Improved point scale climate projections using a block bootstrap simulation and quantile matching method

Abstract

Statistical downscaling methods are commonly used to address the scale mismatch between coarse resolution Global Climate Model output and the regional or local scales required for climate change impact assessments. The effectiveness of a downscaling method can be measured against four broad criteria: consistency with the existing baseline data in terms of means, trends and distributional characteristics; consistency with the broader scale climate data used to generate the projections; the degree of transparency and repeatability; and the plausibility of results produced. Many existing downscaling methods fail to fulfil all of these criteria. In this paper we examine a block bootstrap simulation technique combined with a quantile prediction and matching method for simulating future daily climate data. By utilising this method the distributional properties of the projected data will be influenced by the distribution of the observed data, the trends in predictors derived from the Global Climate Models and the relationship of these predictors to the observed data. Using observed data from several climate stations in Vanuatu and Fiji and out-of-sample validation techniques, we show that the method is successful at projecting various climate characteristics including the variability and auto-correlation of daily temperature and rainfall, the correlations between these variables and between spatial locations. This paper also illustrates how this novel method can produce more effective point scale projections and a more credible alternative to other approaches in the Pacific region.

Appendix: The quantile matching functions for rainfall

A dry day is defined as any day with less than θ_min (>0) mm of rainfall. Often θ_min = 1 mm/day. This is to exclude small amounts of precipitation that may occur as dew or frost. We know the empirical CDF of rainfall, ${F}_{m_b}$, the projected and fitted quantiles and dry-day proportions $\hat{p}_{m_f}$, and the minimum precipitation threshold. Based on these data, we wish to construct a mapping so that we can predict the future CDF of rainfall, $\hat{F}_{m_f}.$ The mappings we use are called the quantile matching functions.

Now we have estimates of the qth conditional quantiles of the future CDF:

$$\hat{\theta}_{m_f}(q)=\hat{F}_{m_f}^{-1}\left(\hat{p}_{m_f}+(1-\hat{p}_{m_f})q\right)\quad \hbox{for}\quad q=0.1,0.5,0.9. $$

(2)

These values are mapped from the three values in the base period:

$$Q_{m_b}(q) \equiv F_{m_b}^{-1}\left(\hat{p}_{m_f}+(1-\hat{p}_{m_f})q\right) ,\quad q=0.1,0.5,0.9, $$

(3)

respectively. These values were chosen as they have the same quantile levels in the base month as $\hat{\theta}_{m_f}(q), \quad q=0.1,0.5,0.9$, has in the future month (Fig. 9). This is what is done in Fig. 1 for the special case of a continuous variable where $\hat{p}_{m_f} = 0$. For rainfall, however, there is a discontinuity in the distribution function at zero which needs to be removed when constructing the quantile matching function. This is achieved by setting:

$$\theta_{\rm min}=\hat{F}_{m_f}^{-1}\left(\hat{p}_{m_f}\right) \quad \hbox{and}\quad 0=\hat{F}_{m_f}^{-1}\left(0\right), $$

(4)

and then using linear interpolation in between these values as shown in Fig. 9. Since θ_min is the minimum observable rainfall, making such a change has no effect on the distribution of data, but it is important for the simulation of data. The quantile values corresponding to these in the base month are:

$$F_{m_b}^{-1}\left(\hat{p}_{m_f}\right) \quad \hbox{and}\quad 0=F_{m_b}^{-1}\left(0\right), \hbox{respectively.} $$

(5)

The five values given in expressions (2–5) are used to construct the quantile matching function (Fig. 10a). We use linear interpolation to approximate quantiles in between these values, and beyond the 0.9 quantile we use the 45° line to extrapolate the extreme quantiles. The rest of the future CDF shown in Fig. 9 is then constructed from the quantile matching function. The quantile matching function is used to project a rainfall value, Y _t , in the base month to a corresponding value, $Y_{{tm}_{f}}$, in the future month (Fig. 10). If, however, Y _t  = 0, it is replaced by a random value between 0 and θ_min before it is projected. This allows for the possibility that $\hat{p}_{m_f} < p_{m_b}$ in which case some days that are dry in the base period will become rain days when projected.

Fig. 9

Rainfall cumulative distribution functions. This shows how the conditional quantile, $\hat{\theta}_{m_f}(q)$, in a future month, m _f , is mapped from the point $Q_{m_b}(q) = F_{m_b}^{-1}\left(\hat{p}_{m_f}+(1-\hat{p}_{m_f})q\right)$ in the base month, m _b

Fig. 10

Quantile matching functions for rainfall. These functions map the empirical distribution of the observed data in a base-period month to either a the projected distribution, or b the fitted distribution for the same month. Note that $Q_{m_b}(q) \equiv F_{m_b}^{-1}\left(\hat{p}_{m_f}+(1-\hat{p}_{m_f})q\right)$. If Y _t  = 0 its value is replaced by a random value from the grey region prior to mapping

An analogous procedure is used to determine $Y_{{tm}_{f}}$, in the base month (Fig. 10b). Finally, the error corrected projected value of rainfall is:

$$\tilde{Y}_{tm_f} = \left\{\begin{array}{ll} Y_{tm_f} + Y_t - Y_{tm_b}, & \hbox{if} \quad Y_{tm_f} + Y_t - Y_{tm_b} > \theta_{\rm min} \,\hbox{and} \\ 0, & \hbox{otherwise.} \end{array}\right. $$

(6)