Short-duration rainfall-intensity estimates and other design aids for regions of sparse data

Journal of Hydrology 1 (1963) 3-28; © North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

SHORT-DURATION RAINFALL-INTENSITY ESTIMATES AND O T H E R D E S I G N AIDS F O R R E G I O N S OF SPARSE DATA BRIAN M. REICH Drakensberg Conservation Area, Estcourt, Natal

Received 3 September, 1963

Abstract. Maps and homographs are developed which facilitate the estimation of shortduration rainfall-intensities for any point in South Africa. Other generalizations available for making similar estimates in other countries with even fewer pluviograph records are presented. Information on the depth-area relations, time-sequence, and other rainfall frequency characteristics have been drawn together for the convenience of designers on small watersheds. Objectives of discussion Maximum rainfall intensities prevailing for durations of 15-, 30-, 45-, and 60-minutes, and their frequency of recurrence form the main subject of this review. Estimates of such rainfall intensities are needed by engineers entrusted with the design of contour drains and dams on farms, road culverts, airfield drainage, storm sewers in urban areas, and flood or sidiment control dams in headwater catchments. Because individual works are often relatively small and inexpensive, the cost of overdesign is frequently neglected. When the total number of such structures being built throughout a country is considered, however, the provision for unrealistically large run-off rates represents a considerable national waste. Likewise, underestimation of shortduration storm potential can lead to large losses incurred in the widespread failure of small works. Engineers in the newly developing countries are obliged to make the maximum use of minimal hydrological records. Thus an account of the author's experience in applying American findings to another country should provide a useful guide in many other parts of the world. For simplicity's sake, the presentation will be divided into two main parts. The first part outlines the highlights in the development of a procedure for estimating short-duration rainfall anywhere in South Africa. The second part presents a review of additional and more recent concepts which may aid engineers in similar and related investigations where even less observed data are available.

4

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SOUTH AFRICAN STUDY This study 1) was completed in 1959 at the Iowa State University. In general the average annual rainfall of South Africa varies from about forty inches in the east to about five in the west. Small areas along the mountainous rim approach one hundred inches of average annual precipitation. Most of the country receives summer rainfall, predominantly from convective thunderstorms. The southwest corner receives its rainfall in the winter. Introduction

A pluviograph shows how rainfall intensity varies with time throughout a storm. Average rainfall intensities read off pluviographs for various durations form the basis of an intensity-duration study. Only fifteen stations throughout the country provided records which were suitable and of sufficient length for an intensity-duration-frequency study. Fig. 1 shows the location of the recording raingauges in respect to the zones o f different rainfall type. The numbers signify the years of record available at each ., .....

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6

n.M. REICH

contrasts sharply with the figure of 855 square miles per gauge for the United States of America. Some European countries have even denser gauge networks. The South African data analysed had an average length of record of fifteen years. Four of these stations had records shorter than eight years. Fortunately it was possible to use daily rainfall maxima and other climatic statistics, available at another 210 stations, in order to depict the storm behaviour in the tracts between the recording raingauges. This was not simply a matter of expressing the maximum for each short duration as a specified percentage of the 24-hour maximum. Fig. 2 displays some annual maxima for short durations as percentages of the corresponding 24-hour precipitations. The large variability is obvious. Similar plots for recording gauges in zones of different rainfall type displayed very different envelope and median values. Thus more elegant methods were required in relating short-duration rainfall maxima to common climatic parameters. Satisfactory empirical relations have fortunately been established in the United States of America, where they have been shown to apply in widely different climates. Their detailed discussion will be postponed until the data from (he fifteen South African rainfall recorders have been fully exploited 3). Analysis of results from recorder stations

The South African data were studied without the assumption that American behaviour would be repeated. Techniques of analysis and methods of expressing the results, however, were borrowed from recent American practice. This examination of short-duration rainfall observed in South Africa provided a basis for testing whether or not the American empirical relations, which will be discussed later, were valid in the new country. FREQUENCY ANALYSIS METHOD

After consideration of six alternatives the Gumbel extreme-value method was selected as the most suitable for analysing the recorder data. The United States Weather Bureau 3) finds this method highly satisfactory for this type of data. Fig. 3 shows the results of this analysis for one recording raingauge. Each of the five graphs contains the annual maxima for one particular duration. The complete Gumbel paper on which the data were originally plotted is shown to a larger scale in Fig. 4. The observed annual maxima are plotted and the best-fitting straight line is computed objectively by a simple mathematical procedure4). The rainfall extremes read off this line at various return periods will, for simplicity, be referred to here as Gumbel values. Ifthe scatter about the Gumbel lines had been unsatisfactorily large,

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To say that a storm feature has a frequency of once in twenty years is synonymous with saying that the event has an average return period of twenty years. Fig. 5a is a reproduction of the rainfall-frequency diagram developed 16

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IMPROVEMENT IN FITTING SHORT DURATION MAXIMA OBSERVED AT A TYPICAL SOUTH AFRICAN STATION; PRETORIA: =-ONTOAMERICAN DIAGRAM

b - O N T O DIAGRAM DEVELOPED FOR SOUTH AFRICA

Fig. 5. Typical effect of altering abscissa scale on the rainfall-frequency diagram. by the U.S. Weather Bureau for the entire continental United States. By joining the 2-year and 100-year rainfall maxima one can read off this diagram those for intermediate return periods. To see if this diagram was valid for South, Africa the 2-, 5-, 10-, 25-, 50-, and 100-year maxima for the 30-minute, 1-hour and 24-hour durations were read from the Gumbel plots for the twelve most reliable recorder stations. This information plotted on Fig. 5a shows the slight but consistent underestimate obtained for all the South African recording stations. Furthermore, there is a theoretical justification, which has been expounded elsewhere 1), for altering the return period scale

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of Fig. 5a. An alteration to the abscissa scale was made by computing the average deviation-to-slope ratio. Fig. 5b shows the improved estimation

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which this makes possible for the three durations. Similar improvements were wrought for all twelve stations. This therefore formed the basis for the rainfall-frequency diagram for South Africa, which is portrayed in full detail in Fig. 6. RAINFALLJDURATION RELATIONSHIP

Untill very recently ~) the generalized rainfall-depth-duration relationship for U.S.A. was presented as two separate drawings T). The one drawing required a knowledge of expected precipitations for the six-hour and twentyfour-hour durations. By laying a straight-edge through these two values an estimate was obtained of the storm precitation for intermediate durations. In a similar fashion, the second drawing enabled one to estimate rainfall intensities or depths for durations between 20 minutes and 6-hours provided one could estimate the one-hour and six-hour amounts. Hershfield and Wilson s) after examining thousands of extreme-value rainfall distributions pointed the way to combining these two separate drawings into one. Thereby it became necessary only to know the one-hourly and twenty-four-hourly values as the basis for estimating precipitations of other durations. Such a combination formed the basis of Fig. 7. Except for the intensity scale for a 15 minute duration, this figure is therefore based upon American finding.

S H O R T - D U R A T I O N RAINFALL-INTENSITY ESTIMATES

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There was consequently a need to test its two-year, 30- and 45-minute rainfall maxima against those actually observed at the South African recorder stations. For all fifteen stations the Gumbel values differed randomly and by less than five percent from the intensities derived from the rainfall duration diagram, Fig. 7. No 15-minute scale was available on the American prototype rainfallduration diagram. It was therefore necessary to add a scale which would best fit the 15-minute results yielded by the South African recorders. The graduations of this scale and its distance from the 30-minute scale were set by inspection. Finally, the estimates made from the 15-minute" scale were

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checked against 2-year Gumbel values obtained from the recorders. In relation to observed values, the estimates were: identical for five stations; about five percent too low for three stations; and an average of four percent too high for four good stations. In addition, Fig. 7 gave over-estimates of 10.9, 12.5 and 13.4 percent for three stations with less than eight years of record. These evidences were well within the limits of accuracy and led to the acceptance of the 15-minute scale as giving results which, being high, were biased slightly towards safety. Mention should be made of the dotted lines on Fig. 7. They are merely copies of the ordinates for the 2-, 6-, and 12-hour durations on the combined American drawing. Their validity has not been checked for South Africa. They are not used in the procedure developed here for estimating short duration rainfall intensity.

12

B.M. R~ZCH Estimating rainfall-intensity where recorders were absent

Two methods were available for obtaining an estimate of the 2-year/l-hour rainfall maxima from common climatological information. The first method is described by Hershfield, Weiss and Wilsong). It was used in the coastal regions of North Africa by the U.S. Weather Bureau 10). Its basic multiple 2.4

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Fig. 8. Empirical relation for estimating the 2-year/l-hour rainfall in the absence of pluviograph records. correlation was, however, obtained from the study of 134 widely scattered first-order United States Weather Bureau stations having an average length of record of forty years. The parameters required for this method are: mean annual precipitation, mean annual number of thunderstorm days, mean of annual series of maximum daily precipitation amounts, and the mean annual number of days with rain.

SHORT-DURATION RAINFALL-INTENSITY

ESTIMATES

13

The second method was described by Hershfield and Wilson 8) and was later extended by Hershfield to include a wider range in its two independent variables, (in personal correspondence). Fig. 8 shows this relation which is seen to depend upon the mean of the annual series of maximum daily precipitation amounts, and on the mean annual number of thunderstorm days. The 2-year/l-hour rainfalls were estimated for some 210 South African stations by both of the above empirical methods. The ratio between the two estimates was plotted on a map and showed no consistent regional trend. This ratio had a mean of 1.1335, while its standard deviation was 0.1815. Fig. 8 gave somewhat higher values than the one used in North Africa. This, and the following reasons, lead to its estimates being used for the rest of this study. COMPARISON OF EMPIRICAL TO GUMBEL VALUES Consistent agreement was found between the empirical estimates and the Gumbel values for the twelve stations at which comparison was possible. The twelve-station average of the North African method exactly equalled the average of the Gumbel values. Fig. 8 averaged 2.6 percent higher estimates for the 2-year/I-hour maximum precipitations. Because it was considered safer to err on the high side, the values derived from Fig. 8 were employed in the remaining work. M A P OF 2-YEAR/l-HoUR MAXIMA

In order to obtain complete coverage, the 210 point values of the 2-year/lhour maximum precipitations obtained from Fig. 8 were plotted onto the map of South Africa. Smooth isopluvial lines were then drawn between the point estimates (Fig. 9). In areas where orographic influences are absent such smoothing eliminates sampling error exhibited by individual station records. In areas of unbroken topography, such as the large central and northern portions of the country, a more reliable estimate of the 2-year/l-hour maximum precipitation can therefore be made from the smoothed isopluvials than from a nearby station's record. Difficulty is encountered in preparing a map for areas where orographic influences are known to exist. One or two degrees inland from the northeastern boundary a series of high 2-year/l-hour maximum precipitation values occur on Fig. 9. These high values occur along a narrow belt of mountainous terrain possessing a high mean annual rainfall. Drawing of the 45- and 50-millimeter isopluvials in this locality was influenced by the pattern of high mean annual rainfall. Similarly, the shape of the 15-millimeter isopluvial enclosing Groot Drakenstein was patterned after the relatively

14

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35

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how to estimate the maximum short-duration rainfall intensity of a chosen return period expected at any particular point in South Africa. REQUIRED MAPS AND CtlARTS

The following are essential for the estimation process: Fig. 9. Map of 2-year/l-hour maximum precipitation; Fig. 10. Map of2-year/24-hour maximum precipitation; Fig. 7. Rainfall-duration diagram; Fig. I I. Mapofratioofl00-yearmaxim,tmto2-ycarmaximumprecipitation Fig. 6. Rainfall-frequency diagram: tot adjusting rainfall intensities to correspond with the required return period between 2-years and 100-years. ESTIMATION PROCEDURE

The following steps should he followed in estimating the expected maximum rainfall intensity of given duration and return period:

SHORT-DURATION RAINFALL-INTENSITY ESTIMATES

17

1. Find Xz/h the 2-year/l-hour maximum precipitation, for the required locality from Fig. 9. 2. Find X~./24, the 2-year/24-hour maximum precipitation, for the required locality from Fig. 10. 3. In Fig. 7 join the values for X2/1 and X2/24 on their respective vertical scales with a straight edge. The point where the straight edge intersects the required short-duration axis is noted as I2/s mm/hr. 4. Fig. 11 is consulted to obtain the ratio of the 100-year maximum to the 2-year maximum, Xloo/X2, for the required locality. 5. Is/s, obtained in step 3, is multiplied by Xloo/X2 to obtain a value of Iloo/s mm/hr. 6. The rainfall intensity for the duration " S " and a specified return period, T, are obtained from Fig. 6. The hundred-year and two-year intellsities, I100/s and I2/s, are located along their respective axes and a straight edge is set along them. Where the straight edge intersects the line for the required return period the required intensity IT/S is read, either in millimeters per hour or inches per hour. WORKED EXAMPLE

The problem is to estimate the maximum rainfall intensity over a thirtyminute duration for a 50-year return period for Victoria West (31-1/2°S, 23°E). By proceeding according to the steps outlined above, the following solution is obtained: 1. From Fig. 9, Xz/1 -----22 mm 2. From Fig. 10, X2/24 = 39 mm 3. Set a straight edge on Fig. 7 to join the ordinates of 39 and 22 on the 24hour and 60-minute duration scales, respectively. The straight edge intersects the 30-minute duration line at 36 mm per hour, which is the intensity 12/3o. 4. From Fig. 1 I, Xloo/X2 = 2.5 5. Whence the 100-year intensity for a 30-minute duration is: 1100/30 ----~ 2.5 × 36 = 90 mm per hour. 6. Apply a straight edge to Fig. 6, through the ordinates 90 and 36 on the 100-year and 2-year return period scales. The straight edge intersects the 50-year line at 81. Therefore the required rainfall intensity is 81 mm per hour (3.19 inches per hour). INFORMATION

AND CONCEPTS

AVAILABLE FOR OTHER

INVESTIGATIONS

If it is necessary to make estimates of rainfall intensity in countries possessing even less pluviograph records than South Africa, the frequency analysis

18

B.M. REICH

of pluviograph data from many United States stations and the correlation of their descriptive parameters with climatic statistics provide the necessary empirical relationships. Some such aids will be described below. Other aspects of design problems involving the rainfall-intensity-frequency regime will also be mentioned.

Generalized relationships in rainfall-intensity-frequency regime Fig. 4 shows details of the complete Gumbel paper on which rainfallintensity-frequency (or rainfall-depth-frequency) analyses are made4). A straight line can be fitted through the plotted series of annual maxima, regardless of whether or not they were associated with tropical storms 14). The important thing to notice about Fig. 4 is that the best-fitting sttaight line through the series of annual maxima will effectively pass through its mean on the ordinate marked "mean". This is seen to have a return period of about 2.3 years. Thus it has been found empirically that the two year rainfall can be obtained, without the use of Gumbel paper, by multiplying the arithmetic mean of annual series of extreme rainfalls by 0.95 s). The approximate equality between these 2-year and mean values and the modal value, about which Gumbel lines of different slope pivot, has led to the popularity of the 2-year value z, 7,11). RATIO BETWEEN MAXIMA FOR VARIOUS DURATIONS AND FOR CORRESPONDING CLOCK-HOUR AMOUNTS

Before observed rainfall statistics are included in an intensity-frequency analysis or used as a guide in the selection of design values one precaution must be taken. Tabulated data are often given with respect to some arbitrary time of day. Thus the maximum observational day rainfall may be listed for days starting at 8 o'clock one morning and continuing to 8 o'clock the following morning. If a heavy rainfall started after 8 o'clock one evening and finished before that time next evening, neither observational day would record as much rain as actually occurred in the most intense 1440 minutes. The observational day data should be multiplied by 1.13 to correct this bias 6). Maxima obtained l¥om clock-hour data should be multiplied by the same figure to obtain maximum 60-minute amounts. PARTIAL DURATION VERSUS ANNUAL SERIES

The transformation factors for changing to and from the partial duration series and the annual maximum series have been listed in many places 11,8,6).

19

SHORT-DURATION RAINFALL-INTENSITY ESTIMATES

Values will not be repeated here. Mention is simply made to check for consistency when various methods are being compared or combined. The two series converge and are virtually identical beyond the 10-year return period. AVERAGE RATIOS BETWEEN FREQUENT MAXIMA AND THOSE OF LONGER RETURN PERIOD

After an analysis of 200 United States stations for durations of one and twenty-four hours it was suggested s) that the 100-year maxima were about 2.2 times the 2-year values. Use of such a generalization cannot be recom.5.0

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20

B . M . REICH

RATIOS FOR OBTAINING SHORT DURATION AMOUNTS

The latest United States study ~) employs a very simple means to obtain the rainfall maxima for the durations of 30-, 15-, 10-, and 5-minutes. All that is necessary is to multiply the l-hour rainfall by the ratios: 0.79, 0.57, 0.45 and 0.29 respectively. This provides a much simpler means of estimating short duration intensity than does Fig. 7. The particular advantages of these ratios is that they only require the use of the 1-hour estimate, whereas the older nomographical methods need the 24-hour estimate as well. Since these ratios were considered as satisfactory for 200 United States stations in widely different climates, their universal applicability becomes a possibility. Combination of this simplification with the methods s, 9) described in the first part of this paper for obtaining the 1-hour rainfall maxima provides an almost universally applicable tool for estimating short-duration rainfall-intensities. SOME EXISTING ESTIMATES THROUGHOUT THE WORLD

A very generalized presentation 15) of l-hour 2-year rainfall is reproduced in Fig. 13. The order of magnitude and broad behaviour patterns which it suggests agree with more detailed studies for the United States of America 6), South Africa a), North Africa 10), New South Wales (Australia)le), and elsewhere.

Other hydrometeorological characteristics needed in applying short-daration rainfall-intensity to design The short-duration rainfall-intensities which can be estimated by the above methods would be directly applicable to estimating peak flows from the old "Rational Formula", Q = clA. Other methods 36) are available for the hydrologic design of spillways, culverts, airfield drainage and so forth. The shape of the entire hydrograph can be predicted 17) if the designer can adequately represent the design storm with respect to its time and space distribution. It is therefore considered appropriate to bring together some pertinent information here. TIME DISTRIBUTION OF RAINS PRODUCING FLOODS FROM SMALL WATERSHEDS

It has been noticed is) that large floods observed on small watersheds, located in eleven widely spaced states in the U.S.A., occured almost exclusively in the summer months. Those watersheds, which were all smaller than four square miles, yielded their maximum floods in response to intense

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22

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rain which usually lasted less than an hour. According to his personal judgement of local conditions, the hydrologist in other parts of the world may also wish to associate his design floods with convective rainstorms. Thus the typical mass curve of one-hour thunderstorm rainfall like those reproduced zg) in Fig. 14 should be valuable in many countries. These were derived from thunderstorm and other intense rainfall observed in 374 storms. They came from widely spaced stations throughout the United States, which possessed a diversity of climates. It should be noted that each of the mass curves in Fig. 14 represents the accumulation of rainfall with time at a point. Because different points throughout a watershed experience similar mass curves lagged in time, with respect to each other, the areal average would grow more slowly than does the mass curve for point rainfall of the corresponding amount. The larger the watershed the more pronounced the straightening of the mass curve would become. Until adequate data can be collected in a newly developing country, according to established proceduresZ0), Fig. 14 should form an acceptable guide to designers.

SHORT-DURATION

23

RAINFALL-INTENSITY ESTIMATES

Another conservative method is available in the absence of adequate pluviograph records. The maximum rainfall amounts expected for the durations of 5, 10, 15, 30, 60 minutes can simply be assumed to envelop the design storm sequence. Since the maxima for all durations are in fact not likely to occur in the same storm this synthesized mass-curve will give an added element of safety. S M A L L - A R E A SPATIAL DISTRIBUTION

Part of the depth-area diagram tentatively proposed by the Weather Bureau 6) for anywhere in continental U.S.A. is shown in Fig. 15. From it one can see the steep gradient which is considered to apply to rainfall of one hour and shorter durations. Actually the American raingauge network, in common with that in many other countries, does not generally permit realistic evaluations for areas smaller than 50 square miles. Special networks in Illinois 21) involve four highly instrumented small areas, having a gauge density as high as 1.7 per square mile. These have afforded some understanding of small-area relations, which probably also apply to other regions I.~

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of flat topography and similar climate. These studies 2z) showed that the average rainfall depth, Y inches, was related to the area enveloped, X square miles, by equation (1). Y = a -- bX ½ (1) where a and b are constants representing maximum storm point rainfall and mean rainfall gradient. Equations were proposed to relate b to the storm duration and to the areal mean rainfall itself. Some of their basic data are reproduced in Fig. 16. It illustrates the complexity of the problem. Court~a) recently compared depth-area formulae proposed by various researchers. An arid zone study 24) gave a very steep decline in rainfall as one moved out from the storm center within a distance of five miles.

25

SHORT-DURATION RAINFALL-INTENSITY ESTIMATES

INSUFFICIENT DENSITY OF THE SAMPLING NETWORK

Intense thunderstorm rainfall often covers an elliptical area, about three miles wide and about nine miles long2X). It is therefore likely that the intense centers of many storms will not be recorded by national raingauge networks. Studies based upon such national networks are therefore likely to underestimate storm rainfall at a point over small areas. The intensive field surveys in Illinois suggested that values derived from the national climatological network should be increased in the ratios 1.23, 1.19, and 1.10 for point estimates, 10 square miles and 500 square miles respectively.

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Design return period as a function o f desired life of a structure and the probability o f it not failing.

26

B.M. RmCH

Design lifetime and probability of not failing Much of the preceding discussion has concerned the frequency, or return period, of rainfall. It is important to add a brief interpretation of "return period" with respect to the required safety of a structure. The return period obtained from Gumbel paper, or from the other common methods of rainfall-frequency analysis, is actually an average return period. Considerable variations may occur in history. For instance, the 10-year/l-hour rain may be exceeded three times within ten consecutive years, but in the following 40 years it may be exceeded only twice. Although the average return period for the 50-year record is 10 years, a structure would have been subjected to this design flood more frequently in the first ten years of its life than may have been expected. For the design of spillways it is not satisfactory to equate return periods to the expected life of a structure. It is necessary to decide on the "desired lifetime of a structure" and the "probability of not failing" within that lifetime. On Fig. 17 these two quantities fix the return period which should be used for each design. Thus, if one requires a probability of 40 % that a spillway will not be overloaded during its desired lifetime of 10 years, then it should be designed for a flood with a 12-year return period. On the other hand, if more stringent safety requirements set the probability of not failing at 80 % during the same desired lifetime of 10 years, then a 46-year return period should be used for the design. It can thus be realised that estimates with return periods exceeding 100 years can be required for structures whose desired life-times are less than 100 years.

Probable maximum precipitation There are rare occasions when designs for small watersheds require a knowledge of how big the most severe storm could be in a certain region. Statisticians may object to the inclusion of the word "probable" in the title, on the valid grounds that probability should mathematically always be associated with a number from 0 to 1. Nevertheless, the phrase and its abbreviation, PMP, are established in hydrometeorological literature. The concept has nothing to do with probability as encountered in rainfall-frequency analysis. PHYSICAL CONCEPT

The probable maximum precipitation is defined as the largest rainfall that a station is ever likely to experience for a particular duration. In the first twenty years of this concept, it was estimated by atmospheric physicists

SHORT-DURATION RAINFALL-INTENSITYESTIMATES

27

according to the moisture that could be precipitated from the atmosphere. This required the a s s u m p t i o n o f a physical model a n d estimates of several meteorological parameters.

STATISTICAL APPROACH THROUGH EVALUATIONOF SERIES OF EXTREME VALUES The earlier approach to P M P required a specialized investigation that could n o t be justified for inexpensive projects. Hershfield 2s) has recently suggested a n o t h e r a p p r o a c h which is based simply on a statistical analysis of local extreme value data. T h i s e m p i r i c a l method should be cheap to apply in m a n y countries.

References 1) Reich, B. M., Probable maximum precipitations for short durations in the Union of South Africa. Unpublished M. S. Thesis, Iowa State University, 1959 2) - - , South African J. of Agric. Sc.,4 (1961) 589 3) D. M. Hershfield and M. A. Kohler, J. Geophys. Research 65 (1960) 1737 4) E. J. Gumbel, Statistical theory of extreme values and some practical applications. U.S. Dept. of Commerce, National Bureau of Standards, Applied Mathematics Series 33, 1954 5) A. F. Jenkinson, Quarterly J. Royal Meteorologic Society fll (1955) 158 6) D. M. Hershfield, Rainfall frequency atlas of the United States for durations from 30 minutes to 24 hours and return periods from 1 to 100 years. Washington, D.C., U.S. Weather Bureau, Hydrologic Services Division, Cooperative Studies Section, Technical Paper 40, 1961 7) U.S. Weather Bureau, Rainfall intensity frequency regime: Part 5 - Great Lakes Region. Tech. Paper No. 29-5, 1960 8) D. M. Hershfield and W. T. Wilson, International Union of Geodesy and Geophysics, International Association of Scientific Hydrology, General Assembly of Toronto, 1 (1957) 499 9) D. M. Hershfield, L. L. Weiss, and W. T. Wilson, Proc. Am. Soc. Civ. Eng. 81 No. 744 (1955) 1-6 10) U.S. Weather Bureau, Rainfall intensities for local drainage design in coastal regions of North Africa longitude i IW to 14E for durations of 5 to 240 minutes and 2,5 and 10 year return periods. Hydrologic Services Division, Cooperative Studies Section, Washington, D.C., (mimeographed manuscript) 1954 1I) W. T. Wilson and D. M. Hershfield, Agric. Eng. 39 (1958) 344, 353 12) South African Weather Bureau, Climate of South Africa: Part 3 - Maximum 24-hour rainfall. W. B. 21, Pretoria, 1956 13) L. L. Weiss, U.S. Weather Bureau Monthly Weather Review 83 (1955) 69 14) D. M. Hershfield and W. T. Wilson, J. Geophys. Research 65 0960) 959 15) U.S. Army Engineer School, Student Reference: drainage. E. 011, 1961 16) J. Stewart, J. of the Soil Conservation Service of New South Wales 16 (1960) 231 17) B. M. Reich, J. of South African Instn. of Civ. Eng., 1962 18) - - - - , Design hydrographs for very small watersheds from rainfall. P h . D . Dissertation. Colorado State University, 1962

28

n . M . REICH

19) U.S. Weather Bureau, Thunderstorm rainfall: Parts 1 and 2. Hydrometeorological Report No. 5, 1945 20) -----, Manual for depth-area analysis of storm precipitation. Cooperative Studies Technical Paper No. 1, 1946 21) G. E. Stout and F. A. Huff, Studies of hydrometeorological factors influencing severe rainstorms on small watersheds. Paper presented to the 10th Hydraulics Div. Conference of Am. Soc. Civ. Eng., in Champaign-Urbana, !11., 1961 22) F. A. Huff and J. C. Neill, Illinois State Water Survey, Bulletin No. 44, 1957 23) A. Court, J. Geophys. Research 66 (1961) 1823 24) D. A. Woolhiser and H. C. Schwalen, Arizona Agric. Exp. Stn., Technical Paper 527, 1960 25) D. M. Hershfield, Proc. Am. Soc. Civ. Eng. 8 7 : H Y 5 (1961) 99 26) B. M. Reich, Annotated bibliography and comments on the estimation of flood peaks from small watersheds. CER60BMR52. Civil Engineering Department, Colorado State University, 1960