The simulation plan assumes three “process classes” that contribute on different time scales, and in differing degrees, to the variability expressed at each of the quinary catchments. These are the forced or anthropogenic component, a “natural” annual-to-decadal component and a subannual component, including the seasonal cycle as well as day-to-day variations. The first two of these are modeled at annual time resolution, the trend inferred using the CMIP5 ensemble and the annual-to-decadal component simulated by the VAR model. Subannual variability is generated by preferentially resampling the observational data in 1 year blocks. The k-NN approach utilized for this step allows for shifts in daily rainfall statistics that may come about as a result of climatic changes. A possible increase in interannual precipitation variability with global temperature was investigated, but was not corroborated by the CMIP5 ensemble and is not modeled.
4.1. Trends, Past and Future
 Local trends are modeled as functions not of time but of global mean temperature, the motivating idea being that trend should represent a response to anthropogenic forcing, rather than simply a shift in the mean level with time. The global mean surface temperature, suitably computed, has been shown to be an effective proxy for the forced climate response [Ting et al., 2009]. Note that various features of the regional climate, including atmospheric circulation, may change as the planet warms. The global mean temperature may be thought of as an index of this warming.
 Recent work suggests that the observed global temperature record may be contaminated to some degree by internal decadal variations [Ting et al., 2009; DelSole et al., 2011]. For this reason the signal is computed here by averaging over the CMIP5 ensemble. The individual GCM simulations are low-passed using a fifth-order Butterworth filter [Smith, 2003] having half power at a frequency of 0.1 yr−1, then averaged. (Results are not sensitive to the precise method of filtering.) Ensemble averaging has the effect of attenuating unforced climate fluctuations, since these are uncorrelated from model to model, while enhancing that part of the signal that the individual GCMs have in common, namely, the response to anthropogenic forcing. The smoothing reduces both residual interannual variability and the effects of short-lived transients such as volcanic eruptions. For further discussion of this procedure see Greene et al. [2011a].
 When a local record is regressed on the smoothed multimodel mean signal the fitted values, representing the trend, take the form of a scaled, shifted version of the regressand. Three such fits appear as the trend lines in Figure 3, which shows the annualized regional (i.e., catchment-averaged) observational series. Note that this procedure implicitly removes any remaining additive bias in the multimodel mean temperature record. Detrending is accomplished by subtracting the fitted values. The regression coefficients are used, in conjunction with the future multimodel mean temperature signal, for forward projection, now in terms of the response to future anthropogenic forcing.
Figure 3. The three regional time series on which simulations are based. Dashed lines are the fitted trends, from regression on the low-passed CMIP5 multimodel mean global temperature record.
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 In going from the 20th to the 21st century regional temperature and precipitation variables, as simulated in the CMIP5 ensemble, behave quite differently, as shown in Figure 4. Regional temperature projects consistently on the global mean, as evidenced by the uniform slope across centuries (Figure 4a). Precipitation (Figure 4b) exhibits considerably greater variability, but 20th century values (from the “historical” simulations) do not trend significantly, while the 21st century regression (based on the RCP4.5 experiments) is significant at . Observed regional precipitation also lacks any significant trend for the period of record, 1950–1999. (To facilitate comparison with the observational record, the 20th century values are shown only for this period; the “historical” data actually extend through 2005, the RCP4.5 simulations beginning in 2006.) Because of this difference in behavior, the trend component is treated differently for the temperature and precipitation variables.
Figure 4. Scatterplots of (a) regional mean temperature and (b) precipitation against global mean temperature, CMIP5 ensemble means. Annual mean values are shown for 1950–1999, corresponding to the observational period of record, and from 2006, when the RCP4.5 simulations begin, through 2065; the region is 30°–35°S, 17°–23°E.
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 Future trends are generated at the catchment level. For maximum and minimum temperatures, this is accomplished by regressing the annualized catchment records on the smoothed multimodel mean signal, as described above, then applying the resulting coefficients to the future multimodel mean temperature record. This enforces a consistent relationship between 20th and 21st century behavior, with respect to the global mean.
 The temperature response differs among catchments and between Tmax and Tmin (catchment means of and , respectively). Forward projection in this manner thus implies a rather complex set of changes in surface temperature gradients over time, while the more rapid increase of Tmin, compared with Tmax, leads to a mean reduction of the diurnal temperature range (DTR). Divergent temperature tendencies could eventually evoke compensating behaviors, such as small-scale circulation adjustments that would act to reduce local gradients. However the reduction in DTR could represent a shift toward a new equilibrium state [see, e.g., Braganza et al., 2004].
 Because of this complexity, and because the simulations under discussion extend just a few decades into the future, we do not attempt to include compensating mechanisms for temperature trends in the simulation model. This could be done, for example, by relaxing catchment trends toward a common mean, insuring that local gradients do not become unrealistically large. However there is some spatial dependence in temperature trends, for which an additional level of modeling would be required.
 Annualized catchment-level precipitation records are similarly regressed on the smoothed multimodel mean, but the resulting coefficients are utilized in a different manner. To begin with, the regional 21st century precipitation response differs among GCMs. The distribution of this response is shown in Figure 5, along with a Gaussian fit. Trends are computed in log space, the coefficients then representing the fractional change in regional precipitation (shown in Figure 5 as percentage change) per degree of global temperature increase. The distribution has mean per degree warming, with a standard deviation of . (These values refer to the 2006–2065 period in the RCP4.5 experiment; the 2000–2005 interval, belonging to the historical simulations, has an intermediate character and its future precipitation trend is interpolated between 20th and 21st century values.) Three of the 14 models become wetter with warming temperatures, suggesting a nonnegligible probability (∼17% in terms of the fitted Gaussian) of such an outcome. Note that absolute GCM precipitation values are not utilized in these computations, bypassing a potential source of bias.
Figure 5. Distribution of the regional precipitation response to global mean temperature change in the CMIP5 ensemble utilized here. Regression is carried out in log space: the abscissa shows the response as the percent change per degree of global temperature increase. The curve is a Gaussian fit to the data.
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 In projecting the precipitation trend, a desired quantile is first specified and the corresponding value calculated using the fitted Gaussian. Catchment-level trends are then computed as
where is the future catchment trend, Tr the quantile-based regional trend, the 20th century catchment trend and is a factor, here set provisionally at 0.5. Thus, the catchment-averaged future trend will correspond to the imposed quantile-based value, while individual catchment trends will scatter around this value according to their 20th century behavior. (Recall that the average 20th century catchment trend is not significantly different from zero.) The degree of scatter, , is at the operator's disposal, but is attenuated here in order that study area precipitation not become overly “disorganized” as the simulations are projected into the future. Catchment-level 20th century trends show no dependence on either altitude or location within the study area, suggesting a significant random component in their dispersion.
 From the physical perspective there is some reason to believe that the Western Cape will dry in coming decades, owing to poleward migration of the dry subtropical belts and midlatitude storm tracks. Indeed, some of this migration has already been observed [Seidel et al., 2008; Yin, 2005]. The phenomenon is also suggested in Figure 11.2 of the IPCC Fourth Assessment Report [IPCC, 2007, p. 869], which shows that the drying projected for the Western Cape is not an isolated regional phenomenon but part of a coherent global pattern.
4.2. Regional Annual-to-Decadal Variability
 The annual-to-decadal component of the simulation model is based on January–December means of the three variables, averaged over the study area (i.e., the series shown in Figure 3, but after detrending). Owing to the winter (JJA) maximum in Western Cape rainfall, time averaging in this way does not bisect the rainy season. Area averaging reduces small-scale “noise” that is uncorrelated across catchments, while enhancing whatever large-scale, quasi-regional signal the catchments share. Climate variability on annual-to-decadal scales may be expected to arise chiefly from large-scale oceanic or coupled ocean-atmosphere processes [Schlesinger and Ramankutty, 1994; Trenberth and Hurrell, 1994; Mantua et al., 1997]. Terrestrial climate variations, conditioned by such processes through atmospheric teleconnections, would tend to exhibit relatively large-scale spatial signatures [Hurrell, 1996; Enfield et al., 2001; Glantz et al., 1991].
4.2.1. Data Attributes
 Tyson et al.  discussed a pervasive 18 year oscillation in southern African climate. The claimed signal, present in both instrumental and paleorecords, was strongest in the 20°–30°S latitude band, but detectable to the southern extremity of the continent. Our regional record was tested for the presence of such a signal, using both singular spectrum analysis (SSA), a technique well suited for detecting quasiperiodic signals in short, noisy time series [Ghil et al., 2002], as well as wavelet analysis [Torrence and Compo, 1998]. Neither method confirms the presence of such an oscillation, as illustrated by the precipitation wavelet spectrum shown in Figure 6a. (Oscillations at the claimed frequency would fall outside the “cone of influence” in this plot but should nevertheless be visible if present.) Likewise, SSA and wavelet analyses of the Tmax and Tmin records (wavelet spectra shown in Figures 6b and 6c) do not suggest the presence of significant oscillatory components. There are many possible reasons for such a discrepancy, including spatial or temporal inhomogeneity of the 18 year signal and the analysis of disparate data sets. Without disputing the claims made by Tyson et al., it is concluded that such an oscillation is not present in the data utilized here.
Figure 6. Wavelet power spectra for the detrended regional series: (a) precipitation, (b) Tmax, and (c) Tmin. Level boundaries correspond to the 25th, 50th, 75th, and 95th percentiles of spectral power, and the solid black contours correspond to the 0.05 red noise significance level. The thick dashed line delineates the “cone of influence,” outside of which edge effects become important. For each variable the panel at the right shows the global wavelet spectrum (solid line) and the 0.05 red noise significance level (dashed line).
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 Lag 1 autocorrelation coefficients for the regional variables are significant at 0.10 and 0.05 for Tmax and Tmin, respectively (one-sided test) but not for precipitation. The Durbin-Watson test for serial autocorrelation is a bit more confident, yielding p values of 0.06 and 0.01 for Tmax and Tmin, respectively. Thus the “decadal” (i.e., persistent) component of the observational record appears to reside in the temperature variables, the precipitation signal being essentially indistinguishable from white noise. The wavelet spectra of Figure 6, despite the presence of some episodic activity in the 10 year band, do not suggest (via the presence of significant peaks in the global spectra) the presence of systematic signal components, i.e., components that differ from AR(1) in character.
 Owing to the well-defined water year, the modeling of seasonal (JJA) values was considered. However the ACRU model, because it includes the memory effects of soil moisture, requires full years of simulated climate. Additionally, water stresses tend to be highest during the summer months, when rainfall is low and evaporation high, while area physiography limits the potential for buffering via the construction of new dams. Thus, behavior outside the rainy season is also of significant interest. Preliminary inspection of the JJA statistics indicates that they do not differ greatly from those derived from annual values.
 A composite scatterplot (Figure 7) shows a negative correlation ( , significant at 0.001) for pr and Tmax, and a stronger positive correlation ( ) for Tmax and Tmin. The first of these may reflect rain-associated cloudiness and/or reduction of the Bowen ratio by surface moistening. Both mechanisms involve a significant insolation component, which may explain the lack of correlation between pr and Tmin. In any event, failure to represent these relationships correctly would likely bias the simulations, and, by extension, the resulting outputs from ACRU.
Figure 7. Scatterplots for the (detrended) regional series. Units are mm d−1 for pr and degrees Celsius for Tmax and Tmin.
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 It has been hypothesized that global warming will bring about an increase in year-to-year precipitation variability, owing to the exponential dependence on temperature of water saturation vapor pressure. Figure 8 shows the CMIP5 ensemble distributions of interannual precipitation variance for the 30°–35°S, 17°–23° E domain for 1950–1999 and 2046–2065, by which time global mean temperature has increased by ∼1.5°C. Almost no change in the distribution of variance is observed, so there would seem to be little justification for modeling such a dependence. The CMIP3 simulations are in accord on this point.
Figure 8. Interannual precipitation variance for the 20th and 21st century simulations. Distributions across the CMIP5 ensemble are shown.
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4.2.2. Statistical Model
 The absence of significant peaks in the three regional spectra (for pr, Tmax and Tmin), together with the serial correlation exhibited by the temperature variables, suggests the deployment of a vector autoregressive (VAR) model, a multivariate generalization of the classical AR model, for the annual-to-decadal component of the simulations. VAR models have been widely utilized in both econometrics (see, e.g., Holden  and other papers in that issue) and climate studies [e.g., Penland and Sardeshmukh, 1995; Newman, 2007], where a VAR model of order unity is known as a linear inverse model.
 A model of the form (2) was fitted to the annualized regional series by least squares, using the dynamical systems estimation (dse) package [Gilbert, 1995] for the R programming language [Ihaka and Gentleman, 1996]. One-step-ahead forecasts are shown in Figure 9, where it can be seen that the fraction of variability accounted for by the predictive component of the model (the first term on the right-hand side of equation (2)) is modest. While some of this predictability arises from autocorrelation, some may also result from lagged cross correlations. To quantify contributions from the latter the Granger causality test [Granger, 1969] was applied. This pairwise test assesses the predictability above and beyond that arising from serial autocorrelation that is contributed by lagged cross-variable dependence. Results suggest a limited degree of additional predictability for precipitation and Tmin based on the inclusion of Tmax and precipitation, respectively, at the preceding time step (p values of 0.17 and 0.14, pooled across variables).
Figure 9. One-step-ahead predictions (dashed lines) for the VAR(1) model. Solid lines show the observed regional series.
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 Although one could perhaps make a case for a model lacking serial dependence, we retain the VAR structure, in part on the accumulated evidence but also because the hydrological significance of this dependence is not known a priori. It should therefore be instructive to compare ACRU outputs with those based on simulations from a model lacking the predictive term. Ultimately the VAR model is also more general, and can better serve as a prototype for application in a diverse range of settings. Neither Akaike's information criterion (AIC [Akaike, 1973]) nor the Bayesian information criterion (BIC [Raftery, 1986]) suggest that there is anything to be gained by moving beyond the complexity of a first-order model.
4.3. Subannual Variations
 Subannual variability is generated by resampling the observations in 1 year blocks, using a modified k-NN scheme [see, e.g., Rajagopalan and Lall, 1999] in which the three-component feature vector consists of a single year's simulated annual means of pr, Tmax and Tmin. The aim is to select, from among the 50 data years, one whose mean annual values approximate this vector. Since year-to-year dependence is already accounted for by the VAR model there is no role in this scheme for a “successor”: A particular year having been chosen from among the candidates, its subannual patterns of variability are appropriated for the year being simulated.
 Experimentation suggested that the use of nearest neighbors provides a reasonable compromise between the generation of sufficient variety in the resultant sequences and the inclusion of too remote candidates (see section 5.2). The Mahalanobis distance metric [Mahalanobis, 1936] is utilized, with weights of assigned to pr, Tmax, and Tmin, respectively, effectively weighting precipitation double the combined weights of the two temperature variables. These weights are based on past results from ACRU implicating precipitation as the most important predictor of runoff, the key application variable, but should be considered provisional subject to further experimentation. (Alternate weights may prove desirable for studies focusing on the summer dry season.)
 A monotonically decreasing resampling kernel, with values , is utilized to select from among the nearest neighbors. The resampling scheme explicitly links the climate change and subannual time scales, in principle enabling the realization of climatically induced changes in daily rainfall statistics, as secular trends in the mean state induce shifts in the population from which resampled statistics are drawn. The three variables are resampled jointly and across the entire study area, preserving spatial coherence (including potential shifts in spatial patterns driven by large-scale mean changes), high-frequency covariation and seasonal cycle shape.
 Figure 10 shows coefficient distributions for regressions of the detrended catchment-level variables on the corresponding regional series. Because the regional signals represent catchment means, the average coefficient for each of the variables is unity, guaranteeing that the catchment-averaged response will reproduce the imposed simulation sequence. The plots give some idea of the degree to which annualized catchment variations follow those of the regional signal, or, put another way, the degree to which the regional signal is expressed at each of the catchments.
Figure 10. Distributions of the coefficient b1 for regressions of the annualized catchment-level variables on the corresponding regional time series.
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 In the downscaling step, simulated annual-level variations are propagated to the catchments using these coefficients. Uncorrelated noise is added to bring variances into agreement, emulating the observed variability. The resulting signals are substituted for intrinsic annual-level catchment variations by adjusting the resampled catchment values, in 1 year blocks. This is done additively for the temperature variables, multiplicatively for precipitation. ACRU is driven by the resulting daily sequences, superimposed on the CMIP5-derived trends.