Supplemental Material to On the internal variability of simulated daily precipitation

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1 /JCLI-D s1 Supplemental Material to On the internal variability of simulated daily precipitation A. Schindler, A. Toreti, M. Zampieri, E. Scoccimarro, S. Gualdi, S. Fukutome, E. Xoplaki and J. Luterbacher October, Minimal record length (MRL) Maps of MRL exceeding 10 years Seasonal maps of locations where MRL exceeds 10 years are given in figure SM1. Figure SM1: As Fig. 1 but for MRL exceeding 10 years: Seasonal maps of locations at which MRL exceeds 10 years for precipitation in climatological seasons; DJF (top left, green), MAM (bottom left, purple), JJA (top right, blue) and SON (bottom right, brown). Colored in grey are grid cells with too few wet days to perform the analysis (lightgrey if the 12-year long record had enough wet days). Simulation study of the MRL test suit To demonstrate the suitability of the series of tests on which the MRL is based, a simulation study has been performed, in which daily precipitation records are represented by gamma distributed samples (Γ(1.5, θ ) with θ {0.5, 0.55, 0.75, 1}). anne.schindler@meteoswiss.ch 1

2 small differences moderate differences large differences 10 a) 10 b) 10 c) 3 seasons d) 10 e) f) 30 seasons temporal mixing temporal mixing temporal mixing Figure SM2: Simulation study of the percentage of rejected null hypotheses (y-axis) by temporal mixing (x-axis) showing the true percentage (grey, dotted line) and the detected percentage with (black, solid line) and without (red, dashed line) FDR. Different sample sizes by rows (a-c) 3 seasons, d-f) 30 seasons) and distributional differences by columns (a+d) small dist. differences, b+e) moderate dist. differences, c+f) big dist. differences). For 600 imaginary grid cells, the influence of record length (3 seasons and 30 seasons), and differences in distributions on the power of the series of tests is investigated. At each grid cell, 100 gamma-distributed samples are drawn, where the first sample is always drawn from a Γ(1.5, 0.5) distribution, while the remaining 99 are either drawn from the same gamma distribution, or from a gamma distribution that differs from it in scale, either hardly (θ = 0.5), moderately (θ = 0.75) or considerably (θ = 1). The proportion of samples drawn from either distribution is called temporal mixing. Spatial mixing denotes the proportion of grid cells at which a temporal mixing was introduced. As expected the series of tests detects distributional differences where the temporal mixing is large (figure SM2). The more the distributions differ between records, the more easily they are detected (figures SM2+SM3). The suit of tests fails to perform adequately, however, if the distributional differences are small and the sample size is small as well (figure SM2+SM3 a) ). As sample size increases, however, the performance of the method improves considerably (figure SM2+SM3 d) ). The falsediscovery-rate reduces the number of false-positive-errors (figure SM3 dashed vs. solid lines). Regional results with the Kolmogorov-Smirnov two sample test Fig. 4 and Fig. SM4 show similar overall patterns. However, with the 2-sample- Kolmogorov-Smirnov test (Smirnov, 1939) fewer significant differences are detected. For instance, for most northern North American grid cells (regions 1 and 2) the MRLs are much shorter for the first half of the year when using the 2-sample-Kolmogorov- Smirnov test compared to the use of the 2-sample-Cramér-von Mises test (Fig. 4). Additional information to the definition of the MRL At each grid cell a set of one hundred records of equal length, d, is sampled from the simulated precipitation time-series. Within each record only wet days (days with more than 0.01 mm of precipitation) are considered for the analysis. If for a specific 2

3 small differences moderate differences large differences 10 a) 10 b) 10 c) 3 seasons d) 10 e) f) 30 seasons spatial mixing spatial mixing spatial mixing Figure SM3: Simulation study of the percentage of rejected null hypotheses (y-axis) by spatial mixing (x-axis) showing the true percentage (grey, dotted line), detected via FDR (black, solid line), detected via point-wise significance (red, dashed line). Different sample sizes by rows (a-c) 3 seasons, d-f) 30 seasons) and distributional differences by columns (a+d) small dist. differences, b+e) moderate dist. differences, c+f) big dist. differences). The results are for a temporal mixing of 0.5 (solid and dashed lines), and additionally in panels a),b)+d) for a 0.01 temporal mixing (with FDR black, dotted line; without FDR red, dot-dashed line). length any sample has less than 10 wet days, the difference in distribution is not tested. If 94 or more tests are performed, the Walker test s p-value (W d for length d) (Fisher, 1929; Wilks, 2006) is calculated to account for multiple testing at grid cell level. At regional or global level, the significant locations are determined with the Benjamini- Hochberg procedure at a false-discovery-rate of 5% (Benjamini and Hochberg, 1995; Benjamini and Yekutieli, 2001; Ventura et al., 2004; Wilks, 2006). The minimal record length is defined for each location: min.record := min{d D W d not significant and W q significant for all q < d} for D = {3,5,7,10,12,14,17,21,25,30,36,44,52,63,76,91,100}. The difference in distribution is assessed with the 2-samples Cramer-von Mises test statistic (Csorgo and Faraway, 1996; Anderson, 1962), adjusted according to equation (15) of Anderson (1962) for comparing with the limiting distribution V (x) of Csorgo and Faraway (1996). The approximation of the distribution of the Cramer-von-Mises statistics needs to be very precise in order to be able to account for multiple testing. For our purpose 40 terms for approximating the limit distribution function are sufficient. Circumventing limits due to computer precision when estimating the gamma function, ( π t 0.5 ) t is calculated instead of the quotient Γ(t+0.5) Γ(t+1). 3

4 Figure SM4: As Fig. 4 but using the 2-sample Kolmogorov-Smirnov test (Smirnov, 1939) instead of the 2-sample-Crame r-von Mises test for detecting distributional differences. The p-values of the two-sided Kolmogorov-Smirnov 2-sample test were approximated based on Durbin (1973) /JCLI-D s1 4

5 2 Daily precipiation as simulated by the CMCC-CM GCM Evaluation of precipitation in historical control run A collection of evaluation studies for precpipitation of the historical control run, which discuss the performance of the CMCC-CM general circulation model, is given in table SM1. Noteworthy is the diagnosed potential of the climate model for studying the North Atlantic Oscillation (Davini and Cagnazzo (2014)), the Asian summer monsoon intra-seasonal oscillations (Sabeerali et al., 2013) and the relationships between the Pacific Ocean and Southwestern U.S. rainfall (Langford et al. (2014)). reference domain comment Sillmann et al. (2013) global land precipitation precip. indices Zappa et al. (2013) North Atlantic North Atlantic extratropical cyclones Davini and Cagnazzo (2014) North Atlantic North Atlantic Oscillation Bellenger et al. (2013) tropical Indo-Pacific ENSO representation Jourdain et al. (2013) tropical Indo-Pacific Indo-Australian monsoon ENSO teleconnection Brown et al. (2013) Western Pacific Monsoon area monthly rainfall to NINO3.4 corr. Sabeerali et al. (2013) Asian summer Monsoon area boreal summer intraseasonal oscillations Huang et al. (2013) Eastern China summer rainfall climatology Rosa and Collins (2013) U.S.A., South East sub-daily MJJA precip. Langford et al. (2014) U.S.A., Southwest low-frequency precip. variability Rupp et al. (2013) U.S.A., Pacific North West mean precip. patterns, JFM precip.-enso corr., variability Table SM1: Selection of evaluation studies of the representation of simulated precipitation in the historical control run of CMIP5 climate models including the CMCC-CM. Average daily precipitation in preindustrial run Seasonal mean of daily precipitation maps shown in figure SM5. Dry regions west of the continents in the subtropics visible throughout seasons. Orthographic impact on daily precipitation visible in the seasonal fields. Highest average daily precipitation of up to 50 mm day -1 occurs in the tropics in JJA. Daily precipitation variability in pre-industrial run Seasonal standard deviation of daily precipitation fields are shown in figure SM6. Lowest values of the standard deviation can be found at the poles, the Sahel zone and west of the subtropical southern hemispheric continents. Highest values occur over the tropical and subtropical Pacific. The coefficient of variation for daily precipitation, i.e., quotient of standard deviation and mean, is shown in figure SM7. It shows lowest variations with respect to the average precipitation in parts of the tropics and southern hemispheric mid-and high-latitudes. 5

6 /JCLI-D s1 Figure SM5: Seasonal mean daily precipitation: DJF (top left), JJA (top right), MAM (bottom left), SON (bottom right). Figure SM6: Seasonal standard deviation of daily precipitation: DJF (top left), JJA (top right), MAM (bottom left), SON (bottom right). 6

7 /JCLI-D s1 Figure SM7: Seasonal coefficient of variation DJF (top left), JJA (top right), MAM (bottom left), SON (bottom right). 7

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