In the training phase, the first-order long-term forced change in X is established using the correlation between X′ and Tglobal′. This relationship is expected to be dependent on the RCM simulation from which the variable of interest is obtained. There are two ways in which this can be managed:

A statistical relationship is quantified for each RCM simulation providing data for the training phase of EPIC. Then, in the “implementation phase” of EPIC (see below), for each ensemble member, a single relationship is randomly selected.

A simple linear correlation between X′ and Tglobal′ is calculated for each of the six RCM simulations and each grid point independently, namely X′(t)=α×Tglobal′(y)+R(t), where X′(t) are the daily anomalies with respect to the 2001–2010 mean annual cycle of X, the Tglobal′(y) are the anomalies of an annual mean global mean surface temperature time series obtained from the AGCM which provided the boundary conditions for the selected RCM simulation, α is the regression coefficient, and R is the residual which is the part of the signal that cannot be explained by the statistical model. In this case, the residuals are used by the stochastic weather generator (see Sect. 3.3) to model higher-order changes in the variability in X which are not captured by Eq. ().

The mean annual cycle of X, which is used to calculate X′, is generated using the same method and time period used to calculate the mean annual cycle of the observational set. X′, rather than X, is used in Eq. (), as the change in the seasonal cycle is of interest. Removing the mean annual cycle removed the need to add additional terms to describe the baseline seasonal cycle.

Because GCM and RCM output provide a much longer time series than observations and extend into a period of greater changes in X, GCM or RCM output are preferentially used in this training phase. In this study, the input to the training phase of EPIC, Tglobal′, is sourced from the AGCM that provided the boundary conditions for the RCM simulation. Both the Tglobal′ time series used in the training phase and later in the implementation phase of EPIC need to be geophysically consistent. This geophysical consistency can be assessed by comparing the Tglobal′ time series obtained from the HadAM3P simulations with the Tglobal′ time series obtained from the CMIP5 AOGCMs that provided the SST boundary conditions for the HadAM3P simulations (which were not used elsewhere in EPIC), as well as with the 190 Tglobal′ time series obtained from MAGICC (Fig. ). There are clear differences between the Tglobal′ time series obtained from the CMIP5 AOGCMs and those obtained from the HadAM3P simulations. This is because the SSTs from the CMIP5 AOGCMs are bias-corrected before being used as the surface boundary conditions for the HadAM3P simulations. The six Tglobal′ time series from the HadAM3P simulations (used in the training phase of EPIC) fall well within the envelope of the 190 MAGICC Tglobal′ time series used in the implementation phase of EPIC, even though MAGICC is emulating a range of CMIP3 models.

Because the fit coefficient, α, is expected to depend on season, it is expanded in two Fourier pairs to account for its seasonality (Eq. ). The resulting α has a smooth seasonal cycle, which would not be the case if each month was fitted independently. When embedded in Eq. (), the resulting equation has five fit coefficients (α0 to α4): X′(t)=(α0+α1×sin⁡(2πd/365)+α2×cos⁡(2πd/365)+α3×sin⁡(4πd/365)+α4×cos⁡(4πd/365))×Tglobal′(y)+R(t). The statistical model is solved using a multivariate least squares regression approach to obtain the fit coefficients. We refer to each such set of five fit coefficients as a tuple; recall that this fit is applied at each grid point and for each available RCM simulation.

An example of a fit of Eq. () to daily maximum surface temperature anomalies is shown in Fig.  for a location in the South Island of New Zealand.

The small annual cycle in the fit, with growing amplitude, results from summertime and wintertime daily maximum surface temperatures exhibiting different correlations against Tglobal′. The inter-annual variation arises from changes in Tglobal as α does not change from year to year. In addition to the long-term forced change, there is significant day-to-day variability. The use of the residuals from such fits in the stochastic weather generator is described in Sect. .

The unitless α coefficient describes a location's sensitivity to changes in annual mean global mean surface temperature. The magnitude of α indicates whether Tmax or Tmin are changing faster (α>1) or slower (α<1) than the global mean surface temperature. Example maps of the α coefficient, over New Zealand, for four selected days throughout the year, are shown in Fig. . This analysis shows that daily maximum surface temperatures over most of New Zealand are warming more slowly than Tglobal. However, high-altitude regions, such as the Southern Alps, indicate a Tmax increasing faster than Tglobal for Southern Hemisphere spring, summer, and autumn.

There is, of course, some uncertainty in α. To account for that uncertainty, a large set of α tuples is derived through a Monte Carlo bootstrapping approach , whereby residuals from the Eq. () fit are randomly sampled and added to the regression model fit to generate multiple statistically equivalent time series, which are then refitted to obtain equally probable α fit coefficients . This approach allows for the incorporation of the uncertainty in the fit of Eq. () into the final ensemble of projections.

Once the Monte Carlo derived sets (just one set if method (2) is used) of α tuples have been obtained, they are used in the implementation phase of EPIC. As described in Sect. , 190 simulations of Tglobal′ can be generated using a SCM. A randomly selected Tglobal′ time series from the 190-member set is used together with a randomly selected tuple of α values to generate a series of maps of Xforced′ using Eq. (), where the forced subscript denotes that these are changes which correlate with Tglobal′.

There might be some concern that the random selection of an α tuple from the available set of tuples for a location could cause the spatial coherence in the forced signal across New Zealand to be lost, as at a nearby location a different tuple could be randomly selected. This was tested for and was found not to be the case, as the multiple instances of tuples (multiple instances of Fig. ) are very similar and consistent (not shown here).

We begin by conducting an EOF analysis on VCSN data that have been detrended by removing the first-order trend and on residuals from the fit of Eq. () to RCM data in the training phase. Where the patterns of variability obtained from EOF analyses of VCSN and RCM diverge is considered to be the cut-off point for where the RCM simulation can be taken to have any integrity with regard to simulating forced changes in weather noise. Visual inspection of the EOF maps derived from VCSN and RCM data suggested that the first four modes of variability are well represented by the RCM simulations (see Fig. ).

It is clear from Fig.  that the RCM EOFs exhibit the same modes of weather variability as seen in the VCSN data up until EOF pattern 4. Together, the first four patterns of variability explain 83.3 % of the total weather variability in the VCSN data and 64.7 % of the variability in the RCM data. It is these four modes of weather variability that evolve with Tglobal′ in our stochastic weather generator.

To compare statistics from the PC time series calculated from VCSN and RCM data, they must share the same underlying weather modes. This is done by projecting the VCSN weather modes (EOFVCSN) onto the RCM data to calculate a pseudo-PC time series. A pseudo-PC time series is calculated in the same way that a standard PC time series is calculated, except that the weather modes are prescribed instead of being calculated from the data. A pseudo-PC time series describes the magnitude of a particular pattern of variability from VCSN, which is present in the RCM data. The VCSN weather modes, rather than the RCM weather modes, were prescribed because the observational data set is more likely representative of patterns of variability seen in New Zealand. The pseudo-PC and VCSNPC time series can be compared as they both describe the same patterns of variability.

In the stochastic weather generator, we consider changes in the following two items:

The amplitude of the weather mode: this is quantified by correlating the associated pseudo-PC time series with Tglobal′ and then using that correlation coefficient (β) to drive a trend in the PC time series obtained from the VCSN-based EOF analysis.

The ability of the method to generate a set of PDFs of the PCsyn_1 to PCsyn_4 time series is demonstrated in Fig. .

The EPIC method corrects for any shortcomings in the ability of the RCM to correctly simulate expected magnitudes of weather variability for these four primary modes and then accommodates these corrections when generating PC time series that evolve into the future.

The stochastic weather generator includes the effects of EOF patterns five and higher but assumes that these modes show no dependence on Tglobal′ as the RCM simulations do not accurately simulate these higher modes of weather variability. The variability of the PC time series often has a strong seasonal cycle. Therefore, for EOF pattern five and higher, synthetic PC time series (PCsyn) are generated using a standard Monte Carlo approach, i.e. randomly selecting values from N(0,σ(d)) – that is, a normal distribution with a mean of 0 and a SD which depends on the day of the year which is being modelled. σ(d) is determined by a linear least squares fit of two Fourier pairs (Eq. ) to the VCSN PC time series. The Fourier pairs model the seasonal cycle in the PC time series. This approach allows selection of extreme PC values that are outside of the range of PC values experienced in the 1972–2013 period, but noting that the PDFs of these PCs do not evolve with time. As with the forced changes in the amplitude and variability of weather modes, the auto-correlation in the PC time series is also quantified and captured in the statistically modelled PC time series.

For a given ensemble member, once synthetic PC time series at daily resolution have been generated, they are used to produce a reconstructed weather field, W, according to W(i,j,t)=∑n=150EOFVCSN_n(i,j)PCsyn_n(t), where i, j, and t represent the latitude, longitude, and time dimensions respectively and n is the nth weather mode.

Since W has been constructed from a linear combination of spatial patterns of variability, each of which is spatially coherent, it retains the property of spatial coherence. The variability evolves as expected under changes in Tglobal′ for the first four modes of variability, as simulated by the RCM, and where extreme conditions, outside the range of the training period, occur with a statistically reasonable frequency due to the stochasticity in the construction of the pseudo-PC time series.

Tmin is modelled identically to Tmax with one small change: days with anomalously low Tmax would be more likely to have anomalously low Tmin. Not accounting for this correlation could result in stochastically modelled Tmin values being higher than the modelled Tmax value for that day. To avoid that, and to capture the correlation between Tmax and Tmin on any given day, the same set of random numbers used to generate the values in the synthetic PCn time series for Tmax for a given day is used to generate the values in the synthetic PCn time series for Tmin. This forces the selection of PCsyn values from the same region of the PDF for both Tmax and Tmin.