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Nonlinear Patterns of North American Winter Temperature and Precipitation Associated with ENSO. Wu, Aiming; Hsieh, William W.; Shabbar, Amir 2005-06-30

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The Nonlinear Patterns of North American Winter Temperature and PrecipitationAssociated with ENSOAIMING WU AND WILLIAM W. HSIEHDepartment of Earth and Ocean Sciences, University of British Columbia, Vancouver, British Columbia, CanadaAMIR SHABBARClimate Research Branch, Meteorological Service of Canada, Downsview, Ontario, Canada(Manuscript received 4 March 2004, in final form 15 September 2004)ABSTRACTNonlinear projections of the tropical Pacific sea surface temperature anomalies (SSTAs) onto NorthAmerican winter (November–March) surface air temperature (SAT) and precipitation anomalies have beenperformed using neural networks. During El Niño, the linear SAT response has positive anomalies centeredover Alaska and western Canada opposing weaker negative anomalies centered over the southeasternUnited States. In contrast, the nonlinear SAT response, which is excited during both strong El Niño andstrong La Niña, has negative anomalies centered over Alaska and northwestern Canada and positiveanomalies over much of the United States and southern Canada.For precipitation, the linear response during El Niño has a positive anomaly area stretching from the eastcoast to the southwest coast of the United States and another positive area in northern Canada, in oppo-sition to the negative anomaly area over much of southern Canada and northern United States, and anothernegative area over Alaska. In contrast, the nonlinear precipitation response, which is excited during bothstrong El Niño and strong La Niña, displays positive anomalies over much of the United States and southernCanada, with the main center on the west coast at around 45°N and a weak center along the southeast coast,and negative anomalies over northwestern Canada and Alaska.The nonlinear response accounts for about one-fourth and one-third as much variance as the linearresponse of the SAT and precipitation, respectively. A polynomial fit further verifies the nonlinear responseof both the SAT and precipitation to be mainly a quadratic response to ENSO. Both the linear andnonlinear response patterns of the SAT and precipitation are basically consistent with the circulationanomalies (the 500-mb geopotential height anomalies), detected separately by nonlinear projection. Across-validation test shows that including the nonlinear (quadratic) response can potentially contribute toadditional forecast skill over North America.1. IntroductionThe impact of the El Niño–Southern Oscillation(ENSO) phenomenon on global climate, especially theNorth American climate variability, has received wideattention since the early 1980s (Trenberth et al. 1998).The extratropical atmospheric response to El Niño innorthern winter is manifested by the Pacific–NorthAmerican (PNA) teleconnection pattern, which can befound simply by linear regression or correlation analy-sis (Wallace and Gutzler 1981; Horel and Wallace1981), and is rather well explained by the linear wavepropagation theory (Hoskins and Karoly 1981).Though the PNA pattern accounts for a considerablepart of the variance of interannual climate fluctuationsover the North Pacific and North America and is re-garded as a major source of skill for seasonal forecasts(e.g., Zwiers 1987; Barnston 1994; Shabbar and Barn-ston 1996; Derome et al. 2001), recent evidence fromobservational studies and numerical models showed theNorth American climate to have asymmetric responsepatterns during the opposite phases of ENSO, suggest-ing that the North American climate responds to ENSOin a nonlinear fashion. For example, compositingmonthly fields of the U.S. surface air temperatureCorresponding author address: Dr. Aiming Wu, Dept. of Earthand Ocean Sciences, University of British Columbia, Vancouver,BC, V6T 1Z4, Canada.E-mail: awu@eos.ubc.ca1736 JOURNAL OF CLIMATE VOLUME 18© 2005 American Meteorological SocietyJCLI3372(SAT), precipitation, and 700-mb height based on thesea surface temperature anomalies (SSTAs) in the cen-tral equatorial Pacific, Livezey et al. (1997) found thatthe fields associated with warm and cold episodes havedifferent spatial structures. The asymmetric patterns ofCanadian SAT and precipitation associated with ENSOwere investigated by Shabbar and Khandekar (1996)and Shabbar et al. (1997). Composited precipitationpatterns over central and eastern North America asso-ciated with the SSTA in different regions of the tropicalPacific were studied by Montroy et al. (1998). The dy-namic foundation for nonlinear atmospheric responseto ENSO was discussed by Hoerling et al. (1997), whereobserved and modeled 500-mb northern Pacific circu-lation anomalies associated with warm episodes areshifted by about 35° eastward relative to their counter-parts during cold episodes.Suppose x denotes the ENSO sea surface tempera-ture (SST) index and y denotes the extratropical atmo-spheric response to ENSO. To properly understand thenonlinear extratropical atmospheric response toENSO, one would need to derive the nonlinear re-sponse function y H11005 f(x). Unfortunately, this nonlinearproblem is oversimplified by using classical linear sta-tistical tools.Composite analysis is a common way to present thenonlinear atmospheric response patterns associatedwith tropical Pacific warm (El Niño) and cold (La Niña)episodes by averaging the data over the years whenwarm episodes occurred and averaging over the yearswhen cold episodes occurred. While the patterns duringwarm and cold episodes are not restricted to be anti-symmetrical, composite analysis does not yield a non-linear response function. “One-sided regression” calcu-lates a linear regression between the SST index and theresponse variable when the SST index is positive andanother linear regression when the SST index is nega-tive (Hoerling et al. 2001), thereby describing the non-linear atmospheric response with a very restrictive non-linear function. The advent of neural network (NN)methods has made the general nonlinear problem trac-table and has led to the nonlinear generalization ofregression, classification, principal component analysis(PCA), singular spectrum analysis, and canonical cor-relation analysis (CCA; see reviews by Hsieh and Tang1998; Hsieh 2004).The linear response to ENSO can be easily obtainedby a linear projection, that is, a linear regression of theatmospheric variables on the ENSO SST index. Thenonlinear projection can be achieved via a NN ap-proach (henceforth a NN projection). The nonlinearprojections of the tropical Pacific SSTA to the North-ern Hemisphere winter 500-mb geopotential height(Z500) anomalies and to the sea level pressure anoma-lies have been performed by Wu and Hsieh (2004a) andWu and Hsieh (2004b), respectively. Compared to thenonlinear canonical correlation analysis (Hsieh 2001),the nonlinear projection has a much simpler NN struc-ture with far fewer model parameters; hence it is easierto obtain robust results from noisy data.In this work, the NN projection method will be fur-ther used to investigate the nonlinear association be-tween ENSO and North American winter SAT andprecipitation. The data and the method are briefly in-troduced in section 2. The nonlinear SAT anomaly pat-terns associated with El Niño and La Niña detected bythe NN projection, and separately by a polynomial fit,are presented in section 3. Similar analysis for the pre-cipitation is presented in section 4. Section 5 relates theSAT and precipitation response to the atmospheric cir-culation (Z500) response to ENSO. Section 6 gives asummary and discussion. The appendix examines thepotential increase in forecast skills by including thenonlinearity.2. Data and methodologya. DataThe monthly SST data on a 2° H11003 2° grid for theperiod 1950–2002 came from the Extended Recon-structed Sea Surface Temperatures, version 2(ERSST2) dataset (Smith and Reynolds 2004). TheSST anomalies were calculated by subtracting themonthly climatology (based on the whole period) fromthe monthly mean SST. The ENSO SST index is de-fined as the standardized first principal component(PC) of the winter (November–March) SSTA over thetropical Pacific (22°S–22°N, 122°E–72°W), which is cor-related with both the Niño-3 and Niño-3.4 indices at0.96. A PC-based ENSO SST index was also used byHoerling et al. (2001) but with the domain expanded to30°S–30°N.The monthly land SAT and precipitation data camefrom the Climate Research Unit (CRU) at the Univer-sity of East Anglia, United Kingdom (Mitchell et al.2003, manuscript submitted to J. Climate; or see http://www.cru.uea.ac.uk/cru/data/hrg.htm). The CRU pro-vided data on a 0.5° H11003 0.5° grid for the period 1901–2002. In this study, only the data after 1950 were used.Similarly, the SAT and precipitation anomalies werecalculated by subtracting the monthly climatology withonly the winter (November–March) data over NorthAmerica used. A notable feature of the precipitationdata is that the precipitation anomalies have muchlarger magnitudes near the coast than inland, which canbe seen from the distribution of standard deviation of1JUNE 2005 W U E T A L . 1737the precipitation (Fig. 1). PCA was used to compressthe data, with the 8 leading SAT PCs and the 12 leadingprecipitation PCs (accounting for 89.4% and 69.6% ofthe total variance of the SAT and precipitation anoma-lies, respectively) retained.The spatial patterns of the three leading PCA modes[also called empirical orthogonal functions (EOFs)] forthe SAT and precipitation anomalies are shown in Fig.2, where the SAT EOF1shows a uniform anomaly signover the whole domain except the southeastern UnitedStates (Fig. 2a), and EOF2reveals SAT anomalies overthe United States and southern Canada in opposition tothe anomalies over Alaska and northern Canada (Fig.2b), whereas the SAT EOF3shows the northeast half ofthe continent to have anomalies opposite in sign to therest (Fig. 2c). The precipitation EOF1(Fig. 2d) showsanomalies in northwestern Canada and Mexico oppos-ing anomalies over other areas of the continent; EOF2displays anomalies over the west coast [from BritishColumbia (BC) to northern California] stretching east-ward, opposing the anomalies in Alaska, northernCanada, and in the southern United States (Fig. 2e),while EOF3shows almost a uniform sign over the do-main except the west coast of the United States withstrong anomalies over the southeastern United States(Fig. 2f).b. The nonlinear projectionA schematic diagram of the multilayer perceptronNN model with one hidden layer is shown in Fig. 3. TheNN has a single input, the SST index (x), which is non-linearly mapped to m intermediate variables, calledhidden neurons (hi, i H11005 1, ···, m), which are then lin-early mapped to l output variables (yj, j H11005 1, ···, l),that is,hiH11005 tanhH20849wix H11001 biH20850, H208491H20850yjH11005H20858iH110051mw˜jihiH11001 b˜j, H208492H20850where wiand w˜jiform an m-element weight vector (w)and an l H11003 m weight matrix (W), respectively, while biand b˜jform bias vectors (b and b˜) of length m and l,respectively. With enough hidden neurons, the NNmodel is capable of modeling any nonlinear continuousfunction to arbitrary accuracy. Starting from randominitial values, the model parameters (in w, W, b, and b˜)are optimized so that the mean square error (MSE)between the model outputs (y) and the leading PCs(yobs) of the SAT (or precipitation) anomalies is mini-mized. There is no time lag between x and yobs.Toavoid local minima during optimization, the NN modelwas trained 30 times with random initial conditions.Among the runs, the solution with the smallest MSEwas chosen and the other 29 rejected.To reduce the possible sampling dependence of asingle NN solution, we repeated the above calculation400 times with a bootstrap approach (Efron and Tib-shirani 1993). A bootstrap sample was obtained by ran-domly selecting (with replacement) one winter’s dataFIG. 1. The standard deviation of North American winter precipitation. The contour intervalis 10 mm; shaded areas indicate values larger than 50 mm.1738 JOURNAL OF CLIMATE VOLUME 18record 53 times from the original record of 53 win-ters. The ensemble mean of the resulting 400 NNmodels was used as the final NN model, which wasfound to be insensitive to the number of hiddenneurons that was varied from two to five in a sensitivitytest. Results from using three hidden neurons are pre-sented here.If the nonlinear mapping function, tanh in Eq. (1), isreplaced by a linear function, the NN model is essen-tially reduced to linear regression.FIG.2.(a)–(c) The three leading EOFs of the North American winter SAT anomalies and (d)–(f) the EOFs of theprecipitation anomalies; solid curves: positive contours, dashed curves: negative contours, and thick curves: zero contours.The contour interval is 0.01 for the SAT and 0.02 for the precipitation, and the EOFs have been normalized to unit norm.The percentage variance explained by each EOF is given in the figure titles.1JUNE 2005 W U E T A L . 17393. The nonlinear patterns of SAT associatedwith ENSOa. The NN projection resultsIn the NN model, since the output PCs are all derivedfrom a single time series (the ENSO index), the atmo-spheric response extracted by the nonlinear projectioncan be presented by a curve in the eight-dimensional(8D) phase space of the SAT PCs, while the linearprojection extracts a straight line in the same 8D space.This curve is parabola-like (Fig. 4) when projected ontothe PC1–PC2plane and the PC2–PC3plane, indicatingthat the SAT response to the ENSO SST index is anonlinear combination of some of its leading PCAmodes. For a specific value of the SST index, the NNsolution gives the eight SAT PCs (the output of the NNmodel; Fig. 3), which can be combined with the corre-sponding EOFs, yielding the SAT spatial anomalies as-sociated with the given SST. As the SST index variesslowly from its minimum value to its maximum value,the SAT response moves smoothly along the 3D curvein Fig. 4 from the end labeled “c” to the other endlabeled “w” (corresponding to the extreme cold andwarm episodes, respectively), with both the spatial pat-tern and amplitude changing; in contrast, the linear re-sponse is restricted along the straight line in Fig. 4,manifesting a variable amplitude but a fixed spatial pat-tern.When the SST index takes on its minimum value (i.e.,strong La Niña), significant negative SAT anomaliesappear over Alaska and northwestern Canada and posi-tive anomalies over the southeastern United States(Fig. 5a). When the SST index takes on its maximumvalue (i.e., strong El Niño), the SAT anomaly center isshifted southeastward to the Great Plains (west of theGreat Lakes) with the sign reversed (to become posi-tive), and with weak negative anomalies appearing overnortheastern Canada and over the southern UnitedStates–Mexico (Fig. 5d). The SAT patterns shown inFigs. 5a and 5d are well consistent with the compositeresults (Hoerling et al. 1997) and the one-sided regres-sion results (Hoerling et al. 2001). The asymmetry be-tween the SAT anomaly pattern in Fig. 5a and that inFig. 5d suggests a considerably nonlinear associationbetween ENSO and the North American winter SAT.The displacement between the negative SATanomaly center in Fig. 5a and the positive SATanomaly center in Fig. 5d is understandable. In Fig. 4, atthe c end, negative PC1concurs with negative PC2andpositive PC3, which when combined with the corre-sponding EOFs (Fig. 2) is favorable for the formationof the SAT anomaly pattern shown in Fig. 5a; whereasat the w end, positive PC1concurs with negative PC2and negative PC3, which is favorable for the SATanomaly pattern in Fig. 5d.Figures 5b and 5e show, respectively, the SATanomalies when the SST index is at half its minimumvalue and at half its maximum value. The SAT anoma-lies decrease in magnitude, but the antisymmetry be-tween Figs. 5b and 5e is much enhanced (relative toFigs. 5a and 5d), suggesting that at these half-minimumand half-maximum SST values the extratropical SATresponse is largely linear.To illustrate the nonlinear response in the SATanomalies to the tropical SSTA, we plotted in Fig. 5cthe difference between the SAT anomalies in Fig. 5aand twice the anomalies in Fig. 5b and similarly in Fig.5f, the difference between the anomalies in Fig. 5d andtwice the anomalies in Fig. 5e. Interestingly, despite thestrong asymmetry between Figs. 5a and 5d, and theconsiderable antisymmetry between Figs. 5b and 5e, theSAT anomalies in Figs. 5c and 5f are in good agree-ment, indicating that regardless of the sign of the SSTindex the nonlinear response has positive SAT anoma-lies appearing over much of the United States andsouth-central Canada and negative SAT anomaliesover Alaska and northern Canada. It is this nonlinearresponse that brings forth the asymmetric SAT anoma-lies during the extreme cold and warm episodes (Figs.5a and 5d).To ensure that the finding above is not the result ofFIG. 3. Schematic diagram of the feed-forward neural networkmodel used to perform the nonlinear projection. The NN has asingle input x, which is nonlinearly mapped to intermediate vari-ables (called hidden neurons) h, which are then linearly mappedto the output variables y. The model parameters are optimized byminimizing the mean square error given by the cost function J H11005H20855|| y H11002 yobs||2H20856, where yobsis the observed data, and H20855···H20856 denotesa sample or time mean. In this study, x is the ENSO SST index,and yobsthe 8 leading PCs of the North American SAT or the 12leading PCs of the precipitation anomalies, with three hiddenneurons used in both cases.1740 JOURNAL OF CLIMATE VOLUME 18fitting to one or two extreme cases, we repeated ourbootstrap calculations but deleting the two winters withthe strongest ENSO warm episodes (1982/83 and 1997/98), and the two winters with the strongest cold epi-sodes (1973/74 and 1975/76) from the data record. Evenwithout the extreme ENSO episodes, the resulting NNprojection yielded basically the same spatial patterns asin Fig. 5 (with somewhat smaller magnitude anomalies),thereby confirming that the nonlinear response foundby our NN projection was a robust result.b. The linear and nonlinear components of theSAT responseThe NN-projected SAT anomalies can be decom-posed into a linear component, that is, the linear pro-jection (the straight line in Fig. 4) and a nonlinear com-ponent—the residual after the linear projection hasbeen subtracted from the NN projection, that is, theeight PCs from the NN projection minus the eight PCsfrom the linear projection. The resulting eight PCs forthe linear and nonlinear components can then be com-bined with the corresponding EOFs to yield the linearand nonlinear responses of SAT to ENSO. The linearand nonlinear components account for 80.4% and19.6%, respectively, of the variance in SAT anomalydata derived from the NN projection; in other words,the nonlinear response has almost one-fourth the vari-ance of the linear response.PCA is then used to separately analyze the linear andnonlinear response fields of the SAT anomalies duringthe period from January 1950 to December 2002 (withonly winter months), where over 99% of the variancefor either data field can be explained by its first PCAmode (the high percentage of variance explained is notsurprising since the SAT anomaly field was generatedoriginally by nonlinearly projecting from a single SSTindex time series). When the PC1of the linear response(the solid line in Fig. 6a) takes on a positive value, theEOF1of the linear response (Fig. 6b) shows positiveFIG. 4. The SAT response to the ENSO SST index extracted by the NN projection shownas a thick curve of overlapping squares in the PC1–PC2–PC33D space. The linear response isshown by a thin straight dashed line. The SAT signal is also projected onto the PC1–PC2,PC2–PC3, and PC1–PC3planes, where the nonlinear projection is presented by the curves of overlap-ping circles, the linear projection by thin solid lines, and the projected data points by thescattered dots. The labels “c” and “w” denote the extreme cold and warm states, respectively.1JUNE 2005 W U E T A L . 1741SAT anomalies appearing over western Canada andAlaska and negative anomalies over northeasternCanada and the southern United States, which can beinterpreted by the classical PNA teleconnection pat-tern. The EOF1of the nonlinear component (Fig. 6c)shows positive SAT anomalies over the United Statesand south-central Canada (with the center located inthe Great Lakes region) and negative anomalies overFIG. 5. The SAT anomalies associated with the (a) minimum SST index and (d) maximum SST index, and with (b)one-half of the minimum SST and (e) one-half of the maximum SST. (c) The SAT anomalies in (a) minus twice theanomalies in (b), and (f) the anomalies in (d) minus twice the anomalies in (e). If the SAT response to the SST index isstrictly linear, then (c) and (f) will show zero everywhere. Contour interval is 0.5°C, and the gray areas indicate statisticalsignificance at the 5% level, based on the distribution of the results from the 400 bootstrap samples.1742 JOURNAL OF CLIMATE VOLUME 18Alaska and northern Canada, strongly resembling theSAT anomaly patterns found in Figs. 5c and 5f. ThePC1of the linear response is synchronous with theENSO SST index, while the PC1of the nonlinear re-sponse (the dashed line in Fig. 6a) has positive valuesnot only during the El Niño winters (1958, 1966, 1973,1983, 1992, and 1998), but also during La Niña winters(1950, 1956, 1971, 1974, 1976, 1989, and 1999, with theyears labeled by the January of the ENSO winter).Hence, regardless of warm or cold episodes, the SAThas the same response pattern as depicted by Fig. 6c,which strengthens the negative SAT anomalies overAlaska and northwestern Canada during cold episodesand strengthens the positive SAT anomalies over theCanadian Plains and the Great Lakes region duringwarm episodes, thereby producing the asymmetric spa-tial patterns of the North American SAT anomaliesduring strong El Niño and strong La Niña (Figs. 5a and5d).The scatterplots between the PC1of the nonlinearresponse and the SST index form a parabola-like curve(shown by the solid circles in Fig. 7), which is well fittedby the polynomial function PCNLH11005H110028.793 H11002 3.683x H110019.318x2H11002 0.051x3H11002 0.102x4H11001 0.003x5, where x is theENSO SST index. From the magnitude of the polyno-mial coefficients, the nonlinear response of the NorthAmerican SAT to ENSO is clearly dominated by a qua-dratic response. For comparison, the scatterplot of thelinear response is well fitted by the straight line PClinH1100526.964x.c. A polynomial fitNow we consider a simple polynomial fit of the SSTindex to the SAT anomaly at each grid point. Let x bethe SST index and xnH11005 xn, then T, the original SATanomaly at a grid point, was fitted by T H11005 a0H11001 a1xˆ1H11001a2xˆ2H11001 ···H11001 aNxˆN, where xˆnis xnnormalized. For 400bootstrap samples and for each spatial point of the SATanomaly field, regression coefficients a0, ..., aNwerecomputed. After ensemble averaging over all bootstrapsamples, anprovided the spatial pattern associated withthe nth order response to the SST index. When testedover independent data (i.e., data not selected in a boot-strap sample), the smallest MSE (averaged over allbootstrap samples) was found when N H11005 2, indicatingoverfitted results when N H11022 2. Hence there is no evi-dence for a cubic or higher-order nonlinear response toFIG. 6. (a) The leading PC of the SAT anomalies from the linear and nonlinear response to the ENSO SST index asextracted by the NN projection, shown by the solid line and dashed line, respectively. (b), (c) The corresponding PCAspatial patterns for linear and nonlinear components, respectively. The contour interval is 0.01, and the spatial modes havebeen normalized to unit norm.1JUNE 2005 W U E T A L . 1743ENSO. With N H11005 2, the ensemble-averaged values of a1and a2are plotted in Fig. 8.The linear term (Fig. 8a) shows positive SAT anoma-lies over western Canada and Alaska and negativeanomalies over the southeastern United States, resem-bling the pattern from the linear projection as shown inFig. 6b. In contrast, the quadratic term (Fig. 8b) showspositive SAT anomalies over the United States andsouthern Canada and negative anomalies over Alaskaand northern Canada, basically consistent with Fig. 6c,as well as Figs. 5c and 5f, confirming that nonlinearresponse of North American winter SAT to ENSO ismainly a quadratic response.4. The nonlinear patterns of precipitationassociated with ENSOa. The NN projection resultsSimilarly, the precipitation anomalies associated withthe ENSO SSTA extracted by the NN projection ismanifested by a curve in the 12-dimensional (12D)phase space of the precipitation PCs, which whenviewed in the PC1–PC2–PC3subspace (Fig. 9), revealsappreciable nonlinearity in the precipitation responsefield to ENSO. When the SST index takes on its mini-mum value (i.e., strong La Niña), significant negativeprecipitation anomalies appear over the Pacific coast ofAlaska and northwestern Canada and over both thesouthwest coast and the east coast of the United States,while positive anomalies appear from southern BC tonorthern California and south of the Great Lakes (Fig.10a). When the SST index takes on its maximum value(i.e., strong El Niño), significant negative anomalies ap-pear over the Pacific coast from northern BC to Alaska,and over the Great Lakes, while the United Statessouth of 45°N (except to the south of the Great Lakes)and Mexico have positive anomalies (Fig. 10d). Com-paring Figs. 10a and 10d, we see that the precipitationanomalies are basically antisymmetrical over central-eastern North America, while the anomaly patterns arein poor antisymmetry in the west (especially the coastalareas), suggesting that considerable nonlinearity occursin the west coast. The precipitation anomalies associ-ated with the half-minimum and half-maximum SSTFIG. 7. Scatterplot between the ENSO SST index and the EOFPC1of the nonlinear SAT response to ENSO, with the small solidcircles revealing a curve. The linear SAT response is shown by thestraight line of open circles.FIG. 8. The SAT anomaly patterns associated with the (a) linear and (b) quadratic terms of the ENSO SST index. Thecontour interval is 0.2°C and the shaded areas indicate statistical significance at the 5% level from bootstrapping.1744 JOURNAL OF CLIMATE VOLUME 18index are shown Figs. 10b and 10e, respectively, withboth the magnitudes of the anomalies and asymmetry(between the two panels) reduced.The nonlinear response in the precipitation anoma-lies to the tropical SSTA is estimated by the differencebetween the precipitation anomalies in Fig. 10a andtwice the anomalies in Fig. 10b (shown in Fig. 10c), andsimilarly the difference between the anomalies in Fig.10d and twice the anomalies in Fig. 10e (shown in Fig.10f). Despite the large difference between Figs. 10a and10d and the smaller difference between Figs. 10b and10e, we see again that the precipitation anomalies inFigs. 10c and 10f agree well with each other, indicatingthat regardless of the sign of the SST index, the non-linear response has significant negative precipitationanomalies over the Pacific coast of Alaska and north-western Canada and positive anomalies over the westcoast of United States and southern BC.Again, the basic patterns in Fig. 10 are unchangedeven if the two winters with the strongest ENSO warmepisodes and the two winters with the strongest coldepisodes were removed from the data record prior tothe bootstrap computation.Although in Fig. 10 the precipitation anomalies arelargely confined over the coastal areas, this does notmean that the weak precipitation anomalies over inte-rior areas can be ignored. A small change of precipita-tion inland (especially the areas with dry climate) canbe important for regional agriculture and human activi-ties. In Fig. 11, the normalized precipitation anomaliesshow significant anomalies stretching far inland. Thenonlinear component of precipitation anomalies stillappears mainly over the west but not focused on thecoast anymore (Figs. 11c,f).b. The linear and nonlinear components of theprecipitation responseSimilarly, by subtracting the linear projection (i.e.,the linear response) from the NN projection, the non-linear component of the precipitation response was ob-tained. This nonlinear component accounts for 26.9%of the variance in the NN-projected precipitationanomaly data, while the linear component accounts for73.1% of the variance. Then PCA was separately ap-plied to the linear and nonlinear response fields of pre-cipitation anomalies for the period 1950–2002, with theleading PC and the corresponding spatial pattern(EOF1) shown in Fig. 12 (the PCA mode 1 explainsover 99% of the variance for either response field).When the PC1of the linear response (the solid line inFIG. 9. As in Fig. 4, but for the precipitation anomalies.1JUNE 2005 W U E T A L . 1745Fig. 12a) takes on a positive value, the EOF1of thelinear response (Fig. 12b) shows a large positive pre-cipitation anomaly area ranging from the east coast tothe southwest coast of the United States and anotherweak positive area in northern Canada. A negativeanomaly area over much of southern Canada andnorthern United States and another weak negative areaover Alaska are also manifested by EOF1. Figure 12bagrees well with Figs. 10b and 10e in pattern.For the nonlinear response (Fig. 12c), its EOF1showspositive anomalies over much of the United States andsouthern Canada, with the main center on the westcoast at around 45°N and a weaker center along thesoutheast coast, and negative anomalies over north-western Canada and Alaska, centering along the Pacificcoast. Figure 12c agrees well with Figs. 10c and 10f.Normalizing the anomalies in Figs. 12b and 12c with thestandard deviation yields anomaly patterns resemblingFIG. 10. As in Fig. 5, but for the precipitation anomalies; contour interval is 10 mm.1746 JOURNAL OF CLIMATE VOLUME 18Figs. 11e and 11c (or Fig. 11f), respectively. Since thePC1of the nonlinear response has positive values notonly during the El Niño winters, but also during LaNiña winters (see the dashed curve in Fig. 12a), regard-less of warm or cold episodes, the precipitation has thesame response pattern as depicted by Fig. 12c, whichcontributes to the asymmetric precipitation anomalypatterns during extreme cold and warm episodes (Figs.10a,d).The scatterplots between the PC1of the nonlinearresponse and the SST index gives a parabola-like curve(not shown), which can be fitted well by the polynomialfunction PCNLH11005H1100287.18 H11002 34.01x H11001 93.23x2H11002 1.15x3H110021.18x4H11001 0.01x5, where x is the ENSO SST index, indi-FIG. 11. As in Fig. 10, but for the normalized precipitation anomalies (obtained from dividing the actual precipitationanomalies shown in Fig. 10 by the corresponding standard deviation as shown in Fig. 1). Contour interval is 0.1.1JUNE 2005 W U E T A L . 1747cating that the nonlinear response of the North Ameri-can precipitation to ENSO is also mainly a quadraticresponse.c. A polynomial fitLet the precipitation anomaly (P) at each spatialpoint be fitted by P H11005 a0H11001 a1xˆ1H11001 a2xˆ2, where xˆ1andxˆ2are, respectively, x and x2normalized (x being theENSO SST index). The ensemble-averaged values of a1and a2at all spatial points give the spatial patterns ofthe linear term and quadratic term, respectively (Figs.13a,b). Using more higher-order terms in the polyno-mial fit leads to overfitting. The precipitation anomalypattern shown in Fig. 13b agrees well with that in Fig.12c, as well as Figs. 10c and 10f, confirming that non-linear response of the North American winter precipi-tation to ENSO is mainly a quadratic response.FIG. 12. As in Fig. 6, but for the precipitation anomalies. Contour interval in (b) and (c) is 0.01.FIG. 13. As in Fig. 8, but for the precipitation anomalies; contour interval is 3 mm.1748 JOURNAL OF CLIMATE VOLUME 185. Relation to the upper circulation anomaliesA similar nonlinear projection of the ENSO SST in-dex onto the Northern Hemisphere winter Z500anoma-lies reveals linear and nonlinear response patterns asshown in Fig. 14. The linear response (Fig. 14a) showsa PNA-like pattern with strong negative anomalies overthe North Pacific, positive anomalies over northwesternCanada, and weak negative anomalies over the south-eastern United States. By geostrophy, over westernCanada and Alaska, warmer air is transferred byanomalous southeasterlies leading to positive tempera-ture anomalies, while over northeastern Canada colderair transferred by anomalous northerlies from the Arc-tic area results in the negative temperature anomalies,and over the southeastern United States the anomalousnortheasterlies bring forth cooler than normal weather(Figs. 6b and 8a). The low pressure cell in the NorthPacific (Fig. 14a) generates onshore flow of moist ma-rine air along California to Mexico, leading to positiveprecipitation anomalies there (Fig. 13a). Meanwhile,the anomalous alongshore flow along the west coast(Fig. 14a) diverts the normal moist westerly flow farthernorth, thereby producing negative precipitation anoma-lies from Oregon to southern BC and positive precipi-tation anomalies farther north along the coast (Fig.13a). The anomalous flow over the southern UnitedStates (Fig. 14a) is also consistent with the positive pre-cipitation anomalies there (Fig. 13a), while the anoma-lous high pressure over northern Canada is consistentwith the negative precipitation anomalies there.The nonlinear Z500response reveals positive anoma-lies over eastern Canada and the United States andnegative anomalies over the Northeast Pacific Oceanand the west coast of Canada and the United States(Fig. 14b). The anomalous southerlies between thenegative anomalies and positive anomalies transferwarmer air from lower latitudes, leading to positivetemperature anomalies over the United States andsouthern Canada, while colder air is transferred fromhigher latitudes to Alaska and northeastern Canada,leading to negative temperature anomalies there (Fig.8b). The anomalous low over the Pacific (Fig. 14b)brings moist air to the west coast from California tosouthern BC and dry Arctic air to Alaska, leading tothe distribution of precipitation anomalies seen in Fig.13b.In brief, the temperature and precipitation anomaliesassociated with the ENSO SSTA can be well explainedby the circulation anomalies extracted separately by theNN projection, confirming the robustness of the non-linear (mainly quadratic) relation between the ENSOSSTA and North American winter climate. This qua-dratic relation is also found in an atmospheric generalcirculation model (AGCM) forced by tropical diabaticheating anomalies, where the relationship between theFIG. 14. Spatial patterns of (a) linear and (b) nonlinear response of the Northern Hemisphere winter Z500anomalies to the ENSOSSTA extracted by the NN projection. Pattern (a) is for El Niño; for La Niña, the sign of the anomalies is reversed. Pattern (b) is forboth El Niño and La Niña. Contour interval is 0.02; the anomalies have been normalized to unit norm (reproduced from Wu and Hsieh2004a).1JUNE 2005 W U E T A L . 1749first PC of Z500anomalies (from an ensemble mean)and the amplitude of the forcing is linear, while thesecond PC of Z500anomalies has a nearly parabolic(quadratic) relationship with the amplitude of the forc-ing (Lin and Derome 2004), consistent with the resultsfound in our work.6. Summary and discussionA fully nonlinear projection of the ENSO SST indexonto the North American winter (November–March)SAT and precipitation anomalies has been achieved us-ing neural networks, which reveals asymmetric atmo-spheric patterns associated with El Niño and La Niña.The NN projection consists of a linear response and anonlinear response. For SAT, the linear response dur-ing El Niño has positive anomalies centered overAlaska and western Canada and weaker negativeanomalies centered over the southeastern UnitedStates (Fig. 6b). In contrast, the nonlinear SAT re-sponse, which is excited during both strong El Niñoand strong La Niña, has negative anomalies centeredover Alaska and northwestern Canada and posi-tive anomalies over much of the United States andsouthern Canada (Fig. 6c), centered around the GreatLakes.For precipitation, the linear response during El Niñodisplays a positive anomaly area stretching from theeast coast to the southwest coast of the United Statesand another positive area in northern Canada, in op-position to the negative anomaly area over much ofsouthern Canada and northern United States and an-other negative area over Alaska (Fig. 12b). In contrast,the nonlinear precipitation response, which is excitedduring both strong El Niño and strong La Niña, dis-plays positive anomalies over much of the UnitedStates and southern Canada, with the main center onthe west coast at around 45°N and a weak center alongthe southeast coast and negative anomalies over north-western Canada and Alaska (Fig. 12c). The normalizedanomalies show more details of the nonlinear precipi-tation response to ENSO (Fig. 11).FIG. 15. Cross-validated correlation skills from the linear model for the (a) SAT and (c) precipitation, and the differenceof skill between the nonlinear model and the linear model (nonlinear skill minus linear skill) for (b) the SAT forecast and(d) the precipitation forecast. Contour interval is 0.1; areas with positive values are shaded.1750 JOURNAL OF CLIMATE VOLUME 18A polynomial fit of the SAT anomalies to the ENSOSST index, and a similar fit of the precipitation anoma-lies, confirm the nonlinear response of the SAT andprecipitation to be mainly a quadratic response toENSO.There is an interesting contrast in the nonlinear SATresponse and the nonlinear precipitation response,namely that the nonlinear response in precipitation ismore focused on the western part of the continent,whereas the nonlinear response in the SAT is wellmanifested in the eastern half. The nonlinear responseaccounts for about one-fourth and one-third as muchvariance as the linear response for the SAT and pre-cipitation, respectively. In some areas, the nonlinearresponse can surpass the linear response and can po-tentially enhance forecast skills (see the appendix).Both the linear and nonlinear response of SAT andprecipitation to ENSO can be respectively explained bythe linear and nonlinear response of the atmosphericcirculation (as seen in the Z500anomalies) to ENSO.Even with the extra skills contributed by the higher-order (quadratic) term of the ENSO SST index (Figs.15b,d), the cross-validated forecast skills from the uni-variate nonlinear model are generally lower than thosefrom multivariate linear models. For instance, usingfive leading PCs of the tropical Pacific SST anomaliesas predictors, a linear CCA model shows higher skillsthan a univariate nonlinear model (using the leadingSST PC and its quadratic term as predictors) for bothSAT and precipitation over most areas of NorthAmerica (figures not shown). In fact, multivariate non-linear regression (MNLR) and nonlinear CCA(NLCCA) models via an NN approach have been de-veloped and applied to forecast the tropical Pacific SSTanomalies (see http://www.ocgy.ubc.ca/projects/clim.pred/). The development of multivariate nonlinearmodels for seasonal climate prediction over extratropi-cal regions is our next step.Acknowledgments. Dr. David Viner of the ClimaticResearch Unit at the University of East Anglia kindlyprovided the surface air temperature and precipitationdata. This study was supported by research and strate-gic grants from the Natural Sciences and EngineeringResearch Council of Canada and a contribution fromEnvironment Canada.APPENDIXCross-Validated Forecast SkillsThe merit of finding the quadratic response can betested by comparing the cross-validated forecast skillsfor the SAT (or precipitation) with and without thequadratic term of the ENSO SSTA. Using the ENSOSST index (x) as a single predictor, the SAT (or pre-cipitation) anomaly at each spatial point reconstructedfrom the 8 leading PCA modes (or the 12 leading PCAmodes for the precipitation) can be predicted with alinear model,y H11005 a0H11001 a1x, H20849A1H20850and a nonlinear model with the quadratic term,y H11005 a0H11001 a1x H11001 a2x2. H20849A2H20850The 53-yr winter data were divided into five equalsegments. Data from one segment were withheld asvalidation data, while data from the other four seg-ments (the training data) were used to build the models.Thus, an independent forecast was made for the periodof the validation data using the models based on thetraining data. This procedure was repeated until all fivesegments were predicted, and the correlation betweenthe predicted SAT (or precipitation) anomaly and thecorresponding observation could be calculated over thewhole record, yielding the a cross-validated skill at thispoint. The skills from the linear model for the SAT andprecipitation are shown in Figs. 15a and 15c, respec-tively, and the difference of skill between the nonlinearmodel and linear model is shown Figs. 15b and 15d forthe SAT and precipitation, respectively. It is not sur-prising to see that positive skills from the linear model(shaded areas in Figs. 15a,c) are basically consistentwith the linear response pattern (Fig. 6b for the SAT;Fig. 12b for the precipitation). The major improvementbrought by the quadratic term for the SAT forecast ismainly over the eastern United States and northeasternCanada (Fig. 15b) and for the precipitation forecast,over the west coast, northwestern Canada, and Alaska(Fig. 15d). The improvement of forecast skill (shadedareas in Figs. 15b,d) is also consistent with the nonlin-ear response patterns found by the nonlinear projection(Fig. 6c for the SAT; Fig. 12c for the precipitation),indicating that using the quadratic response can poten-tially improve the North American climate forecast.REFERENCESBarnston, A. G., 1994: Linear statistical short-term climate pre-dictive skill in the Northern Hemisphere. J. Climate, 7, 1513–1564.Derome, J., and Coauthors, 2001: Seasonal predictions based ontwo dynamical models. Atmos.–Ocean, 39, 485–501.Efron, B., and R. J. Tibshirani, 1993: An Introduction to the Boot-strap. CRC, 456 pp.Hoerling, M. P., A. Kumar, and M. Zhong, 1997: El Niño, LaNiña, and the nonlinearity of their teleconnections. J. Cli-mate, 10, 1769–1786.1JUNE 2005 W U E T A L . 1751——, ——, and T. Xu, 2001: Robustness of the nonlinear climateresponse to ENSO’s extreme phases. J. Climate, 14, 1277–1293.Horel, J. D., and J. M. Wallace, 1981: Planetary-scale atmosphericphenomena associated with the Southern Oscillation. Mon.Wea. Rev., 109, 813–829.Hoskins, B., and D. Karoly, 1981: The steady linear response of aspherical atmosphere to thermal and orographic forcing. J.Atmos. Sci., 38, 1179–1196.Hsieh, W. W., 2001: Nonlinear canonical correlation analysis ofthe tropical Pacific climate variability using a neural networkapproach. J. Climate, 14, 2528–2539.——, 2004: Nonlinear multivariate and time series analysis byneural network method. Rev. Geophys., 42, RG1003,doi:10.1029/2002RG000112.——, and B. Tang, 1998: Applying neural network models to pre-diction and data analysis in meteorology and oceanography.Bull. Amer. Meteor. Soc., 79, 1855–1870.Lin, H., and J. Derome, 2004: Nonlinearity of the extratropicalresponse to tropical forcing. J. Climate, 17, 2597–2608.Livezey, R. E., A. Leetmaa, M. Mautani, H. Rui, M. 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