## 1. Introduction

The hydroclimate of a catchment is regarded as the long-term average climate (precipitation and evapotranspiration) and streamflow across the catchment. Variation around that long-term average is known as hydroclimatic variability. Environmentally sustainable management of land and water resources requires designing and operating systems to at least cope with and, preferably, prosper within present and future hydroclimatic variability.

Existing water resources engineering projects have been designed generally on the assumption of hydroclimatic stationarity (McMahon and Adeloye 2005). For a given catchment this assumption has not always proven correct and now concern is increasing about the potential impacts of hydroclimatic variability outside the range of previous experience on land and water resources management. Overlying the challenge of catchment and water resources management within natural hydroclimatic variability is the likely modification of hydroclimatic variability due to anthropogenic enhancement of greenhouse gas concentrations (Folland and Karl 2001; Milly et al. 2008). These hydroclimatic changes are likely to require a significant planning response for land and water resources design and management. In Australia and southern Africa these concerns are compounded by annual streamflow being more variable than elsewhere in the world, particularly in temperate regions (McMahon et al. 1992; Peel et al. 2001, 2002, 2004). There are two issues that need to be addressed in this context: changes in mean annual flow and changes in interannual runoff variability. We deal with both of these issues in this paper.

There are at least two approaches that can be used to assess the change in annual mean runoff and interannual runoff variability that may result from climate change. The first approach is based on developing a rainfall-runoff model, calibrated for the catchments under consideration, and using generated projections from general circulation models (GCMs) of future precipitation and climate variables to produce sequences of annual runoff from which the annual mean and variability can be computed. In the second approach, analytical equations are developed that relate the mean and variance (or standard deviation) of runoff to a functional relationship of the aridity index (mean annual potential evapotranspiration divided by mean annual precipitation) and to the variances and covariance of precipitation and potential evapotranspiration. We adopted the second approach in this paper as it provides a method that does not require the calibration of a rainfall-runoff model, which is often not a straightforward procedure and normally not readily available to climatologists. It will be shown in the paper that the adopted approach provides a simple and quick method to estimating mean annual runoff and interannual runoff variability with confidence.

In the next section we develop the analytical equations defining the expected value and the variance of annual runoff. This is followed in section 3 by a discussion of functional relationships of annual precipitation–evapotranspiration–runoff that can be used without calibration to estimate runoff characteristics in a climate change scenario. In section 4 we validate the approach and provide an application using projected annual precipitation data generated from the Hadley Centre Global Environment Model version 1 (HadGEM; Johns et al. 2006). The penultimate section discusses and assesses two simpler models and, finally, the results are summarized in section 6.

## 2. Mean annual and interannual variability of runoff

The underlying purpose of this paper is to develop simple models that can estimate the change in mean and interannual variability of runoff based on GCM scenarios of changes in future annual precipitation and temperature. To do this we have drawn on the approach of Arora (2002) who examined inter alia the change in interannual variability of evapotranspiration due to changes in precipitation and potential evapotranspiration.

*R*,

_{t}*P*, and

_{t}*E*are runoff, precipitation, and actual evapotranspiration during year

_{t}*t*. Here

*F*(

*ϕ*) is a functional relationship relating annual actual evapotranspiration to annual precipitation during year

_{t}*t*: and

*ϕ*is the aridity index, defined as in which

_{t}*E*is the potential evapotranspiration during year

_{ot}*t*.

*P*

*R*

*F*(

*ϕ*) is the mean annual value of

*F*(

*ϕ*).

_{t}*g*and

_{x}*g*are the first partial derivatives of

_{y}*g*(

*X*,

*Y*) with respect to

*x*and

*y*, taken at the position of the means of

*X*and

*Y*. Substituting

*R*=

_{t}*X*and

*E*=

_{ot}*Y*in Eq. (8) and rearranging, gives which is an expression of the ratio of the variance of annual runoff to the variance of annual precipitation, and in which cov(

*P*,

*E*) is the covariance of annual precipitation and annual potential evapotranspiration,

_{o}*σ*is the standard deviation of annual runoff, and

_{R}*σ*is the standard deviation of annual precipitation.

_{P}*P*,

*E*) and

_{o}*σ*can be considered negligible and, therefore, if that assumption is valid, we can simplify Eq. (9) to These two assumptions are tested in section 4c.

_{Eo}*P*,

_{t}*F*(

*ϕ*)] can be assumed to be negligible, Eq. (7) reduces to This assumption is tested in section 5a.

_{t}In the next section, simple models of the functional relationship *F*(*ϕ*) are explored.

## 3. Models of *F*(*ϕ*)

The utilization of the above analysis [Eqs. (7), (9), (10), and (11)] is based on the availability of an appropriate functional relationship of the aridity index to estimate the ratio of annual actual evapotranspiration and annual precipitation. There are at least six models, detailed in Table 1, that can be used to define *F*(*ϕ*). These models operate at a mean annual time scale and vary from simple relationships (Schreiber 1904); also see Gardner (2009) who proposed a similar model (Ol’dekop 1911); generalized Turc–Pike (Turc 1954; Pike 1964; Milly and Dunne 2002) to more complex Budyko-like models including Budyko (1974), Zhang (Zhang et al. 2001), and the Fu–Zhang (Fu 1981; Zhang et al. 2004). In Table 1 it is noted that Milly and Dunne (2002) suggested that the Turc–Pike model could be generalized with the introduction of a curve parameter *ν*, for which we adopted *ν* = 2. The *w* parameter in the Zhang model is a plant-available water coefficient and Zhang et al. (2001) found *w* = 1 to be satisfactory. In the Fu–Zhang function, *γ* is an empirical parameter and although it varies from catchment to catchment, we adopted *γ* = 2.6 based on Zhang et al. (2004) who analyzed a worldwide dataset of 470 catchments. They suggested the optimum *γ* for forest catchments is 2.84, for grass catchments is 2.55, and a best fit value for all data is 2.63.

The derivative functional relationships [*F*′(*ϕ*)] for the six models are tabulated in Table 2.

## 4. Application

In this section the six functional relationships tabulated in Tables 1 and 2 are applied to Eqs. (7), (9), and (10) to estimate how changes in mean and variability of annual precipitation and annual potential evapotranspiration affect the mean and variability in annual runoff and reservoir yield.

### a. Data

The following analyses are based on 699 annual streamflow records that are a subset of a larger global dataset reported in detail in Peel et al. (2010). Catchment locations are shown in Fig. 1. All catchments have at least 10 years of annual streamflow data with concurrent annual and monthly catchment precipitation, estimated potential and actual annual catchment evapotranspiration, and annual and monthly catchment temperature. The means and ranges of the annual data characteristics are presented in Table 3.

To estimate values of the catchment aridity index, an estimate of potential evapotranspiration was required for each catchment. Mean annual potential evapotranspiration (*E*_{0}) values for the 699 catchments were estimated as the reference crop estimate of evapotranspiration from the modified Hargreaves equation (Droogers and Allen 2002) as adapted by Adam et al. (2006), which was developed using monthly values of P, T, and daily temperature range (DTR). Again details are given in Peel et al. (2010).

HadGEM precipitation and temperature data, see section 4d, were extracted from the World Climate Research Programme (WCRP) Coupled Model Intercomparison Project phase 3 (CMIP3) multimodel dataset.

### b. Assessing F(ϕ)

To validate the approach adopted, we computed values of the mean [Eq. (7)] and the standard deviation [Eq. (9)] of annual runoff for the 699 catchments based on the 6 functional relationships (section 3), observed precipitation, and estimated potential evapotranspiration, and compared the resultant values with the mean and standard deviation of observed annual runoff.

In Tables 4 and 5 we present a comparison of the ability of the 6 models to estimate the mean and the standard deviation of observed annual runoff for the 699 catchments. The tables list the values of the coefficient, *a*, and exponent, *b*, of the power relationship between the predicted and observed data, the observed and predicted means (Table 4) and standard deviations (Table 5), the root-mean-square error (RMSE), RMSE/observed mean or standard deviation, the index of agreement (IA), and the correlation squared between predicted and observed values. (In the analysis, computed means and standard deviations ≤1 mm were set to 1 mm.) In addition, estimates are computed for three values covering the range of means and standard deviations observed in the data. The index of agreement (IA),^{1} developed by Willmott (1982), is bounded between 0 and 1 (perfect prediction) and allows cross comparisons to be made between two models.

Based on the key measures in Tables 4 and 5, the results suggest that overall the Schreiber model provides the most satisfactory estimates of the computed annual mean and the standard deviation of annual runoff. In Figs. 2 and 3 the computed mean and standard deviations of annual runoffs based on the Schreiber model for the 699 catchments are compared with values from observed data. The plots illustrate the scatter in predictions and the bias in the estimates of the mean and standard deviation. As noted in Tables 4 and 5, these underestimate the 100 mm observed values by 11.8% and 13.9%, respectively. In Figs. 2 and 3 the group of catchments with estimated mean annual runoffs and standard deviations less than 10 mm are predominately located in cold arid climates where the Schreiber model may not perform well. In Fig. 4 the power curves of interannual variability of runoff for all six aridity type models are presented without their data points. If the data points were plotted, their scatter would cover a larger range than that for the Schreiber model (Fig. 3).

### c. Testing the Koster and Suarez (1999) assumptions that cov(P, E_{o}) and σ_{Eo} are negligible and assessing their simple equation to compute σ_{R}

As noted in section 2, Koster and Suarez (`) argued that cov(*P*, *E _{o}*) and

*σ*can be considered negligible. To test these assumptions, we compared the computed values of cov(

_{Eo}*P*,

*E*) and (

_{o}*σ*)

_{Eo}^{2}with the values for (

*σ*)

_{P}^{2}(and with values for (

*σ*)

_{E}^{2}as an additional comparison) for the annual data from the 699 catchments. The results are plotted in Fig. 5. The median value of the ratios of (

*σ*)

_{Eo}^{2}to (

*σ*)

_{P}^{2}is only 0.031 and 90% of values are less than 0.11, suggesting the assumption of Koster and Suarez (1999) that

*σ*can be considered negligible is tenable. However, the assumption that cov(

_{Eo}*P*,

*E*) can be assumed zero is probably not acceptable for this dataset given that the median value of the ratios of cov(

_{o}*P*,

*E*) to (

_{o}*σ*)

_{P}^{2}is −0.11 and 10% of values of the ratio are less than −0.17.

*σ*values: This relationship is plotted in Fig. 6 and it should be noted that of 699 estimates, 118 (16.9%)

_{R}*σ*values are negative. This fact and that Eq. (11) underestimates the observed values by approximately 60% across the range of data (Table 5) suggest that Eq. (11) is not suitable for estimating

_{R}*σ*.

_{R}The above assumptions of Koster and Suarez (1999) were also examined by Arora (2002) in terms of (*σ _{E}*/

*σ*) rather than (

_{P}*σ*/

_{R}*σ*). Arora (2002) concluded that the simplified equation of Koster and Suarez (1999) [our Eq. (12)] was not a good approximation to the complete equation [our Eq. (9)]. To build on the above analysis, we computed for the 699 catchments

_{P}*σ*values using Eq. (9) in conjunction with the Schreiber-modified Hargreaves model, but setting each term to zero as tabulated in Table 5. In the table we observe that setting the second term in Eq. (9) to zero results in a small reduction (~2.5%) in the estimates of standard deviation of annual runoff. However, setting the third term to zero in Eq. (9) in addition to the second term results in a reduction in the estimate of the standard deviation of annual runoff by ~21%. Hence, from this analysis we conclude that

_{R}*σ*can be assumed to be zero but it may be inappropriate to assume cov(

_{Eo}*P*,

*E*) is zero.

_{o}### d. Example application based on HadGEM

To illustrate the application of Eqs. (7) and (9) to assess the impact of changes in the mean and variability of precipitation on runoff characteristics, we use for the 699 catchments the projected annual precipitation time series extracted from HadGEM for 30-yr windows centered at 1985 (denoted as “present climate”) and 2020 (scenario A1b). In the case of Eq. (7), the term cov[*P _{t}*,

*F*(

*ϕ*)]/

*P*

*σ*and

_{P}*ϕ*=

*E*/

_{o}*P*

*E*values, estimated for the present climate, are adopted for the 2020 analysis. For each catchment, we computed

_{o}*P*

*σ*of annual precipitation for the 30-yr windows.

_{P}In Eqs. (7) and (9), potential evapotranspiration is an important variable. To incorporate *E _{o}* into the analysis, it would be inappropriate to apply a simple model that relates

*E*

_{o}to

*T*

*E*, based on temperature, in the analysis that follows. If values for all the variables to estimate potential evapotranspiration were available to us from HadGEM at the time of analysis, we could have used for example the Penman equation to estimate potential evapotranspiration for the present climate and that for the 2020 30-yr period. Nonetheless, by considering only changes in projected precipitation rather than both precipitation and potential evapotranspiration, we are able to illustrate the approach recommended in this paper.

_{o}Applying estimates of *P**σ _{P}* from the HadGEM model along with no change to the estimates of

*E*

_{o}and

*σ*in Eqs. (7) and (9),

_{Eo}*R*

*σ*values for each catchment were estimated for the 30-yr windows centered around 1985 and 2020. As noted earlier in applying Eq. (7), because the cov[

_{R}*P*,

_{t}*F*(

*ϕ*)] term would not be readily available from GCM output, it was set equal to the value computed from observed/estimated data. A summary of the results of the analyses is shown by country (where 10 or more catchments were analyzed for each country) in Table 6 in which each value is the ratio of the specific variable for the 2020 climate with respect to the equivalent present climate (1970–99) value. The table is interpreted as follows. For example, consider Brazil and Zimbabwe. For Brazil, Δ

*R*

*σ*(the standard deviation of annual runoff relative to the present value) will decrease by 10%, and the reservoir yield will increase by 31%. On the other hand, for Zimbabwe, the analysis suggests that Δ

_{R}*R*

*σ*will decrease by 9%, and the reservoir yield will decrease by 25%. The basis of the reservoir yield estimate is described in the next section where the results in Table 6 are discussed further.

_{R}### e. Assessing impact on water resources

*D*is the annual yield or draft (mm yr

^{−1});

*μ*and

*σ*are the mean and the standard deviation, respectively, of annual runoff (mm yr

^{−1}) into the reservoir storage;

*τ*is the hypothetical storage capacity as a ratio of mean annual runoff;

*γ*is the coefficient of skewness of annual runoff;

*ρ*is the autocorrelation of annual runoff; and

*z*is the standardized variate at 100 ×

_{p}*p*% probability of nonexceedance of annual flows.

The G-DG model was applied to the present climate 30-yr time series and to the 2020 30-yr time series as outlined in section 4d. The example analysis was carried out for a hypothetical reservoir associated with each of the 699 catchments in which the capacity was assumed equal to 0.5 × present climate mean annual runoff and for 95% annual time-based reliability. The 0.5 × mean annual runoff storage size was chosen to represent typical reservoirs worldwide [some details are given in McMahon et al. (2007b) and McMahon (2008)]. For the analysis, we assumed that the annual river flows are Gamma distributed and, therefore, we set *γ* = 2*C _{υ}* in Eq. (13), where

*C*is the coefficient of variation of annual flows. The assumption that annual flows are Gamma distributed is supported by McMahon et al. (2007c). In terms of autocorrelation, a constant value equal to 0.125 was adopted, which is the median value of the mean observed values for the 699 catchments. In the analysis, if computed reservoir yields were < 0, they were set to 0.

_{υ}By way of illustration the impact of changes in mean and runoff variability through climate modification of precipitation on reservoir yield based on the 699 hypothetical rivers are presented in Table 6 for 18 countries worldwide. In reviewing these results readers should note that these changes are based on the estimated precipitation of only one simulation for only one of more than 20 GCMs that provide future scenarios of climate and, furthermore, it should be noted that although 699 catchments were available to us, the catchments within a country are not necessarily representative of that country. Nevertheless, we are able to make several general comments. First, in the Southern Hemisphere there is an overall reduction in mean annual runoff (except for Brazil and New Zealand) and in the standard deviation of annual runoff (except for Argentina and New Zealand). This translates to a reduction in reservoir yield for the Southern Hemisphere except for Brazil and New Zealand. Second, countries experiencing predominately cold climates (e.g., Canada and Russia) exhibit increased runoff and annual standard deviation as well as an increase in reservoir yield. Third, based on the 699 catchments, the analysis suggests that the world as a whole may experience a small reduction in mean annual runoff and reservoir yield and a small increase in the standard deviation of annual runoff.

## 5. Simple equations of mean and standard deviation of annual runoff and reservoir yield

### a. Mean and standard deviation of annual runoff

*F*(

*ϕ*) yielding the estimate of mean annual runoff as Adopting only the first term of Eq. (9) and substituting the Schreiber function as

*F*(

*ϕ*) yields the following simple equation to estimate

*σ*:

_{R}Estimates of *R**σ _{R}* based on the simplified Schreiber-modified Hargreaves model [denoted as simplified Schreiber model; Eqs. (14) and (15)] were compared with estimates based respectively on Eqs. (7) and (9), which incorporate the Schreiber-modified Hargreaves model (denoted as the complete Schreiber model). The comparisons are plotted in Figs. 7a,b and details are given in Table 7. Figures 7a,b show that the mean annual runoff and the standard deviation of annual runoff are accurately modeled by the simplified Schreiber model when compared with the complete Schreiber function as defined by Eqs. (7) and (9). Table 7 confirms the high correlations for the means and standard deviations between the two methods with the index of agreement being virtually 1.0 and RMSE being very low, and indicates that the assumptions of setting the covariance term to zero in Eq. (7) and adopting only the first term of Eq. (9) are reasonable.

### b. Assessing reservoir yield ratios based on simplified Schreiber model

In this section we examine whether the simple Schreiber model can provide a satisfactory estimate of reservoir yield in a changing climate. Two analyses as represented by Figs. 8 and 9 are carried out. In Fig. 8 reservoir yields are estimated based on both the simplified and the complete Schreiber models for the 699 catchments and compared with values estimated using observed runoffs. The figure shows that both models underestimate the yield by about 20%. On this basis relative to the complete model the simplified Schreiber model is considered a satisfactory method to estimate reservoir yield.

In Fig. 9, we compare for the 699 catchments reservoir yield ratios based on the 30-yr window around 2020 to the yield estimated for the present climate 30-yr window for the simplified Schreiber model with the ratios based on the complete Schreiber model. There were 33 catchments for which ratio values <0.1 computed by the simplified Schreiber model. These are considered unrealistic and were not included in Fig. 9. Overall the simplified Schreiber model performed satisfactorily, although there are a small number of outliers in the figure that reduces the utility of the approach. Considerable care would need to be exercised in using the results on a catchment by catchment analysis. We recommend that the ratios be combined on a country or regional basis to estimate a median value so that there is less chance that an outlier will mask the estimate of the true impact of the change in climate in a region.

### c. Application of the simplified model

Noting the final comment in the previous section, and in view of uncertainty in aspects of the analysis (see Table 5 for levels of uncertainty in the aridity functions), it is recommended that both the simplified [Eqs. (14) and (15)] and the complete [Eqs. (7) and (9)] Schreiber models proposed herein be not used to estimate the mean, the standard deviation, or the reservoir yield in an absolute sense, but rather used in a relative analysis as presented in Table 6.

## 6. Conclusions

The purpose of this paper was to develop simple models to assess the impact of climate change on the mean and variability of annual runoff, the latter expressed as the standard deviation or variance of annual runoff. This was achieved through an analytical approach utilizing functional relationships of aridity. From our analyses we can draw several conclusions.

- The mean annual runoff was found to be related to mean annual precipitation, a function of the aridity index, and the covariance of precipitation and the aridity function [Eq. (7)].
- Following the lead of Arora (2002) and based on a simple analysis, the ratio of the variance of annual runoff to annual precipitation was found to be related to a function of the aridity index and its derivative, the ratio of the variance of annual potential evaporation to annual precipitation, and the covariance of annual precipitation and annual potential evapotranspiration [Eq. (9)].
- Analysis of observed annual precipitation and potential evapotranspiration data for the 699 worldwide catchments used in this study showed that the variance ratio term (
*σ*/_{Eo}*σ*)_{P}^{2}could be considered negligible, but the covariance term [cov(*P*,*E*)/_{o}*σ*_{P}^{2}] is probably not zero. - Of the six functional aridity relationships (Schreiber, Ol’dekop, generalized Turc–Pike, Budyko, Zhang, and Fu–Zhang), the Schreiber model reproduced the observed mean and the standard deviation of runoff most accurately.
- The simple model of Koster and Suarez (1999)
*σ*=_{R}*σ*−_{P}*σ*is not suitable to estimate_{E}*σ*._{R} - Simple equations based on the Schreiber functional aridity relationship: one for the mean annual runoff and another for the standard deviation of annual runoff [Eqs. (14) and (15)], were developed that allow climate analysts to estimate the impact of future changes in runoff characteristics and reservoir yield performance [Eq. (13)] as a result of changes in annual precipitation. The approach can incorporate potential evapotranspiration values if available.
- It is recommended that the equations developed herein for the mean and standard deviation not be used to estimate the mean, standard deviation, or reservoir yield in an absolute sense, but rather used in a relative analysis as a ratio of values across two periods.

## Acknowledgments

This research was financially supported by the Australian Research Council Grants DP0773016 and LP100100756, and a CSIRO Flagship Collaborative Research Fund grant. Professor Richard Vogel of Tufts University and Dr. Vivek Arora of the Canadian Centre for Climate Modelling and Analysis provided helpful advice. We acknowledge the modeling groups, the Program for Climate Model Diagnosis and Intercomparison (PCMDI), and the WCRP’s Working Group on Coupled Modelling (WGCM), for their roles in making available the WCRP CMIP3 multimodel dataset. Support of this dataset is provided by the Office of Science, U.S. Department of Energy.

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Models of functional relationships, *F*(*ϕ*), based on annual aridity index, *ϕ*.

The *F*′(*ϕ*) (the derivative of the functional relationships).

Hydrologic characteristics of the 699 catchments. Coefficient of variation = *C _{υ}*.

Performance of models estimating the mean annual runoff incorporating the modified Hargreaves equation to estimate the mean annual potential evapotranspiration. Here, *N* is sample size; *a* and *b* are the coefficient and the exponent, respectively, in *y* = *ax ^{b}* (the regression between predicted and observed values); Obs.

*R*

*R*

*R*

^{2}is the correlation coefficient squared. Values in bold indicate the best result across the models.

Performance of models estimating the standard deviation of annual runoff incorporating the modified Hargreaves equation to estimate the mean annual potential evapotranspiration. Variables are the same as in Table 4.

Expected changes in country mean and variability of annual runoff and reservoir yield in 2020 (2005–35) compared with the present climate (1970–99) based on Eqs. (7), (9), and (13) and the complete Schreiber model in conjunction with precipitation estimates from HadGEM. Here *N* is the number of catchments, Δ*R**σ _{R}* is the median ratio of future standard deviation of annual runoff to the present climate standard deviation of annual runoff, and Δyield is the median ratio of 2020 reservoir yield to the present climate reservoir yield.

^{1}

IA is a relative and bounded measure that may be used to compare models. Here IA is defined as (Willmott 1982) * _{i}* and Ob

*are the predicted and observed values, respectively;*

_{i}*N*is the number of paired values; and Pr′

_{i}= Pr

_{i}−

_{i}= Ob

_{i}−