MAXIMUM ESTIMATION

WORLD METEOROLOGICAL ORGANIZATION

TECHNICAL NOTE No. 98

ESTIMATION

OF MAXIMUM FLOODS

Report of a working group of the Commission for Hydrometeorology

WMO -No. 233. TP.126

Secretariat of the World Meteorological Organization • Geneva • Switzerland 1969

NOTE

The designations employed and the presentation of the material in this publication do not imply the expression of any opinion whatsoever on the part of the Secretariat of the World Meteorological Organization concerning the legal status of any country or territory or of its authorities, or concerning the delimitation of its frontiers.

Editorial note: This publication is an offset reproduction of a typescript submitted by the authors.

PREFACE

The preparation of this Technical Note was an exercise in international collabo- ration. The Working Group had been asked to give as many examples from various countries of the worJ,.d as possible. It was perhaps inevitable that the majority of examples would be drawn from those countries whose experts were members of the Working Group. The reader will note, however, that there has been a conscious effort to include references and examples from other countries as well. It was also inevitable that, for solving some problems, more than one technique is presented, reflecting procedures and practices in different countries.

It is hoped that the reader will find this an enrichment of the text rather than a compli- cation.

In addition to the official members of the Working Group, there were several

"unofficial" Working Group members who contributed substantially to the Technical Note. .In particular, Chapter 5 was written by Prof. A. F. Jenkinson, of University College, Nairobi, Kenya.and Section 4.4 by David Rockwell, Corps of Engineers, U.S. Army, Portland, Oregon, U.S.A. The members of the Working Group were Mr. R. Arlery (France), Mr. S. BanerJi (India), Mr. D. J. Bargman (East Africa), Mr. J. P. Bruce (Canada chairman),.Dr. A. G. Kovzel

(U.S.S.R.), Dr. V. Kfiz (Czechoslovakia), Mr. V. A. MYers (U.S.A.).

It is the hope of the WOrking Group that hydrologists and hydrometeorologistsin many countries will benefit from this summary of techniques, both physical and statistical, for estimation of design floods.

J. P. Bruce (Chairman)

CONTENTS

Page Foreword . . . • • . . . • . . . • • . . . • . . . • . • . . . • . . . . VII Summaries (English, French, Russian, Spanish) . . . • . . . • . . . • • . . . VIII CHAPTER 1 - INTRODUCTION

1.1 Introduction... • • . . . • . . . • . • . . . • • • . . . • 1

1.2 Glossary of terms . . . • • . . . • . . . • . . . . 3

CHAPTER 2 - MAXIMUM RAINFALL 2.1 2.2 2.3 2.4 2.5 2.6 Physical models of rainstorms Analysis of storm rainfall data ...•..•.•..•••.•...•....•.• Storm transposition . . . • . . . • . . . • • . • . . . • . . . • • . . . • storm rainfall maximization •...•...•.••.•.••••.•.•...•..••.•• Mapped values of maximum precipitation •...••.•.•••..••••..•....•.••..•... Storm sequences and maximum rainfall for long durations 9 18 35 47 83 101 CHAPTER 3 ~ SNOWMELT CONTRIBUTIONS TO MAXIMUM FLOODS 3.1 I n t r o d u c t i o n . . . . 117

3.2 Maximum snow accumulation •...•••...•••..•••••..•.•. .... ..•.... ...•. ..•.•• 117

3.3 CriticaJ,. snowmelt r a t e s . . . . 126

3.4 Rain on snow e v e n t s . . . .. ... 1)4 CHAPTER 4 - CONVERSION OF CRITICAL METEOROLOGICAL FACTORS TO FLOOD HYDROGRAPHS 4.1 Statement of problem . . . • . . . . 137

4.2 Estimation of runoff volumes... .•..•.. . . . • . . 138

4.3 Time distribution of runoff - unit hydrographs •... ..•. ...•.•.•. 145

4.4 Computer techniques for estimation of hydrographs of maximum floods from 166 meteorological input • . . . • . . . • . CHAPTER 5 - STATISTICS OF EXTREMES 5.1 Introduction and theory . . . • . . . • . . . • . . . . • 183

5.2 Practical applications - the maximum likelihood solution •.... .... ... ... ....

196

5.3 Confidence limits • . • . • . . . • . . . • . . . , .. ...•. 209

5.4 Further applications . . . . • . . . • . . . • . . . • . . . . 213 CHAPTER 6 - STATISTICAL ANALYSIS OF FLOOD FLOWS

6.1 6.2

Introduction

Pre-analysis procedures . . . • . . . • . . . .

229 229

VI CONTENTS

CHAPTER 6 (continued)

6.3 Methods of applying probability distributions •..•.•...•...•..••• 232

6.4 Making use of historical flood data •...•....•••...••••.••...•.•...• 237

6.5 Analyses for rivers with two flood regimes .•...•..••... ...•.••..••.•• 239

6.6 Peak discharge probabilities for ungauged locations •...••.•... ...••...••• 241

CHAPTER 7 - USES OF METEOroLOGICAL DATA IN ESTIMATING FLOOD FREQUENCIES Introduction ..•...•...•..••...••....•..••....••.•.•...•..•..•••••...• 263

263 263 264 265 266 . . . .e . • • • • • • • • • • • • • •

... ...

.

II. Methods of estimating probable maximum runoff according to the maximum intensity of precipitation or snowmelt ...•. 281

FOREWORD

At its second session (Warsaw 1964) the WMO Commission for Hydrometeorology (CRy) established a Working Group to prepare a Technical Note on Estimation of Maximum Floods. The members of the Working Group were Mr. R. Arlery (France), Mr. S. Banerji (India), Mr. D. J.

Bargman (East Africa), Mr. J. P. Bruce (Canada, Chairman), Dr. A. G. Kovzel (U.S.S.R.), Dr.

V. _K~iz (Czechoslovakia) and Mr. V. A. Myers (U.S.A.). The members of the working group had several collaborators and advisers who also contributed substantially to this Technical Note.

In particular, chapter 5 was written by Professor A. F. Jenkinson of University College, Nairobi, Kenya, and section 4.4 by Mr. R. Rockwell, Corps of Engineers, U.S. Army, Portland, Oregon, U.S.A.

It is with great pleasure that I express the gratitude of WMO to the members of the working group and to the other individuals who have assisted the group in the preparation of this Technical Note. In particular, I should like to express a word of thanks to Mr. J.

P. Bruce who, as chairman of the working group, devoted much time and thought to this excel- lent monograph on a very complex subject.

I should also like to take this opportunity to thank Mr. Max. A. Kohler, the former president of the Commission for Hydrometeorology, for his assistance in the arrange- ments for the preparation and publication of this Note.

~ . ...-:--:.'---

(D.A. DAVIES) Secretary-(}eneral

IX

SUMMARY

The aim of this Technical Note is ,t.o supply the reader with information on methods of evaluation of meteorological conditions for estimation of maximum floods.

The first and greater part (chapters 2 to 4) of the Technical Note describes methods for estimating the extremes of rainfall and· snow melt on the basis of physical analysis, and methods for converting these into estimates of extreme flood flows. The Note then treats (chapters 5 and 6) statistical methods and their application to storm and flood events. It gives background on statistical analysis and outlines some tech- niques used· in various countries in flood frequency analysis.. The last chapter describes the use. of meteorological data in estimating flood frequencies~

RESUME

L'objet de la presente'Note technique est de renseignerle lecteur sur les methodes utilisees pour evaluerles conditions meteorologiques dans le but d'estimer les crues maximales.

La plus grande partie de la note (chapitres 2

a

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RESUMEN

El objeto de esta Nota Tecnica es dar cuenta al lector de los metodos de evalua- ci6n de las condicionesmeteoro16gicas, que se utilizan actualmente para calcular por esti- maci6n las crecidas m&ximas.

En la primera y mayor parte de la Nota Tecnica (Cap1tulos 2, 3 y 4) se describen los metodos utilizados para estimar los valores extremos delluvia y nieve fundida fundan- dose en el an~lisisf1sico, as1 como los procedimientos que se aplican para convertir estos valores en estimaciones de las crecidas maximas de las corrientes. A continuaci6n se estu- dian (Cap1tulos 5 y 6) los metodos estad1sticos y sus aplicaciones al estudio de los tempo- rales y crecidas. Se exponen tambi~n ciertos antecedentes relativos al anAlisis estad1stico y se describen algunas tecnicas utilizadas en distintos pa1ses para el analisis de frecuen- cia de las crecidas. En el ultimo cap1tulo se explica'el uso de los datos meteoro16gicos para la estimaci6n de la frecuencia de las crecidas.

1

CHAPTER

INTRODUCTION

Selection of suitable hydrologic design criteria for major

river structures is a problem faced by engineers in all parts of the world.

For many structures and hydrologic problems design criteria should be based primarily on economic considerations. These include the optimum storage capacity of reservoirs, spillway design far concrete structures in remote areas, and carrying capacities of channels, culverts, channel improvement schemes, and storm sewer systems.' In these cases, if the design capacity is exceeded, some damage will result,but it will not be catastrophic in magnitude, nor will loss of ,life likely occur. The' optimum design criteria can be obtained by balancing the costs of

repairing damage on infrequent occasions against the costs of providing for larger floods. For such an e~onomic analysis, flood frequency data are required and may be derived by statistical techniques if sufficient data are available. These statistical methods and their applications to storm and flood events are discussed in ohapters 5 and 6.

However, in some cases a very high degree of safety is required in the design criteria. For example,a dam of earth~fill construction may fail entirely if the dam is over-topped. The spillway of such a dam must then be able to pass the largest flood likely to occur in the lifetime of the structure, if failure is to be avoided. Statistical analyses of the less than 100 years record available in most parts of the world are not able to provide safe enough criteria. It can be sho~vn

that if the structure is to last 100 years, the chance of a lOO-year return period flood occurring within its lifetime is 63%. That is the lOO-year structure, designed for a lOO-year flood, is designed with a 63% chance that its capacity will be exceeded. Extending the analysis further, in order to have only a 5% chance that the structure's capacity . will be exceeded, the engineer must design fora 1950-year return period

flood. The unreliability of estimates of the magnitude of such rare

events by statistical means from relatively short periods of observational records, and the need for very safe design criteria particularly for

structures which are upstream from populated areas, has led to increasing.

use of physical analyses of design floods. Indeed where earth-fill construction is used upstream from urban centres, many engineers are of the opinion that the spillways should be designed to pass the physical upper limits to flood flows which the basin above the dam site is capable of producing. The greater part of this Technical Note (chapters 2-4) is concerned with physical analysis estimates of extreme floods.

Since,. aside from those caused by earthquakes and landslides, major floods are a result of meteorological conditions, such physical analyses start with meteorological studies. In all climatic zones this involves estimation of maximum snow accumulation and melt rates. Where the rainfall studies are directed towards estimation of the physical upper limits to storm rainfall in a basin or region, the resulting_estimates are usually called the "probable maximum storm" or "probable maximum precipitation". vlhen converted into flood flows by one of the methods outlined in chapter 4, the resulting flood is known as the "probab le maximum flood". Another, less widely used set of terms is "standard' project storm and flood". These terms are used to denote the largest

CHAPTER 1

storm that has occurred in a-climatically homogeneous region and is considered to be reasonably typical of that region, and the flood that would result if such a storm was centred on a basin _within this region.

A better understanding of the significance of these terms will be obtained by a study of the methods of estimating the magnitude of these extreme events and the application of such estimates, as outlined in subsequent chapters.

Throughout the Technical Note, examples are given from various parts of the world covering as many of the major climatic zones as proved feasible.

It should be emphasized that the final selection of design criteria for any structure involves economic and even moral and political considerations in addition to those of a hydrologic nature. The job of the hydrologist and hydrometeorologist is to provide the data and analyses needed to permit intelligent assessment of the flood potential of the site in question. It is our hope that this Technical Note will contribute to an improvement in analysis procedures and practices in the world, and to a better understanding of the importance of hydrological analyses in safe and efficient design of river structures.

3

1.2 Glossary of Terms

A selection of technical terms employed in sections 2.1 to 2.6 is listed below for the convenience of the reader, with either their corresponding definition or a paragraph reference to a definition found in the text. In general, the terms defined here are common to all of meteorology while the terms defined in the text belong primarily to

the specialties of rainfall maximization or analysis.

adiabatic chart - a thermodynamic diagram employed by meteorological services

to plot atmospheric temperature vs. pressure. Contains saturated adiabats, and other curves.

barrier - a mountain range which partially blocks the flow of warm humid air from its oceanic source region to a basin under study. (2.3.4.2).

convergence - the tendency of the horizontal component of the wind to converge in cyclones and other pressure systems, fOJ::..cing upward vertical motion of the air and thereby, if sufficient moisture is present, releasing precipitation. Convergence is the negative of divergence and, in meteorology, unless explicitly stated otherwise, refers to - • V

h where V

h is the horizontal component of the wind.

depth-area curve - a curve depicting maximum values of storm precipitation over various area sizes during _so~e specified duration. (2.2.8.4)

depth-duration-area values - maximum values of storm rainfall for fixed durations and area sizes. Presented either in tabular form or as a set of depth-area curves for various durations. (2.2.2)

depth-duration curve - a curve depicting maximum values of storm precipitation during various durations over a specified area. (2.2.8.3)

IOOO-rob. dew point - see wet-bulb potential temperature.

envelope - the curve constructed in the process of envelopment.

envelopment - the analysis procedure of estima.ting the maximum values of a weather element by fitting a smooth curve (usually by eye) to the highest data points plotted on a graph or map. Other considerations than the data themselves may influence the shape of the curve between data points.

hyetograph - a plot in chronological sequence of the increments of precipitation either at a station or averaged over a designated area, in equal time periods of an hour, six hours, or a day depending o.n the scale of interest. (2.2.8.2)

CaAPTER 1

5

isohyet-area graph - a curve derived from an isohyetal chart depicting the isohyetal values vs. the area enclosed within each isohyet. (2.2.8.5) lapse rate - the rate at which temperature in the atmosphere changes in the vertical; either aT/ah or- aT/ap where T is temperature, h height, and P pressure.

mass curve - a plot of accumulated depth of precipitation at a point or averaged over a desired area against time. Also see paragraph 2.2.6.

mixing ratio - the dimensionless ratio of mass of water vapor to mass of dry air with which it is mixed

w

=

where w is mixing ratio, p atmospheric pressure, e vapor pressure, and .622 is the ratio of the molecular weight of water to the average molecular weight of dry air. Also given in gm kg-I, and is then 1000 times above value. Similar to specific humidity.

moisture maximization - the process of adjusting storm precipitation upward to a theoretical value that would have pertained if the moisture content of the air had been at the maximum for the location and season but other storm conditions had remained unchanged.

precipitable water - the total atmospheric water vapor contained in a vertical column extending between two specified levels, or if unspecified, from the surface to the top of the atmosphere. Expressed as the depth of liquid water of equal mass over an area equal to the cross-sectional area of the column. Also called liquid equivalent of water vapor.

1

gp

f

where w

=

units fulfilling this formula are cm for w, mb for P, g kg-l

for q,

-2 -3

cm sec for g, and gm cm for d g

=

=

rain profile - a form of isohyet-area graph in which the area enclosed

by an isohyet is replaced by the radius of a circle of equal area. (2.2.8.6) rawin - a method of measuring upper-air winds by tracking a balloonborne target \vith radar, or radio direction-finder. Possesses the advantage over the earlier visual tracking of pilot balloons in that observations are not limited by clouds or precipitation.

saturation adiabat - a curve on a thermodynamic diagram depicting the saturation adiabatic lapse rate.

saturation adiabatic lapse rate - the theoretical rate at which the

temperature of a rising saturated air parcel decreases, with the following assumptions. (1) adiabatic, that is no heat exchange by radiation or conduction between particle and environment. (2) water vapor in excess of saturation immediately condenses to liquid. (3) latent heat released by condensation warms the air. The saturation adiabatic lause rate is normally closely approximated in clouds of marked vertical development such as cumulus, cumulonimbus, or deep layers of altostratus.

sequential maximization - reducing the observed elapsed time between

storms to develop a hypothetical severe precipitation sequence. (2.4.1.5)

*

CHAPTER 1

spatial maximization - reducing the distance between precipitation storms or storm bursts to develop a hypothetical severe precipitation sequence.

(2.4.1.5).

specific humidity - the dimensionless ratio of the mass of water vapor to the total mass of humid air.

q

=

P

where q is specific humidity, P atmospheric pressure, e vapor pressure, and .622 is the ratio of the molecular weight of dry air. Specific humidity is also given- in g k -1, and is then 1000 times the above value. For most practical purposes may be interchanged with mixing ratio.

composite maximization - developing hypothetical severe precipitation events by joing together storms or storm bursts. Comprised of sequential maximiza-

tion and spatial maximization.

storm transposition - moving a storm from its place of occurrence to a basin under study in representation of a possible future storm at the latter

location.

wet-bulb potential temperature - the temperature an air parcel would have if cooled dry adiabatically from its initial state to saturation, and thence

1

brought to 1000 mb. by a saturation-adiabatic process. The wet-bulb potential temperature is constant along a saturation adiabat, and thereby may be used as a label for such a curve. Same as 1000-mh. dew point in many hydro- meteorological writings.

CHAPTER 2 MAXIMlThi RAINFALL

2.1 Physical Models of Rainstorms

2.1.1.1 Two rainstorm models are described here, a general model and a model for orographic rainfall on the windward side of mountain ranges.

Further details on the first model relating to moisture maximization of storms are found in chapter 2.4. A list of definitions of terms used in chapters 2.1 to 2.6 is found in section 2.1.4.

The convergence model

2.1.2.1 The convergence model focuses attention on the following three properties of precipitation storms: (a) humid air converges quasi- horizontally toward the storm area; (b) the humid air rises; (c) the humid

air cools by adiabatic expansion, forcing water vapor in excess of saturation from the gaseous to the solid or liquid form. This general model applies to all scales of storms from the individual thunderstorm to the large-area rain associated with a tropical or extra-tropical cyclone.

2.1.2.2 The theoretical interrelationship of convergence, vertical motion,and condensation is known. To whatever precision either the convergence at the various levels in -the atmosphere or the vertical motion should be kno\vn or assumed, averaged over some definite time and space, the other could be calculated to equal precision from the principle of continuity of mass. The yield of precipitation by'adiabatic cooling of air of a certain water vapor content is also knmvn to a high degree of precision. Observations confirm that the theoretical saturated

10

ESTIMATION OF MAXIMUM FLOODS

adiabatic lapse rate of temperature of ascending saturated air from which precipitation yield is calculated is closely approximated in

deep, precipitating clouds. The higher the specific humidity, the greater the precipitation yield for a given decrease in pressure. Thus the model clarifies the concept that intense rainfall over a basin results from the combination of intense rate of convergence of air (or maximum vertical motion) and high water vapor content. The extreme rainfall would result from the extreme combination.

2.1.2.3 There is a problem in estimating maximum

rainfall with th"e convergence model. Maximum water vapor content of the air can be estimated with acceptable reliability for all seasons for most parts of the world by appropriate interpretation of climatological data. But there is neither an empirical nor a satisfactory theoretical basis for assigning maximum values to either convergence or vertical motion. Direct measurement of these variables has been elusive. The solution to this dilemma is to use observed

rainfall as an indirect measure of convergence and vertical motion.

Extreme rainfalls are the indicators of maximum rates of convergence and vertical motion in the atmosphere. The convergence and vertical motion are jointly called the precipitation "mechanism".

2.1. 2.4 Extreme "mechanisms" from extreme storms are then transposed to basins under study without the necessity of calculating the magnitude of the convergence and vertical motion explicitly.

Rather, the observed rains in storms are adjusted to values over the basin by attention to the following questions. (a) Can each

observed storm be transposed to the study basin, that is, can the

. .

"mechanism" which produced the storm be shifted to the basin? The

answer to this is found i n a synoptic meteorology approach, discussed in chapter 2.3 on storm transposition. (b) Dpon transposing an

observed storm to the study basin, what is the maximum moisture content of the air that the transposed mechanism could be expected to operate upon to produce precipitation? How much would the precip- itation in these circumstances exceed that observed in the actual storm? This adjustment is calculated from the phvsics of the moist adiabatic process and is discussed in chapter 2.4.

Cc)

sufficient number of intense rainstorms must be transposed and adjusted to the basin and the resulting adjusted storm rainfall

magnitudes enveloped. The difficult question of what is "sufficient"

is discussed at the end of chapter 2.4.

2.1.2.5 The most simplified technique for carrying out the process described in the preceding paragraph is to divide the precipitation in a storm (in tilillimeters) by the precipitable water in the surrounding air (also in millimeters) and obtain a dimensionless ratio that is a measure of the efficiency with which the "mechanism" produces precipitation from water vapor. Various names have been applied to this ratio. In some reports of the D.S.

Weather Bureau (24),(30), this is called a P/M ratio, standing for

"precipitation/moisture." A "P/H ratio" thus determined is related to a specific duration, location, and area of the rainfall value

used for IIp''. P/M ratios may be smoothed and enveloped geographically, seasonally, and over storm duration, to obtain characteristic maximum values. Multiplication of a maximum ratio of this nature by the

12

ESTIMATION OF MAXIMUM FLOODS

expected maximum precipitablewater then yields expected maximum precipi.tation.

2.1.2.6 The "plM ratio^{l l} procedure has been applied primarily to using extreme observed point rainfaLl-s to estimate maximum rainfall for small-size basins. The more detailed -con- sideration of transposition and maximization of storm precipitation in sections 2.3 and 2.4, respectively, is based on the same general principles.

2.1. 2 The orographic model

2.1.2.1 Many important water-development projects are in mountainous areas where precipitation is much heavier than over adjacent lowlands. The increase in precipitation on the windward slope of a mountain chain is accounted for by two effects. One is simple lifting of the wind stream by the mountain slope. The other is the stimulation of convection in an unstable air mass by an initial small lift. Both effects may be present in a single storm. The

latter, the triggering effect, dominates in quiescent flows in tropical regions while the former, the direc·t lifting effect,

dominates where winds are strong, as in typhoons and in well-developed extra-tropical cyclones. The orographic model assumes laminar flow of the wind stream and accounts for the precipitation from the direct lifting effect. The simultaneous stimulated convective precipitation, if present, must be accounted for _sepa~ately (see para. 2.4.7.4).

2.1.2.2 The orographic model considers the flow of air in a vertical plane at right angles to a mountain chain or ridge! It is 1vhat is termed a IItwo~dimensional" modeL The plane has an "x- coordinate" in direction of flow and "z-coordinate" in the vertical.

The flow may represent an average over a few kilometers or tens of kilometers in the transverse, or "y", direction, 1iJhich does not

~pear explicitly in the model. The wind at ground level moves along the ground. The slope of the air streamlines decreases upward, becoming horizontal at some great height, called the nodal surface.

An example of this model is shown in figure 2.1.

2.1.2.3 The orographic model provides a means of computing the precipitation that would result from laminar flow, where the inflovl speeds and specific humidities at various levels above the foot of the mountain are known or assumed. The model is tested in observed storms and then applied to estimate maximum rainfall. The steps are:

(a) Adopt a simplified ground profile from topographic maps.

(b) Assign· a pressure to the nodal surface.· 300 mb. is recommended for mountains of greater than 2,000 meters elevatio.n.

(c) Divide the flow into equal layers, as shown in figure 2.1.1, each bounded by two streamlines

(d) (e)

Note the, location of the freezing level.

*

From rawin observations, or by estimate, assign a mean inflow speed, V, to each layer. V is the component toward the mountain where

V

=

Vt being the total 1vind speed and a the angle between 1vind direction and orientation of the cross section.

(f) From radiosonde observations, or by estimate, assign a mean inflow specific humidity to each layer.

*

~t-3 H~

t-3H

~

~

S

~

g

tn +=>-I-'

----

~

-

I

s...---

800

1;0 I ^{. ; 0 -}

900

I.. ^I

5001 ..

I _~<:::

,>- I

4001~-F';;:~

^{- - - -}

I- ^.

^I

300n ) J , , , )11, • i i i i i i "i i ) l i

o

1= x,-I

I

L.1

,,,,,,,,,,,,,,",,»>>m"'''''''''''' I I

1000

~ ^6oor:t1 ~---:---.::~::s~_-.--~_..:.-_~

~ ~ ^I

::::> 0

Cl) ....

Cl) LLZ

w -I

D:': .

a..

700 -- ^---

Figure 2.1 - Schematic diagram of orographic model

(g) Calculate the rate of precipitation generation within the layer from

where

R

t

x

(2.2)

R rainfall in centimeters in time t in hours.

V wind speed in km/hr. at inflow.

P depth of layer in millibars at inflow.

q ,q

=

respectively. q is found from qa on an adiabatic chart by proceeding e

up a moist adiabatic from the inflow pressure to the outflow pressure atcenter of the layer.

g acceleration of gravity (980 cm sec).-2 p density of water

=

X horizontal distance from foot to crest of mountain, km. The total precipitation is the sum of that generated in the several layers.

2.1.2.4

calculated distribution of the precipitation along the slope is obtained by constructing trajectories of the precipitation - rain or snow -

from point of formation down to the ground (figure 2.1). Each segment of a trajectory is the vector sum of the wind and the assumed terminal

*

velocity of the raindrop or snowflake as in figure 2.2. Snow falls

*

Acceptable averages are about

6

1.5

(24

53-54).

16

much more slowly than rain ~nd drifts greater distances as it descends.

The freez.ing level is use.d to divide the snow zone from the rain zone.

Figure 2.1 shows, as an additional refinement, an intermediate wet snow zone of intermediate terminal velocity.

2.1.2.5 Having constructed precipitation traject"ories, the specific humiditiesare determined at the intersection of the trajectories with the midpoint of each layer. This may be done by reference to a

specific humidity vs. pressure curve reconstructed on the graph, as in figure 2.1, if a single moist adiabat characterizes the humidity distribution of the entire inflow column. Otherwise the specific humidities are scaled from an adiabatic chart

**

Possible calculations of precipitation yield byeqtiation 2.1 with various specific humidity differences are:

Specific humidity difference gives

Total precipitation formed' in layer.

Precipi tation falling on wind'vard slope.

Precipitation "spill over" to lee slop,e.

Precipi tation reaching ground betvleert C and D.

The appropriate X must be used in each calculation.

**

SNOW TERMINAL VelOCITY

RAIN TERMINAL VELOCITY

SNOW

TRAJECTORY

RAIN

TRAJECTORY

Figure 2.2 - Construction of raindrop and snowflake trajectories

18 ESTIMATION OF MAXIMUM FLOODS

2.2 Analysis of Storm Rainfall Data

2.2.1 The need for volumetric rainfall data. Rainfall is measured, and tabulated in the usual climatological records, at

isolated points on the surface of the earth. Floods, however, result from substantial volumes of rain spread out over a substantial fraction of a basin or all of it. Thus any appraisal of storm rainfall for the purpose of estimating flood magnitudes is concerned with rainfall volumes, expressed as average depths (in millimeters or inches) over specified sizes of area (in square kilometers or square miles) falling in specified intervals of time.

2.2.2 Depth-duration-area values. Point rainfall measure- ments are commonly accepted as presenting the average depth over a few squalre kilometers. For larger areas,valumetric storm rainfall values are obtained by an integratic;m of point rainfall values. Usually, the largest. values of precipitation averaged within selected sizes of area and in selected durations within a storm are abstracted from the complete array of such depth-duration-area values (commonly abbreviated DDA values) and are presented in graphical or tabuclar form as the

principal end product of the analysis.

2.2.3 Treatment of analvsis of storm rainfall data in

.

this Note. This section of the Technical Note is restricted to discussing the p-urposes and characteristics of storm rainfall data in the DDA. form, as the WMO is issuing a separate manual describing in detail the procedures for computing such values. The purposes and characteristics of DDA analyses can be clarified by a review of some of the history of their development. Certain developments in the United States of America are reviewed in sections 2.2.4 and 2.2.5

because they illustrate approaches to problems that have been found

wid~ly applicable.

2.2.4 Development of Methods

2.2.4.1 Floods provoke needs. In March 1913, a great flood struck the Miami River in the State of Ohio of the D.S.A. with a loss of more than 360 lives. The physical damage by the flood was very great in that highly industrialized valley of. some 15,000 sq. km.

To prevent any recurrence of such flood damages, the valley residents and the State Government of Ohio developed a cooperative enterprise, the Miama Conservancy District, to design and construct flood control works.

To understand the flood risks to which the Miami Valley was exposed and to compare costs with benefits for various flood control plans, the Conservancy District felt, according to one of their reports (21, page 1) "it was necessary to determine not only the largest flood that could ever possibly' occur, but also, so far as

possible, the frequency of all smaller floods which would cause damange."

It was soon realized that examination of the 20 years of discharge measurements on the Miami River, supplemented by historical experience

going back another 80 years, did not clarify the relative magnitude of the maximum flood that could "ever possibly occur" in comparison to the 1913 flood. Nor did the rainfall records within the Miami Basin, which extended back a few more years than the discharge records, offer much additional guidance. Clearly what was needed was a climatology of rainfall volumes derived from data outside the Miami Basin

as well as within. This the Conservancy District set out to compile.

The engineers of the District were interested not only in the known

20

maximum volumes of 'storm rainfall, but also in the fYeq-uency, the' seasonal variation, and the geographical distribution of storm rainfall volumes.

2.2.4.2 Storm selection. The first task in establishing the climatology of rainfall volumes in a region is to select pertinent storms. This the Miami Conservancy District did by reviewing all rainfall records of the D.S. Weather Bureau, as well as other sources, and listing all storms fulfilling a particular criterion in the United States east of the Rocky Mountains (east of 103⁰W) during the years 1892-1916; A criterion appropriate to their needs was that for a storm to be selected, each of five or more adjacent precipitation observing stations should experience a three-day precipitation total of at least 6 inches (152 mm.). One hundred and sixty such storms were found; later 120 additional storms for the years 1917-1933 were

added. Approximately 70 of the largest storms were subjected to the depth-duration-area analysis described below.

2.2.4.3 Conservancy District method of DDA ~nalvsis. With three variables - depth, duration and area - it is necessary to fix either duration or area and then consider the concomitant variation of the other two. As the rainfall data were already broken into duration increments - daily values - the Miami Conservancy District chose to depict the depth-area variation pertaining to fixed duration increments. The steps for each storm, after assembling the rainfall data, \\Tere:

1. Determine the day of greatest average precipitation, consecutive two days of greatest average precipitation, and so on to five consecutive davs.

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liE

MAXIMUM 3 DAYS

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....-..-...'\

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MAXIMUM DAY

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Figure 2.3 - Isohyeta1 maps, storm of March 24-27,· 1913. From (21) ^I\)I-'

22 ESTIMATION OF MAXIMUM FLOODS

2. Plot a map of the one-day preciuitation, a seoarate ~ap of the two-dav urecipitation, etc., to five davs. Construct isohyets on these precipitation charts. The resulting five charts for the March 1913 storm are duplication in figure 2.3 from the Conservancy District report (21). Larger-scale charts, of course, ,-,ere used in the original computation.

A uroblem here was that observational hours were not uniform. Some stations made a daily measurement in the morning,

others in the evening, and a few, the principal meteorological stations, at midnight. No simple manner was found to adjust for this and amounts reported on a given day were collected on one map. The longer the duration (e.8" the thre-, four-, and five-day maps) the less the significance of this time discrepancy.

3. ~easure the areas enclosed bv each isohvet on each map with a planimeter, and calculate the average denth within the area enclosed by each isohyet. The exact manner of making this calculation is explained in reference (26), in abbreviated form in references (16 and 37), and will be covered in detail in the \~10 manual on DDA analvsis.

4. Plot curves of area vs. avera~e depth within the area. One curve is obtained from each isohyetal map. If an isohyetal map has two or more centers, thus having two or more average rainfalls for a narticular area size, the larger of the values of the rainfall is ulotted. Thus the maximum volumetric characteristics of a five- day storm are _sum~arized in five curves. These are shmvn for the 1913 storm in figure 2.5, adapted fro~ (21).

The Hiami Conservancy District T\7ork, the first comuilation

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CINCINNATI, OHIO" I _{7.5 IN.}

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(RECO-RDER)

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Figure 2.4 - Mass curves of rainfall, M~rch 24-27, 1913.

Cincinnati il:? a recorder st3.tion. The other two curves are based on daily meaxurements at

1

5

patterned after Cincinnati.

24

ESTIMATION OF MAXIMUM FLOODS

of volumetric rainfall of the United States, was successful in

indicating the broad outlines of storm magnitudes and their seasonal and geographical variations and in forming the background (along with other considerations) against which to judge the severity of the 1913 flood and estimate the possibility of a larger flood.

2.2.5 Later developments in the U.S.A.

The assumption of greater responsibility in flood control by the Federal Government of the U.S.A. with passage of the Flood

Control Act of 1936 led to the development of a National Storm Study Program under the primary sponsorship of the Corps of Engineers, the federal agency with the most direct responsibilities for flood control.

Under this program about 600 storms throughout the United States have been analyzed in a uniform manner and summary sheets distributed to g9vernment agencies and the engineering _pro~ession (7). Other countries with similar needs have inaugurated similar programs;

for example, Canada (4).

2.2.5.1 The mass curve method. In the new program much of the Miami Conservancy District analysis method was retained but certain changes were made. Because the Miami River is not a flashy stream and most flood-producing storms there last at least a day, a one-day time breakdown of rainfall was adequate - or at least almost adequate - for studying the flood characteristics of that basin; furthermore, rainfall data was available only in one-day

increments. But for smaller basins, and more rapidly flowing streams, a breakdmvn into smaller time units is essential for determing storm hydrograph characteristics. Six-hour time increments meet most needs, and 1vere adopted as standard for DDA analysis. To provide

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Figure 2.5 - Maximum depth-area curves for various durations.

~ame storm as figure

2.3.

2

100,000 10,000

1,000 500

200

o

" " ,

100

AREA (SQUARE MILES) \J1^I\.)

26 ESTIMATION OF MAXIMUM FLOODS

such a time breakdown in storms yet to come a network of recorder stations ~vas established, comprising about 25 percent of the total recording and non-recording gauge net,vork. (Net'vorks are discussed in W}10 Technical Note No. 25(38).)

To estimate 6-hr. DDA values from point rainfall values, all or many of which are read but once a daY,requires both space .and time interpolation. In the mass curve method the time inter- polation is accomplished by the mass curve, described in paragraph 2.2.6. A mass curve is constructed for each rainfall station.

By way of compensation for the labour in constructing all the mass curves, the procedure. requires the construction of .only one isohyetal map, based on precipitation amounts accumulated

for the total duration of the storm. Fon computational purposes, the map is divided into zones, each containing an isohyetal center·.

The total storm precipitation over each zot:le, expressed as depth....area values, is divided into time increments in proportion tio the mass curve distributions averaged for groups of stations. A detailed procedure for this is explained in (26) and will be contained in

the WMO manuaL

2.2.6 Mass curves

A mass curve is a plot of accumulated depth (or "mass") of precipitation vs. time. Examples are shown in figure 2.4.'Plotting of key mass curves at rainfall centers is a convenient method fo~

, depicting the time distribution of the precipitation. Another use of mass curves is to provide the time interpolation needed in depth- duration-area analysis by the method described in paragraph 2.2.5.1,

,^.

to break daily precipitation measurements into smaller time increments.

In constructing mass curves for such an analysis, the analyst considers all possible clues. These clues include comparison with adjacent

recorder mass curves, noting of any times of beginning and ending of precipitation or miscellaneous corrnnents (such as "rain heaviest in the afternoon") on observational forms, and weather maps. When the rainfall can be associated with synoptic features that are depicted on weather maps, these in turn give clues to the time distribution of the rainfall 'and progression of rainfall centers through the storm area. These techniques have been surrnnarized in a report (22).

One mass curve of figure 2.4 depicts the trace from a recorder (Cincinnati). The 'other two mass curves, from stations with daily measurements at 7a.m. and 5.p.m. respectively, are

constructed by taking the recorder chart observation as a guide.

2.2.7 Isohyetal charts

2.2.7.1 Flat terrain. In flat terrain isohyets are generally drawn smoothly, interpolating between stations. The interpolation should not be excessively mechanical.

2.2.7.2 Mountainous terrain. In mountainous regions

the simple interpolation technique would yield unsatisfactory isohyets.

Yet to prepare a valid isohyetal pattern in a mountainous region is not easy. One commonly used procedure is the isopercental technique, excellent under certain limited conditions stated in the next paragraph.

This method requires a base chart of either mean annual precipitation, or preferali1.y mean precipitation for the season of the storm, such as winterr summer, or monsoon months. In this method the ratio qf

28 ESTIMATION.OF MAXIMUM FLOODS

the storm precipitation to the mean annual or mean seasonal precipi- tation (base precipitation) is plotted at each station. Isolines are drawn smoothly to these numbers. The ratios on the lines are then multiplied by the original base chart values at a large number of points to yield the storm isohyetal chart. Thus the storm

isohyetal gradients and locations of centers tend to resemble the features of the base chart, which in turn is influenced by terrain.

The first requirement for success of the isopercental technique is that a reasonably accurate mean annual or mean seasonal precipitation chart be available as a base. The base chart is of more value if it contains precipitation stations in addition to

those reporting in the storm than if both charts are drawn exclusively from data observed at the same stations. The value of the base chart is also enhanced, in regions where the runoff of streams is a large percentage of the precipitation, if the precipitation shown on the chart has been adjusted not only for topographic factors, but also adjusted to agree with seasonal streamflow. In regions where a large percentage of the precipitation evaporates adjustment to runoff volumes would be of dubious value.

An additional requirement for success of the isopercental technique is that most of the annual or seasonal precipitation in the region result from storms with relatively the same wind direction, and from storms with minimal convective activity. Under these

circumstances an individual storm will have a strong resemblance to the mean chart, as the latter is an average of kindred storms.

In the Tropics with the dominance of convective activity and with lighter winds, the isopercental technique is of less value

~n analysis of an individual storm than in middle latitude locations that meet the other requirements.

If the isopercental technique is of limited application because of the above problems, often the best that can be done to construct an isohyetal chart in a mountainous region is to overlay the storm isohyetal map on a topographic map - 1:1,000,000 is generally a good scale - and make a conjecture of the probable topographic influences on the rainfall in ungauged regions. An

intimate knowledge of the meteorological aspects of the storm

rainfall in the region will assist greatly in this kind of estimation.

However; a little experience will convince anyone of the necessity for more precipitation gauges at relatively inaccessible high- elevation sites throughout most of the world for an adequate definition of the rainfall regime for hydrologic purposes.

2.2.8 Presentation depth-duration-area data

2.2.8.1 DDA arrays. The table or graph of maximum depth- duration-area values is the most common method of summarizing the volumetric characteristics of both real storms and hypothetical storms for design. Figure 2.5 is such a graph. Another example is found in figure 2.6, which illustrates the now common use of a 6 -hour time unit. There are other methods of presenting DDA information suitable for certain purposes. The main tvpes of curves are described below.

2.2.8.2 Hveto~raph. A hyetograph is a plot in chronolog- ical seauence of the increments of preci~itationeither at a station or averaged over a designated area, in equal time periods of an hour, six hours, or a day depending on the scale of interest. An

30

ESTIMATION OF MAXIMUM FLOODS

MAXIMUM DEPTH - AREA CURVES (AVERAGE FOR AREA SPECI FI ED)

6

4

2 8 24

14

12 22

18 20

16

10

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100,000 10,000

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Figure 2.6 - Maximum depth-duration-area values. Storm of January 12-16, 1961, centered at Seymour Falls, British Columbia, Canada. From

(4).

estimate of probable maximum precipitation (PrW) over a basin may be broken down in a similar manner. (The sequencing requirements for

the latter are discussed in section 2.5). Hyetographs are commonly a prelude to runoff calculations. An examnle is found in fieure 2.7.

2.2.8.3 Depth-duration curve. A depth-duration curve shows maximum values of storm precipitation for various durations either over a fixed area, such as

a

PMP ~alues for fixed basins are commonlv presented by a depth-duration curve.

It might be noted that in estimating the probable maximum precipitation that will lead to the maximum flood over a basin the sequence of operations is normally reversed from the analysis of a historical storm; namely, the depth-duration curve is worked out first and then broken down into a hyetograph.

2.2.8.4 Depth-area curve. A depth-area curve shows maximum values of storm precipitation over various area sizes

either for some fixed duration or for the total storm. (The individual curves of figures 2.5 and 2.6 are depth-area curves. Collectivelv they form the DDA array).

The depth-area curve for an intense thunderstorm is found in figure 2.8, together with curves for the same storm of the types next described.

32 ESTIMATION OF MAXIMUM FLOODS

6P 26

6P 6A 25

6P 6A 24

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Figure 2.7 - Hyetograph (upper) and maximum depth-duration curve (lower) for recorder station of figure 2.4.