Overestimated global warming in the past 20 years

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1 SUPPLEMENTARY INFORMATION DOI: /NCLIMATE1972 Overestiated glbal waring in the past 20 years Jhn C. Fyfe 1, Nathan P. Gillett 1, Francis W. Zwiers 2 CMIP5 Mdels The 37 CMIP5 dels used, with the nuber f runs available in parenthesis, are: ACCESS1-0(1), ACCESS1-3(1), BNU-ESM(1), CCSM4(6), CESM1-BGC(1), CESM1-CAM5(3), CMCC-CM(1), CMCC-CMS(1), CNRM-CM5(10), CSIRO-Mk3-6- 0(10), CanESM2(5), FGOALS-s2(3), FIO-ESM(3), GFDL-CM3(5), GFDL-ESM2G(1), GFDL-ESM2M(1), GISS-E2-H(5), GISS-E2-H-CC(1), GISS-E2-R(6), GISS-E2-R- CC(1),HadCM3(10), HadGEM2-AO(1), HadGEM2-CC(1), HadGEM2-ES(1), IPSL- CM5A-LR(6), IPSL-CM5A-MR(3), IPSL-CM5B-LR(1), MIROC-ESM(3), MIROC-ESM- CHEM(1), MIROC5(5), MPI-ESM-LR(3), MPI-ESM-MR(3), MRI-CGCM3(3), NrESM1-M(3), NrESM1-ME(1), bcc-cs1-1(3) and bcc-cs1-1-(3). All the CMIP5 del utput was fred by erging histrical siulatins up t 2005 with RCP4.5 siulatins fr 2006 t RCP4.5 siulatins included greenhuse gas cncentratins very siilar t thse bserved ver this perid and aersl precursr eissins bradly cnsistent with best bservatinal estiates. Observed slar irradiance variatins were specified up until 2008 fllwed by a repeating slar cycle which apprxiately reprduces bserved slar irradiance variatins ver this perid 1. 1 Canadian Centre fr Cliate Mdelling and Analysis, Envirnent Canada, Victria, BC V8W 2Y2, Canada 2 Pacific Cliate Ipacts Cnsrtiu, University f Victria, Victria, BC, V8W 3R4, Canada NATURE CLIMATE CHANGE 1

2 Statistical Analysis In ur analysis tw statistical representatins fr trends are estiated either fr bservatins r fr individual del siulatins. The frs f these representatins depend n the assuptins cncerning the exchangeability f infratin between the dels and the bservatins. Trend representatin assuing exchangeability between dels In this case a reasnable trend representatin is: (1) M ij = u + Eintij + Edi, i = 1,..., N, j = 1,..., Ni and (2) k O = u + Eint + Esap, k =1,..., N. k Here M ij and f the bservatins. O k are trends calculated fr single runs, r single btstrap saples u and u are the true, unknwn, deterinistic trends due t external frcing in the delled and bserved wrlds, where u is the trend cpnent that is cn t all dels (in the liit, as the cllectin f exchangeable dels grws infinitely large). N prir assuptin that ade. Eint ij and Eint are perturbatins t u u = is M ij and Ok respectively due t internal variability. These are different fr each del run, but are essentially identical fr each resaple f the bservatins. Ed i is the perturbatin t M ij that is intrduced by del errr in del i. We assue that these perturbatins are exchangeable. Esap k is the perturbatin t O k intrduce by the k-th btstrap 2

3 resapling. N is the nuber f dels, N i is the size f the enseble fr del i, and N is the nuber f bserved recnstructins. Trend representatin assuing exchangeability between dels and bservatins In this case a reasnable trend representatin is: (1) M ij = u + Eintij + Eexchi, i = 1,..., N, j = 1,..., Ni and (2) O k = u + Eint + Eexch N +1 + Esap k, k = 1,...,N. Here Eexch is the deviatin fr the true, underlying (frced) trend in the N +1 bservatins that is exchangeable with the deviatins i Eexch, i = 1,..., N f the dels fr the cn cpnent f the del trend. That is, the deviatins are assued t be exchangeable, but the tw true trends sae. The ther cpnents are defined as abve. u and u need nt be the Null hypthesis Fr either trend representatin, a reasnable estiatr f u u is O. M.., where. replacing a subscript indicates averaging ver that subscript. In bth cases, the null hypthesis u = u H 0 : culd be tested in a nuber f ways. One apprach wuld be t ake distributinal and independence assuptins fr the individual, nn-deterinistic cpnents f (1) and (2), r (1) and (2), and subsequently derive a distributin fr O. M.. under H 0. Instead, we pt fr a resapling apprach, thereby aviding distributinal assuptins. In this apprach, the equatins abve 3

4 are used a guidance t ensure that the epirical distributin fr O. M.. includes the surces f uncertainty described in either equatins (1) and (2), r (1) and (2). Epirical distributin assuing exchangeability between dels In this case an epirical estiate f the distributin fr O. M.. under H 0 is cnstructed as fllws: a) Select a saple f N bserved recnstructins with replaceent and average t btain a O. realizatin. b) Select a saple f dels with replaceent and fr each selectin, draw ne run at rand fr that del s available enseble f siulatins, and then average ver thse N N runs t btain a M.. realizatin. c) Select, at rand, a single del i fr dels with ulti-run ensebles, and then select, at rand, a single run j fr that del s enseble. Calculate the difference M ij M i. between the trend in that single run and the ean f the trends fr that del s enseble. This difference is an estiate f the deviatin in the j-th trend fr del i that is induced by internal variability. Since the del i enseble is generally sall, the deviatins are saller than wuld be representative f an infinitely large replicatin f runs fr del i, and s t cpensate fr that lss f variance, ultiply the difference M ij M i. by [ N /( 1)] i N i 1/ 2. Finally, calculate a-b+c cputed in the steps abve and repeat the resapling prcedure any ties t build a distributin fr a-b+c under the null hypthesis that u u = 0 and the assuptin f del exchangeability. Fr this distributin we cpute p-values where a p-value is the prbability f ccurrence f a trend at least as large as that fund under the null hypthesis f equal underlying trends. Here we nte that the saller the p-value the strnger the evidence against the null hypthesis. 4

5 The ratinale fr this prcedure is a fllws: 1) Variatins in O. fr step a) represent the effect f sapling uncertainty in the bservatinal estiate f That is, the uncertainty that results fr the Esap k ter in equatin (2). 2) u. Variatins in M.. fr step b) represent the uncertainty in the ultidel estiate f u that arises fr the chice f exchangeable dels used t btain M.. [i.e., fr the Ed i ter in equatin (1)] and fr internal variability [i.e., fr the Eint ij ter in equatin (1)]. Step (c) is used t estiate the uncertainty in O. that arises fr internal variability. Since a single realizatin f internal variability is cnfunded with O., c) is cnstructed by estiating single realizatins f internal variability as they were realized in dels. This can nly be dne using dels with ulti-eber ensebles. An iplicit assuptin is that sapling uncertainty in O. is independent f uncertainty due t internal variability and als independent f uncertainty in M... Trend distributin assuing exchangeability between dels and bservatins In this case an epirical estiate f the distributin fr O. M.. under H 0 is cnstructed as fllws: a) As abve. b) As abve. c) Select a saple f N + 1 entities with replaceent fr the pl f N + 1entities cnsisting f N dels plus the bservatins (as the additinal entity in the pl). Fr each eber f the saple f entities, draw an enseble eber at rand fr that entity's available enseble. Fr these entities P i, i = 1,...,( N + 1) calculate P. and then select a single Pi at rand and calculate P i P.. Finally, calculate a-b+c as cputed in the steps abve and repeat the resapling prcedure any f ties t build a distributin fr a-b+c under the null hypthesis that u u = 0 and the assuptin f 5

6 exchangeability between dels and bservatins. The ratinale fr this apprach is the sae as abve, except cpnent c) nw includes uncertainty fr tw surces. Fr this distributin we again cpute p-values that assess hw unusual the discrepancy is between the bserved and ean f del trends. Nte that the p-values are necessarily dependent upn the assuptins that are used t cnstruct reference distributins fr O. M.. under the null hypthesis. The assuptin f exchangeable between dels and bservatins, which is a strnger assuptin than del exchangeability, sets a re stringent criterin fr rejecting the null hypthesis at a given significance level (i.e., the discrepancy between bserved and ean del trends needs t be larger) than the del exchangeability assuptin. 6

7 0.4 AMO signal in glbal teperature Teperature analy ( C) HadCRUT4 CMIP Year Suppleentary Figure 1 Atlantic Multidecadal Oscillatin (AMO) signal in siulated and bserved glbal ean surface teperature. The AMO index was cputed fr nthly ean sea surface teperature averaged ver the Nrth Atlantic (i.e. 25 N-60 N and 75 W-7 W) with trends reved and sthed with a sliding 121-nth average. ENSO, COWL and vlcanic signals 2-3 were reved fr the glbal ean surface teperature, which was then regressed against the AMO index t give the tieseries shwn abve. The % range f bserved estiates is shwn with red shading and the % range f siulated estiates is shwn with grey shading. The years 1993 and 1998 are indicated with vertical dashed lines. 7

8 References 1 Van Vuuren, D.P., Ednds, D., Kainua, J., Riahi, M. & Weyant, J. Cliatic Change, 109, di: /s z (2011). 2 Thpsn, W.J., Wallace, J.M., Jnes, P.D. & Kennedy, J.J. J. Cliate, 22, (2009). 3 Fyfe, J.C., Gillett, N.P. & Thpsn, D.W.J. Gephys. Res. Lett., 37, L16802, di: /2010GL (2010). 8