Adaptive and local analysis of climate data

-0-Adaptive and local analysis ofclimate dataWu Zhaohua'，Huang Norden E，Wallace John M’(1. Department of Earth, Ocean and Atmospheric Science, Florida State University , Tallahassee,Florida 32306, USA; 2. First Institute of Oceanography, SOA, Qingdao, Shandong 266061, China; .3. Department of Atmospheric Sciences , University of Washington, Seattle , Washington 98195, USA)Abstract: This paper focuses on how to extract physically meaningful information from climate data, with em-phases placed on adaptive and local analysis. It is argued that many traditional statistical analysis methods withrigorous mathematical footing may not be efficient in extracting essential physical information from climate da-ta; rather, adaptive and local analysis methods that agree well with fundamental physical principles are more ca-pable of capturing key information of climate data. To ilustrate the improved power of adaptive and local analy-sis of climate data, we also introduce briefly the empirical mode decomposition and its later developments.Key words: statistical analysis of data; adaptive analysis of data; locality of analysis; empirical mode decom-position; ensemble empirical mode decomposition; global warming trend; multidecadal climate variabilitydecadal timescale, we are facing the task of under-1 Introductionstanding the“hiatus”of global warming in the last 15The ultimate goals of climate data analysis are toyears. Numerous previous studies，including those re-extract hidden coherent physical and dynamical infor- ferred in Wu et al ", reached dramatically differentmation in climate data collcted from observation, ex- conclusions on the origin of the“hiatus” from diffe-perimentation and modeling，and to improve our unrent analyses of the same or different data using diffe-derstanding of climate system evolution and predictrent analysis methods. A reconciliation of these con-future climate. The essential information obtained clusions through improved analysis of climate data isthrough climate data analysis serves to the formationurgently needed. This improvement can come fromof various hypotheses and theories of climate systemtwo aspects: a. adopt adequate analysis methods toevolution and provides justifications for the correct- analyze data of different characteristics; b. developness of these hypotheses and theories. This goal calls new methods if current analysis methods are insuffi-for the focus of climate data analysis being placed oncient.extracting key physical information instead of obtai-3 Adaptive and local analysis of climatening mathematical or statistical quantities from data.data2 Current challengesClimate data are often nonstationary, reflectingThis workshop devoted to the understanding ofnumerous nonlinear interactions and nonstationary ex-the ocean' s role in multidecadal climate variability. .ternal forcings associated with climate system evolu-The earth' s climate system contains many interactingtion. In addition to that, these data are often collectedcomponents，such as atmosphere, ocean and land,in a noisy environment and hence contain noise.etc, and is subjected to nonstationary natural and an-When such data are analyzed, the problems of nonlin-thropological external forcings, resulting in its varia-earity, nonstationarity and noise need to be properlybility and changes of numerous timescales. On multi-handled. In these situations, the adequacy of a dataReeived 18 December 2013中国煤化工MYHCNMH G .36 ENGINEERING SCIENCES-φ--0-ding the assumptions behind a method and its advan- system cannot alter the reality that has already hap-tage and disadvantage becomes important.pened. Suppose that the evolution of a physical sys-In common practces of climate data analysis, atem involves nonlinear and nonstationary change, thefocus is often placed on understanding variability and physics behind that evolution in a given period maychanges of a particular timescale, such as the multi-be unique to that period. The derived physical inter-decadal timescale mentioned above. In such practi-pretation based on the analysis of the data should alsoces，a decomposition of climate data based on its be unique for that period if it reflects the true physicstimescales is often involved. When a decomposition behind a system evolution, unless the physics repeatsmethod is applied to data, often, there are hidden as-again and again or remains the same. From this argu-sumptions of data to be analyzed. For example, Fouri-ment，one can continue to infer that the analysis ofer transform assumes that the future evolution willre- the data extended to the prior or/and later periodspeat the past. This assumption is usually not consis- should not change that physical interpretation of thetent with the reality. In addition to that, when a selec-tion of a complete and orthogonal set to decompose cess cannot be determined due to the shortness of thedata is made，the preference of one complete and or- data of that period. In this sense, an analysis methodthogonal set (such as trigonometric functions) to oth-must satisfy the“ temporal locality” requirementers (wavelets) is not based on physical reasons, for based on the physical intuition and it is expected thatthe suitability of a particular set is not a known a such a method would provide a better chance to re-priori. In this sense, the selection of a complete andveal true physics. More detailed arguments can beorthogonal set is quite subjective. Often, different se- found in Wuet al I5].lections lead to different physical interpretations of4 Empirical mode decomposition and itsthe data.ensemble versiondata determine the basis that is used to decomposeThe empirical mode decomposition (EMD) wasthemselves，i.e.， using adaptive basis. There are adeveloped to meet the requirements of adaptive andfew approaches that have already contributed to thelocality in later 20th century间Using“natural wave-understanding of many important climate phenomena.forms”and amplitude-frequency modulated pure OS-For example, empirical orthogonal function (EOF)cillatory components definedanalysis was introduced and applied to extract climateany one-dimensional series is decomposed into a summodes [2However, to define adaptive basis of data,of oscillatory components of different timescales.some a priori determined constraints need to be ap-EMD is intuitively simple: Suppose there is a time se-plied. In EOF，the covariance matrix of temporal-ries composed of two mono- components of differentspatial data is used to obtain the orthogonalWhile EOF analysis avoids using a priori basis, it in-frequencies and amplitudes at any time, the time se-ries can be essentially recognized as the high frequen-vokes a new mathematical (but not physical) quanti-cy component riding on the low frequency compo-ty, the covariance matrix, to determine a completenent. In such a time series, if all the maxima and min-and orthogonal basis ，in which physical consid-ima are connected with smooth lines (called uppererations are not integrated into the determination ofand lower envelopes， respectively) separately， onethe posteriori basis. Since covariance matrix is de-may intuitively find that the mean of these two enve-rived from data over the global temporal/spatial dolopes at any temporal location is almost identical tomain, often, the results from EOF analysis are quitethe lower frequency component. Therefore ，subtrac-sensitive to the selections of spatial or/and temporalting the mean of the envelopes from the overall signaldomain. With above concerns in mind, a natural ques-will naturally isolate the riding wave of high frequen-tion following is whether it is possible to define adap-cy from the low frequency wave on which it is riding.tive basis that does not invoke seemingly unreaso-In this way, waves of high frequency and low fre-nable mathematical constraints for scientific data anal-quency are naturally separated in terms of naturalysis.vaves. For more complicated signal x(t) ，such aA related question is whether we can integratephysical constraint into a data analysis method. As weseparation process中国煤化i:vel toTYHCNMHGVol. 12 No.2,Apr. 2014 37-φ--0-extracriding waves from high frequenciesfrequencies, i.e.， x(t) can be expressed asfore，the added noise collates the portion of the sig-nal of comparable scale in one IMF, significantly re-x()= Re|Sa,()e +r。(1) ducing the chance of mode mixing and leading to the :where，Re[] represents the real part of“.”; a,(1)stability of decomposition. As the EMD is a timeand w(t) are instantaneous amplitude and frequencyspace analysis method, the white noise is averagedof jth riding wave components , formally called intrin-out with sufficient number of trials. In this sense, thesic mode fiunction (IMF), respectively; e is the baseadded noise plays a role in mimicking multiple obser-vations of a phenomenon recorded by a single obser-of natural logarithmic function; i=、FI ; ro is the re-vation and serves as a“ catalyst”in the decompositionmainder of x(t) after n riding waves are extracted.that leads to stable and more physically interpretableFrom the procedures described above, it is clearresults. An EEMD example of the global- mean sur-that EMD does not invoke any significant mathemati-face temperature (GST) anomaly is displayed in Fig.l.cal constraints such as a priori selected functions .(e.g. trigonomtric functions in Fourier transform and5 Trend and multidecadal variabilitymother wavelet in wavelet analysis), and its basic in-The partitioning between cycles and time-gredient is the natural wave forms obtained adaptive-varying secular trend of a time series and the interpre-ly from data themselves. In this sense, the EMD is a .tation of time-varying trends have long been dauntingmethod that decomposes the complicated signal inproblems，as exemplified by the statement of Stockterms of the basic ingredients of the physical world.and Watson :“ One economist’s‘trend’ can beThe inclusion of amplitude and frequency modulationanother's‘ cycle’.”The most widely used method ofin a component provides capability of reflecting the determining the trend in a dataset is to draw the leastcomplication of the physical world caused by nonlin-squares best fit straight lines as adopted in the Fourthear interactions and nonstationary external forcings.Asessment Report (AR4) of the Intergovermmental .In addition, the locality of EMD satisfies a fundamen-Panel on Climate Change (IPCC). In reality, the ratetal physical principle: If a component extracted fromof increase of GST in response to the cumulativethe data does reflect the physical processes operatingbuildup of long-lived greenhouse gases and the chan+at a given time, then they should be temporally localging rates of emission of aerosols are time-dependent.quantities and the corresponding physical interpreta-Representing secular trends in GST in terms of lineartion within specified time intervals should also nottrends is ad hoc and not necessarily physically realis-change with the addition of new data， because thetic. A more informative representation is an intrinsi-subsequent evolution of a physical system cannot al-cally- determined monotonic curve， having at mostter the reality that has already happened.one extremum within a given time span ..In practice, EMD uses the extrema informationThe EMD/EEMD provides a mean to naturallyto separate the riding natural wave and its reference,separate GST in terms of oscillatory components ofand the extracted natural waves are often highly sensi-different timescales and a time- varying secular trend.tive to any noise. To overcome this drawback, ensem-As shown in Fig.2, the resulted multidecadal compo-ble empirical mode decomposition (EEMD)7.8， anent and time- varying trend fit extremely well to thenoise assisted data analysis method, was developed.raw GST time series. As proved by Wu et al "，the .In EEMD，the decomposition consists of sifting anmultidecadal component and the secular trend indeedensemble of white noise- added data and treats thereflected the physics of the global climate change, es-mean of the corresponding components from thepecially accelerated warming associated with accele-EMD of data with adding dfferent realizations of rated greenhouse gas accumulation in the earth' sat-noise as the final result. The effect of the added whitemosphere.noise in EEMD is to provide a dyadic filtering refe-中国煤化工YHCNMH G .38 ENGINEERING SCIENCES-φ--0-wwwBRnfumhmwWWwWWwwwinfuumnrwninR1850190019501900，19502000(a) GST and residuals(b) EEMD componcntsFig.1 Components (Cr~Cx and Rs) of GST anomaly decomposed using EEMDries in the frequency domain. Part II : Application to the study of0.8- - - Raw GST tine seriestropical wave diturbances [D]. Journal of Applied Meteorology,1972, 11(6): 893-900.0.6-- The sum of secular trendand multidecadal component4] Thompson David W J, Wallace John M. Annular modes in the ex-tratropical circulation. Part I: Month-to-month variability [J] Jour-ρ0.2nal of Climate, 2000, 13(5): 100-1016.[5] Wu Zhaohua, Huang Norden E, Chen Xianyao. Some conside-0-0.2rations on physical analysis of data [J]. Advances in Adaptive Da--04Hta Analysis, 2011, 3(1/2): 95-113.[6] Huang Norden E, Shen Zheng, Long Steven R, et al. The empiri--0.6fcal mode decomposition method and the Hilbert spectrum for non--0.8-linear and non- stationary time series analysis [J]. Proceedings ofthe Royal Society A, 1998, 454(1971): 903-995.[7] Huang Norden E, Wu Zhaohua. A review on Hilbert Huang trans-Fig.2 Reconstruction of the raw GST time series usingform: Method and its aplicaions to geophysical studies [I]. Re~views of Geophysics, 2008, 46(2): 1-23.secular trend only and the sum of secular trend and[8] Wu Zhaohua, Huang Norden E. Ensemble empirical mode de-multidecadal componentcomposition: A noise-assisted data analysis method [J]. AdvancesNote:Adopted from Wu et al (2011)in Adaptive Data Analysis, 2009, 1(1): 1-41.[9] Stock James H，Watson Mark W. Variable trends in economicReferencestime series 小. Journal of Economic Perspectives, 1988, 2(3):[1] Wu Zhaohua, Huang Norden E, Wallace John M, et al. On the147-174.time- varying trend in global- mean surface temperature 小Cli-[10] Wu Zhaohua, Huang Norden E, Long Steven R, et al. On themate Dynamics, 2011, 37(3/4): 759-773.trend, detrending and variability of nonlinear and nonstationary[2] Wallace John M, Dickinson Robert E. Empirical orthogonal repre-time series [J]. Proceedings of the National Academy of Scien-sentation of time series in the frequency domain. Part I: Theoreti-ces of the United States of America, 2007， 104(38): 14889-cal considerations [D]. Journal of Applied Meteorology, 1972, 1114894.(): 887-892.[3] Wallace John M. Empirical orthogonal representation of time se-中国煤化工MHCNMHG"Vol. 12 No.2,Apr. 2014 39-φ--0-AuthorWu Zhaohua, male, born in 1966 in Fenghua, Zhejiang, earned his BS degree in Atmospheric Sciences atNanjing University and PhD at the University of Washington, USA. In January 2009, he joined the Departmentof Meteorology (currently part of the Department of Earth, Ocean and Atmospheric Science) of the Florida StateUniversity as an assistant professor. He will become a tenured associated professor in August 2014. Dr. Wuworks in two interwoven scientific fronts: atmospheric/climate dynamics and data analysis method develop-ment, with his developed methods being applied to numerous fields of natural, social and medical sciences, aswell as to many branches of engineering. Dr. Wu is currently an editor of Journal of the Atmospheric Sciences ofthe American Meteorological Society and one of the founding editors of journal Advances in Adaptive Data Anal-ysis published by World Scientific. His accomplishment and impact in scientific research can be found at http://scholar.google.com/citations?user=fquaTrMAAAAJ&hl=en&oi=ao. He can be reached by E-mail: zwu@fsu.eduFoundation item: US National Science Foundation Grant (No. AGS- 1139479)(cont. from p.35)[30] Large W G, Yeager S G. On the observed trends and changes innal of Climate, 2012, 25: 6123-6135.global sea surface temperature and air- sea heat fluxes [J]. Jour-Drijfhout Sybrens S, graduated from the University of Utrecht Netherlands). At present he is a professor ofclimate physics at the University of Southampton. He still works for 1 day a week at the Royal Netherlands Mete-orological Institute. His research has focused on the role of the ocean in climate, with emphasis on the AMOC.He has published more than 80 papers in international journals， about 50 over the last 10 years. He can bereached by E-mail: S.S.Drijfhout@soton.ac.uk中国煤化工YHCNMH G .40 ENGINEERING SCIENCES-φ-