The nonlinear relationship between summer precipitation in China and the sea surface temperature in preceding seasons: A statistical demonstration

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1 PUBLICATIONS Journal of Geophysical Research: Atmospheres RESEARCH ARTICLE Key Points: Nonlinear relation has been found between seasonal rainfall and concurrent SST We focus on seasonal rainfall and preceding SST Statisticalmethodisusedto demonstrate the lagged and nonlinear relations Correspondence to: E. Lu, Citation: Lu, E., et al. (2015), The nonlinear relationship between summer precipitation in China and the sea surface temperature in preceding seasons: A statistical demonstration, J. Geophys. Res. Atmos., 120, 12,027 12,036, doi: / 2015JD Received 5 AUG 2015 Accepted 17 NOV 2015 Accepted article online 19 NOV 2015 Published online 14 DEC 2015 The nonlinear relationship between summer precipitation in China and the sea surface temperature in preceding seasons: A statistical demonstration Er Lu 1, Hongxing Chen 1,2, Juqing Tu 1, Jinbo Song 1, Xukai Zou 3, Bing Zhou 3, Hui Li 1, Wenyue Cai 3, Yun Chen 4, Xianyan Chen 3, Qiang Zhang 3, Haishan Chen 1, and Zhihong Jiang 1 1 Key Laboratory of Meteorological Disaster of Ministry of Education, Nanjing University of Information Science and Technology, Nanjing, China, 2 Meteorological Bureau of Guangdong Province, Guangzhou, China, 3 National Climate Center, CMA, Beijing, China, 4 National Meteorological Center, CMA, Beijing, China Abstract Previous studies show that the seasonal precipitation over land may have strong nonlinear relationships with the concurrent sea surface temperature (SST). In this study, we demonstrate that summer precipitation also has a strong nonlinear relationship with preceding SSTs, which are more robust than the linear relationships. A strategy is employed to demonstrate the robustness of the nonlinear relation. With the 60 year observed precipitation and SST, we use data of 50 years to fit the linear and nonlinear rainfall-preceding SST relations. The SSTs used include all the samples that are from each of the global ocean grids over each of the preceding time periods considered. The fitted relations are then used to predict the rainfall of the remaining 10 years with the SST data. From all the predictions that use a large number of SST samples, we choose the ocean grid cell and the season that give the minimum prediction error. Results indicate that for an individual station, it may well be that the linear relation is better than the nonlinear relation. However, overall, the nonlinear method is better than the linear method as assessed from all stations in the country. This statistical analysis demonstrates that, when treated nonlinearly, the rainfall-preceding SST relationship can be much strengthened American Geophysical Union. All Rights Reserved. 1. Introduction Previous studies have demonstrated that monthly and seasonal rainfall over land is often driven by nonlinear responses to the sea surface temperature (SST) [e.g., Hoerling et al., 1997; Montroy et al., 1998; Hoerling et al., 2001; Su and Neelin, 2003; Wu and Hsieh, 2004; An and Jin, 2004; Lin and Derome, 2004; Wu et al., 2005; Wang and Hendon, 2005; Poweretal., 2006; Hsieh et al., 2006; Chung et al., 2014]. For example, Montroy et al. [1998] found nonlinearities in the monthly teleconnections between tropical Pacific SST anomalies and the precipitation over central and eastern North America. Wu et al. [2005] presented the nonlinear patterns of North American winter precipitation, which are associated with the El Niño Southern Oscillation (ENSO). Wang and Hendon [2005] showed that the rainfall of Australia has a nonlinear response to the SST during the ENSO. Hoerling et al. [2001] demonstrated the robustness of the nonlinear atmospheric response to the opposite phases of the ENSO. In particular, Hsieh et al. [2006] studied the nonlinear response of precipitation to the ENSO index and found that the nonlinear response is predominantly quadratic. They suggested that with a quadratic polynomial fit (i.e., y = ax 2 + bx + c), the spatial patterns of the nonlinear teleconnections can be well reflected. Furthermore, they found that nonlinear teleconnections can propagate perturbations over greater distances, into regions where the classical linear teleconnections are insignificant. It was therefore expected that the incorporation of the nonlinear teleconnections may lead to enhanced skills in the seasonal climate predictions. Different from a linear relation, the nonlinear relation between rainfall and SST implies that both very warm and very cold SSTs can lead to rainfall maxima (or minima). The rainfall-sst relations displayed in the above studies are all examples of concurrent relations. Since SSTs are required to be from preceding seasons for seasonal prediction of summer precipitation, we hope to know whether there is also a robust nonlinear relationship between the summer precipitation and the SST of the preceding seasons. The statistical methods currently used in the operational service in China for the seasonal prediction of precipitation with the SST [e.g., Zhao, 2001; Zhao and Liu, 2003; Shen, 2004; Chen, 2010] are LU ET AL. NONLINEAR RAINFALL-PRECEDING SST LINKAGE 12,027

2 Table 1. The Six Preceding Seasons Considered in This Study ID of Season Season 1 preceding spring (MAM) 2 preceding winter (DJF) 3 preceding fall (SON) 4 preceding winter-spring 5 preceding fall-winter 6 preceding fall-winter-spring almost all based on linear relations between precipitation and preceding SST [e.g., Wu and Yang, 1991; Zhang and Ju, 1999; Ge, 2001; Li and Su, 2005; Gong and He, 2006; Li et al., 2007; Liang et al., 2008; Wei and Wang, 2010; Zhou et al., 2011; Liu and Li, 2011; Cai et al., 2012; Chen et al., 2012]. If the nonlinear relation between summer rainfall and preceding SST is better than the linear relation, we should pay more attention to the nonlinearity in the seasonal prediction of summer rainfall. In previous studies, the preceding SSTs linked to the precipitation in China can be from the Pacific Ocean [e.g., Wang et al., 2002; Chen et al., 2005; Yao and Zhang, 2006; Chen, 2008; Liang et al., 2008], including the equatorial eastern Pacific[e.g.,Li and Shou, 2000], western Pacific[e.g.,Liu and Li, 2011], the East China Sea and its adjacent seas [e.g., Cai et al., 2012], and the North Pacific[e.g.,Sun and An, 2003; Li and Su, 2005; Wei and Wang, 2010; Zhou et al., 2011]. The SSTs can also be from the Indian Ocean [e.g., Wang et al., 1998; Li et al., 2007; Chen et al., 2012] and even the Atlantic Ocean [e.g., Bai, 2001]. These suggest that, in general, the summer precipitation in the country canbeaffected,moreorless,bythesststhatarefromalltheplacesintheglobaloceans[e.g.,yu et al., 2001]. The strategy we employ in this study to demonstrate the robustness of the nonlinearity in the lagged relation is to fit the linear and nonlinear relations with observed data of multiple years and do separate predictions with the relations and the observed data of additional years. The fittings and predictions are performed for the rainfall of each station. For a given station, the fitting and prediction are made with every available SST sample, which comes from all the global ocean grids and all the preceding seasons. The seasons considered include the preceding spring (March-April-May (MAM)), winter (December-January-February (DJF)), fall (September-October-November (SON)), and their consecutive combinations (Table 1). The linear and nonlinear predictions with the smallest prediction error are then compared, with the same rule for comparison, based on various considerations. The data used in this study are introduced in section 2. The detailed methodologies, including the fittings, predictions, and the comparisons, are presented in section 3. In section 4, the nonlinearity in the lagged relation is demonstrated through comparing the best results of the two methods. Summary and discussion are given in section Data The precipitation data used in this study are the daily observation of the 160 stations in China during the 60 years from 1951 to 2010, obtained from the National Climate Center (NCC) of China ( index.php?channelid=43&wchid=5). The summer (June-July-August (JJA)) rainfall is the summation of the daily data. The locations of the 160 stations and their station numbers used in this study are shown in Figure 1a. The sea surface temperature used in the study is taken from the NOAA Extended Reconstructed Sea Surface Temperature (ERSST) v3b [Xue et al., 2003; Smith et al., 2008], a global monthly sea surface temperature analysis starting from 1854 with a horizontal resolution of 2 2. We use the data from 1950 to Methodologies 3.1. Fitting the Relations With Data of 50 Years In each test that uses the data of the 60 years, we choose five decades as the training (fitting) period and the remaining one decade as the verification (prediction) period. The observed rainfall and SST of the 50 years are used to fit the linear and nonlinear relations. In this study, similar to Hsieh et al. [2006], we also use quadratic polynomial fit toreflect the nonlinearity. The quadratic relation can be expressed as P ¼ at 2 þ bt þ c; (1) where P is the summer precipitation and T is the SST of a preceding season. Note that instead of using the P and T, we may also use their anomalies to establish a quadratic relation. LU ET AL. NONLINEAR RAINFALL-PRECEDING SST LINKAGE 12,028

3 Figure 1. (a) The locations of the 160 stations in China and the station numbers (i) used in this study and (b) the climatic mean summer (JJA) rainfall (unit: cm) of the 60 years from 1951 to Denote the observed precipitation as P(i, n), where i is the station number (varying from 1 to 160) and n is the year (from 1 to 50). The SST can be denoted as T(k, j, n), where j is the ocean grid number (from 1 to J, the number of the total available ocean grids, or part of them based on the significance of the fittings, as will be mentioned below) and k is the number for the preceding seasons (from 1 to 6). With the data of P(i, n) and T(k, j, n), the coefficients a, b, and c regressed from equation (1) are thus functions of station i, preceding season k, and ocean grid j and can be denoted, respectively, as ai; ð k; jþ; bi; ð k; jþ; and ci; ð k; jþ: (2) 3.2. Predicting Rainfall of Remaining Years With Relations Fitted The regressed relation (i.e., the three coefficients) based on the data of the 50 years is used to predict the precipitation of each of the remaining 10 years (the verification period). The predicted precipitation (ep) is thus a function of station i, preceding season k, ocean grid j, and year m (from 1 to 10) and can be expressed as ep ði; k; j; m Þ ¼ ai; ð k; jþ½tk; ð j; mþš 2 þ bi; ð k; jþtk; ð j; mþþci; ð k; jþ: (3) LU ET AL. NONLINEAR RAINFALL-PRECEDING SST LINKAGE 12,029

4 3.3. Finding Best Results From Predictions That Use Numerous SST Samples The mean absolute error of the rainfall prediction is computed using the observations from the 10 years of data not included in the training group. We then find the best prediction result (i.e., the one that has the least mean error), from the predictions that use a large number of the SST samples. In other words, the task here is to find the best result, no matter where the SST is from, over the global oceans, and which season it is from, in all the six preceding ones. The least error can be expressed as k¼1;6 Bi ðþ¼ Min Ave ep ði; k; j; mþ Pi; ð mþ : (4) j¼1;j m¼1;10 The fitting and prediction of the linear method can be done similarly. In equation (1), let a = 0, then P = bt + c. With the observed data of P(i, n) and T(k, j, n), we can determine the coefficients, as in equation (2), of b(i, k, j) and c(i, k, j). Following the steps in equations (3) and (4), we finally obtain the least error of the linear method, which is denoted as B 0 (i) Comparing the Best Results of the Linear and Nonlinear Methods For a given station i, we compute the difference metric Di ðþ¼b 0 ðþ Bi i ðþ (5) to examine whether the least error of the nonlinear method is smaller than that of the linear method. The relative difference of the least error between the linear and nonlinear methods can be assessed with Ri ðþ¼di ðþ=b 0 ðþ: i (6) To eliminate the possible randomness, in addition to the least error, we may also take the next several minimum errors and compare each of them between the linear and nonlinear methods. For example, for both the two methods, we may remove their first five best results and compare their sixth best results. The difference of the corresponding error between the linear and nonlinear methods is denoted as D 6 (i). 4. Comparing the Best Prediction Results of the Linear and Nonlinear Methods 4.1. The Best Results of the Two Methods To assess the prediction error of summer rainfall, the distribution of the observed climatic mean of summer precipitation is provided in Figure 1b. West China is climatically dry. In East China, summer rainfall decreases from south to north. Floods occur frequently in the regions from south China to the Yangtze River basin and occur from time to time in north China and the northeast. The data of 50 years are used to fit the relations, and the data of the other 10 years are used to verify the relations. We first use all SST samples, which include the SST in every available ocean grid, to do the fitting and prediction, and calculate the B 0 (i) and B(i), the 10 year mean least error of the predicted precipitation from the observation by use of the linear and nonlinear methods. With all the grids in the global oceans and the six preceding seasons, there are totally about 60,000 SST samples. Considering that for many SST samples, among all the samples, the precipitation of the 50 years fitted with the SST may not be significantly correlated with the observation, we remove these SST samples and use the remaining SST samples, with which the correlation of the fitted precipitation with the observation can be significant at a 95% confidence level with the Student s t test, to predict the rainfall of the 10 years. In the linear fitting, there are only ,000 significant SST samples, 10 20% ofthetotal, formostofthe stations. By contrast, in the nonlinear fitting, there are 12,000 24,000 significant SST samples, which are 20 40% of the total, for most of the stations (figures not shown). So the number of the significant SST samples for the nonlinear fitting is much greater than the number of the samples for the linear fitting, an indication that the nonlinear rainfall-sst relation is better than the linear rainfall-sst relation. Using the significant SST samples, the least errors of the predicted precipitation from the observations, the B 0 (i)and B(i), for the linear and nonlinear methods, are calculated. Figure 2 shows the distribution of the B(i), the result of the nonlinear method, with the first 50 years being the training period and the later 10 years being the verification period. The result of the linear method, the B 0 (i) (figure not shown), is compared to the climatic summer precipitation (Figure 1b). The comparison shows that although it is the least error obtained LU ET AL. NONLINEAR RAINFALL-PRECEDING SST LINKAGE 12,030

5 Figure 2. The distribution of B(i), the 10 year ( ) averaged least error (mm) of the nonlinearly predicted rainfall from the observation, found from the predictions that use significant SST samples. from a large number of SST samples, the error of the linearly predicted precipitation is still 20 30% of the climatic mean summer rainfall for most of the stations, suggesting that there is a room for the linear method to improve A Comparison Between the Best Results of the Two Methods With the least errors B 0 (i) and B(i) being obtained (for the fittings of the first 50 years and predictions of the later 10 years), we now examine the D(i), the difference between the linear and nonlinear methods. The differences corresponding to all samples and significant samples are calculated, and comparison shows that their spatial distributions are similar in characteristic. Figure 3 presents the difference from the significant samples. There are 19 stations where the best result of the linear method outperforms the best result of the nonlinear method (negative difference, stations marked in blue). Nevertheless, for the rest of the 141 stations, the differences are positive (in red). This suggests that, overall, the best prediction results of the nonlinear method are better than that of the linear method. Figure 4 shows the distribution of the R(i), the relative difference of least error between the linear and nonlinear methods that use significant SST samples. Relative to the linear method, the nonlinear method can reduce the least error by about 20 30% around the Yangtze River basin and by up to 45% in a station located in the middle of the Yangtze River basin. To be sound in statistics (avoid the possible randomness), in addition to the least error, a few of the best results of the two methods are also compared. Figure 5 shows the distribution of the D 6 (i), the difference of the sixth minimum errors in the linear and nonlinear predictions that use significant SST samples. It suggests that after removing their first five best results from the two methods, the result of the nonlinear method is still better than that of the linear method. Comparisons are also made for other best results, with the number of the best results removed varying from 1 to 10, between the two methods. These comparisons all indicate that overall, assessed from all the stations, the best results of the nonlinear method are better than that of the linear method Further Comparisons Between the Best Results of the Two Methods In order to avoid the arbitrariness in selecting the training and verification periods, we may alternatively choose to predict the precipitation of the 10 years in other decades, with the relations fitted from the data LU ET AL. NONLINEAR RAINFALL-PRECEDING SST LINKAGE 12,031

6 Figure 3. The distribution of D(i), the 10 year ( ) averaged difference of least error (mm) between the linear and nonlinear methods, from predictions that use significant SST samples. of the corresponding remaining 50 years. Table 2 shows the 10 year period used for the predictions and the corresponding number and percentage of the stations where the best result (indicated by the least prediction error) of the nonlinear method is better than that of the linear method. In the test described above, with predictions being made for the 10 years of the sixth decade ( ), the result of the nonlinear method is better than that of the linear method for 141 out of the 160 stations (accounting for 88.1%). In the four tests with predictions being made for the 10 years of the second, third, fourth, and fifth decades, the number (percentage) of the station where the nonlinear method outperforms the linear method is 151 (94.4%), 154 (96.3%), 158 (98.8%), and 158 (98.8%), respectively, and thus, the performances of the nonlinear method are also better than that of the linear method. In the test with prediction being made for the 10 years of the first decade ( ), the performance of the nonlinear method is relatively poor but there are still more than half of the stations exhibiting better results. The reason of the performance of this test will be investigated. Nevertheless, we can still reach the conclusion, from this consideration, that overall, the nonlinear relation is stronger than the linear relation. In sum, to be robust for the demonstration of the nonlinearity, various aspects have been considered in the above comparisons, which are based on the fair rule to compare both the best results of the linear and nonlinear methods, the two different approaches. From the results of these comparisons, we can conclude with confidence that overall (as assessed statistically from the 160 stations), the summer rainfall in China possesses better nonlinear relation than linear relation with the preceding SST. 5. Summary and Discussion Studies on the rainfall over North America and Australia showed that the precipitation may have a robust nonlinear response to the concurrent SST, and the relation is predominantly quadratic. The goal of this study is to investigate whether, when we link the summer rainfall in China to preceding SST, the rainfall and SST can also possess a robust nonlinear relation. The main task of this study is to develop a strategy to demonstrate the robustness of the nonlinearity in the rainfall-preceding SST relation. As suggested by the previous studies on summer precipitation in China, the rainfall can be affected, more or less, by the SSTs that are from all the places in the global oceans during all the preceding seasons. In this LU ET AL. NONLINEAR RAINFALL-PRECEDING SST LINKAGE 12,032

7 Figure 4. Same as Figure 3 except for R(i), the 10 year ( ) averaged relative difference of least error (%) between the linear and nonlinear methods. study, we use all the SST samples that are from each of the global ocean grids during each of the preceding seasons, as well as part of the samples with which the rainfall fitted can be significantly correlated with the observed, to compare the overall behaviors of the linear and nonlinear methods. The same game rule is used for the comparisons. That is, we compare both the best results of the linear and nonlinear methods. These best results are chosen from the predictions that use all the SST samples or the Figure 5. Same as Figure 3 except for D 6 (i), the 10 year ( ) averaged difference of the sixth best results of the two methods. LU ET AL. NONLINEAR RAINFALL-PRECEDING SST LINKAGE 12,033

8 Table 2. The 10 Year Period Used for the Predictions (With the Remaining 50 Years Being Used for the Fittings) and the Corresponding Number and Percentage (Over the Total 160 Stations) of the Stations Where the Averaged Prediction Error of the Nonlinear Method Is Smaller Than That of the Linear Method 10 Year Period Number of stations Percentage of stations significant samples. With the data of 60 years, we predict the rainfall of the 10 years in one of the six decades with the lagged relations fitted from the data of the remaining 50 years. The averaged least errors, as well as the first a few minimum errors, are compared between the linear and nonlinear methods. Since the training and verification are over different time periods, the performance of the nonlinear relation may not necessarily be better than that of the linear relation. For an individual station, it is possible that the linear method can provide a better result than the nonlinear method, as illustrated by the stations marked blue in Figures 3 and 5. Nevertheless, overall, in most of the stations (marked red in the plots) from all the 160 in the country, the results of the nonlinear method are better than that of the linear method. Therefore, we may conclude that through considering the nonlinearity, the rainfall-preceding SST relationship can be strengthened. The major characteristic of the nonlinear relation we emphasize in this study is that both very warm and very cold SST can lead to the rainfall maxima (or minima). This is different from the linear relation. We thus can simply use quadratic polynomial to reflect the nonlinearity. Hsieh et al. [2006] also used quadratic fit to stress nonlinearity in their study. Such nonlinear characteristic widely exists in the climate. For example, Hu et al. [2012] pointed out that both too large and too small thermocline slopes as well as too strong and too weak wind stress may lead to the suppression in the ENSO variability. As pointed out in section 1, the prediction performed in the present study is just a strategy to demonstrate the nonlinearity. It can be expected that through taking into account the nonlinearity characteristic revealed in this work, the skill of the seasonal prediction of summer rainfall may be enhanced. Special methods for predicting the summer rainfall with the preceding SST will be developed in our next study. For the purpose of including all the possibilities, the SSTs used in this study are taken two-dimensionally from both space (ocean grids) and time (preceding seasons). Thus, for adjacent precipitation stations, the SSTs that Figure 6. The ocean grid and preceding season of the significant SST sample used to gain the least prediction errors in the nonlinear fitting ( ) and prediction ( ) of the precipitation of each station. The number in the plot is the station number (i, see Figure 1a). The position of the number indicates the location of the SST. The color of the number reflects the preceding season (see Table 1). LU ET AL. NONLINEAR RAINFALL-PRECEDING SST LINKAGE 12,034

9 Figure 7. The scatterplots of the summer rainfall in Lijiang station (i = 115) and the SST in preceding fall-winter-spring at ocean grid (148 E, 32 N), which is used in the nonlinear fit. (a) The quadratic curve is fitted with the 50 ( ) samples during the fitting period (blue). (b) This fitted curve is used to illustrate whether and how the 10 ( ) samples during the prediction period (red) can follow the relation. are finally chosen to gain the least prediction errors may come, in a scattered fashion, from different ocean regions and different preceding seasons. Figure 6 shows the ocean grid and the preceding season of the SST finally used (to gain the least prediction errors) in the nonlinear fitting (from the first 50 years) and prediction (for the last 10 years) of the precipitation of each station. It is interesting that with the nonlinear method, there are only a few SSTs that are from the tropical oceans. Most of the SSTs used are over the ocean areas along 30 N and 30 S as well as the higher latitudes. In the operational predictions, we may consider a fixed preceding season and use the SST averaged over a certain ocean region. Figure 7 shows an example (station Lijiang) of the relation between the rainfall and the SST, which is finally used in the nonlinear method, during the training and verification periods. Figure 7a suggests that although the summer rainfall and the SST in preceding period (fall-winter-spring) during the fitting period are not linearly correlated at a 95% confidence level with the t test, they can be significantly correlated when using the quadratic fit. Figure 7b indicates that the rainfall and the SST during the prediction period can well follow the nonlinear relation fitted. As mentioned above, the purpose of this study is to demonstrate the nonlinearity in the rainfall-preceding SST relation with data and the strategy designed. As to why the very warm and the very cold preceding SSTs can both lead to the heaviest (or the very light) summer rainfall, we need to investigate the physical mechanisms in the atmosphere-ocean system and explore how the preceding SST can gradually affect the atmospheric circulation and finally the summer rainfall. The present study is statistically for all the stations, not for a specific station (or region). However, the analysis of the mechanism, responsible for the nonlinear lagged linkages, needs to be performed individually for each of the stations (or regions) we concern. Recently, we examined the atmospheric and SST anomalies associated with the 2011 severe drought over the Yangtze River basin [Lu et al., 2014]. Our preliminary results from the numerical experiments indicate that the rainfall over this basin has nonlinear responses to the preceding SST anomalies over certain ocean regions. Statement of Data Access The precipitation data used in this study can be obtained from the National Climate Center (NCC) of China ( The sea surface temperature used in the study can be obtained from the NOAA Extended Reconstructed Sea Surface Temperature (ERSST) v3b ( There is no restriction on the data access. LU ET AL. NONLINEAR RAINFALL-PRECEDING SST LINKAGE 12,035

10 Acknowledgments This study was supported by the China Special Fund for Meteorological Research in the Public Interest (Major projects) (grant GYHY ), the National Basic Research (973) Program of China (grant 2012CB955900), the National Natural Science Foundation of China (grants , , and ), the Sino-US Center for Weather & Climate Extremes at Nanjing University of Information Science and Technology, and the Priority Academic Program Development of Jiangsu Higher Education Institutions. The anonymous reviewers and Ruby Leung, the Editor, are thanked for their constructive suggestions. The precipitation data used in this study were provided by the National Meteorological Center of China Meteorological Administration. The data of sea surface temperature were provided by the Extended Reconstructed Sea Surface Temperature (ERSST) of the National Oceanic and Atmospheric Administration (NOAA). References An, S. I., and F. F. Jin (2004), Nonlinearity and asymmetry of ENSO, J. 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Monthly, 37, LU ET AL. NONLINEAR RAINFALL-PRECEDING SST LINKAGE 12,036