The hydrology of areas of low precipitation - L'hydrologie des régions à faibles précipitations (Proceedings of the Canberra Symposium, December 1979; Actes du Colloque de Canberra, décembre 1979): IAHS-AISH Publ. no. 128.

Hydrological characteristics of arid zones

T. A. McMAHON Monash University, Clayton, Victoria, Australia

Abstract. Based on an analysis of approximately 70 annual flow records and peak discharge series from six arid zones, hydrological characteristics were determined. Interrelationships among characteristics and comparisons with generalized humid region features were examined. Predicta­bility of hydrological behaviour in the arid zone was also discussed. It was concluded that: (1) arid zone streams are considerably more variable than humid region ones, (2) hydrological character­istics based on humid region data should not be extrapolated to arid zones, and (3) except for the North American arid zone where up to 40 per cent regulation is possible, maximum potential streamflow regulation for the Australian, east Mediterranean and southern African arid zones is generally less than 10 per cent of mean annual flow.

Les caractéristiques hydrologiques des zones arides

Résumé. Se basant sur une analyse d'environ 70 relevés de débit annuel et de séries de débit maximal de six zones arides, on a déterminé certaines caractéristiques hydrologiques. Les relations entre ces caractéristiques et leurs comparaisons avec les traits généraux des régions humides ont été considérées. Le caractère prévisible d'un comportement hydrologique de la zone aride a égale­ment été étudié. On en a conclu que: (1) les cours d'eau des zones arides subissent beaucoup plus de variations que celles des régions humides, (2) les caractéristiques hydrologiques établies suivant les données des régions humides ne doivent pas être extrapolées aux zones arides, et (3) à l'excep­tion des zones arides de l'Amérique du Nord où une régulation atteignant 40 pour cent est pos­sible, la régulation potentielle maximale du débit des zones arides de l'Australie, des régions est-méditerranéennes, et de l'Afrique du sud est généralement moins de 10 pour cent de la moyenne annuelle de débit.

NOTATION

A cs es cv C"„ EM F g h h K MAR MR N NA O Q Q Q\QO

q Qi

Australian zone coefficient of skewness Caspian Sea zone coefficient of variation Cv for small unit area east Mediterranean zone drainage area [km2] coefficient of skewness of logarithms of peak annual discharges Hurst exponent index of variability Hurst's sample estimator of h mean annual runoff [mm] moment ratio number of years of data or number of data items North American zone other zone peak annual discharge mean annual flow specific peak discharge for 100-year flood specific mean peak discharge [m3s-1km~2] specific mean log peak discharge in absolute units

105

106 T. A. McMahon ÔMAX largest observed annual flow in station record <2MIN smallest observed annual flow in station record R range of cumulative departures of annual flows from mean ri lag one serial correlation coefficient r2 coefficient of determination s standard deviation of annual flows SA southern African zone SE standard deviation of annual flows x annual flow volume x mean annual flow 0i = Atl//4

ix i = z'th moment about mean

INTRODUCTION

The main purpose of this symposium is to describe the hydrology of low rainfall areas and to examine the effectiveness of existing methods for predicting their hydrological behaviour. This paper outlines for these areas some flow volume and flood character­istics and relates these to features from humid regions.

For convenience, the area of study is denoted by the term 'arid zone' and is shown in Fig. 1. The zone includes all regions with average annual precipitation less than 500 mm and average annual potential évapotranspiration greater than 800 mm. Figure 1 was constructed from maps by Korzun et al. (1974).

This study is based on only 2060 station-years of streamflow records distributed among six continental zones as set out in Table 1 and Fig. 1. The six zones are desig­nated as Australia (A), Caspian Sea (CS), east Mediterranean (EM), North America (NA), southern Africa (SA) and other (O).

Record lengths given in Table 1 are short relative to that considered desirable for adequate estimation of statistical parameters (Rodriguez-Iturbe, 1969). However, it is believed that while individual station parameters may be in considerable error, those based on averaging the parameters from a group of stations within a zone should

FIGURE 1. The arid zones.

Hydrological characteristics of arid zones 107 TABLE 1. Arid zone streamflow data

Item

Number of drainage basins (Number used in flood analysis)

Average length of flow records [years] (Ranges of record lengths)

Average area of basins [km2] (Range of basin areas)

Continental arid zones:

A

16

(14)

40

(14-89)

86 000

( 4 4 0 -570 000)

CS

3

(3)

35

(23-49)

270 000

(180 0 0 0 -450 000)

EM

13

(12)

27

NA

22

(18)

26

(16-37) (18-50)

380

( 3 3 -1090)

5900

( 5 7 -44 000)

SA

17

(16)

20

(11-38)

4300

( 3 4 -23 000)

O

1*

(0)

52

1 840 000

* Nile River at Aswan Dam.

provide an adequate estimate of average characteristics, although for several of the regions spatial correlation is probably high.

No papers dealing with global hydrological characteristics of the arid zone were found in the literature. However, several deal with the hydrology of specific arid zone regions (e.g. Rosenberg, 1971) and others have included one or more arid zone streams in continental (e.g. McMahon, 1973, 1977) or global studies (e.g. Yevdjevich, 1963; Kalinin, 1971).

The analyses of arid zpne hydrological characteristics that follow are considered under two headings — flow volumes and peak discharges. The flow volume analyses are based mainly on annual data although some monthly flows are analysed; the flood analyses utilize the peak discharge in each year.

FLOW VOLUMES

Examining annual precipitation—runoff relations for the arid zone stations, it is noted that mean annual runoffs do not exceed about 75 mm. However, in each zone one or two stations are included in the study with runoffs slightly in excess of this amount. Overall, the mean runoffs [mm] for the stations are A 21, CS 84, EM 25, NA 30, SA 32 and O 45. Although mean runoffs are necessary to establish the total water resources of a region, other parameters like variability and skewness are also important flow attributes. These are discussed in the following sections.

Seasonality is important too. Table 2 shows the seasonal flows expressed as a per­centage of mean flow for four arid zones. Considerable variation occurs among the areas. The east Mediterranean streams exhibit seasonality with more than 80 per cent of runoff occurring in winter whereas for Australia and North America the season of maximum flow yields only double the runoff of the minimum season. On the other hand, arid zone streams of southern Africa are slightly more seasonal than the latter two areas as about half the streamflow occurs in summer.

TABLE 2. Seasonal flows in arid zones (percentage of mean flow per season)

Zone Autumn Winter Spring Summer

Australia iast Mediterranean forth America louthern Africa

29 2

17 25

21 81 28 18

14 16 35

9

36 1

20 48

108 T. A. McMahon

FIGURE 2. Coefficients of variation of annual flows.

Flow parameters The most useful measure of hydrological variability is the coefficient of variation, Cv, defined as the standard deviation divided by the mean. Values based on annual flow data are plotted in Fig. 2. Mean Cv for each zone is A 1.27, CS 0.34, EM 1.25, NA 0.65, SA 1.14 and O0.16.

For the arid zone as a whole the mean value is 0.99 compared with the following continental values: Australia 0.67 (McMahon, 1973), North America 0.3, Europe 0.2 and Asia 0.2 (Yevdjevich, 1963; Kalinin, 1971). For Australia and North America where Cvs for the arid zone and continental areas are available, it is concluded that the variability of the arid zones is about double that of the continental areas as a whole.

As a general rule it has been found that areas of low precipitation (hence low runoff) exhibit high variability. Such a relationship is examined in Fig. 3 where, for the arid zone, Cv is plotted against mean annual runoff (MAR). The relationships established by Kalinin (1971) and McMahon (1973) for non-arid areas are also shown. Several points about Fig. 3 can be made:

(1) Australian and east Mediterranean zones exhibit similar relationships. (2) The more northern drainage basins of the southern African zone (apex of

triangular symbol pointing upwards in Fig. 3) are nearly twice as variable as the southern ones.

(3) Overall the North American zone is considerably less variable than the other arid zones.

(4) Extrapolation of the Cv versus MAR relationship from temperate regions to arid zones overestimates Cv for the arid zones.

Kalinin (1971) reviewed the literature dealing with the effect of drainage area on the coefficient of variation of annual runoff and concluded 'that the size of drainage area

Hydrological characteristics of arid zones 109

o l L I

FIGURE 3. Relation between coefficient of variation of annual flows and mean annual runoff.

(being the runoff integrator) indirectly, through other factors, reduces its variability' (p. 114). He derived the following relation:

Cv = CVo(l - 0.0008F°-s)0-5

where CVo = Cv for a small unit area (say less than 1000 km2), and F = drainage area [km2]. This relationship which is plotted in Fig. 4, where CVo was evaluated as the average of Cvs for arid zone drainage areas of less than 1000 km2, fits the data satis­factorily. A semilogarithmic least squares curve

Cv = 1.26-0.07 log!,,/7

(N = 72, r2 = 0.03, SE = 0.42 per cent) was also derived from the data. The regression coefficient is not significantly different from zero at the five per cent level of significance.

One measure of shape of a probability distribution is the coefficient of skewness (Cs). Values for each zone are plotted in Fig. 5. The average value for the arid zones is 1.8 which is several times larger than the average skewness for humid regions. For example, for Australia as a whole the median value is about 1.1 (McMahon, 1973) whereas for other continents it is only about 0.4 (Yevdjevich, 1963). Except for the North American zone where average Cs is 1.6, other arid zones show higher values (A 2.2, EM 2.1 and SA 2.0).

Distribution types A moment ratio (MR) diagram is useful in determining how well a data set fits various theoretical distributions. An example is included as Fig. 6 where the third moment is plotted against the fourth moment (0i versus /32). (3i and j32 are defined as follows:

where nt = X(x—x)'/N. Several points are noted about the plotted data.

(1) Most streams exhibit distributions that are considerably different from normal. (2) Nearly all streams lie away from the lines representing the lognormal, gamma

and Weibull distributions.

110 T. A. McMahon

m-(2)-

Kalinin (1971) X A Least squares fit ° *-"S

• EM + NA

X V SA x * O

100 100000 1 000 000 1000 10000

Drainage area Ckm2)

FIGURE 4. Effect of drainage area on coefficient of variation of annual flows.

FIGURE 5. Coefficients of skewness of annual flows.

Hydrological characteristics of arid zones 111

Pi

FIGURE 6. Moment ratio diagram of annual flows.

(3) Many streams exhibit very large /3i and j32 values.

With the exception that the arid zone data exhibit more extreme values, the above points are similar to those observed for two sets of non-arid zone streams, one for northern Australia (McMahon, 1977), and the other for the Missouri drainage basin in North America (Joseph, 1970). Curves for these two studies are compared in Fig. 6 with.the j3i—132 curve resulting from this analysis. The figure shows that the arid zone data are slightly more skewed than the non-arid data used in the other two studies.

Another moment ratio relationship is that between Cs and Cv which is plotted in Fig. 7. Kalinin's (1971) relationship based on non-arid zone streams (curve 1) and the theoretical curves for a two parameter lognormal distribution (curve 2) and a gamma distribution (curve 3) are shown. A least squares fit of the data

Cs = \.13Clvm

(N = 72, r2 = 0.67, SE = +43 per cent, - 3 0 per cent) is plotted as curve (4). This latter curve is only slightly different to the non-arid zone curve (1) and the theoretical gamma relationship. However, individual zones appear to follow a squared rather than the linear relationship observed for the consolidated data. Thus again the problem of extrapolating from non-arid zone data to arid zones is highlighted.

Persistence Another important hydrological characteristic is persistence which is the effect of one event on a following event in a time series. One measure of this is the serial correlation coefficient. Lag one serial correlation coefficients (r{) for the arid zone streams without broken records (50 in all) are plotted in Fig. 8 - the average value is 0.03. This is considerably lower than the average world value of 0.15 for non-arid zone streams calculated by Yevdjevich (1963). Only four of the 50 values were found to be different from zero at the five per cent level of significance. Higher lags were also

112 T. A. McMahon

4

3

2

1

s

0.5

0.4

0.3

0.2

X A o es

~ o EM

- f NA

- 0 SA * O

/

*

/

/A

CD

(2)

(3) (4)

/ /

4 /+*«

^ / +

+ +

/ ( 2 ) / X

/ ' + i + \J + &JJS + A/y

V

Kalinin (1971) C S =3C V +C^

cs =

i i

2CV

squares fi

i

- ( 3V

\Jyy Tm

X

i 0.2 0.4 0.6

C„

FIGURE 7. Relation between coefficients of skewness and variation for annual flows.

FIGURE 8. Lag one serial correlation coefficients of annual flows.

found to be little different from zero, further suggesting that basin carry-over storage is not significant in arid zones.

Serial correlation of low order lag represents the short memory effect of a time series process. It has been suggested by Mandelbrot and Wallis (1968) that long-term

Hydrological characteristics of arid zones 113

0.8

0.6 -

0.4

0.2 -

-0.2

-0.4

. + X o • + A V *

V

o o V

* + +j-X 1 1

+*+x o y

X

•

• •

A

cs EM NA

SA

O

A V

+ X ., A

» 2 i-j-,

x+ A

+ •

+

X

X „ A 3

A i

A " * *

X

X 4 l

cs

FIGURE 9. Relation between lag one serial correlation and coefficient of skewness of annual flows.

effects may be examined by the Hurst coefficient which is defined by h in the following equation (Hurst et al, 1965):

range of cumulative departures from mean

standard deviation of original series = (0.5N)h

A number of measures to estimate sample values of h are given in the literature (see McLeod and Hipel, 1978) but in this study Hurst's procedure is used in which h for annual flows is estimated by K:

K iog(RA)

'log (N/2)

where R = range of cumulative departures of annual flows from the mean. For the 72 streams, the average value of K was 0.68 ± 0.08. This value is not significantly different to the average value observed by Hurst et al. (1965) of 0.72 ± 0.08 for many different geophysical time series.

Other interrelationships and characteristics For drainage basins with a large water carry-over from year to year, Klemes (1970) postulated that large values of r, should be associated with low values of Q . As expected for the arid zone no such relation is exhibited (see Fig. 9).

The relations between arid and non-arid zone streams are examined further in Figs. 10 and 11 where the largest (<2MAX)

ar)d smallest (<2MIN) observed annual flows expressed as ratios of mean annual flow ( 0 are plotted against annual Cv. In Fig. 10 Kalinin's (1971) envelope curve (for non-arid zone streams) is shown along with a curve representing the arid zone data. The latter curve is of the form

<2MAX/Q = 4 . 9 C ^ 5 4 + 1

The agreement between the two curves is very good.

114 T. A. McMahon

X o e

+ A V

*

-

A es EM NA SA

O

" - 0 " + jjir

/

fe ^

+ V ©A

V

+

(2)

/en

// / / A

/ / 8 X + V X A

«X. X A A „ * vx

x? ®x

x*

(1)

X

*

A

X o

X

A

— Kalinin (1971)

— Envelope

Cv

FIGURE 10. Largest observed annual flows as a function of coefficient of variation.

(1) O

\ \ \ o

\ + \ \ + . \ +

\ "•

\ * \

\

+ + +

H- + +

} -tt-

+

+

+ A

X A O CS ® EM + NA

0 SA * O

CD- Kalinin (1971)

" \ X ' vXx A < V v + X

^ — ® V - » f ^ ^ ^ X-?-®-

FIGURE 11. Smallest observed annual flows as a function of coefficient of variation.

The plot of the smallest annual flows versus Cv in Fig. 11 shows that historically, at least, annual cease-to-flow conditions for arid zone streams are rare in North America yet in other zones A, EM and SA several stations have each recorded years of no flow. These results are a function of length of record and so the percentage of years that cease-to-flow conditions occurred may be a more reliable indicator of annual low flows. The percentages are A 3.8%, EM 4.5%, NA 0%, and SA 1.6%. Again these results suggest that the streams of the North American arid zone are more reliable than those found in the other catchments. Kalinin's (1971, p. 67) relation for non-arid zone streams is also shown in Fig. 11. It is noted from the data and Kalinin's curve that the value of Cv at which cease-to-flow conditions begin is approx-mately 0.8. However, there appears to be some difference between the envelope of

Hydrological characteristics of arid zones 115

10 25 ' 50 75 90

Percent of time indicated flow equalled or exceeded

FIGURE 12. Generalized monthly flow duration curves for arid zones.

data and Kalinin's curve. This possibly results from the shorter records available for the arid zone streams.

Low flow conditions were examined further using monthly flow duration curves. Generalized curves, based on six arid zone streams in Australia, four in east Mediter­ranean, five in North America and six in southern Africa, were prepared and are plotted in Fig. 12. They show clearly the relative variability of low flows among the four zones. North America stands out as the region with the streams contributing most baseflow to overall yield. This conclusion is consistent with our previous observa­tions about the low relative variability of NA streams compared with those in the other arid zones. There is remarkable consistency between EM and SA streams, however, the Australian ones yield slightly higher low flows.

The only comparative continental data that are available for non-arid zone streams are from the writer's study of Australian drainage basins (McMahon, 1973), the results from which are plotted also in Fig. 12 as curve (5). Relative to the arid zone curve (1), the generalized continental curve at 75 per cent exceedance exhibits baseflow that is about one order of magnitude larger than the arid zone streams. This feature of arid zone hydrology can probably be attributed to the higher but less variable rainfall over the continent as a whole compared with that over the arid zone.

PEAK DISCHARGES

No studies dealing primarily with the flood characteristics of arid zones were found in the literature. Consequently we initially examine the relation between peak annual discharge and mean annual runoff. This is followed by studies of variability and skewness, distribution of flood peaks, and finally the relation of extreme floods to variability. For all analyses, the peak discharge in each year is the basic item of data.

116 T. A.

j ID-

McMahon

(1)

X A

o cs a EM + NA 0 SA * O

Least squares f i t

o + A

o°

+

_ 1 _ J

FIGURE 13. runoff.

10 100

MAR (mm)

Relation between specific mean peak annual discharge and mean annual

In Fig. 13 specific mean peak discharge (q) in units of m3s-1km~2 is related to mean annual runoff (MAR) for 63 arid zone streams for which flood data were available (Table 1). The least squares relationship is:

^ = 3.3xlCr3(MAR)a83

(N = 63, r2 = 0.2, SE = +300 per cent, —75 per cent). Some continental trends are evident.

(1) Relative to MAR, east Mediterranean streams produce higher mean annual floods than the mean floods from the other arid zones.

(2) In contrast, Australian streams produce considerably smaller floods, about an order of magnitude lower.

Often mean annual floods have been related to drainage area. For the arid zone streams this relation is shown in Fig. 14 and described by:

q=1.22F-°A2

{N = 63, r2 = 0.44, SE = +22Q ?fc cent,-69 per cent). The slope of this relation appears to be about VA times as steep as that found in a number of studies for non-arid zone drainage basins reviewed by Alexander (1972). This suggests that flood magnitude is affected more in arid zones by drainage area (or related factors) than it is in higher precipitation regions.

Hydrological characteristics of arid zones 117 1

0.1

T

0.01

0.001

A

a

o

•

•

+

A

• A

• • ^ ^ - > A +

-* ! + +- + +

+x^^4^

X

1

X

. A V

x \ ^

S7

X

jares fit

I

X

X

X

X

1

X o

• + A

o

*o

A CS EM NA

SA

O

o

(1)

X

X

I

100 100000 1000 10 000 Drainage area (km2)

FIGURE 14. Effect of drainage area on specific mean peak annual discharge.

1 ooo ooo

X X X X 0.6 0.8 0 0.2 0.4

FIGURE 15. Index of variability of annual peak discharges.

Variability and skewness With increased attention now being given to analysing flood data in the logarithmic domain rather than using natural flows, variability and skewness of log-transformed data are examined. In Fig. 15 the index of variability (Iv) of peak annual discharges (the standard deviation of the logarithms of flows) is plotted; average values for the arid zones are found to be A 0.65, CS 0.21, EM 0.74, NA 0.38 and SA 0.58. Overall, the /„ values are considerably larger than the typical values of 0.3^0.6 for dry areas

118 T. A. McMahon and 0.1—0.4 for wet regions (Institution of Engineers, Australia, 1977). For example, based on an unpublished study (T. Kneen, unpublished data, 1979) of 67 streams in Victoria (a humid region), the average Iv was found to be 0.39.

In Fig. 16 the index of variability of annual flood peaks for each stream is plotted against the respective specific mean log peak discharge (#/); qt in m3 s_1 km""2 is defined as

^ = ( 1 0 ^ 0 ) / F

where log Q = mean of logarithms of peak annual discharges. A number of points follow from Fig. 16.

1.0

0.8

0.6

0.4

0.2

0

X o

+ A V

X

-

A CS EM NA

SA

X X

+

X

X X

X

V

X

X

o

+

I

•

v A

+

• • X

# A

4 x A

A +

A A

+

. A

A

0.0001 0.001 0.01 0.1

cif (m3/s/km2) FIGURE 16. Relation between index of variability of peak annual discharges and specific mean log peak discharges in absolute units.

(1) Overall there appears to be no relation between variability (/„) and mean flood (<?/).

(2) Continentally, the flood characteristics are distinctive. The peak discharges of the east Mediterranean streams are twice as variable as those of North America. On the other hand, the Australian streams exhibit larger ranges of mean peak dis­charges and variabilities than either zone. A large range in variability is also observed in the southern African data.

In the log-domain, one measure of the shape of the distribution of peak annual discharges is the coefficient of skewness (g). The mean and standard deviation (in parentheses) of g values for each zone are as follows: A -0.89 (1.0),CS 0.19, EM -1.6 (1.0), NA 0.37 (0.7) and SA -0.76 (1.0). Twenty-eight of the 63 streams have values that are significantly different from zero at the five per cent level of significance. Based on an analysis of 1450 streamfiow records in the United States, Hardison (1974) found that average values ranged from 0.6 on the Atlantic coast to —0.5 for the eastern mid-continent to 0.2 on the west coast. In Australia for New South Wales and Victoria average skews are —0.5 and —0.3 respectively (Boyd, 1978; T. Kneen, unpublished data, 1979). Thus except for North America and the Caspian Sea region, the arid zones are more negatively skewed than less arid regions.

Distribution types The effect of the logarithmic transformation on the distribution of peak annual discharge is significant. For untransformed data, the |3i—132 relation is similar to that

Hydrological characteristics of arid zones 119 for annual flows shown in Fig. 6, but for peak discharges the range of ft and ft is larger. However, after taking a logarithmic transformation, skewness and kurtosis are considerably reduced. Comparative moment ratio results for non-arid zones are not available.

Comparisons dealing with the relation between Cs and Cv for peak annual dis­charges are plotted in Fig. 17. Theoretical and empirical curves are also shown. Kalinin's (1971) curve is based on his world study of non-arid zone streams. Overall,

FIGURE 17. Relation between coefficients of skewness and variation of peak annual discharges.

the Cs—Cv arid zone data differ markedly from the generalized temperate curve. Up to a value of Cv equal to unity, Cs of streams from humid regions is greater than Cs of arid zone areas. Beyond Cv = 1, the opposite features are observed. Also the LN2 distribution (curve 2) overestimates Cs for the whole range. The picture presented by Cs—Cv data is very similar to that observed for the case of annual volumes shown in Fig. 7.

Extreme floods and variability Kalinin (1971, Table 15) presented a generalized relation between extreme discharge and Cv of peak annual discharge based on non-arid zone streams. This relation is super­imposed on Fig, 18 which shows the estimated peak discharge for the 100-year flood, expressed as a ratio of mean annual flood, versus Cv of peak annual discharge. The 100-year flood estimates were based on an analysis assuming the flood data are distri­buted as log Pearson III. Overall, the arid zone data fit Kalinin's generalized curve satisfactorily. Specific continental trends are not evident. The large scatter in the relation probably results from the short length of records used for many of the stations.

PREDICTION OF HYDROLOGICAL BEHAVIOUR

In the previous sections we have described briefly some hydrological characteristics of low rainfall areas and commented on the relationship between these characteristics and those from humid regions. In this section we consider how the observed character­istics affect our prediction of hydrological behaviour.

But first consider the problem of data. In Australia, the arid zone (as defined in this paper) covers about 75 per cent of the continent and contains, on the average, one

120 T. A. McMahon

x A o cs ® EM + NA $ SA

qioo

+

( D • Kalinin (1971 ) +

+

0 0.5 1.0 C 1-5 2.0

FIGURE 18. Relation between 100-year flood and coefficient .of variation of peak annual discharges.

streamflow measuring station (with at least 15 years of data) per 350 000 km2. In contrast the humid regions contain one station per 3200 km2. Although lack of development and of population are important reasons for this situation, physical and logistic difficulties of streamgauging are also of significance. For the arid region, French and Roberts (1975) have noted some of these difficulties based on their Australian experience:

(1) Flood events are irregular and of short duration. (2) There is no regular rainy season and hence it is not economical to wait at

a gauging station for a flood. (3) High stream velocities (of the order of 4 m/s) are typical. (4) High debris loads on flood wave fronts occur. (5) Gauging station controls are sandy*and unstable. (6) Drainage basin boundaries are indefinite. (7) Overbank flow frequently occurs. (8) Underflow is often a large proportion of small flood events.

This picture suggests that where good quality data are available, they should be subjected to detailed analysis. Consequently, representative basin programmes for determining hydrological characteristics in arid zones are of upmost importance. An essential aspect of these programmes covers the development of rainfall—runoff models for synthesizing streamflows at ungauged sites.

Another method for predicting hydrological behaviour and hence the potential for water resources development is to use stochastic time series models. The results of a number of analyses described in this paper — Figs. 2, 5, 6, 7, 8, 9 and 11 — are of direct relevance. Srikanthan and McMahon (1978) have shown that the difficulty in stochastic modelling increases directly with increase in variability of streamflow. Furthermore, highly variable streams are often associated with cease-to-flow conditions which add to the problem of data generation. It has been found that the more common procedures for handling zero flows are inadequate for highly variable data. As noted in Figs. 2, 5, 6, 7 and 11, annual flows of arid zone streams are generally highly variable and skewed, and excepting those in the North American zone, often contain zero flows. Thus the difficulties encountered with stochastic data generating pro­cedures are particularly severe for arid zone streams.

Hydrological characteristics of arid zones 121 The peak discharge analyses presented in Figs. 13-18 illustrate not only that the

variability of arid zone data is greater than humid area discharges but also within-zone differences. Absolute variabilities and skews are higher than those observed for humid regions, thus implying larger errors in flood estimates. However, no other special features about the peak flow characteristics of arid zones relative to humid zones are evident.

Throughout the analyses comparisons between generalized results and relationships for humid regions and those for arid zones have been made. Generally for yield analysis it has been observed that results based on humid region data should not be extrapolated to arid zones. Extrapolated results tend to overestimate both variability and yield. Few generalized relationships were available for humid area flood data, but those that were available tended to show characteristics different to the arid zone ones.

The final observation concerning the prediction of hydrological behaviour relates to the potential level of effective development of the surface water resources of the arid zone. Four characteristics — MAR, Cv, Cs and rx of annual volumes — are impor­tant here. But most significant is Cv which is considerably higher for the arid zone than humid regions. This effectively means relatively larger reservoir capacities than those required in humid regions (storage capacity is approximately proportional to annual Cv squared) or, alternatively, less regulated yield. Two studies — Lof and Hardison (1966) in North America and McMahon (1978) in Australia - illustrate this effect. In North America, the maximum potential streamflow regulation in the humid region approaches 90 per cent yet for the arid zone this drops to about 40 per cent because streamflow variability and evaporation are considerably higher in the arid zone than in humid regions. For Australia the maximum humid region value is also about 90 per cent but in much of the arid zone the maximum potential regulation drops to less than 10 per cent. Essentially, this difference between North America and Australia in minima results from the higher flow variability of the Australian arid zone over that observed in North America (Fig. 2). It is anticipated that the Australian result is typical of the potential for streamflow regulation in the east Mediterranean and southern African arid zones.

CONCLUSIONS

Based on analyses of 72 annual flow records and 63 peak flood series from six arid zones including Australia, Caspian Sea, east Mediterranean, North America and southern Africa, hydrological characteristics of flow volumes and peak discharges were deter­mined. In addition, interrelationships among the variables were examined and results for the arid zones were compared with generalized characteristics of humid regions. The question of how the observed characteristics affect the predictability of hydro-logical behaviour in the arid zone was also discussed.

The main conclusions were as follows:

(1) Variability of arid zone streams expressed as the coefficient of variation of annual flows is about double that for continental areas as a whole.

(2) The effect of drainage basin area on annual Cv is insignificant. (3) The distribution of annual flows of arid zone streams is considerably more

skewed than the flow distribution of humid streams. (4) On average, the lag one serial correlation coefficient is very close to zero. (5) North American streams are less variable and yield relatively more low flow

than other arid zone streams. (6) Australian streams produce considerably smaller mean annual flood peaks

than the peaks observed in other arid zones. (7) The variability of peak floods (expressed as the standard deviation of logarithms

of peak discharges) for Australia, east Mediterranean and southern Africa is about

122 T. A. McMahon double the value found for humid region streams. North American streams do not follow this general trend.

(8) Skews of logarithms of peak annual discharges were found to be, on average, negative for Australian, east Mediterranean and southern African streams, but positive for North American ones.

The effect of these results on the predictability of hydrological behaviour in arid zones was considered from several points of view:

(1) Arid zone data. Because of inadequate arid zone data, it is essential that available data be subjected to detailed analysis.

(2) Rainfall—runoff models. In view of (1) above, the development of rainfall-runoff models for synthesizing arid zone streamflow data is essential.

(3) Stochastic data generation. It was concluded that for arid zone streams, diffi­culties with stochastic data generation procedures will be severe.

(4) Peak discharge analysis. Although more variable, peak discharges of arid zone streams present no unique features for analysis.

(5) Humid to arid zone extrapolation. Hydrological characteristics based on humid region data should not be extrapolated to arid zones.

(6) Surface water resources development. Based on North American and Australian analysis, the potential for arid zone streamflow regulation is very low. At best, up to 40 per cent regulation may be realized in North America, but this falls to below 10 per cent for much of the Australian arid zone. The latter figure would be typical of the potential in the east Mediterranean and southern African zones.

Acknowledgement. The writer wishes to sincerely thanK Dr V. Klemes for reviewing the draft manuscript.

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