Global streamflows – Part 1: Characteristics of annual ...Global streamflows – Part 1: Characteristics of annual streamflows Thomas A. McMahon a,*, Richard M. Vogel b, Murray

Global streamflows – Part 1: Characteristics of annualstreamflows

Thomas A. McMahon a,*, Richard M. Vogel b, Murray C. Peel a,Geoffrey G.S. Pegram c

a Department of Civil and Environmental Engineering, The University of Melbourne, Victoria 3010, Australiab Department of Civil and Environmental Engineering, Tufts University, Medford, MA, USAc Civil Engineering Programme, University of KwaZulu-Natal Durban, South Africa

Received 15 February 2007; received in revised form 20 July 2007; accepted 3 September 2007

KEYWORDSGlobal hydrology;Global streamflowdata;Global streamflowcharacteristics;Global rivers

Summary This is the first of three related papers that summarizes streamflow character-istics of a set of 1221 global rivers. The rivers are well distributed world-wide, are un-impacted by upstream reservoirs or diversions for the period of data collection and haveat least 10 years of continuous monthly and annual streamflow data. The following keyfeatures of annual flows are examined: mean, variability and skewness, distribution type(Gamma or Lognormal), flow percentiles and dependence. High and low frequency persis-tence is examined through the Empirical Mode Decomposition technique. Low flow runlength, magnitude and severity are also explored, where severity is based on ExtendedDeficit Analysis. It has been observed elsewhere that there are large differences in hydro-logic characteristics between Australia and southern Africa in contrast to the rest of theworld. This issue is tested further in this paper. The range of analyses and results pre-sented herein also form a suite of empirical evidence that future unified theories ofhydrology at the catchment scale must be able to adequately describe.ª 2007 Elsevier B.V. All rights reserved.

Introduction

This is the first of a series of three papers that describes thehydrologic characteristics of a unique global data set, which

consists of historical annual and monthly streamflow recordsof 1221 rivers. With the likelihood of climate change havingan impact on future streamflow, we develop and discuss keycharacteristics of both unregulated rivers (this paper) andhypothetically regulated ones (paper 2, McMahon et al.,2007a). We also examine the variations in hydrologic char-acteristics on a country and on a climatic basis in paper 3(McMahon et al., 2007b). In addition, the global data set en-ables us to examine, on a world-wide basis, climate change

0022-1694/$ - see front matter ª 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.jhydrol.2007.09.002

* Corresponding author. Tel.: +61 3 8344 7731; fax: +61 3 8344 6215.E-mail addresses: [email protected] (T.A.

McMahon), [email protected] (R.M. Vogel), [email protected] (M.C. Peel), [email protected] (G.G.S. Pegram).

Journal of Hydrology (2007) 347, 243–259

ava i lab le a t www.sc iencedi rec t . com

journal homepage: www.elsevier .com/ locate / jhydrol

impacts on gauged and, by inference, ungauged catch-ments. Further, it allows us to explore broad differencesin regional hydrologic features and to make world-widehydrologic comparisons of surface water resources.

Although paper 3 (McMahon et al., 2007b) of this seriesconcentrates largely on continental, regional and climaticdifferences in hydrologic characteristics of the annualstreamflow time series, we have deliberately included inthis paper observed differences between hydrological re-sponses in Australia–southern Africa (ASA) and the rest ofthe world (RoW).

We had four objectives in carrying out this project.Firstly, there have been a number of papers highlightingthe hydrologic differences between ASA and RoW and dis-cussing the reasons for the differences (McMahon et al.,1992; Peel et al., 2001; Peel et al., 2002; Peel et al.,2004a). With this enhanced data set, we wish to reviewthese results and evaluate them more fully. Moreover, wewanted to extend the analysis to include reservoir capacityestimates and also reservoir performance at continental,country and regional climatic scales (using the Koppen clas-sification) (Koppen, 1936). As far as we can determine, anal-ysis relating to reservoir capacity and performance at theselarge geographic scales has not yet been addressed. Thesecond objective was to develop empirical relationships be-tween hydrologic characteristics (for both the natural flowsand hypothetically regulated rivers) based on the globaldata set and relevant independent parameters. Such rela-tionships would be useful to water managers concerned withexploring surface streamflow characteristics by continent,region, country or climate zones. Furthermore, the rela-tionships could be used to estimate surface streamflowcharacteristics of ungauged rivers. The third objective ofthis project is to make available to modellers of global run-off, streamflow characteristics that are based on measuredvalues which can be used for comparison with model esti-mates. Furthermore, the analysis provides a global charac-terization of hydrologic characteristics which grid-basedmodels must be able to reproduce. The fourth objective re-lates to the search for a unified theory of hydrology at thecatchment scale (Sivapalan, 2005). In presenting the annualand monthly characteristics of streamflow globally, we pro-vide a range of empirical evidence that future theories ofhydrology at the catchment scale must be able to ade-quately describe. This ties in with adopting a top-down ap-proach to our analysis which, in the context of our globaldata set of streamflows, involves learning how surfacestreamflow characteristics vary across a range of scalesfrom small catchments to continents and within a range ofclimate zones. Using the hypothesis of top-down modellingnamely that ‘‘. . . at the whole-catchment scale much ofthe complexity needed to model hydrological response at fi-ner scales is unnecessary. . .’’ (Post et al., 2005), we exploreand develop relationships that will be of assistance to bothwater managers and hydrologic modellers.

Following this introduction we review the key literaturerelevant to this paper. Details of the historical streamflowdata are discussed in Section ‘Annual streamflow data’.‘What probability distribution function?’ is examined in Sec-tion ‘What probability distribution function?’ followed by adiscussion of the key statistical characteristics in Section‘Mean, variability and skewness’. In Section ‘Flow percen-

tiles’ we present an analysis of flow percentiles which is fol-lowed in Section ‘Dependence’ by an examination of auto-correlation in annual data. Section ‘Persistence characteris-tics’ examines persistence through Empirical Mode Decom-position (EMD). The final section describes analyses of lowflow run length and magnitude, then severity using Ex-tended Deficit Analysis (EDA). Some conclusions are pre-sented in Section ‘Summary and conclusions’.

Literature

As far as we are aware there is no other series of papers thathas captured, assessed and reported on such a range of sur-face hydrologic characteristics world-wide as is attemptedhere. Most papers and reports that addressed global runoffdealt mainly with mean or median flows and in only a fewcases with variability, skewness and/or auto-correlation.

One of the earliest attempts to compute global meanrunoff was by Ayushinskaya et al. (1977) which was pub-lished about the same time as a number of other atlasesof the world water balance (Korzun et al., 1974; Van DerLeedon, 1975; UNESCO, 1977), the most comprehensivebeing the report by Baumgartner and Reichel (1975) andby UNESCO (1978). These papers and reports concentratedon the mean annual water balance although several dis-cussed streamflow variability. A recent report by Shikloma-nov and Rodda (2003) presents an up-to-date analysis ofworld water resources and, inter alia, includes informationof streamflow variability and skewness for selected rivers,countries and continents.

Kalinin (1971) carried out the first detailed analyses ofriver flows world-wide, but only 8 of his 137 catchmentswere from the southern hemisphere. McMahon (1975,1977, 1979) carried out the first analyses of southern hemi-sphere rivers along the lines adopted by Kalinin and de-scribed some interesting deviations from Kalinin’sconclusions. These analyses were extended to a world-widedata set by McMahon and his colleagues over the next twodecades (see for example, McMahon et al., 1992; Peelet al., 2001; Peel et al., 2004a).

An extensive report of continental comparisons ofstreamflow characteristics by McMahon et al. (1992) utilizesthe world-wide data set of annual flows and peak dischargesnoted in the previous paragraph. The report describes thevariability, skewness, distribution types and auto-correla-tion of both annual flows and annual peak discharges. Inaddition, it examines the time series structure of annualflows. Finally, it explores inter-continental comparisonsalong with those relating to seasonal flow regime stratifica-tion (as defined by Haines et al. (1988)) and to climate usingthe Koppen classification (Koppen, 1936). Of the many con-clusions in McMahon et al. (1992), the key one relates to thehypothesis that there are major differences in terms of run-off characteristics between the northern and southernhemispheres; however, that hypothesis could not be con-firmed. What was found, however, was that the annualcharacteristics of streamflow for Australia and southernAfrica are different from those found for streams in the restof the world. This observation has been examined by Peelet al. (2001, 2004a) who concluded that the variability ofannual runoff for temperate Australia and temperate andarid southern Africa was higher than for other continents

244 T.A. McMahon et al.

across similar climatic zones. They further concluded thatfor temperate areas the distribution of evergreen anddeciduous vegetation was a potential cause of the observedvariability differences, along with differences in the vari-ability of annual precipitation, percentage of forestedcatchment area, seasonality of precipitation and mean an-nual daily temperature range.

Peel et al. (2004b, 2005) analyzed a slightly larger dataset than that used herein. They found that, for annualstreamflow, run lengths of years below the median weresimilar across all continents except for the Sahel region innorthern Africa where a distinct bias towards longer lengthswas evident. However, in terms of relative magnitude of thedeficits below the median flow, Australia and southern Afri-ca exhibited values that were much larger than elsewhere.These issues are reviewed in more detail herein and in thethird paper in this series.

Both Yevdjevich (1963) and Kalinan (1971) suggested thatthe annual time series of world rivers were normally distrib-uted, a feature not found by McMahon’s (1975, 1977, 1979)analysis. Markovic (1965) studied five distributions (Normal,Lognormal 2 and 3 parameter, Gamma 2 and 3 parameter)and concluded that for rivers in western United States andsouthwestern Canada, all five probability distribution func-tions describe the annual flows satisfactorily, though morerecently, Vogel and Wilson (1996) found that the Gamma 2distribution was the only two parameter distribution whichwas satisfactory for US rivers. This aspect is explored in Sec-tion ‘What probability distribution function?’, because theshape of the tail of the probability distribution describingflow volumes has a critical influence on the reliability of res-ervoir systems.

Annual streamflow data

The global data set consists of 1221 unregulated rivers with10 years or more of continuous historical annual andmonthly flows. The water year was the basis of the annualflow volumes. Fig. 1 shows the locations of the rivers andit is noted that they are reasonably distributed globally,although there are some regions (for example, arid regions– Mediterranean North Africa, the Middle East, South Wes-

tern Africa, central Australia and tropical regions – centralAmerica, Indonesia, non-coastal Brazil, Peru, Bolivia, Ecua-dor) with little or no data.

The data set was initially collated by the first author dur-ing the 1980s and the details were reported several years la-ter (McMahon et al., 1992), with subsequent additions andrevisions to the data set since that time (Peel et al.,2001, 2004a). Considerable effort has gone into ensuringthe data are free of errors, are not impacted by major waterwithdrawals from the streams and the flow values are notaffected by upstream reservoirs. However, except for obvi-ous transcription errors and those related to catchmentareas and location, measurement errors particularly relat-ing to the adequacy of rating curves have not been ad-dressed. Such a task is well beyond a project of thismagnitude.

Fig. 2 shows for the global data set a plot of length ofcontinuous record and catchment area. The catchmentareas vary from 115 to 65 200 km2 (10th percentile to 90thpercentile) with a median area of 1670 km2 and with lengthsfrom 15 to 58 years (10th percentile to 90th percentile) anda median of 29 years. Because of the difficulty in collatingmonthly and annual streamflow data without omitting partsof the records, we were unable to utilize a common periodin our analyses. This limitation may affect some results andneeds to be kept in mind in their interpretation. In themain, the period from 1950 to 1980 is best represented inthe data set.

As a further introduction to the data, Table 1 sets outsome key hydrologic features of the rivers. The rivers covera very wide range of hydrologic regimes, with mean annualrunoff varying from a minimum of 0.373 mm to a maximumof 5370 mm. Great variability is also observed in the annualcoefficient of variation (Cv) from 0.062 to 2.97. In Table 1the statistics for the coefficient of skewness (c) and lag-one serial correlation coefficient (auto-correlation) (q) ofannual flows are based on both a minimum of 10 and 25years of annual flow records respectively, the latter shownin parentheses. Again, large variations in these parametersis evident with skewness varying from �2.2 to 6.1 andauto-correlation ranging from �0.48 to 0.90 for 25 yearsor more data.

Figure 1 Location of streamflow stations (1221) with 10 or more years of continuous streamflow data (plate carree projection).

Global streamflows – Part 1: Characteristics of annual streamflows 245

What probability distribution function?

A perennial challenge in hydrology is to choose an appropri-ate probability distribution function (pdf) to describestreamflow data. Over the past fifty or so years many papersand reports have dealt with this question in relation to an-nual streamflows – Yevdjevich (1963), Markovic (1965),Lof and Hardison (1966), Kalinin (1971), McMahon et al.(1992), Vogel and Wilson (1996) and others. We do not planto traverse details in this paper but suffice to say that forEurope and portions of North America the normal pdf is of-ten adequate but the 2-parameter lognormal (LN2) andGamma are more generally applicable. The normal distribu-tion was found to provide a poor approximation to the dis-tribution of annual flows in ASA.

To explore this issue further, we use L-moment diagramsto assess whether the global annual flows tend towards aLognormal (LN2 and LN3) or a Gamma (or Pearson III) distri-bution. To do this we developed theoretical relationshipsbetween L-Cv and Cv and L-Skewness and the coefficientof skewness (c) for these distributions, which are describedin the Appendix.

Based on the theoretical analysis in the Appendix, tradi-tional L-moment diagrams for annual average streamflowsbased on the global dataset of 1221 rivers are provided in

Fig. 3. What one observes from these two figures is that itis nearly impossible to distinguish whether the observationsarise from either a Lognormal or a Gamma pdf. As an inno-vation, in Fig. 4 we show relationships between (a) Cv andL-Cv and (b) Skewness and L-Skewness for the same dataset. In each of these figures it is quite apparent that a Gam-ma pdf provides a much better fit to the observations thanthe LN2.

It is well known that ordinary product moments arebiased downward (Vogel and Fennessey, 1993) which iswhy the estimated values of both Cv and Skewness tend tobe a bit lower than the theoretical curves in Figs. 4a andb, respectively. However, even given this bias, Fig. 4 en-ables us to distinguish the tail behaviour between the Log-normal and Gamma probability distribution functions moreeffectively than through the use of traditional L-momentdiagrams as shown in Fig. 3. It is possible that the downwardbias in estimates of both Cv and Skewness still impedes ourability to discern which distribution is best, though there issome evidence here that a Gamma distribution function fitsthe global historical annual streamflows better than a Log-normal distribution function. This result is consistent withother recent investigations using L-moment diagrams andvery large datasets (Vogel and Wilson, 1996). Fig. 4 indi-cates that a promising avenue for future research involves

0

50

100

150

200

1 100 10,000 1,000,000Catchment area (km2)

Record length(years)

Figure 2 Length of continuous record versus catchment area.

Table 1 The key hydrologic features of the 1221 rivers used in this paper

Area (km2) N (years) l (mm) Cv c q

Median 1610 29 339 0.314 0.552 0.103(0.561)a (0.119)

Maximum 4,640,000 185 5370 2.97 6.14 0.936(6.14) (0.895)

Minimum 0.360 10 0.373 0.0619 �2.22 �0.756(�2.22) (�0.482)

Area is catchment area (km2), N is the number of years of continuous annual or monthly data, l is the mean annual runoff (mm), Cv is thecoefficient of variation of annual flows, c is the coefficient of skewness of annual flows and q is lag-one serial correlation coefficient.a Values in parentheses are computed using 25 or more years of historical data (729 rivers).


plotting two different unbiased measures of Cv, skewness,or perhaps kurtosis against each other to discriminate tailbehaviour.

Mean, variability and skewness

Fig. 5 shows the relationship between mean annual flow andthe standard deviation of annual flows for the 1221 globalrivers. We note the separation of the two curves represent-ing ASA and RoW. For example, for a river with a mean an-nual flow (l) of 1000 · 106 m3 the ratio of the standarddeviation (r) of flow of such a river in ASA would be1.78 · rRoW where rRoW is the standard deviation of a typicalriver in the RoW. The global equation defining the relation-ship between r and l was fit using weighted least squares(WLS) regression which resulted in

r ¼ 0:752l0:876 ð1Þn ¼ 1221; R2 ¼ 0:940; se ¼ þ72%;�42%

where l is the mean annual streamflow (106 m3), r is thestandard deviation of the annual flows (106 m3), n is the

number of rivers in analysis, R2 is the square of the correla-tion coefficient and se is the standard error of estimate. Theweightings in the WLS procedure used here and elsewhere inthis paper are historical record lengths.

We have also related l to catchment area for ASA andRoW by the following empirical relationships which were ob-tained using WLS regression:

lASA ¼ 1:013 Area0:727 ð2Þ

n ¼ 263; R2 ¼ 0:60; se ¼ þ290%; �74%; p ¼< 0:001

lRoW ¼ 1:526 Area0:818 ð3Þ

n ¼ 958; R2 ¼ 0:78; se ¼ þ199%; �67%; p ¼< 0:001

where the subscripts ASA and RoW refer to Australia–south-ern Africa and rest of the world respectively, Area repre-sents the drainage area in km2 and p is the probability ofthe value of R2 occurring by chance. Substituting a typicalrange of catchment areas (10 and 100,000 km2) into Eqs.(2) and (3), rivers in the RoW yield between about 100%and 300% more annual flow than rivers in ASA.

0

0.2

0.4

0.6

0.8

1

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1L-Skewness

L-Cv

ObservationsLN2Gamma

-0.2

0

0.2

0.4

0.6

0.8

1

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1L-Skewness

L-Kurtosis

ObservationsLN3Pearson type III

Figure 3 Relationships between L-Cv, L-Skewness and L-Kurtosis: (a) L-Cv versus L-Skewness, (b) L-Kurtosis versus L-Skewness.


0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1L-Cv

Cv

ObservationsLN2Gamma

-1

0

1

2

3

-0.2 0 0.2 0.4 0.6 0.8L-Skewness

Skewness

ObservationsLN3Pearson Type III

Figure 4 Product-moment versus L-moments: (a) Product-moment Cv versus L-Cv, (b) Coefficient of skewness versus L-Skewness.

1

100

10,000

1,000,000

1 100 10,000 1,000,000Mean annual streamflow (106 m3)

Standard deviation of annual

streamflows(106 m3)

ASARoW

ASAσ = 1.017 μ0.906

R2 = 0.93

RoWσ = 0.508 μ0.914

R2 = 0.95

Figure 5 Standard deviation of annual flows versus mean annual flow. Solid line = RoW, broken line = ASA.


The next two figures explore the relationship betweenthe Cv of annual flows and mean annual runoff (MAR) inmm (Fig. 6) and catchment area (Fig. 7). Similar figureshave been produced elsewhere (see for example, McMahonet al., 1992) but are reproduced here for completeness. InFig. 6, Cv decreases with an increase in MAR (which can beconsidered a surrogate for mean rainfall). The global curvefit using WLS is

Cv ¼ 1:828 MAR�0:299 ð4Þ

where Cv is the coefficient of variation of annual stream-flows and MAR is the mean annual runoff (mm). No statisticsare provided here as the values would be spurious, given themean is included in both sides of the equation.

The generalised relationship of Kalinin (1971) was alsoapplied to the data set and the resulting curve superim-posed on Fig. 6 is

Cv ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5:82 MAR�1 þ 0:206

pð5Þ

and it clearly provides a poor fit to the data, particularly forhigh mean annual runoff where it overestimates annual Cv.

The relationship between Cv and catchment area is givenin Fig. 7. Based on the entire global data set, there is littleassociation between the two variables but again the differ-ence between the ASA region and the RoW is clear. Theequation fit using WLS with all the data is

Cv ¼ 0:447 Area�0:0384 ð6Þn ¼ 1221; R2 ¼ 0:024; se ¼ þ85%; �46%

where Area is catchment area (km2). The generalised rela-tionship between Cv and catchment area proposed by Kal-inin (1971) is also applied to the data set and is clearlydifferent from Eq. (6) (see Fig. 7).

0.1

1

10

1 10 100 1,000 10,000Mean annual runoff (mm)

Annual Cv ASARoWKalininASA

Cv = 2.55 μ-0.271

RoWCv = 1.16 μ-0.243

Figure 6 Coefficient of variation of annual flows versus mean annual runoff. Solid line = RoW, broken line = ASA.

0.1

1

10

0 10 1,000 100,000 10,000,000Catchment area (km2)

Annual CvASARoWKalinin (1971)ASA

Cv = 0.513 Area0.0335

R2 = 0.015

RoWCv = 0.325 Area-0.0191

R2 = 0.008

Figure 7 Coefficient of variation of annual flows versus catchment area. Solid line = RoW, broken line = ASA.


In this context we note that large Cv’s imply larger reser-voir storages or lower system yields. For example, the fol-lowing equation based on the Gould–Dincer Gammamethod (see McMahon and Adeloye, 2005) for estimating re-quired over-year storage capacity can be used to explorethis further.

S ¼z2p

4ð1� aÞC2vl

1þ q1� q

ð7Þ

where S is the required reservoir capacity to meet a draft ofa (target draft divided by mean annual flow), zp is the stan-dardized Gamma variate at 100p% probability of non-excee-dance, i.e. probability of not being able to meet the draft,Cv is the coefficient of variation of annual inflows, l is themean annual inflow, and q is the annual lag-one serial cor-relation coefficient.

From Eq. (7) we can compare two rivers, regions or con-tinents by taking the ratio of the two estimated reservoircapacities, assuming the same draft and failure conditionsin systems A and B, leading to

SASB¼ CvA

CvB

� �2 lA

lB

1þ qA

1� qA

1� qB

1þ qB

ð8Þ

To compare required reservoir capacities in ASA with thosein the RoW, a common catchment size of 10,000 km2 isadopted. From Eqs. (2) and (3), lASA = 0.312 lRoW whereinflows l are per unit catchment area. Also we note thatqASA � qRoW (see Fig. 9), thus

SASASRoW

¼ 0:312CvASA

CvRoW

� �2

ð9Þ

Noting that CvASA � 2.6 CvRoW , (from Fig. 7 forarea = 10,000 km2), then for these specific conditions, res-ervoirs necessary to meet over-year storage requirementsin the ASA continental region need to be a little more thantwice as large as those in the RoW for the same level of reg-ulation (i.e. for the same target draft and reliability), be-fore adjusting for net evaporation losses, which wouldexacerbate the difference.

Flow percentiles

In Fig. 8, the empirical 10th and 90th exceedance percen-tiles of annual flow values of each record are plotted againstthe mean annual runoff. The differences between the twocontinental areas are evident. A plot relating the percentilevalues to catchment areas (not included here) exhibits sim-ilar differences. The 10th percentile values in Fig. 8 for ASAare 22% larger than the equivalent values for a catchmentwith MAR of 500 mm located in the RoW. For the 90th per-centile flows ASA flows are typically 28% smaller than theRoW percentiles, again for a catchment with 500 mm of an-nual runoff. These differences relate directly to the rela-tively larger variance observed for ASA rivers compared tothose for the RoW (Figs. 5–7).

Dependence

In examining the auto-correlation of annual streamflowsof the global rivers, only values that are statistically sig-nificantly different from zero are considered. Fig. 9 showsa plot of estimates of the statistically significant lag-oneserial correlation q against catchment area. Of the 1221rivers in the global data set, 249 values were significantlydifferent from zero at the 5% level, that is, about fourtimes as many as would be expected by chance. Elevenof the rivers exhibited statistically significant negativeauto-correlations. The average magnitude of the statisti-cally significant auto-correlations (absolute values) is0.441. The differences in average absolute values be-tween ASA and RoW are negligible, 0.424 and 0.445respectively.

A relationship between statistically significant positive qvalues and the independent variables record length N andcatchment area (km2) was fit using WLS regression resultingin

q ¼ 0:832N�0:281 Area0:0344 ð10Þðn ¼ 249; R2 ¼ 0:22Þ

0

1

2

3

4

1 10 100 1,000 10,000Mean annual runoff (mm)

Percentile value as ratio of

mean annual flow

90th percentile ASA90th percentile RoW10th percentile ASA10th percentile RoW

Figure 8 10th and 90th percentile values of annual flows (as ratio of mean annual flow) versus mean annual runoff (mm). Solidline = RoW, broken line = ASA.


This overall relationship may be statistically significantalthough the residuals are not normally distributed. Never-theless, it suggests that lag-one serial correlation is inver-sely related to record length and positively related toarea. Approximately 90% of the variance in the relationshipis accounted for by the record length term. The positiverelationship of q with area, although weak and not statisti-cally significant, is consistent with our understanding thathigh positive auto-correlation maybe related to water carry-over in catchments from year to year. Yevdjevich (1964) ar-gues that dependence is not only a function of carryoverstorage (see also Klemes, 1970) but is also affected byinconsistencies and inhomogeneities in the data. However,in Yevdjevich’s annual runoff data set, estimates of annualq increased with increases in record length, although henoted that this conclusion should be accepted very cau-tiously. In summary, our analysis suggests that q is inverselyrelated to record length and that it is positively, but weakly,related to catchment area. This issue is discussed further inSection ‘Persistence characteristics’.

Also observed in Fig. 9 are values of q for some rivers thatappear to be excessively high. After closer examination ofthese values and reviewing their spatial location we notethat rivers with high values of q generally: (1) are in regionswhere permanent snow cover is likely to be contributing tothe streamflow (e.g., Iceland, northern Canada, Chile/Argentina); (2) have natural lakes within the catchment(e.g., tropical East Africa); (3) experience a significanttrend in streamflow during the period of record (e.g., Sa-hel); or (4) have a short record length (sampling variability).

The 11 rivers with significant negative auto-correlationsare widely distributed and show no clear spatial pattern. Areview of their flow characteristics and other potentiallyexplanatory variables including climate, as defined by theKoppen classification, identified a common link as beinglength of data. Fig. 10 shows the relationship between thehigh negative q values and N fit using WLS regression. Thehigh correlation (R2 = 0.754) between the two variableswould suggest that generally the high negative q valuesare mainly the result of sample size and sample variability

-0.8

-0.6

-0.4

-0.2

00 10 20 30 40

Length of data (years)

ρ ρ = −1.284 + 0.577 log10 N

Figure 10 Relationship between the statistically significant negatively lag-one serial correlations and record length.

-1

-0.5

0

0.5

1

10 1,000 100,000 10,000,000Catchment area (km2)

RoWASA

ρ

0.1

Figure 9 Statistically significant lag-one serial correlation of annual flows versus catchment area.


rather than some other factor. This explanation is plausiblegiven that, unlike positive correlations, there is no physicalexplanation why large negative correlations should occur atthe annual level.

Fig. 11, a plot of statistically significant positive and neg-ative q against the coefficient of skewness of annual flows,shows that auto-correlation is not correlated with skewness(weighted least squares analysis R2 for +q = 0.018 and�q = 0.000). Based on a Monte Carlo modelling study and areview of the eight most negatively skewed series of annualstreamflow from 140 rivers compiled by Yevdjevich (1963),Klemes (1970) postulated that negatively skewed annualstreamflows are likely to occur in catchments with largecarry-over water storage and, hence, high positive auto-cor-relations. Fig. 11 would suggest otherwise especially asmany of the large positive auto-correlations are associatedwith positive skewness.

Persistence characteristics

Inter-annual characteristics, i.e. high frequency features,were considered in the previous section where lag-one serialcorrelations of annual flows were discussed. In this sectionwe examine low frequency characteristics by using Empiri-cal Mode Decomposition (Huang et al., 1998) to examinethe proportion of variance in the historical annual time ser-ies of flows from the global data set which is due to low fre-quency fluctuations. However, to ensure there are sufficientdata to define the characteristics we have restricted theanalysis to 595 rivers that have at least 30 years of historicaldata.

EMD is an adaptive form of time-series decompositionfor non-linear and non-stationary data, and thereforeappropriate for time-series of annual streamflows. Itdecomposes a time series into a set of independent intrin-sic mode functions (IMFs) and a residual component. If theIMFs and residual are summed together they form the ori-ginal time series. Depending on the nature of the time-series being studied, the IMFs and the residual componentmay exhibit linear or non-linear behaviour (amplitude andfrequency modulation).

Application of EMD analysis is illustrated in Fig. 12 usingthe Zambesi River at Victoria Falls. The figure shows the his-torical time series, three IMFs and the residual. We chosethis example as it is a difficult river to model yet the resid-ual shows the trend very clearly. EMD analysis allows one todetermine the average period for each IMF and the varianceassociated with each IMF and the residual. These values areshown in Table 2. Because of the major increase in theZambesi flows (�70% after the 1940s and 1950s) we observea very high proportion of variance, approximately 47%, ac-counted for in the residual. We note that the concept of‘‘period’’ is misleading in the EMD analysis of non-stationarytime series. Each IMF is likely to exhibit both amplitude andfrequency modulation. We therefore use the term ‘‘averageperiod’’ with this caveat.

Table 3 is a summary of the results of applying EMD to asubset of 595 global rivers with 30 or more years of annualflows. A minimum of 30 years was chosen to ensure thatthere was sufficient data for the shorter record lengths toadequately define at least the first two IMFs. The tableshows that approximately 43% of the variance is accountedfor by IMF1 which has an average period of 3.12 years. IMF2accounts for a further 19% of the variance, thus the intra-decadal component defined here as the IMFs with periodsless than 10 years accounts for 60% of the variance. Theremaining variance is contained in the inter-decadal compo-nent, of which a significant proportion of the variance iswithin the residual or trend component. Results of our intra-and inter-decadal component analysis are consistent withDettinger and Diaz (2000) who, using a high pass filter witha seven year cutoff, found that 61% of the variability of an-nual streamflow was due to interannual (<7 years frequency)fluctuations.

Fig. 13 shows that the ratio of variance due to inter-dec-adal (low frequencies on the annual time scale) fluctuationsrelative to total variance is directly correlated to lag-oneserial correlation (based on WLS R2 = 0.45, p < 0.001). Asthe proportion of total variance due to inter-decadal fluctu-ations decreases, the lag-one serial correlation decreasesand tends toward zero (random). Conversely, the lag-oneserial correlation increases as the proportion of total vari-

-1

-0.5

0

0.5

1

-3 -2 -1 0 1 2 3 4Annual coefficient of skewness

ρ

Figure 11 Statistically significant positive and negative lag-one serial correlation of annual flows versus coefficient of skewness ofannual flows.


ance due to inter-decadal fluctuations and the trend compo-nent increases. For the plotted data in Fig. 13, there is anegligible difference between the ASA rivers and those forthe RoW (the separate relationships are not shown in thefigure).

It was observed earlier that auto-correlation is onlyweakly correlated with catchment area, but as observedin Fig. 13, auto-correlation is strongly correlated with theratio of inter-decadal fluctuations to total variance(R2 = 0.45, p < 0.001). Thus large values of q are associatedwith large values of the ratio which, in turn, implies that alarge trend component or strong inter-decadal fluctuation isassociated with high positive auto-correlations. The cross-correlations in Table 4 among these variables for 120 globalrivers (those with 30 or more annual flows and statisticallysignificant positive auto-correlations) further support theseobservations.

Low flow run length, magnitude and severity

Low flows in the context of this paper relate to values belowthe median of a historical series. This discussion is based onan analysis of the annual time-series for the 1221 global riv-ers. We have purposely not used the term drought to referto low flow sequences, as it has been argued elsewhere(McMahon and Finlayson, 2003) that drought is a constructof society, so we will continue in this paper to use the termlow flows for a series of streamflows below the median re-cord value. For each river we computed the frequency ofrun lengths below the median, as well as the run magnitudeand run severity.

An illustration of run length is presented for the ZambesiRiver at Victoria Falls in Fig. 14. Fig. 14a shows the time ser-ies and Fig. 14b plots the resulting frequency of run lengthsequal to or below the median for the same river. In under-standing run lengths, the shape of the run length frequencydistribution is important and can be described by a skewnessmetric as (Peel et al., 2004b)

g ¼

PLi¼1

fi‘3i

N � 1ð11Þ

where g is a metric of skewness of the run length frequencydiagram, fi is the frequency of run length ‘i, L is the longest

-20,000

0

20,000

40,000

60,000

80,000

1924 1934 1944 1954 1964 1974

106 m3

Historical annual flowsIMF1IMF2IMF3Residual

Figure 12 EMD analysis of annual streamflow (1924–1977) for Zambesi River at Victoria Falls showing the observed data, the threeIMFs and the residual trend.

Table 2 Average period and variance of IMFs for ZambesiRiver at Victoria Falls

Averageperiod(years)

Variance(as ratioof samplevariance)

Varianceas % of

PIMFs +

residual

IMF1 3.6 0.384 31IMF2 6.8 0.217 17IMF3 15.4 0.067 5Residual 0.579 47P

IMFs + residual 1.247 100

Table 3 Mean average period and mean variance for riversfrom global data set with 30 years or more of annualstreamflows

Item (numberof rivers)

Averageperiod(years)

Variance(as ratioof samplevariance)

Variance as% of

PIMFs +

residual

IMF1 (595) 3.12 0.592 43.3IMF2 (594) 7.13 0.256 18.7IMF3 (426) 15.5 0.162 11.9IMF4 (85) 29.6 0.109 8.0IMF5 (5) 53.8 0.038 2.8Residual 0.209 15.3P

IMFs + residual 1.366 100.0


observed run length and N is the length of historical data.For the Zambesi River annual flows, g = 68.0.

To assess whether a record contains run lengths that arelonger or shorter than expected we follow the procedure ofPeel et al. (2004b) in which the run-frequency relationshipin the historical series is compared with that for a first-orderlinear autoregressive, AR(1), model. The AR(1) model is auseful comparator as 91% of world rivers were found (inthe analysis of 720 rivers by McMahon et al. (1992)) to beeither white noise (77%) or AR(1) (16%). Furthermore,AR(1) does not capture long-term persistence in the formof decadal or multi-decadal fluctuations in runoff records(Thyer and Kuczera, 2000).

To test whether the run length behaviour at each sta-tion was different from an AR(1) model the run lengthskewness (g) of the frequency of run lengths wascomputed for each station along with the 90% confidenceinterval of g, based on the record length and lag-oneserial correlation at each station following Peel et al.(2004b). A total of 170 stations (14% of the global dataset) have g values significantly different, at the 10% levelof significance, from an AR(1) model, which is similar tothe 13% found in Peel et al. (2004b) using a slightly largerand earlier form of the current dataset. Globally and forASA and RoW the proportion of stations exhibiting non-AR(1) behaviour is not statistically significantly differentfrom the 10% expected. Like Peel et al. (2004b), theproportion of stations exhibiting longer runs below themedian (9%) than expected from an AR(1) model wasgreater than the proportion of stations exhibiting shorter

runs below the median (5%). This analysis supports thehypothesis that the AR(1) model adequately representsthe range of lag-one serial correlations observed world-wide.

To compute run magnitude we use a simple metric de-fined as follows (Peel et al., 2005):

Rm ¼

Pnji¼1

Mij

nj

~qð12Þ

where Rm is the relative magnitude and is equal to the sumof the deficits below the median of the historical annualstreamflows for each run length divided by the run length,nj is the number of runs of length j, Mij is the ith run mag-nitude of run length j where i = 1,2, . . ., nj and ~q is the sam-ple median.

Fig. 15 shows the relative magnitude Rm of streamflowdeficits versus run length, averaged separately for the ASAand RoW data sets. Two features are clear from the fig-ure. First, for both sets of data, the relative magnitudesincrease slowly with run length. This means that for reser-voirs with long critical periods, storages will need to berelatively larger than those with shorter critical periods.Second, the relative magnitude of ASA rivers is approxi-mately double the rivers in the RoW. This feature is con-sistent with our earlier observation that the annualvariability of ASA rivers is more than double the variabilityof RoW rivers.

Run severity is the third characteristic we examine in thissection. Here, severity of a streamflow record is defined asthe product of run length and magnitude. A neat measure ofthis characteristic is the flow deficit relative to the mean fora given recurrence interval. It is a specific property of atime series and is equivalent to estimating the size of a res-ervoir to meet a specific target draft. Briefly, Extended Def-icit Analysis estimates the deficit for a given recurrenceinterval of providing a given draft at a given level of reliabil-ity (Pegram, 2000). Details of the technique which is set outin McMahon and Adeloye (2005) were slightly modified; de-tails of this modification are given in McMahon et al.(2007c).

0

0.2

0.4

0.6

0.8

1

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Lag-one serial correlation coefficient

Inter-decadalfluctuations/total

variance

Figure 13 Ratio of inter-decadal variance to total variance calculated using EMD analysis versus annual lag-one serial correlationcoefficient of the record.

Table 4 Cross-correlation matrix (as R2) of pairs of vari-ables: auto-correlation, ratio of intra-decadal fluctuations tototal variance, record length and catchment area

Ratio N Area

q 0.199 0.067 0.025Ratio 0.011 0.004N 0.0001


-20,000

-10,000

0

10,000

30,000

40,000

1920 1930 1940 1950 1960 1970 1980

Annual streamflow -median flow

(106 m3)

0

1

2

3

4

0 5 10 15Run length (years)

Frequency

Figure 14 (a) Time series of annual streamflow relative to the median for Zambesi River at Victoria Falls. (b) Frequencydistribution of run lengths equal to or below median (N = 54 years) for Zambesi River at Victoria Falls.

0.00

0.10

0.20

0.30

0.40

0.50

1 2 3 4 5 6 7 8 9 10Run Length

RelativeMagnitude

ASAROW

Figure 15 Relative magnitude versus run length for Australia–southern Africa and rest of world. (N = 1221, with minimum of 10rivers per run length).


The importance of the standard deviation of flows in suchan analysis is illustrated dramatically in Fig. 16 where defi-cits computed by EDA (for 99% reliability and 75% draft ra-tio) are plotted as a function of the standard deviation ofannual flow. The equation representing this relationship fitusing WLS is

Def ¼ 1:746r0:986 ð13Þn ¼ 641; R2 ¼ 0:892; se ¼ þ106%; �51%; p ¼< 0:001

where Def is the deficit (106 m3) and r is the standard devi-ation of annual streamflows (106 m3).

We note once again the clear difference between therivers of Australia–southern Africa and those for the restof the world. For an annual r = 1000 · 106 m3, the deficitin ASA is 67% larger than the equivalent deficit for RoWrivers.

In summary, there were no differences in terms of runlength between ASA and ROW, while there were significantdifferences in terms of run magnitude and severity. Peelet al. (2005) showed that run magnitude and, conse-quently, run severity are strongly related to interannualvariability (Cv). The ASA and RoW difference in run magni-tude and severity observed here is the result of differ-ences in the interannual variability of streamflowsobserved earlier.

Summary and conclusions

As a result of the analyses described in this paper we offerthe following points.

1. Although there are many reports dealing with waterbalance studies of large catchments worldwideincluding their distribution by country and climate,there are few reports or papers that address both

the regulated and the unregulated characteristics ofannual streamflows of global rivers. This is Part 1 ofthree papers that considers these issues.

2. The data set of annual and monthly flows for 1221 riv-ers, adopted in this project, is unique in that consid-erable effort has been made to ensure the data arefree of major errors and the streamflows are notimpacted by major water withdrawals nor by reser-voirs upstream.

3. Based on a theoretical analysis of the relationshipbetween product-moments and L-moments associatedwith the measures of Cv and skewness, it was shownthat the Gamma probability distribution function(pdf) appears to fit the annual streamflows betterthan the Lognormal pdf for the 1221 rivers. This resultis consistent with other recent studies using largedatasets from very heterogeneous regions (Vogel andWilson, 1996).

4. Useful global relationships were established amongthe following variables: mean annual flow, meanannual runoff, standard deviation of annual flows,coefficient of variation of annual flows and catch-ment area. The differences in annual streamflowcharacteristics between Australia–southern Africaand the rest of the world are highlighted in theanalysis.

5. Based on the approximate Gould–Dincer Gamma res-ervoir capacity yield model which considers only over-year storage, we found that reservoirs in Australia–southern Africa need to be roughly twice the capacityof those in the rest of the world assuming a commoncatchment size (10,000 km2) and the same draft andfailure conditions.

6. Our analysis suggests that q is inversely related torecord length and that it is positively, but weakly,related to catchment area.

1

100

10,000

1,000,000

1 100 10,000 1,000,000Annual standard deviation (106 m3)

EDA deficit(106 m3)

ASARoW

Figure 16 Extended Deficit values (estimates of storage required to provide a relative target draft of 75% with 99% annualreliability) versus standard deviation of annual flows. Solid line = RoW, broken line = ASA.


7. It is postulated that the statistically significant nega-tive auto-correlation observed in 11 of the 1221 riversin the data set is the result of sample size and sam-pling variability rather than some physical explana-tion. The magnitude of these values was stronglynegatively correlated with record length.

8. Low frequency oscillations in the annual flow recordswere studied by Empirical Mode Decomposition. Ouranalysis showed that 60% of the variability in annualstreamflow is accounted for by intra-decadal variabil-ity. The remaining variance is contained in the inter-decadal component, of which a significant proportionof the variance (more than one-third) is within theresidual or trend component. It was also observedthat as the proportion of total variance due to intra-decadal (high frequencies on the annual time scale)fluctuations increases, the lag-one serial correlationdecreases and tends toward zero (random).

9. For 14% of rivers, the run length behaviour was differ-ent, at the 10% level of significance, to that expectedfrom an AR(1) process and there was no differencebetween Australia–southern Africa and the rest ofthe world.

10. Our analysis of the relative magnitude of streamflowdeficits shows that relative magnitude increases withrun length and that the relative magnitudes of Austra-lia–southern Africa rivers are approximately doublethat for rivers in the rest of the world.

11. Run severity which is defined as the product of runlength and magnitude was estimated using ExtendedDeficit Analysis. For an equivalent hypothetical riverand given streamflow variability, the deficit in flowin Australia–southern Africa is approximately 67%more than for a similar river from the rest of theworld.

Acknowledgement

The authors are grateful to the Department of Civil and Envi-ronmental Engineering, the University of Melbourne and theAustralian Research Council Grant DP0449685 for financiallysupporting this research. Our original streamflow data setwas enhanced by additional data from the Global RunoffData Centre (GRDC) in Koblenz, Germany. Streamflow datafor Taiwan and New Zealand were also provided by Dr.Tom Piechota of the University of Nevada, Las Vegas. Pro-fessor Ernesto Brown of the Universidad de Chile, Santiagokindly made available Chilean streamflows.

We are also grateful to Dr. Senlin Zhou of the Murray-Dar-ling Basin Commission who completed early drafts of thecomputer programs used in the analysis. Two anonymousreviewers provided very helpful comments.

Appendix A. Frequency distribution of annualstreamflows

The following describes the methodology for plotting theo-retical relations between L-Cv and Cv and between L-Skew-ness and Skewness for the Lognormal and Gammaprobability distribution functions (pdfs). Here the annual

flows are denoted X and their logarithms are denotedY = ln(X)

Lognormal case

L-Cv versus Cv

Stedinger et al. (1993, Eq. 18.2.13) report the first andsecond L-moments for the Lognormal pdf as a function ofthe distribution parameters

k1 ¼ exp ly þr2y

2

!ðA1Þ

k2 ¼ exp ly þr2y

2

!erf

ry

2

� �¼ 2exp ly þ

r2y

2

!U

ryffiffiffi2p� �

� 1

2

� �

ðA2Þ

where ly and r2y are the mean and variance of the logarithms

of the annual flows.erfðxÞ ¼ 2ffiffi

ppR x

0 expð�t2Þdt and U (x) is the cumulativeprobability density function of a standardized normal vari-ate. For a Lognormal variable the coefficient of variationof X, denoted Cv, is given by

Cv ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiexpðr2

yÞ � 1q

ðA3Þ

Now, we obtain the L-Cv as a function of the ordinary prod-uct moment Cv by combining Eqs. (A1), (A2) and (A3) withthe definition L-Cv = k2/k1,

L-Cv ¼ erf

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiln 1þ C2

v

q2

24

35

¼ 2 U

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilnð1þ C2

v

qÞffiffiffi

2p

0@

1A� 1

2

24

35 ðA4Þ

L-Skew versus Skewness

There are no simple expressions for L-moment ratios aboveL-Cv for the Lognormal pdf, however, Hosking and Wallis(1997) provide the following approximation which hasaccuracy better than 2 · 10�7 for jkj 6 4, corresponding tojL-Skewj 6 0.99

L-Skew ¼ A0 þ A1k2 þ A2k

4 þ A3k6

1þ B1k2 þ B2k

4 þ B3k6

ðA5Þ

where k = �ry. The values of the coefficient Ai and Bi in Eq.(A5) are reproduced in Table A1.

The ordinary product moment skewness c for a 2-param-eter Lognormal variable is given by

c ¼ 3Cv þ C3v ðA6Þ

The ordinary skewness c in Eq. (A6) can be related to the L-Skewness in Eq. (A5) using the fact that

�k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilnð1þ C2

vÞq

ðA7Þ

The relationship between L-Skewness and c is obtained bythe simultaneous solution of Eqs. (A5), (A6) and (A7) usinga numerical root-finding algorithm.


Gamma case

L-Cv versus Cv

Stedinger et al. (1993) report the first and second L-mo-ments for the Pearson type III pdf as a function of its param-eters as

k1 ¼ fþ ab

ðA8Þ

k2 ¼Cðaþ 0:5ÞbCðaÞ

ffiffiffipp ðA9Þ

where C() is the Gamma function. A Gamma pdf is simply atwo parameter form of the Pearson type III pdf, when thelower bound parameter f = 0. Stedinger et al. (1993) also re-port that the ordinary product moment skew is twice thecoefficient of variation and is related to the model parame-ter a as follows

c ¼ 2Cv ¼2ffiffiffiap ðA10Þ

Combining Eq. (8) with f = 0, Eqs. (9) and (10) with the def-inition of L-Cv we obtain a relationship between L-Cv and or-dinary Cv:

L-Cv ¼C2v � C 1

C2vþ 1

2

� �C 1

C2v

� � ffiffiffipp ðA11Þ

L-Skew versus Skewness

Hosking and Wallis (1997, Eqn. A.84) report an expressionfor the L-Skewness of a Pearson type III variable as

L-Skew ¼ 6I1=3ða; 2aÞ � 3 ðA12Þ

where Ix(p,q) denotes the Incomplete Beta function definedas

Ixðp; qÞ ¼Cðpþ qÞCðpÞCðqÞ

Ztp�1ð1� tÞq�1dt ðA13Þ

and 0 > a >1. Combining Eq. (A12) with (A10) yields therelationship between L-Skewness and Skewness for a Gam-ma pdf:

L-Skew ¼ 6I1=34

c2;8

c2

� �� 3 ðA14Þ

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Global streamflows – Part 1: Characteristics of annual ...Global streamflows – Part 1: Characteristics of annual streamflows Thomas A. McMahon a,*, Richard M. Vogel b, Murray