, is know as

Wmin. Both the W-index and φ index vary from storm to storm.

In estimating the φ index for design purposes, in the absence of any other data, a φ index value of

0.10cm/h can be assumed.

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7.6 Evaporation and transpiration

Evaporation: - may be defined as the process by which the liquid water is converted into vapor. The

following types are distinguished:

(a) Evaporation in to atmosphere from:

− From water surface reservoir, lakes, ponds, river, channels etc. and

− Bare soil , Es

− Wet crop, Ewet it also referred to as evaporation of intercepted water, Ei

(b) Transpiration, Et Transpiration is the process by which the water vapor escapes from the living

plant through stomata of living plants and inters the atmosphere.

(c) Evapotranspiration, E a combination of evaporation and transpiration. In its broader sense,

evapotranspiration or consumptive use is the total water lost from a cropped (or irrigated) land due to

evaporation from the soil and transpiration by the plants or used by the plants in building up of plant

tissue. Potential evapotranspiration is the evapotranspiration from the shore-green vegetation when

the roots are supplied with unlimited water covering the soil. It is usually expressed as a depth (cm,

mm) over the area.

Evaporation from water surface and soil are of great importance in hydro - metrological studies. The

factors affecting evaporation are air and water temperature, relative humidity, wind velocity, surface

area (exposed), barometric pressures and salinity of the water; the last two have a minor effect.

Methods of Estimating Evaporation from free water surface:

Methods:

1) Water budget methods

2) Energy budget method

3) Mass transfer method

4) Empirical formulas

5) Evaporation measurements using pans (direct method of measuring

evaporation)

Methods of Estimating Transpiration:

Botanists derived various methods for measurement of transpiration and one of the widely used

methods is photometer.

Photometer: Consists of a closed watertight tank with sufficient soil for plant growth with only the

plant exposed; water is applied artificially till the plant growth is complete. The equipment is

weighed in the beginning as (W1) and at the end of the experiment as (W2). Water applied during the

growth (W) is measured and the water consumed by the transpiration (Wt) is obtained as:

Wt = (W1 + W) - W2

The experimental values (from the protected growth of the plant in the laboratory) have to be

multiplied by a coefficient to obtain the possible field results.

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Transpiration ratio is the ratio of the weight of water absorbed (through the root system), conveyed

through and transpired from a plant during the growing season to the weight of the dry matters

produced exclusive of roots.

Transpiration ratio = Weight of water transpired

Weight of dry matter produced

Methods of Estimating Evapotranspiration:

The following are some of the methods used in estimating evapotranspiration.

(i) Experimental methods (using field experimental plots, Tanks and Lysimeter experiments,

Installation of sunken (Colarado tanks);

(ii) Theoretical method

(iii) Empirical method -these include the empirical methods such as Thornth Waite method,

Blaney- Criddle, Pen-man, Evaporation index method (Hargreaves & Christiansen

The following methods of evapotranspiration estimation are combination of empirical, analytical and

theoretical approaches derived on the basis of regional relationship between measured ET and

climatic factors.

SN Estimation Method Corresponding equation

1 FAO Balnney-Criddle Method

ET0 = C × P (0.46T + 8)

2 FAO Radiation Method

ET0 = C ×(W⋅Rs) ; Rs = (0.25 + 0.50 n/N) RA

3 FAO Penman Method

( ) ( ) ( )[ ]adno eeufwRWCET −−+⋅= 1

(Radiation term) (Aerodynamic term)

4 Hargreave's Class

A Pan Evaporation Method

ET0 = 0.0023 × Ra (T + 17.8) × TD0.50

5 FAO Pan Evaporation Method ET0 = kp⋅Epan

6 FAO Penman-Monteith Method ( ) ( )

( )2

2

034.01

273

900408.0

V

eeVT

GR

ET

asn

++∆

−+

+−∆=

γ

γ

7 Thornthwaite Method a

e

mfo

T

TRET

=

1062.1

Where:

C in (1) = adjustment factor which depends on the min relative humidity, sunshine hours and

daytime wind estimates

C in (2) = Adjustment factor which depends on mean humidity and daytime wind condition

C in (3) = Adjustment factor to compensate for the effect of day and night weather

conditions

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Introduction to hydrology (CE-3601) Tessema B; Mekdim M;Belay,Suleman 66

T = Mean daily temperature in oC over the month considered

P = Mean daily percentage of total annual day time hours obtained from the relevant

table for a given month and latitude

Rs = Solar radiation in equivalent evapora on mm/d

W in (2) = the temperature and altitude dependant weighing factor

W in (3) = Temperature related weighing factor for the effect of radiation on PET

(1-w) = a temperature and elevation related weighing factor for the effect of wind and

Humidity on PET

Rn = Net radiation in equivalent evaporation mm/d

f (u) = Wind related function

(ed - ea) = Difference between the saturated and actual vapour pressure, in mbar

Ra = Extraterrestrial radiation in mm/d

TD = Difference in max and min mean temperature in oC

Epa = Pan evaporation in mm/d and presents the mean daily value of the period

considered

kp = Pan coefficient

Rn in (6) = Net radiation at crop surface (MJ/ m2day)

G = Soil heat flux (MJ/ m2day)

V2 = Wind speed measured at 2m height (m/s)

(ed - ea) = Vapour pressure deficit (kPa)

∆ = Slope of vapour pressure curve (kPa oC

-1)

γ = Psychometric constant (kPa oC

-1)

900 = a conversion factor

Rf = reduction factor 32 000000675.00000771.001792.04923.0 eee TTTa +−+=

In (7) 514.1

5

= m

e

TT

Page 67: Final Handout (Intro to Hydrolgy

Adama university, Department of civil Engineering and Architecture

Introduction to hydrology (CE-3601) Tessema B; Mekdim M;Belay,Suleman 67

EXAMPLE PROBLEM: Use Blaney-Criddle method to calculate consumptive use (PET) for rice

crop grown from January to March at latitude 220 N from the following data taken from a nearby

observatory.

Solution:

Mean monthly sunshine hours for latitude of 220N for the months of January, February and March

will be read from table.

Table: Blaney-Criddle Method of Computation of Consumptive use of Rice Crop for example above:

Month Mean monthly

temp (Tm)

Monthly % (P) of day

time Hours from Table

4.9

Monthly

Consumptive

Use factor (F)

PET (4) x (5)

(1) (2) (3) (4) (6)

January 12 7.62 10.37 11.40

February 16 7.20 11.12 12.23

March 24 8.40 16.04 17.64

For col. (4) for F (January) = (0.0457 Tm + 0.8128) x P

= (0.0457 x 12 + 0.8128) x 7.62 = 10.37 cm

F (February) = (0.0457 x 16 + 0.8128) x 7.2 = 11.12 cm

F (March) = (0.0457 x 24 + 0.8128) x 84 = 16.04 cm

ASSIGNMENT QUESTION:

1. Compute the monthly and yearly potential evapotranspiration (PET) for the place located 23039N

and using Thronthwaite formula from the following data.

Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Temp.

(oC)

18.5

21.30

26.68

30.34

30.88

27.72

29.29

28.67

28.70

26.91

23.26

19.34

Monthly

heat index(i)

7.25

9.04

12.99

15.68

16.15

15.45

14.47

14.09

14.02

12.85

11.02

7.61

Month January February March

Mean temperature 0C 12 16 24

Rainfall (mm) 8 20 16

P(%) from table: 7.62 7.20 8.40