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WORLD METEOROLOGICAL ORGANIZATION

TECHNICAL NOTE No. 98

ESTIMATIONOF 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

1969, World Meteorological Organization

NOTE

The designations employed and the presentation of the material in this publication do notimply the expression of any opinion whatsoever on the part of the Secretariat of the WorldMeteorological Organization concerning the legal status of any country or territory or of itsauthorities, or concerning the delimitation of its frontiers.

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

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 countriesof the worJ,.d as possible. It was perhaps inevitable that the majority of examples would bedrawn from those countries whose experts were members of the Working Group. The reader willnote, however, that there has been a conscious effort to include references and examplesfrom other countries as well. It was also inevitable that, for solving some problems, morethan 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. .Inparticular, 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 hydrometeorologistsinmany countries will benefit from this summary of techniques, both physical and statistical,for estimation of design floods.

J. P. Bruce (Chairman)

v

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 Introduction................................................................. 117

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

3.3 CriticaJ,. snowmelt rates...................................................... 126

3.4 Rain on snow events.................................................... .. ..... 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 166meteorological input ......................................................

CHAPTER 5 - STATISTICS OF EXTREMES

5.1 Introduction and theory .................................................. 183

5.2 Practical applications - the maximum likelihood solution .... .... ..... ... .... 1965.3 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.

........................................................

......................- .Small impervious areasMultiple influences in streamflow frequencies for natural basins ..........

Historical series method

Historical series method for very large basins ....................

Joint probability method

7.17.2

7.3

7.4

7.57.6

Annexes

I. Procedures Used in U.S.S.R. for Computation of Maximum Discharge ofSnowmelt Floods with Little or No Hydrometric Data ..... ........ 269

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

VII

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. Themembers 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 hadseveral 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 ofthe working group and to the other individuals who have assisted the group in the preparationof 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, theformer 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 onmethods of evaluation of meteorological conditions for estimation of maximum floods.

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

RESUME

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

La plus grande partie de la note (chapitres 2 a 4) decrit les methodes appliqueepour estimer les valeurs extrgmes de la hauteur des precipitations et de l' eau de fonte deEneiges sur la base d'une analyse physique, ainsi que les methodes utilisees pour convertirces valeurs en estimations des debits maximaux decrue. La note traite ensuite (chapitreset 6) des methodes statistiques et de leur application a Itetudedes averses et des crueEElle expose les fondements de Itanalyse statistique et decrit brievement certaines tech-niques utilisees dans divers pays pour analyser la frequence des crues. Le dernier chapitrexplique comment il est fait usage des donnees meteorologiques pour estimer la frequencedes crues.

PESIOME

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B STOM pa3.n;eJIe .n;aIOTC.a: TaKlKe OCHOB~ CTaTHCTHtleCKOrO aHaJIHSa HOIIHC~BaIOTeH HeKOTop~e MeTo.n;H, HCIIOJI:b3yeM~e BOT.n;eJI:bH~X CTpaHax IIpH aHaJIH3eIIOBTopHeMOCTK IIaBO,ll;KOB. IToCJIe.n;HHH rJIaBa IIOCBHmeHa IIpOOJIeMe HCIIOJI:b30BaHH.R:MeTeOpOJIOrHtleCKHX .n;aHHHX .n;JIJI o~eHKH IIOBTopHeMOCTH rraBo.n;KOB.

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 describenlos metodos utilizados para estimar los valores extremos delluvia y nieve fundida fundan-dose en el an~lisis f1sico, as1 como los procedimientos que se aplican para convertir estosvalores 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 estad1sticoy 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 meteoro16gicospara la estimaci6n de la frecuencia de las crecidas.

1

CHAPTER I

INTRODUCTION

1.1 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

2 ESTIMATION OF MAXIMUM FLOODS

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

4 ESTIMATION o.FMAXIMUM FLOODS

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 - Vh

where Vh

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 = .622 ep-e

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.

1gp f qap

where w = precipitable water, g is acceleration of gravity, q specific

humidity, P pressure, and a density of liquid water. One set of consistent


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

-2 -3cm sec for g, and gm cm for d -2 -3g = 980 cm sec ,d = 1.0 gm cm

probable maximum precipitation - "The theoretical greatest depth of

precipitation for a given duration that is physically possible over a

particular drainage area at a certain time of year."*

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)

* from Glossary of Meteorology, American Meteorological Society, Boston,Nass., USA, 1959.

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.

eq = .622 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.

9

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


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

CHAPTER 2

answer to this is found ina 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

11

adiabatic process and is discussed in chapter 2.4. Cc) Hhat assurance

is there that a maximum "mechanism" has been introduced by this indirect

'process of transposing and adjusting rainstorms? To ensure this, a

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


expected maximum precipitablewater then yields expected maximum

precipi.tation.

2.1.2.6 The "plM ratioll 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 direct 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.

CHAPTER 2

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

13

(d)

(e)

Note the, location of the freezing level.

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

V = Vt cos a (2.1)

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.

* Defined in section 1.2

~t-3H

~t-3H

~

~

SH

~~ogtn

I-'+=>-

----~ -Ias.....---8001;0 I .;0 -

900 I.. IAIR STREAMLINE )I -

5001 .. I ~- I

4001~-F';;:~ ---- I-~ ~ .~ I300 n)J, , , )11, i i i i i i" i i)l i

o

1= x,-II

L.1 ;0 ,,,,,,,,,,,,,,",,>>m"'''''''''''' I I. ,;;;;;,;;, ;;1000 .~ 11"" "". . X DISTANCE ~

~ 6oor:t1 ~-------:---.::~::s~_-.--~_..:.-_~~ ~ I::::> 0Cl) ....Cl) LLZw -ID:': .a.. 700 -- ---------

Figure 2.1 - Schematic diagram of orographic model

CHAPTER 2

(g) Calculate the rate of precipitation generation within the

layer from

15

where

Rt 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.

-1q ,q = specific humidity of air in gm kg at inflow and outflml7

a e

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

up a moist adiabatic from the inflow pressure to the outflow pressure

atcenter of the layer.

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

-3p density of water = 1.0 gm cm

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 Distribution of precipitation along slope. The

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

* Terminal velocities vary with raindrop and snowflake dimensions.Acceptable averages are about 6 meters per second for raindropsand 1.5 meters per second for snowflakes (24 p. 53-54).


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 by lifting the inflow

air of each layer to the appropriate pressures. The air is lifted

dry adiabatically to saturation, then along a saturated adiabat.**

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 "spillover" to lee slop,e.

Precipi tation reaching ground betvleert

C and D.

The appropriate X must be used in each calculation.

** Defined in section 1.2.

SNOWTERMINALVelOCITY

RAINTERMINALVELOCITY

CHAPTER 2

SNOWTRAJECTORY

RAINTRAJECTORY

17

Figure 2.2 - Construction of raindrop and snowflake trajectories


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

CHAPTlm 2

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.

19

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 1030 W) 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.

$'

I\)

(')

liE'1:lt-3

~

MAXIMUM 3 DAYS"l-V')wI-' 4w0:::(9

........

Figure 2.5 - Maximum depth-area curves for various durations.~ame storm as figure 2.3. Adapted from (21).

2

100,00010,0001,000500200o )~ "" , I , , I I I , , I I I100

AREA (SQUARE MILES) I\.)\J1


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.

CHAPTEli 2 21

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 winter r 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

CHAPTER 2

~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~itation either 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

29


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

6

4

2

8

24

14

12

22

18

20

16

10

o~OO,OOO100,00010,0001,000100

..

__126 Hours

1--72 Hour: -- ....... :""00 ~1...... 0/SI'- ~;._48 tours~~

-.' 1'000 "_36 Hours "I ,"'- "-30 Hours , "1"- , ", , "-24 ~ours "- , \.

" "-- " \. \., '" "", ,...... " '\.~18 Hours , I", " '\."1"-...... ",

"'..... ,r--12 Hours

,"f- , I" "- '", '"" ""- I" ,""- '"'"" "

::::=::-6 HO\Irs..... , - I

I',-"- 1"-, .....

,'0

VILU

:t:UZ

z

L&-

o

zoI--

CHAPTER 2

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 river basin, or at a single

station. It is constructed from the hyetograoh (or its tabular

counterpart) by plotting the most intense 6-hr. amounts at 12 hours,

etc., to the total storm depth at the total storm duration. An

example is given in figure 2.7.

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.

31


6P26

6A6P25

6A6P24

6A

I I I ., I I I I I I I

- -- -

I- -

- -

f- -r ""lI

3

o

(j)2w~UZ-Z--clt-3

~

16144 6 8 la 12AREA (lOO'S KM 2 )

2o I I I --I I I I I I

RAIN PROFILE

2 4 6 8 la 12 14 16AREA (lOO'S KM 2 )

200

150.........~

$~ 100>-:c0Vl

50

o I I I I I I I I ,

o I 1 I 1 i I I ---I I I I --s.12 4 6 8 la 12 14 16 18 20 22

EQUIVALENT RADIUS (KM)

Figure 2.8 - Depth-area curve, isohyet-area curve, and rain profilefor thunderstorm at Vallecito, California, U.S.A.,Juiy 18, 1955. Storm duration is less than two hours ww

34 ESTIMATION OF MPJCIMUMFLOODS

2.2.8.5 Isohyet-area graph. The areal variation of storm

precipitation, for a given duration, is sometimes portrayed by plotting

isohyetal vahies, vs. the area inscribed by the isohyets. Thus 50

mm opposite the area encompassed by that isohyet, etc. (The depth-

area curve requires plotting the average depth within the 50 mm

isohyet vs. its area, etc.) This type of curve does not yield

volumes of rainfall directly but is useful in comparing the areal

extent of storms and other features. The isohyet-area graph goes

to zero ata definite area bounded bv the outer limit of precipitation,

not a property of the depth-area curve. (Example in fig; 2.8).

2.2.8.6 Rain profile. A rain profile is a variation of

the isohyet-area graph in which the isohyetal values are plotted

agai~st eguival~nt radius instead of area. The equivalent radius,

r is the radius of a circle containing the same area as the isohyet,e'

and is approximatel an average radius for the isohyet. It is

calculated by re = A!rr, where A is the area within the isohyet. Rain

profiles .areuseful for studying the areal characteristics of intense

local storms. Interest in rainPTofiles began long ago when engineers- - ,

faced the problem of design of storm dr:ains with point rainfall as

their primary basic data. Fruhling in 1894 proposed a-formula

to describe the rain profile of typical storms innorth~central

Europe (11). Other efforts to define rain prOfiles by niflthematical

formulae have been summarized comprehensively by Court (8~.;

(Example in fig. 2.8).

2.2.9 Radar

Radar has given man his first mapped view of storm rainfall-

by direct observation. Efforts are being pursued in many countrie~

CHAPTER 2 35

to adapt radar to yield, on an areal basis, quantitative measures of

the precipitation that is falling. These efforts are faced Hi th the

problem that, for a given number of raindrops, the radar reflectivitv

varies as the sixth power of their diameters while the rainfall volume

is prolJortional to the cube of the diameters. Also eaual masses

of frozen and liquid precipitation have greatlvdifferent reflectivities.

Another, and less difficult problem, is that radar presents

an instantaneous picture of reflectivity, ~"hilefox hydrologic

purposes the rainfall is needed over some. interval of time, at least

an hour and often much longer . Hethods are being developed to inte-

grate the radar return over time automaticallv and present an inte-

grated picture (numerical values) of 'vhat the radar has seen durinq

some interval of time (39,40).

Pending solution of these problems, radar ~'lhen available

is a qualitative aid to extending isohyets from gau8ed into ungauged

areas and to interpolating between stations.

2.3 Storm Transposition

Definitions

2.3.1.1 The outstanding rainstorms in a region surrounding

a basin are a very important part of the historical evidence on which

an estimate of maximum rainfall over the basin is based. Mcving these

storms to the study basin is called storm-translJosition.

2.3.1.2 Transposition limits is the name given to the outer

boundary of a region throughout which a storm mav be transDosed with

only minor modifications to its rainfall magnitudes. The area ~ithin

the transposition limits has similar, but not identical, climatic

and topographic characteristics throughout. :fore restricted


transposition limits may b~ defined if a region has a long record

of a reasonably dense network of precipitation stations and has

experienced several severe torms. Where the record of storms is

more limited, either from lack of observing points of lack of

occurrence of severe storms with the period record then more generous,

though necessarily less precise,transposition limits must be accepted.

2.3.1. 3 A transposition adjustment is a ratio by ,.,hich

the precipitation magnitudes in a storm are multiplied when it is

transposed to compensate for variations in climatic or topographic

conditions.

Discussion of transposition

2.3.2.1 Transposition is not peculiar to estimates of

maximum rainfall but, in fact, is inherent in most uses of climatolog-

ical data. Meteorological elements of all kinds are measured at

individual stations around the world. The climate of intervening

areas is inferred from these observations; that is, the observations

are transposed by interpolation between points. Transposition is

particularly important in estimation of maximum rainfall because of

the necessity of extracting as much information as possible from the

known major storms. The record of major storms is limited not only

by the number and spacing of observational points but by chance

occurrence of these extreme events.

Steps in transposition

2.3.3.1 The storm. The first step in transposing a storm

is to identify clearly when and where the heaviest rain in a storm

fell and the approximate causes in terms of synoptic meteorology. An

isohyetal chart, one of two key mass curves, and weather maps serve

37CHAPTER 2

these purposes. The isohyetal chart may be a simple one since its

primary function is to identify the storm location. Routinely

available weather maps may be sufficient to identify the storm causes,

particularly if the precipitation is closely associated with either

a tropical or an extratropical cyclone. In other instances a detailed

analysis may be necessary to identify causes.

In the Tropics it is often difficult to associate

precipitation clearly with features on the available weather maps.

2.3.3.2 Region of influence of storm type. The second

step is to delineate the region in which the meteorological storm

type identified in step I is both common and important as a producer

of precipitation. This is accomplished by survey of a long series

of weather charts. The daily Northern Hemisphere weather charts

(23) are suitable for this purpose over much of the Northern Hemisphere

outside the Tropics. Tracks of tropical and extratropical cyclones

are generally available in published form to indicate the regions

in which these storms are frequent.

2.3.3.3 Topographic controls. The third step is to

delineate topographic limitations on transposability. Coastal

storms are transposed along the coast, but only a limited distance

inland. Inland storms are so placed that major mountain barriers

do not block the inflow of moisture from the sea unless this circum-

stance was present in the original location of the storm. Transposition

behind moderate and small barriers is taken care of by storm adjustment

(see below). Some limitation is placed on latitudinal transposition

in order not to involve excessive changes in air mass characteristics.


2.3.3.4 Final step. The final step in transposition is

to apply transposition adjustments discussed in the next section.

Transposition adjustments

2.3.4.1 Moisture adjustment for location. The moisture

available in the atmosphere for production of precipitation is an

'important factor in the maximum precipitation that may be expected

in different regions. The extreme demonstration of this is a

comparison of precipitation in polar regions with tropical regions.

It is customary in transposing storms to apply an adjustment for

moisture. This is derived from charts of enveloping dew point

values, reduced to a common elevation. Such dew point maps are

discussed in section 2.4, on maximization. The dew points are

converted to precipitable water in a saturated pseudo-adiabatic

atmosphere from the ground to some great height by figure 2.9. The

transposition adjustment is then the ratio of the precipitable water

for the enveloping dew point at the transposed location to that

where the storm occurred.

r

(1)

(2)

where

RI observed precipitation in a storm, for a particular duration

and size of area.

R2

= precipitation adjusted for transposition.

r transposition adjustment.

Wl

precipitable water in a saturated pseudo-adiabatic atmosphere

from ground to some great height, corresponding to maximum

surface dew point at location of storm occurrence.

1

1101009080.."..... I i i I ill I i i j , 0- - - - - 30 40 50 60 70

900

lOOO-mb DEW POINT M.ID LATITUDE

- ~ - - -.- - . - - 17.5C 20C 22.5C 25C 27.5C J.ITROPICAl300 i 1 11 i i I 1 1 1 i i d 1 r I 11

9

8400, I I I I / / / / ~

7

500 6

1) 5 EE ~

;:;:; 600Z

0::4 0:::::>

V) i= .0V) 'ljW t-3

3 ..... ~wI\)

800l //////// U-2

PRECIPITABLE WATER (mm)

Figure 2.9 - Precipitable water in saturated pseudo-adiabaticatmosphere between 1000 mb. and indicated pressure.

VJ\D


W2

same as Wl

at transposed location.

The entire stormdepth-duration-area array of rainfall is multiplied

by this ratio. The transposition adjustment for moisture may be

either greater or less than unity, depending on whether the trans~

position is toward, or away from, the source of moisture.

2.3.4.2 Barrier adjustment. Another occasion for a

transposition adjustment is placement of a storm behind a barrier.

By barrier is mean a mountain range lying between the basin under

study and the sea, in the direction from which moisture normally

reaches the basin. This is a common situation because basins upstream

from suitable dam sites are often rimmed by mountains. Transposing

storms from plans behind extremely high mountain barriers is dubious

because of the dynamic influence on the storm of the mountains.

However, lesser barriers, up to approximately 1000 meters in elevation,

are regarded as decreasing the storm potential by a certain percent.

This is applied as a transposition adjustment. The mountain range

blocks off a certain fraction of the moist inflow into the transposed

location. The storm is decreased by the ratio of percipitable water

in a column at- the mountain crest to precipitable water at the foot

of the mountain on the windward side:

(3)

(4)

where

Rl

= observed preci~tation in a storm not behind a barrier.

R3

= adjusted precipitation at a transposed location behind a barrier.

b = transposition adjustment for barrier.W

I= same as in equation (2).

CHAPTER 2

W3 precipitable water in a saturated pseudo-adiabatic atmosphere

from top of barrier to same great height.

Wl and W3 may correspond to the same or different dew points, depending

on distance of mountain from storm location, reduced to sea level.

If different, ~hen b is a combined adjustment for geographical location

and for barrier.

2.3.4.3 Recent studies have made a distinction between

storm types in applying barrier adj ustments (3.2). Intense local

thunderstorms of short duration that are important for basins of a

few square kilometers in size can draw in the moist air that was

lying over a large interior region rimmed by mountains before the

storm began. High atmospheric moisture can occur at all levels in

such a location b~ evaporation into the air of previous precipitation

as it fell. For such storms, the barrier adjustment may be omitted

altogether. If a moisture adjustment for location is required, WI

and W2

of equation 2 corresponding to enveloping dew point values

observed in the interior region are used.

Storms covering larger areas and associated with general

cyclonic activity are observed to receive a sustained inflow of

moist air from the sea during the storm. For these storms the

barrier adjustment described in the preceding paragraph is applied.

2.3.4.4 Elevation adjustments. Other topographic adjust-

ments for transposition of storms are less well-defined than those

for placement behind barriers. An increase in elevation decreases

the moisture that may be contained in a column of the atmosphere.

However, many storms receive most of their moisture in a strong low-

level flow 1 to 1.5 km. deep and this is not necessarily affected ~y

41


elevation changes. Foothills tend to stimulate convection

and increase rainfall, while the windward side of mountain

slopes provides the mechanical lift. These effects in releasing

precipitation may more than compensate for the decrease in

precipitable water with higher terrain.

2.3.4.5 In view of these conflicting techniques on

precipitation magnitudes by topography, the United States Weather

Bureau usually follows these practices in maximum rainfall studies

in middle latitudes:

1. Transpositions of large-area storms into generally mountainous

areas of less than 1,000 meters elevation from adjacent flat regions

are generally made with no elevation adjustments, it being assumed

the stimulation of precipitation referred to above offsets the

decrease in moisture with h~gher elevations (31).

2. In regions of high mountains and steep slopes the attempt is

made in comprehensive studies to divide storm precipitation into

two parts: that due to orographic effects and that due to the storm

processes in the atmosphere that would be about the same without the

mountains (25, 32). (The latter component is called "convergence

precipitation" in the references.) Only the latter component is

elevation~adjusted for decreasing moisture. In a recent report (32)

an empirical elevation adjustment for this component was adopted

which decreases precipitation with elevation about half as fast as

a full precipitable water adjustment would require.

3. Intense loc~l thunderstorms are not elevation-adjusted for

transposition to locations where the ground elevation ranges up to

about 1500 meters. That violent thunderstorms occur in mountainous

CHAPTER. 2

regions with greater frequency than over adjacent valleys is well

knovTn. Above 1500 meters the decrease in available moisture becomes

over-riding and an elevation adjustment is applied for transposition

based on precipitable water. In making such adjustments, the effective

elevation of the ground at the place of occurrence of the storm and

in the transposed position are employed rather than the precise point

elevations, to allow for the fact that a thunderstorm draws in moisture

from some distance away. The effective elevation is either

the average ground elevation over some tens of square kilometers

surrounding a location, or the average elevation over a specified

sector five to ten kilometers long in the downhill direction only.

4. On broad, gradually sloping plains, such as the Plains region

extending from Texas and Oklahoma northward, the relocation adjust-

ment for transposition is applied as described in paragraph 2.3.4.1

but no explicit additional adjustment for elevation is made. However,

elevation change of more than 700 meters is generally avoided.

Local or regional studies of available storm precipitation

should influence any elevation adjustments. For example, it is not

known prior to study of a particular tropical region whether the

most intense precipitation from the deep moist air mass occurs at

low elevations from ready triggering of convection or at higher

elevations from other effects.

2.3.4.6 Climatological adjustments. Other factors besides

topography and moisture effect storm magnitudes. The action of these

factors is suggested by such climatological charts as mean annual

precipitation, maximum observed values of point precipitation, and

heavy rainfall frequency charts. An example of the latter would be

43

44 ESTTMATION OF M,\XIMUM FLOODS

a six-hour rainfall 'with a mean recurrence interval of five years.

A climatological chart is indicative of geographical variations of

prohablemaximum precipit.ation provided both theclimatological

chart and the PMP are strongly influenced by the same storm _,type.

For example, ifmonsoon rains contribute most of the annual

precipitation and also offer the greatest threat of extreme rain

to a basin, then the mean annual precipit'ationand large-area PMP

will generally have a similar distribution. On the ether hand,the

mean annual precipitation ~may be but slightly correlated with

extreme rain from tropical depressions if these are rare, or with

rain from intense thunderstorms over small areas. Frequency maps

of point precipitation (such as 6-hr., ID-yr.) should correlate

better with the latter.

Often a climatological chart fulfilling the similarity-

of-sterm-type requirement will show a larger'percentage variation

from point to point than does the PMP because it is influenced by

storm frequency as well as storm potential.

2.3.4.7 The hydrometeorologistpreparing estimates of

PMP is confronted with the follmving dilemma. Climatological charts

of the types referred to may provide the best available guide to the

spatial variation of the PMP, especially in regions where moisture

variations are slight and are not predominant control on maximum

precipitationmagnitudes. Yet there is no explicit theory available

on which to hase a quantitative interrelationship between PMP and

the climatological chart. In fact the strength of the climatological

chart is that it depicts the real action of the atmosphere in

distributing rainfall independent of theories (except for statistical

CHAPTER 2

theory in constructing frequency maps) and integrates many factors

not all of which can be identified.

2.3.4.8 The recommended procedure for using climatological

charts as guides to transposition is:

(a) Select the climatological chart that is most strongly

influenced by storms of the type to be transposed.

(b) Calculate a tentative transposition adjustment ratio,

r', from:

45

r' = F IF2 1 (5)

where Fl and F2

are the climatological rainfall values at the location

of storm occurrence and the transposed location, respectively.

(c) Regard r' as the outer limit of the transposition adjustment

and subjectively adjust to a value, r, closer to 1.0 on the basis

of judgment. (Increase r' if less than 1.0, decrease if more than

1.0). The full adjustment, r', would more often be used as a

transposition adjustment to develop some lesser category of design

storm such as "Standard Project" storm than to develop PMP.

Reference distance procedure for moisture adjustment

2.3.4.9 An alternate procedure for moisture adjustment for

relocation to that described in paragraph 2.3.4.1 is to use as an

index of the moisture in a storm, not the observed surface dew points

at the center of the storm, but rather such dew points at some

distance from the storm as much as several hundred kilometers, in

the direction from which the moist air enters the storm. This

procedure is particularly appropriate with winter cyclones. The dew

points are from the warm sector of the cyclone regardless of whether

the precipitation occurs there or, more typically, north of the


,TRANSPOSED STORM CENTER

, "sTORM CENTER~(IN PLACE)

.,/REFERENCE.-l""'"r DISTANCE

MAXIMUM DEW POINT~l./(READ FROM MAXIMUM'V DEW POINT MAPS)

~STORM DEW POINT(READ FROMWEATHER CHARTS)

Figure 2.10 - Use of reference distance in storm transposition

CHAPTER 2 47

warm front with cold air at the surface. In this circumstance

the surface dew points near the storm are not representative

of the moisture flowing into the storm. At the transposed location

the same referenced distance is laid out on the same bearing from

the transposition point. This indicates where to scale the maximum

dew points from the maximum dew point chart for calculating the

transposition and maximization adjustment. See Figure 2.10.

Examples of transposition

2.3.5 Figures 2.11 and 2.12 illustrate transDostion

limits applied to storms in the course of studies in the United

States. Included are notes as to the reasons for establishing

the indicated transposition limits. In the study of a particular

basin, it is of course not necessary to establish transposition

limits completely around a storm but only in the direction of

the basin.

2.4 Storm Rainfall Maximization

2.4.1 Introduction

2.4.1.1 There are three princiDal methods of storm rain-

fall maximization: statistical, physical and composite. Statistical

methods are discussed in chapters 5 and 6.

2.4.1.2 Physical Method. rne physical method of maximization

is applied to individual storms and is used in combination with trans-

position and envelopment. References 3, 10, 12, 15, 18, 20 dnd 36

are survey papers which describe this method or some aspect of it.

The physical method is based on the model described in section 2.1.

In that model, the most vital element of a rainstorm is a cloud

system into which air converges radially at lower levels, rises to

~:>. .. 29 __ 8.5010 31 33 30 39 42 45 49 53 57 01 00 '71 70 81 87

...6..0.0._. 3' 34 3 7 "0 43 46 50 54 58 6 2 67 72 2:L 82 88590 32 34 37 40 43 47 51 55 59 03 68 73 78 84 9058.0- _32..... 35 -3.8_.4L. .44 _!.L8---5.l.__55. hO- ...oll _ 09. 74 8.0 85. 9157J 32 35 38 41 45 48 52 50 01 05 70 75 81 87 93

-5.6..0- .33 30 _ .3.9._.A.2.-105 ...1L'L.:. s.3.--'iL_.6.l._....6.t> 71 77 1i2 88 94550 33 30 39 42 40 49 53 58 02 07 72 78 83 90 9054 33 ~_~_4:L...!l6. ~ ....54 58 6.3 -.1... ..5.1-.5 5 60 04 .. 70-.15 B.:l _8.7- 93 100510 34 37 40 44 48 51 50 00 65 70 70 82 88 95 102

.--50:: 3-4 .37 -4l~4.4 48- 52.56. ..b.l-_-6b.- 71 .:1.1 .-8.-3 8.9 96.1Q.3.49. 80 88 95 10.3 112 12127: 30 40 44 48 52 57 02 08 74 81 88 95 104 112 12220C 3t.. 40 44.. 4.8. . 52 _. 51.. 6.2. 08 71l 81 88 90 104 113 122L5~ 3U 40 44 48 52 57 02 08 74 81 88 96 104 113 1222..4.: 36 40 4L...4.B_~ . .5.L-b2_.. 6.B-.li.-.. 8L-IllL 96 10lLl1L.123230 3L 40 44 48 ;2 57 02 08 74 B1 88 90 104 113 123~2C 36 40 44 48 52 5L 6.2 6.8. . 'Ill 81 88 96. 104 113 12321;) 30 40 44 48 52 57 02 08 74 81 88 90 105 114 12'

'""- 36. 40 44 48 52 5'7 h'2 6.13.._..110 81 88 90 105 114 123

63


not as great. For comparison, specific humdity differences along

moist adiabats between 900 mb. (1 km.) and 400 mb. (7 km.) are

shown in figure 2.13, curve El, and the relative variation of this

difference in figure 2.14, curve E. However, inflow levels and

outflow levels are less readi1y'specified in this type of storm

than in extreme thunderstorms. Further, in large-area storms

much of the more intense precipitation is frequently the result of

thunderstorms and associated convective activity. In view of the

uncertainties, the U.S. Weather Bureau has applied the precipitab1e

water ratio adjustment of formula (4) to large-area storms as well

as thunderstorms.

2.4.2.8 Orographic storms. The maximization of

orographic storm precipitation by the orographic model of paragraph

2.1.3 is described in section 2.4.7. In maximization by the

orographic model the moisture adjustment is applied implicitly by

processing air along sloping streamlines, each with its own moist

adiabatic temperature variation and its own decrease in pressure

over the span of the windward face of the mountain.

2.4.3 Dew Points

2.4.3.1 Moisture maximizationof a storm requires

identification of two saturation adiabats. One typifies 'the

vertical temperature distribution in the storm to be maximized,

with the greatest weight given the time and place of the heaviest

precipitation. The other is the warmest saturation adiabat that

could be expected in a storm at the same place and season. It is

tBcessary to identify these two saturation adiabats with some

indicator, and the conventional 1ab1e in meteorology for saturation

CHAPTER 2 65

adiabats is the wet-bulb potential temperature. An alternate

identifier is the 1000-mb. dew point. Surface dew points in the

inflowing tropical air in or near a storm identify the storm

saturation adiabat. The moist adiabat corresponding to either

the highest dew point of record at the location and season, or

dew point of some specific return period such as 25 or 50 years,

is considered sufficiently close to the warmest probable saturation

adiabat. Both the storm and maximum dew points from higher elevatioI

stations are reduced to 1000 mb. along the moist adiabat on which

they lie at their respective pressures to obtain the wet-bulb

potential temperature. Ensuring paragraphs give further specifications

on the use of dew points in this manner as the basis for moisture

adjustment of storms.

2.4.3.2 Maximum dew points. Where surface dew point data

are available, a satisfactory method for obtaining the maximum

moisture index is to survey along record at several stations. All

high values for each station are plotted against date and a smooth

seasonal envelope drawn as illustrated in figure 2.15. Monthly

values are then read from these graphs at the 15th day of each

month, adjusted by the saturation adiabatic to 1000 mb. and plotted

m monthly maps. Smooth enveloping isopleths are drawn on the maps.

Figures 2.16 and 2.17 show maximum dew point charts constructed in

this way for selected dates in West Pakistan (17) and the United

States (35) .

. 2.4.3.3 Synoptic limitations on maximum dew points. Certain

precautions are advisable in the dew point maximization procedure.

First, the maximum dew point charts are intended to be an index of

0\0\

26

NOTE: VALUE ON JULY 20 IS UNDERCUT IN ACCORDANCEWITH PARAGRAPH 2.4.3.4.

25

ITe......I-zoA.

~ 24 1w .o

23

~8

H

~H

~ 1 ~

~Hs=~l:I:jt""00l::;jtr.l

5 15 25 5 15 25 5 15 25 5 15 25 5 15 255 15 255 15 25

22' .'.! ..'.~_'.. '_'_J_~,.~_' ..'I .'.'_~_~ I I ~_'_~.' I I I.. '..~' " L.'..L'" !..'.". I I ~.'. ' ..'_1 '_' I "'" I I I L ~.' -'.' ..'_' 'l_'_!.~ "

Figure 2.15 - Enveloping maximum dew points at a station

30-JUNE

CHAPTER 2

rvco IS-JULY

67

HIGHEST PERSISTING 12-HOUR IOOO-MS

DEWPOINTS-DEGREES FAHRENHEIT

Figure 2.16 - Highest persisting 12-hr. lOOO-mb. dew points (oF)in West Pakistan. Selected dates. From (17).


\\~:--

--~

Figure 2.11 - Highest persisting l2-hr. lOOO-rob. dew points (oF) inthe United States. Selected months. From (35).

CHAPTER 2

moisture in storms. In certain places and seasons characterized

by ample sunshine, sluggish air circulation, and numerous lakes,

rivers and swamps, a local high dew point may result from local

evaporation of moisture from the surface and not represent a large

volume of a tropical air mass. Such values can be discounted in

constructing the maximum dew point charts. This problem is most

aggravated in the Tropics but it is also present at higher latitudes

in summer, where daily insolation equals tropical values.

2.4.3.4 To control this local modification of dew points,

the analyst inspects the surface weather charts for the dates of the

highest dew points and eliminates those in which the station is clearly

in an anticyclonic or fair weather situation rather than a

cyclonic circulation with tendencies toward precipitation.

2.4.3.5 l2-hr. persisting dew points. Another problem

with high dew points has to do with observational techniques. The

most common method of measuring dew point is with a psychrometer. If

the wet-bulb of this instrument is not sufficiently moistened and

ventilated, its temperature will not be depressed sufficiently below

the dry bulb. A calculated dew point from such contaminated data

is incorrectly high. Assuming such errors are committed only

occasionally, there is merit in basing maximum dew point values

on two or more consecutive observations rather than on a simple

individual reading. The D.S. Weather Bureau uses the highest

persisting 12-hr. dew point, that is the highest value equaled or

exceeded at all observations during 12 consecutive hours. For

example, the following is a series of dew points observed at

6-hour1y intervals. The highest persisting 12-hr. dew point is 24C.

69


22 23 24 26 24 20 21

2.4.3.6 Average maximum dew points. Another method of

obtaining smoothing in maximum dew points is to average over six

or twelve hours. The maximum average 6-hr. dew point in the above

series is 25.00 C (two consecutive observations) while the maximum

average 12-hr. value (3 consecutive observations) is also 25.00

C.

2.4.3.7 Single observation maximum dew point. Single

observation dew point maximums may be used as the maximum moisture

index provided the record is examined for dubious values and the

synoptic test of paragraph 2.4.3.3 is applied. These tests should

be applied in any event, but are particularly necessary to appraise

single observation maximum dew points.

2.4.3.8 Storm dew point. To select the saturation adiabat

representing the observed storm moisture, the highest dew points in

the warmest airmass flowing into the storm are identified on surface

weather charts. This determination may be made in the rain area

but not necessarily so. Dew points at stations between the rain

area and the sea should also be considered. This tolerance

is to insure that the dew points are in the warmest airmass involved.

In some storms, particularly storms related to warm fronts, surface

dew points in the rain area may represent only a shallow layer of

cold air and not the temperature distributions in the convective

clouds that are releasing the rain. Figure 2.18 illustrates

schematica11y a weather map on which the storm dew point determination

is made. On each consecutive weather map for the duration of a

storm the maximum dew point is average over several stations as

CHAPTER 2

14

H EA VY RAI N AR.EA

71

16

2423

24

19

Figure 2.18 - Determination of maximum dew point in a storm.Representative dew point for this map time isaverage of values in boxes

ESTIMATION OF MAXIMUM FLOODS

illustrated in the figure. Occasionally for lack of data it is

necessary to rely on the dew point at only one suitably located

station.

2.4.-3~9 The maximum dew points, one value per map from

each consecutive weather map, determined as described in paragraph

2.4.3.8, forma serie~. The representative storm dew point is then

abstracted from this series by the same rules followed ,in determining

climato10gical maximum dew points, be it single observation, average,

or persisting maximum.

2.4.4 Combihedrlli:ixirnization and tran~positionadjustment for moi~ture

2.4.4.1 Where a storm is both maximized for .mo;isture

and transposed with'a:-n{oisture'a-djustment, the two adjustmeIl;ts,may

be combined into a single ratio. Using a precipita.ble water type_

of adjusi~ent, the transposition adjustment is:

The maximization ratio is:

Obviously the combined adJ'ustment ratio, r ' ist m'

..~

jI

(5)

(4)

r'tm

wx

Ws

(6)

w , W , and Ware respectively the precipitable water correspondingt x s

to the maximum wet-bulb potential temperature at transposed locati9n,

the maximum where the storm occurs, and the representative storm

value. The transposition adjustment, rt

, may be less or greater

than 1.0, depending on whether the transposition is toward

CHAPTER 2

greater or less moisture; the maximization ratio, rm

, is never

less than 1.0.

2.4.5 Maximization from precipitable water measurements

2.4.5.1 Intelligent smoothing is required in developing

reliable maximum values of most hydrometeorological variables. The

representative maximum storm dew points are smoothed areally by

averaging several stations on each weather map (par. 2.4.3.8). The

climatological maximum dew points are smoothed or enveloped

areally by constructing isopleths on maps on which the basic data

are station values. Time smoothing is provided by the procedure

of figure 2.15. Vertical smoothing is discussed in the next

paragraph.

2.4.5.2 Long records of radiosonde observations open up

the possibility of adjusting storms for moisture, by measurements

of atmospheric water vapour integrated through a layer, rather than

by surface dew point alone. Both the maximum atmospheric moisture

and the storm moisture would be derived from the radiosonde observa-

tions. This method has not been fully developed because of (a)

the considerable added expense in processing the upper air data

records, (b) the lesser density of radiosonde stations as compared

with surface stations, and (c) the lack of radiosonde stations in

early years of climatological records. It would seem desirable

to explore this method, and base the moisture adjustment on

precipitable water integrated through the atmosphere ~vith emphasis

on the bottom 1000 to 2000 meters, the most significant inflow

layer in storms.

73


2.4.6 Wind maximization

2.4.6.1 Iand maximization is most commonly used in

mountainous regions wi th storm types "lhich it can logically be

considered that if the strength of the wind blowing against the

mountain range in the observed storm had been increased, the

precipitation would have been increased in proportion. The

most direct way to apply a wind adjustment is to compare the

total daily air movement (wind) at a coastal station, or other

station suitably located between the storm and the sea, with the

maximum daily air movement in a considerable period of record at

the same place and season. Only days with wind direction from a

sector appropriate to storms are considered in arriving at this

maximum value. The wind maximization ratio is then the ratio of

the maximum air movement to the storm air movement.

2.4.6.2 Wind maximization.may also be applied to large-

area long-duration storms in regions not necessarily mountainous

but far enough away from warm seas that the inflow of moisture

during the storm is an important limiting factor o

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