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Testing a Four-Dimensional Variational Data Assimilation Method Using an Improved Intermediate Coupled Model for ENSO Analysis and Prediction


doi: 10.1007/s00376-016-5249-1

  • A four-dimensional variational (4D-Var) data assimilation method is implemented in an improved intermediate coupled model (ICM) of the tropical Pacific. A twin experiment is designed to evaluate the impact of the 4D-Var data assimilation algorithm on ENSO analysis and prediction based on the ICM. The model error is assumed to arise only from the parameter uncertainty. The "observation" of the SST anomaly, which is sampled from a "truth" model simulation that takes default parameter values and has Gaussian noise added, is directly assimilated into the assimilation model with its parameters set erroneously. Results show that 4D-Var effectively reduces the error of ENSO analysis and therefore improves the prediction skill of ENSO events compared with the non-assimilation case. These results provide a promising way for the ICM to achieve better real-time ENSO prediction.
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  • Balmaseda M. A., D. L. T. Anderson, and M. K. Davey, 1994: ENSO prediction using a dynamical ocean model coupled to statistical atmospheres. Tellus A, 46( 4), 497- 511.10.1034/j.1600-0870.1994.00012.xad425656b8c02ba4213874692a4d12f6http%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1034%2Fj.1600-0870.1994.00012.x%2Fabstracthttp://onlinelibrary.wiley.com/doi/10.1034/j.1600-0870.1994.00012.x/abstractABSTRACT The predictability of El Niño/Southern Oscillation (ENSO) events is addressed by means of statistical and dynamical schemes. The statistical schemes are based on principal oscillation pattern (POP) analysis of various observed and model ocean fields: these statistical predictions establish a lower limit for the predictability of such a system. For the dynamical predictions, an ocean model of intermediate complexity is coupled to several statistical surface wind stress models. In these coupled models, the atmospheric anomalies are a linear response to the oceanic fields: several combination of fields are considered, such as SST and heat content. The spatial features of predictability are discussed. Predictions seem to be better in the central Pacific. In the western and eastern Pacific, the predictability skill scores are poorer, possibly due to deficiencies in the ocean thermodynamics and in the coupling. The model predictions exhibit a pronounced seasonal dependence, with spring and summer being less predictable. Best results are obtained with seasonally-dependent predictors.
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    Bjerknes J., 1969: Atmospheric teleconnections from the equatorial Pacific. Mon. Wea. Rev., 97, 163- 172.10.1175/1520-0493(1969)097<0163:ATFTEP>2.3.CO;29f0298dd45e510c14c05703bfaea0d37http%3A%2F%2Fwww.rand.org%2Fpubs%2Fpapers%2FP3882.htmlhttp://www.rand.org/pubs/papers/P3882.htmlAbstract The “high index” response of the northeast Pacific westerlies to big positive anomalies of equatorial sea temperature, observed in the winter of 1957–58, has been found to repeat during the major equatorial sea temperature maxima in the winters of 1963–64 and 1965–66. The 1963 positive temperature anomaly started early enough to exert the analogous effect on the atmosphere of the south Indian Ocean during its winter season. The maxima of the sea temperature in the eastern and central equatorial Pacific occur as a result of anomalous weakening of the trade winds of the Southern Hemisphere with inherent weakening of the equatorial upwelling. These anomalies are shown to be closely tied to the “Southern Oscillation” of Sir Gilbert Walker.
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    Dommenget D., D. Stammer, 2004: Assessing ENSO simulations and predictions using adjoint ocean state estimation. J.Climate, 17( 22), 4301- 4315.10.1175/3211.18523642a3793eb91fd550539508fe52bhttp%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F2004JCli...17.4301Dhttp://adsabs.harvard.edu/abs/2004JCli...17.4301DSimulations and seasonal forecasts of tropical Pacific SST and subsurface fields that are based on the global Consortium for Estimating the Circulation and Climate of the Ocean (ECCO) ocean-state estimation procedure are investigated. As compared to similar results from a traditional ENSO simulation and forecast procedure, the hindcast of the constrained ocean state is significantly closer to observed surface and subsurface conditions. The skill of the 12-month lead SST forecast in the equatorial Pacific is comparable in both approaches. The optimization appears to have better skill in the SST anomaly correlations, suggesting that the initial ocean conditions and forcing corrections calculated by the ocean-state estimation do have a positive impact on the predictive skill. However, the optimized forecast skill is currently limited by the low quality of the statistical atmosphere. Progress is expected from optimizing a coupled model over a longer time interval with the coupling statistics being part of the control vector.
    Evensen G., 1994: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res., 99, 10 143- 10 162.10.1029/94JC00572ef8053eecb8d37c88057c7928546f3a5http%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1029%2F94JC00572%2Fcitedbyhttp://onlinelibrary.wiley.com/doi/10.1029/94JC00572/citedbyCiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): . A new sequential data assimilation method is discussed. It is based on forecasting the error statistics using Monte Carlo methods, a better alternative than solving the traditional and computationally extremely demanding approximate error covariance equation used in the extended Kalman filter. The unbounded error growth found in the extended Kalman filter, which is caused by an overly simplified closure in the error covariance equation, is completely eliminated. Open boundaries can be handled as long as the ocean model is well posed. Well-known numerical instabilities associated with the error covariance equation are avoided because storage and evolution of the error covariance matrix itself are not needed. The results are also better than what is provided by the extended Kalman filter since there is no closure problem and the quality of the forecast error statistics therefore improves. The method should be feasible also for more sophisticated primitive equation models. The computati...
    Galanti E., E. Tziperman, M. Harrison, A. Rosati, and Z. Sirkes, 2003: A study of ENSO prediction using a hybrid coupled model and the adjoint method for data assimilation. Mon. Wea. Rev., 131( 11), 2748- 2764.10.1175/1520-0493(2003)131<2748:ASOEPU>2.0.CO;29d8bcf9d-0573-49cd-b642-da998fc4dd6dccebf642d91bd910019a2920cf3664c2http%3A%2F%2Fconnection.ebscohost.com%2Fc%2Farticles%2F11280262%2Fstudy-enso-prediction-using-hybrid-coupled-model-adjoint-method-data-assimilationrefpaperuri:(8a2aadab711a909de65725230d4dc3c2)http://connection.ebscohost.com/c/articles/11280262/study-enso-prediction-using-hybrid-coupled-model-adjoint-method-data-assimilationAn experimental ENSO prediction system is presented, based on an ocean general circulation model (GCM) coupled to a statistical atmosphere and the adjoint method of 4D variational data assimilation. The adjoint method is used to initialize the coupled model, and predictions are performed for the period 1980-99. The coupled model is also initialized using two simpler assimilation techniques: forcing the ocean model with observed sea surface temperature and surface fluxes, and a 3D variational data assimilation (3DVAR) method, similar to that used by the National Centers for Environmental Prediction (NCEP) for operational ENSO prediction. The prediction skill of the coupled model initialized by the three assimilation methods is then analyzed and compared. The effect of the assimilation period used in the adjoint method is studied by using 3-, 6-, and 9-month assimilation periods. Finally, the possibility of assimilating only the anomalies with respect to observed climatology in order to circumvent systematic model biases is examined. It is found that the adjoint method does seem to have the potential for improving over simpler assimilation schemes. The improved skill is mainly at prediction intervals of more than 6 months, where the coupled model dynamics start to influence the model solution. At shorter prediction time intervals, the initialization using the forced ocean model or the 3DVAR may result in a better prediction skill. The assimilation of anomalies did not have a substantial effect on the prediction skill of the coupled model. This seems to indicate that in this model the climatology bias, which is compensated for by the anomaly assimilation, is less significant for the predictive skill than the bias in the model variability, which cannot be eliminated using the anomaly assimilation. Changing the optimization period from 6 to 3 to 9 months showed that the period of 6 months seems to be a near-optimal choice for this model.
    Han G. J., W. Li, Z. J. He, K. X. Liu, and J. R. Ma, 2006: Assimilated tidal results of tide gauge and TOPEX/POSEIDON data over the China seas using a variational adjoint approach with a nonlinear numerical model. Adv. Atmos. Sci.,23, 449-460, doi: 10.1007/s00376-006-0449-8.10.1007/s00376-006-0449-8869cea61dc14065bc241deda79959e53http%3A%2F%2Fwww.cnki.com.cn%2FArticle%2FCJFDTotal-DQJZ200603012.htmhttp://d.wanfangdata.com.cn/Periodical_dqkxjz-e200603012.aspxIn order to obtain an accurate tide description in the China Seas, the 2-dimensional nonlinear numerical Princeton Ocean Model (POM) is employed to incorporate in situ tidal measurements both from tide gauges and TOPEX/POSEIDON (T/P) derived datasets by means of the variational adjoint approach in such a way that unknown internal model parameters, bottom topography, friction coefficients and open boundary conditions, for example, are adjusted during the process. The numerical model is used as a forward model. After the along-track T/P data are processed, two classical methods, i.e. harmonic and response analysis, are implemented to estimate the tide from such datasets with a domain covering the model area extending from 0 to 41N in latitude and from 99E to 142E in longitude. And the results of these two methods are compared and interpreted. The numerical simulation is performed for 16 major constituents. In the data assimilation experiments, three types of unknown parameters (water depth, bottom friction and tidal open boundary conditions in the model equations) are chosen as control variables. Among the various types of data assimilation experiments, the calibration of water depth brings the most promising results. By comparing the results with selected tide gauge data, the average absolute errors are decreased from 7.9 cm to 6.8 cm for amplitude and from 13.0 to 9.0 for phase with respect to the semidiurnal tide M2 constituent, which is the largest tidal constituent in the model area. After the data assimilation experiment is performed, the comparison between model results and tide gauge observation for water levels shows that the RMS errors decrease by 9 cm for a total of 14 stations, mostly selected along the coast of Mainland China, when a one-month period is considered, and the correlation coefficients improve for most tidal stations among these stations.
    Han G. J., X. R. Wu, S. Q. Zhang, Z. Y. Liu, I. M. Navon, and W. Li, 2015: A study of coupling parameter estimation implemented by 4D-Var and EnKF with a simple coupled system. Advances in Meteorology ,2015, doi:10.1155/2015/530764.101fe06ebd928cd1dd194033abaa3400http%3A%2F%2Fdownloads.hindawi.com%2Fjournals%2Famete%2Faip%2F530764.pdfhttp://downloads.hindawi.com/journals/amete/aip/530764.pdf
    Houtekamer P. L., H. L. Mitchell, 1998: Data assimilation using an ensemble Kalman filter technique. Mon. Wea. Rev., 126( 3), 796- 811.d3f8c1aa877d5eac5d73e53c0fdac200http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F1998mwrv..126..796h/s?wd=paperuri%3A%28a746c15529ef81420dfd1d8ebc7aecc8%29&filter=sc_long_sign&tn=SE_xueshusource_2kduw22v&sc_vurl=http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F1998mwrv..126..796h&ie=utf-8&sc_us=8718907562637087200
    Kalnay E., 2003: Atmospheric Modeling,Data Assimilation and Predictability. Cambridge University Press, 342pp.10.1198/tech.2005.s32617d7fbb1594d0cade45315e4ab692760http%3A%2F%2Famstat.tandfonline.com%2Fdoi%2Fabs%2F10.1198%2Ftech.2005.s326%3FjournalCode%3Dutch20http://amstat.tandfonline.com/doi/abs/10.1198/tech.2005.s326?journalCode=utch20Atmospheric modeling, data assimilation and predictability Eugenia Kalnay Cambridge University Press, 2003 : hbk : pbk
    Keenlyside N., R. Kleeman, 2002: Annual cycle of equatorial zonal currents in the Pacific. J. Geophys. Res., 107( C8), 8- 1.10.1029/2000JC0007112aab7aa7e1ba6d8ebb183276b198bf4ahttp%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1029%2F2000JC000711%2Fabstracthttp://onlinelibrary.wiley.com/doi/10.1029/2000JC000711/abstractObservational (Tropical Atmosphere-Ocean array) data on the annual cycle of upper ocean zonal currents on the equator are analyzed using a simple dynamical ocean model in order to investigate underlying dynamics. The model, by treating linear and nonlinear terms semi-independently, allows a separation of various linear and nonlinear effects. The model focuses on linear dynamics of low-order baroclinic modes. By realistically simulating the vertical structure of annual cycle, the model shows that linear dynamics determines the vertical and meridional structure of the annual cycle. Nonlinearity is weak and only important in the undercurrent, where it provides a simple mechanism for the annual cycle: mean meridional advection of the annual cycle north of the equator onto the equator, with the boreal springtime surge in the undercurrent being a direct result of a surge centered at 2N. Model results show that annual variations in zonal currents are out of phase across the equator, surging in the corresponding spring. This behavior is a response to trade wind variations, which are also equatorially antisymmetric, and is generated by the second meridional mode Rossby wave.
    Keenlyside N., M. Latif, M. Botzet, J. Jungclaus, and U. Schulzweida, 2005: A coupled method for initializing El Niño Southern Oscillation forecasts using sea surface temperature. Tellus A, 57( 3), 340- 356.10.1111/j.1600-0870.2005.00107.x37496007-66d9-492d-a03b-06354e5d86533e30b3eff5293d5d3daaeb1ee9b760b6http%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1111%2Fj.1600-0870.2005.00107.x%2Ffullrefpaperuri:(007affb0aa8d24f5bb59c1f7fe1610d6)http://onlinelibrary.wiley.com/doi/10.1111/j.1600-0870.2005.00107.x/fullABSTRACT Top of page ABSTRACT 1.68Introduction 2.68Model description and experimental design 3.68Coupled ENSO variability 4.68Assimilation scheme 5.68Ensemble member generation 6.68Hindcast results 7.68Discussion and conclusions 8.68Acknowledgments References A simple method for initializing coupled general circulation models (CGCMs) using only sea surface temperature (SST) data is comprehensively tested in an extended set of ensemble hindcasts with the Max-Planck-Institute (MPI) climate model, MPI-OM/ECHAM5. In the scheme, initial conditions for both atmosphere and ocean are generated by running the coupled model with SST nudged strongly to observations. Air–sea interaction provides the mechanism through which SST influences the subsurface. Comparison with observations indicates that the scheme is performing well in the tropical Pacific. Results from a 500-yr control run show that the model's El Ni09o Southern Oscillation (ENSO) variability is quite realistic, in terms of strength, structure and period. The hindcasts performed were six months long, initiated four times per year, consisted of nine ensemble members, and covered the period 1969–2001. The ensemble was generated by only varying atmospheric initial conditions, which were sampled from the initialization run to capture intraseasonal variability. At six-month lead, the model is able to capture all the major ENSO extremes of the period. However, because of poor sampling of ocean initial conditions and model deficiencies, the ensemble-mean anomaly correlation skill for Ni09o3 SST is only 0.6 at six-month lead. None the less, the results presented here demonstrate the potential of such a simple scheme, and provide a simple method by which SST information may be better used in more complex initialization schemes.
    Kirtman B. P., S. E. Zebiak, 1997: ENSO simulation and prediction with a hybrid coupled model. Mon. Wea. Rev., 125( 10), 2620- 2641.10.1175/1520-0493(1997)125<2620:ESAPWA>2.0.CO;29ede8d553e559520af01c7256d92eb6bhttp%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F1997MWRv..125.2620Khttp://adsabs.harvard.edu/abs/1997MWRv..125.2620KNot Available
    Kleeman R., A. M. Moore, and N. R. Smith, 1995: Assimilation of subsurface thermal data into a simple ocean model for the initialization of an intermediate tropical coupled ocean-atmosphere forecast model. Mon. Wea. Rev., 123, 3103- 3114.10.1175/1520-0493(1995)123<3103:AOSTDI>2.0.CO;20aa5ba3a190add777b13bf5ca1b116b6http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F1995MWRv..123.3103Khttp://adsabs.harvard.edu/abs/1995MWRv..123.3103KTo promote SE maturation, the influence of different media components on different developmental stages was quantitatively evaluated. Advanced maturation was achieved with a sequence of culture media (prematuration medium and maturation medium) that contained various carbohydrates, organic nitrogen compounds and plant growth regulators. Application of lactose, BA, L-glutamine and casein hydrolysate in the prematuration medium enhanced the total number of SEs and promoted advanced differentiation. The highest number of late torpedo stage SEs was observed on maturation medium supplemented with 200 mM lactose and 29 mM sucrose. Lactose and sorbitol favoured SE maturation up to the early cotyledonary stage. With application of PEG and high ABA concentrations (20–40 μM), only early torpedo stages were formed. The number of late torpedo stage SEs was significantly higher on hormone free media or with lower ABA concentrations (0–5 μM). Formation of early and late cotyledonary SEs was significantly enhanced by adding BA in the maturation medium: neither Zeatin nor 2iP were effective. In addition, low sucrose concentrations in the proliferation medium (29 mM compared to 58 mM) also favoured the formation of cotyledonary SE in the maturation medium.
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    McCreary J. P., 1981: A linear stratified ocean model of the equatorial undercurrent. Philos. Trans. Roy. Soc.London, 298, 603- 635.10.1098/rsta.1981.0002b66aa9fd7e3c7fc9e041372902540081http%3A%2F%2Fwww.jstor.org%2Fstable%2F36691http://www.jstor.org/stable/36691A linear stratified ocean model is used to study the wind-driven response of the equatorial ocean. The model is an extension of the Lighthill (1969) model that allows the diffusion of heat and momentum into the deeper ocean, and so can develop non-trivial steady solutions. To retain the ability to expand solutions into sums of vertical normal modes, mixing coefficients must be inversely proportional to the square of the background Vaisala frequency. The model is also similar to the earlier homogeneous ocean model of Stommel (1960). He extended Ekman dynamics to the equator by allowing his model to generate a barotropic pressure field. The present model differs in that the presence of stratification allows the generation of a baroclinic pressure field as well. The most important result of this paper is that linear theory can produce a realistic equatorial current structure. The model Undercurrent has a reasonable width and depth scale. There is westward flow both above and below the Undercurrent. The meridional circulation conforms to the 'classical' picture suggested by Cromwell (1953). Unlike the Stommel solution, the response here is less sensitive to variations of parameters. Ocean boundaries are not necessary for the existence of the Undercurrent but are necessary for the existence of the deeper Equatorial Intermediate Current. The radiation of equatorially trapped Rossby and Kelvin waves is essential to the development of a realistic Undercurrent. Because the system supports the existence of these waves, low-order vertical modes can very nearly adjust to Sverdrup balance (defined below), which in a bounded ocean and for winds without curl is a state of rest. As a result, higher-order vertical modes are much more visible in the total solution. This property accounts for the surface trapping and narrow width scale of the equatorial currents. The high-order modes tend to be in Yoshida balance (defined below) and generate the characteristic meridional circulation pattern associated with equatorial Ekman pumping.
    McCreary J. P., Jr., 1983: A model of tropical ocean-atmosphere interaction. Mon. Wea. Rev., 111( 2), 370- 387.10.1175/1520-0493(1983)111<0370:AMOTOA>2.0.CO;2d2a0c5bdf3b0fcd33a1875395d28fa10http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F1983MWRv..111..370Mhttp://adsabs.harvard.edu/abs/1983MWRv..111..370MA model is used to study ocean-atmosphere interaction in the tropics. The model ocean consists of the single baroclinic mode of a two-layer ocean. Thermodynamics in the upper layer is highly parameterized. If the interface is sufficiently shallow (deep), sea surface temperature is cool (warm). The model atmosphere consists of two wind states that interact with the ocean according to the ideas of Bjerknes. When the eastern ocean is cool, the trade winds expand equatorward in the central Pacific, simulating an enhanced Walker circulation (WC). When the eastern ocean is warm, the trade winds expand eastward, simulating an enhanced Walker circulation (WC) there. For reasonable choices of parameters, the model oscillates at all time scales associated with the Southern Oscillation.
    Mu M., W.-S. Duan, D. Chen, and W. D. Yu.2015: Target observations for improving initialization of high-impact ocean-atmospheric environmental events forecasting. National Science Review, 2, 226- 236.10.1093/nsr/nwv0217bc9d266fbd90a28233b34be5aca941bhttp%3A%2F%2Fwww.cnki.com.cn%2FArticle%2FCJFDTotal-NASR201502019.htmhttp://www.cnki.com.cn/Article/CJFDTotal-NASR201502019.htmIn this paper, we emphasize the importance of accurate initial conditions in predicting high-impact ocean-atmospheric environmental events, such as El Nin?o-Southern Oscillation(ENSO), Indian Ocean Dipole(IOD), tropical cyclone(TC), and Kuroshio large meander(KLM), by reviewing recent progresses toward target observations for improving the initialization of these events forecasting. Since ield observations are costly and will never be dense enough to fully cover the vast space of these events, it is necessary to develop methodologies that guide the design of eicient and efective observation strategy. Of particular interest is a method called conditional non-linear optimal perturbation(CNOP), which has been shown to be very useful in determining the sensitive areas for target observations applicable to the predictions of ENSO, IOD, TC, and KLM. Further studies are needed to understand the predictability of these events under the inluence of climate change, and to explore the possibility of implementing ield programs of target observations. hese studies are challenging but are crucially important for improving our forecast skill of the high-impact ocean-atmospheric environmental events, and thus for disaster prevention,climate change mitigation, and sustainable socio-economic development.
    Navon I. M., X. Zou, J. Derber, and J. Sela, 1992: Variational data assimilation with an adiabatic version of the NMC spectral model. Mon. Wea. Rev., 120, 1433- 1446.5e5f2109318126c6b6d30ab7b81fb2e5http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F1992MWRv..120.1433N/s?wd=paperuri%3A%2852a87aab4fca961e5f08157ebd70f5a3%29&filter=sc_long_sign&tn=SE_xueshusource_2kduw22v&sc_vurl=http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F1992MWRv..120.1433N&ie=utf-8&sc_us=17387289179188643444
    Neelin J. D., 1990: A hybrid coupled general circulation model for El Niño studies. J. Atmos. Sci., 47( 5), 674- 693.10.1175/1520-0469(1990)047<0674:AHCGCM>2.0.CO;2460c3954-c2a0-4c94-8e02-6ad56e1cb1282f169ccbea16f7a23ccb9243f3eba3f0http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F1990JAtS...47..674Nrefpaperuri:(315bac627d2201962ce3b3ba72cbf97e)http://adsabs.harvard.edu/abs/1990JAtS...47..674NA model is developed for tropical air-sea interaction studies, which is intermediate in complexity between the large coupled general circulation models (coupled GCMs) coming into use and the simple two-level models with which pioneering El Niño-Southern Oscillation studies were carried out. The model consists of a stripped-down tropical Pacific ocean GCM, coupled to an atmospheric model which is sufficiently simple that steady state solutions may be found for low level flow and surface stress, given oceanic boundary conditions. This hybrid coupling of an ocean GCM to a steady atmospheric model permits examination of the nature of interannual coupled oscillations in the absence of atmospheric noise. Tests of the atmospheric model against an atmospheric GCM simulation of El Niño anomalies are presented, and the ocean model climatology is examined under several different conditions. Experiments with the coupled model exhibit a variety of behaviors within a realistic parameter range. These indicate a partial bifurcation diagram in which the coupled system undergoes a Hopf bifurcation from a stable climatology, giving rise to sustained El Niño-period oscillations. The amplitude, period and eastward extent of these oscillations increase with the strength of coupling and the El Niño-period oscillation itself becomes unstable to a higher frequency coupled mode which coexists with it and may affect predictability. The difference between these flow regimes may be relevant to results found by other investigators in coupled GCM experiments.
    Peng S. Q., L. Xie, 2006: Effect of determining initial conditions by four-dimensional variational data assimilation on storm surge forecasting. Ocean Modelling, 14( 1), 1- 18.10.1016/j.ocemod.2006.03.0054f53a8ca-3ac1-405c-a2e7-7a52990bdaa0217fab128de6cfdfb647c8f60454c843http%3A%2F%2Fwww.sciencedirect.com%2Fscience%2Farticle%2Fpii%2FS1463500306000382refpaperuri:(5611df0f4273c0ce8a122baea3eb203a)http://www.sciencedirect.com/science/article/pii/S1463500306000382A tangent linear model and an adjoint model of the three-dimensional, time-dependent, nonlinear Princeton Ocean Model (POM) are developed to construct a four-dimensional variational data assimilation (4D-Var) algorithm for coastal ocean prediction. To verify and evaluate the performance of this 4D-Var method, a suite of numerical experiments are conducted for a storm surge case using model-generated "pseudo-observations". The pseudo-observations are generated by a nested-grid high-resolution numerical model which is coupled to an inundation/drying scheme that is not included in the original POM. The 4D-Var algorithm based on POM is tested thoroughly for both code accuracy and the potential application in storm surge forecasting. The assimilation cycles lead to effective convergence between the forecasts and the "observations". Assimilating water level alone or together with surface currents both lead to significant improvements in storm surge forecasts within and several hours beyond the data assimilation window. It is worth noting that, assimilating water level alone produces improvements in storm surge forecasts that are comparable to those by assimilating both water level and surface currents, suggesting that optimizations of water level and surface currents are linked through the 4D-Var assimilation cycles. However, it is also worth noting that, the benefit resulting from the reduction of initial error in water level and/or surface currents through data assimilation decreases rapidly in time outside the assimilation window. This suggests that determining initial conditions of water level and/or surface currents via data assimilation is only effective within and a few hours beyond the assimilation window for storm surge forecasting. Thus, alternative data assimilation approaches are needed to improve the accuracy and lead time in operational storm surge forecasting.
    Philand er, S. G. H., R. C. Pacanowski, N. C. Lau, M. J. Nath, 1992: Simulation of ENSO with a global atmospheric GCM coupled to a high-resolution tropical Pacific Ocean GCM. J.Climate, 5( 4), 308- 329.0a5a47b5-5055-4b19-8dab-fe4bbd9000af1e77c53b77a55ca4ff760a46f302f7b8http%3A%2F%2Fadsabs.harvard.edu%2Fcgi-bin%2Fnph-data_query%3Fbibcode%3D1992JCli....5..308P%26db_key%3DPHY%26link_type%3DABSTRACT%26high%3D04030refpaperuri:(6e22e6cd45045a7ea502d0e5f4a40d3d)/s?wd=paperuri%3A%286e22e6cd45045a7ea502d0e5f4a40d3d%29&filter=sc_long_sign&tn=SE_xueshusource_2kduw22v&sc_vurl=http%3A%2F%2Fadsabs.harvard.edu%2Fcgi-bin%2Fnph-data_query%3Fbibcode%3D1992JCli....5..308P%26db_key%3DPHY%26link_type%3DABSTRACT%26high%3D04030&ie=utf-8&sc_us=18190680938444575723
    Reynolds R. W., N. A. Rayner, T. M. Smith, D. C. Stokes, and W. Wang, 2002: An improved in situ and satellite SST analysis for climate. J.Climate, 15( 13), 1609- 1625.1ecd9ceedf9ede6f75f40eecf88c95f5http%3A%2F%2Fwww.bioone.org%2Fservlet%2Flinkout%3Fsuffix%3Dbibr36%26dbid%3D16%26doi%3D10.1080%252F19425120.2012.675985%26key%3D10.1175%252F1520-0442%282002%290152.0.CO%253B2/s?wd=paperuri%3A%28ba996f056a7c404ef100fd71ca28ec61%29&filter=sc_long_sign&tn=SE_xueshusource_2kduw22v&sc_vurl=http%3A%2F%2Fwww.bioone.org%2Fservlet%2Flinkout%3Fsuffix%3Dbibr36%26dbid%3D16%26doi%3D10.1080%252F19425120.2012.675985%26key%3D10.1175%252F1520-0442%282002%290152.0.CO%253B2&ie=utf-8&sc_us=8005906790852054181
    Rosati A., K. Miyakoda, and R. Gudgel, 1997: The impact of ocean initial conditions on ENSO forecasting with a coupled model, Mon. Wea. Rev., 125( 5), 754- 772.d74e234c-134d-4ea4-b7bd-230b25a6d61c10195e773ddacdd8d5379678e869ee0chttp%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F1997MWRv..125..754Rrefpaperuri:(7f466904c11d40f13ef77520610868c1)/s?wd=paperuri%3A%287f466904c11d40f13ef77520610868c1%29&filter=sc_long_sign&tn=SE_xueshusource_2kduw22v&sc_vurl=http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F1997MWRv..125..754R&ie=utf-8&sc_us=114775675340534505
    Sugiura N., T. Awaji, S. Masuda, T. Mochizuki, T. Toyoda, T. Miyama, H. Igarashi, and Y. Ishikawa, 2008: Development of a four-dimensional variational coupled data assimilation system for enhanced analysis and prediction of seasonal to interannual climate variations. J. Geophys. Res., 113( C10), C10017.10.1029/2008JC004741a1d777f84ec09b95feb692fcf26878dfhttp%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1029%2F2008JC004741%2Ffullhttp://onlinelibrary.wiley.com/doi/10.1029/2008JC004741/full[1] A four-dimensional variational (4D-VAR) data assimilation system using a coupled ocean-atmosphere global model has been successfully developed with the aim of better defining the dynamical states of the global climate on seasonal to interannual scales. The application of this system to state estimations of climate processes during the 1996–1998 period shows, in particular, that the representations of structures associated with several key events in the tropical Pacific and Indian Ocean sector (such as the El Ni09o, the Indian Ocean dipole, and the Asian summer monsoon) are significantly improved. This fact suggests that our 4D-VAR coupled data assimilation (CDA) approach has the potential to correct the initial location of the model climate attractor on the basis of observational data. In addition, the coupling parameters that control the air-sea exchange fluxes of mass, momentum, and heat become well adjusted. Such an initialization using the 4D-VAR CDA approach allows us to make a roughly 1.5-year lead time prediction of the 1997–1998 El Ni09o event. These results demonstrate that our 4D-VAR CDA system has the ability to enhance forecast potential for seasonal to interannual phenomena.
    Tang Y. M., W. W. Hsieh, 2001: Coupling neural networks to incomplete dynamical systems via variational data assimilation. Mon. Wea. Rev., 129( 4), 818- 834.10.1175/1520-0493(2001)1292.0.CO;2e98d16f2f292c7b5681c58fa87b557a1http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F2001MWRv..129..818Thttp://adsabs.harvard.edu/abs/2001MWRv..129..818TInvestigates how a neural network model can be coupled to a dynamical model via variational assimilation. Development of a hybrid neural-dynamical Lorenz model; Results from simple hybrid model experiments; Determination of neural network parameters via variational assimilation.
    Tang Y. M., J. Amband an, and D. K. Chen, 2014: Nonlinear measurement function in the ensemble Kalman filter. Adv Atmos. Sci.,31(3), 551-558, doi: 10.1007/s00376-013-3117-9.10.1007/s00376-013-3117-9085f062099febbd343b493200c768080http%3A%2F%2Fwww.cqvip.com%2FQK%2F84334X%2F201403%2F49301875.htmlhttp://d.wanfangdata.com.cn/Periodical_dqkxjz-e201403006.aspxOn the above basis, we present two modified Kalman gain algorithms. Compared to the current Kalman gain algorithm, the modified ones remove the above assumptions, thereby leading to smaller estimated errors. This outcome was confirmed experimentally, in which we used the simple Lorenz 3-component model as the test-bed. It was found that in such a simple nonlinear dynamical system, the modified Kalman gain can perform better than the current one. However, the application of the modified schemes to realistic models involving nonlinear measurement functions needs to be further investigated.
    Wang B., X. L. Zou, and J. Zhu, 2000: Data assimilation and its applications. Proceedings of the National Academy of Sciences of the United States of America, 97( 21), 11 143- 11 144.10.1073/pnas.97.21.1114311027322a6601f9cb5cae987731f38f5aab9d1c6http%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpubmed%2F11027322http://med.wanfangdata.com.cn/Paper/Detail/PeriodicalPaper_PM11027322In data assimilation, one prepares the grid data as the best possible estimate of the true initial state of a considered system by merging various measurements irregularly distributed in space and time, with a prior knowledge of the state given by a numerical model. Because it may improve forecasting or modeling and increase physical understanding of considered systems, data assimilation now plays a very important role in studies of atmospheric and oceanic problems. Here, three examples are presented to illustrate the use of new types of observations and the ability of improving forecasting or modeling.
    Weaver A. T., J. Vialard, and D. L. T. Anderson, 2003: Three and four dimensional variational assimilation with a general circulation model of the tropical Pacific Ocean: Part I: formulation, internal diagnostics, and consistency checks. Mon. Wea. Rev., 131, 1360- 1378.c1278724-f065-4da4-bc67-9d596729cbcce5725c23b5e641faaf8ae5f1f50688bdhttp%3A%2F%2Fmy.safaribooksonline.com%2Fbook%2Fbiotechnology%2F9781598298444%2Fbasic-concepts%2Fconsistency_checksrefpaperuri:(db79805794ab8e00a9982413e2a3d1c0)/s?wd=paperuri%3A%28db79805794ab8e00a9982413e2a3d1c0%29&filter=sc_long_sign&tn=SE_xueshusource_2kduw22v&sc_vurl=http%3A%2F%2Fmy.safaribooksonline.com%2Fbook%2Fbiotechnology%2F9781598298444%2Fbasic-concepts%2Fconsistency_checks&ie=utf-8&sc_us=7918370634512938217
    Wu X. R., S. Q. Zhang, Z. Y. Liu, A. Rosati, T. L. Delworth, and Y. Liu, 2012: Impact of geographic-dependent parameter optimization on climate estimation and prediction: Simulation with an intermediate coupled model. Mon. Wea. Rev., 140( 12), 3956- 3971.10.1175/MWR-D-11-00298.1e60529c9db41d4d2e2804f6c0d5c80c5http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F2012MWRv..140.3956Whttp://adsabs.harvard.edu/abs/2012MWRv..140.3956WNot Available
    Wu X. R., W. Li, G. J. Han, S. Q. Zhang, and X. D. Wang, 2014: A compensatory approach of the fixed localization in EnKF. Mon. Wea. Rev. , 142, 3713- 3733.10.1007/BF00323184ea368eec1ae38624e7e06685fdf2b620http%3A%2F%2Fwww.researchgate.net%2Fpublication%2F273205738_A_Compensatory_Approach_of_the_Fixed_Localization_in_EnKFhttp://www.researchgate.net/publication/273205738_A_Compensatory_Approach_of_the_Fixed_Localization_in_EnKFNot Available
    Wu X. R., G. J. Han, S. Q. Zhang, and Z. Y. Liu, 2016: A study of the impact of parameter optimization on ENSO predictability with an intermediate coupled model. Climate Dyn. ,46, 711-727, doi:10.1007/s00382-015-2608-z.10.1007/s00382-015-2608-z0bbc07b0ef94b83f7a4d2e3f2a4bdccehttp%3A%2F%2Flink.springer.com%2F10.1007%2Fs00382-015-2608-zhttp://link.springer.com/10.1007/s00382-015-2608-zModel error is a major obstacle for enhancing the forecast skill of El Ni09o-Southern Oscillation (ENSO). Among three kinds of model error sources—dynamical core misfitting, physical scheme approximati
    Wyrtki K., 1975: El Niño-the dynamic response of the equatorial Pacific Ocean to atmospheric forcing. J. Phys. Oceanogr., 5( 4), 572- 584.c9899457-0dbd-421c-aae4-bccf5c62346df566ceef6c3c171977672c3b2f2e549dhttp%3A%2F%2Fci.nii.ac.jp%2Fnaid%2F10013127141%2Frefpaperuri:(435f5e0c9edbaff14bfe2e016ff4babe)http://ci.nii.ac.jp/naid/10013127141/El Nino-The dynamic response of the equatorial Pacific Ocean to atmospheric forcing. WYRTKI K. J. Phys. Oceanogr 5, 572-584, 1975
    Zebiak S. E., M. A. Cane, 1987: A model El Niño-Southern oscillation. Mon. Wea. Rev., 115, 2262- 2278.10.1038/302295a0fbff0f50-8c26-4685-a6ef-b4f44b04cc05c65dd6c79eaaa702eee2a149cb0bddbbhttp%3A%2F%2Fwww.researchgate.net%2Fpublication%2F244948692_1987_A_model_El-Nino-Southern_Oscillationrefpaperuri:(02aa79a86a12d930aec389c613c4f943)http://www.researchgate.net/publication/244948692_1987_A_model_El-Nino-Southern_OscillationAt intervals that vary from 2 to 10 yr sea-surface temperatures and rainfall are unusually high and the tradewinds are unusually weak over the tropical Pacific Ocean. These Southern Oscillation El Niño events which devastate the ecology of the coastal zones of Ecuador and Peru, which affect the global atmospheric circulation and which can contribute to severe winters over northern America, often develop in a remarkably predictable manner. But the event which began in 1982 has not followed this pattern.
    Zhang R. H., C. Gao, 2015: Role of subsurface entrainment temperature (Te) in the onset of El Niño events,as represented in an intermediate coupled model. Climate Dyn., 1-19, doi: 10.1007/s00382-015-2655-5.10.1007/s00382-015-2655-5e37a52da624d4c240e1c735b1d05e68ehttp%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%2Fs00382-015-2655-5http://link.springer.com/article/10.1007/s00382-015-2655-5An improved intermediate coupled model (ICM) is described for use in ENSO-related modeling in the tropical Pacific, with ten baroclinic modes included in the vertical and horizonatally varying stratification taken into account. One crucial component of the model is the way in which the subsurface entrainment temperature in the surface mixed layer (T e ) is explicitly used to determine the sea surface temperature (SST) variability. An optimized procedure is developed to depict T e using inverse modeling from an SST anomaly equation and its empirical relationship with the sea surface pressure variability. The coupled system realistically produces interannual variability associated with ENSO cycles, with a dominant 4-year oscillation. The onset and development of El Ni09o events from this ICM are examined in view of the well-known delayed oscillator paradigm; an example is given for the evolution of La Ni09a conditions in model year 2 to El Ni09o conditions in year 4. Right after a La Ni09a event (e.g., in year 2), there is a clear signature of reflections at the western boundary in early year 2, with related equatorial signals propagating eastward along the equator into the eastern basin in middle year 2. However, these reflected signals on the equator do not directly lead to an onset of an El Ni09o event at that time. Instead, approximately 1-year delay, a major El Ni09o event is seen to develop in the following year (late year 3), at a time when there is no reflected signal explicitly from the western boundary, indicating that the origin of the El Ni09o event cannot be directly ascribed to the reflection processes. Instead, Kelvin waves in the ocean that actually triggers the El Ni09o event in early year 3 are generated by interior wind anomalies near the date line that are associated with the first appearance of warm SST anomalies off the equator. Persisted T e anomalies off the equator in the western tropical Pacific initiate the warm SST anomaly near the date line along the North Equatorial Countercurrent region, which induces wind anomalies and an ocean–atmosphere coupling, leading to the El Ni09o event in year 4. The relevance of these ICM-based results to other onset mechanisms of El Ni09o and observations is also discussed.
    Zhang R. H., S. E. Zebiak, R. Kleeman, and N. Keenlyside, 2003: A new intermediate coupled model for El Niño simulation and prediction. Geophys. Res. Lett.,30(19), doi:10.1029/2003 GL018010, 19.
    Zhang R. H., R. Kleeman, S. E. Zebiak, N. Keenlyside, and S. Raynaud, 2005a: An empirical parameterization of subsurface entrainment temperature for improved SST anomaly simulations in an intermediate ocean model. J.Climate, 18, 350- 371.10.1175/JCLI-3271.11a96302493af47d3b03534dc4da31a4fhttp%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F2005JCli...18..350Zhttp://adsabs.harvard.edu/abs/2005JCli...18..350Z&nbsp; &nbsp; An empirical model for the temperature of subsurface water entrained into the ocean mixed layer (T-e) is presented and evaluated to improve sea Surface temperature anomaly (SSTA) simulations in an intermediate ocean model (IOM) of the tropical Pacific. An inverse modeling approach is adopted to estimate T-e from an SSTA equation using observed SST and simulated Upper-ocean currents. A relationship between T-e and sea surface height (SSH) anomalies is then obtained by utilizing a singular value decomposition (SVD) of their covariance. This empirical scheme is able to better parameterize T-e anomalies than other local schemes and quite realistically depicts interannual variability of T-e, including a nonlocal phase lag relation of T-e variations relative to SSH anomalies over the central equatorial Pacific. An improved T-e parameterization naturally leads to better depiction of the subsurface effect on SST variability by the mean upwelling of subsurface temperature anomalies. As a result, SSTA simulations are significantly improved in the equatorial Pacific a comparison with other schemes indicates that systematic errors of the simulated SSTAs are significantly small-apparently due to the optimized empirical T-e parameterization. Cross validation and comparisons with other model simulations are made to illustrate the robustness and effectiveness of the scheme. In particular it is demonstrated that the empirical T-e model constructed from one historical period can be successfully used to improve SSTA simulations in another.
    Zhang R. H., S. E. Zebiak, R. Kleeman, and N. Keenlyside, 2005b: Retrospective El Niño forecasts using an improved intermediate coupled model. Mon. Wea. Rev., 133, 2777- 2802.10.1175/MWR3000.1d7e01395-11cf-4052-9d08-8d6ef0786c74c56515f86939e580924fd4743b4cc8fchttp%3A%2F%2Fwww.researchgate.net%2Fpublication%2F242443365_Retrospective_El_Nio_Forecasts_Using_an_Improved_Intermediate_Coupled_Modelrefpaperuri:(d31ce0f823315f5ab6f96a7ab6efe803)http://www.researchgate.net/publication/242443365_Retrospective_El_Nio_Forecasts_Using_an_Improved_Intermediate_Coupled_ModelAbstract A new intermediate coupled model (ICM) is presented and employed to make retrospective predictions of tropical Pacific sea surface temperature (SST) anomalies. The ocean dynamics is an extension of the McCreary baroclinic modal model to include varying stratification and certain nonlinear effects. A standard configuration is chosen with 10 baroclinic modes plus two surface layers, which are governed by Ekman dynamics and simulate the combined effects of the higher baroclinic modes from 11 to 30. A nonlinear correction associated with vertical advection of zonal momentum is incorporated and applied (diagnostically) only within the two surface layers, forced by the linear part through nonlinear advection terms. As a result of these improvements, the model realistically simulates the mean equatorial circulation and its variability. The ocean thermodynamics include an SST anomaly model with an empirical parameterization for the temperature of subsurface water entrained into the mixed layer ( T e ), which is optimally calculated in terms of sea surface height (SSH) anomalies using an empirical orthogonal function (EOF) analysis technique from historical data. The ocean model is then coupled to a statistical atmospheric model that estimates wind stress ( ) anomalies based on a singular value decomposition (SVD) analysis between SST anomalies observed and anomalies simulated from ECHAM4.5 (24-member ensemble mean). The coupled system exhibits realistic interannual variability associated with El Niño, including a predominant standing pattern of SST anomalies along the equator and coherent phase relationships among different atmosphere-搊cean anomaly fields with a dominant 3-yr oscillation period. Twelve-month hindcasts/forecasts are made during the period 1963-2002, starting each month. Only observed SST anomalies are used to initialize the coupled predictions. As compared to other prediction systems, this coupled model has relatively small systematic errors in the predicted SST anomalies, and its SST prediction skill is apparently competitive with that of most advanced coupled systems incorporating sophisticated ocean data assimilation. One striking feature is that the model skill surpasses that of persistence at all lead times over the central equatorial Pacific. Prediction skill is strongly dependent on the season, with the correlations attaining a minimum in spring and a maximum in fall. Cross-validation experiments are performed to examine the sensitivity of the prediction skill to the data periods selected for training the empirical T e model. It is demonstrated that the artificial skill introduced by using a dependently constructed T e model is not significant. Independent forecasts are made for the period 1997-2002 when no dependent data are included in constructing the two empirical models ( T e and ). The coupled model has reasonable success in predicting transition to warm phase and to cold phase in the spring of 1997 and 1998, respectively. Potential problems and further improvements are discussed with the new intermediate prediction system.
    Zhang R. H., A. J. Busalacchi, and D. G. DeWitt, 2008: The roles of atmospheric stochastic forcing (SF) and oceanic entrainment temperature (Te) in decadal modulation of ENSO. J.Climate, 21, 674- 704.10.1175/2007JCLI1665.1d1946e4e-ceb5-41ef-84b7-a1bc6e177ace783fd267ae7027dd4a7b6bb9b9c59466http%3A%2F%2Fwww.researchgate.net%2Fpublication%2F241204450_The_Roles_of_Atmospheric_Stochastic_Forcing_%28SF%29_and_Oceanic_Entrainment_Temperature_%28Te%29_in_Decadal_Modulation_of_ENSO%3Fev%3Dauth_pubrefpaperuri:(8be4a6c0b0b8608c3cdc672ed004e8a0)http://www.researchgate.net/publication/241204450_The_Roles_of_Atmospheric_Stochastic_Forcing_(SF)_and_Oceanic_Entrainment_Temperature_(Te)_in_Decadal_Modulation_of_ENSO?ev=auth_pubThe El Ni01±o09恪癝outhern Oscillation (ENSO) has been observed to exhibit decadal changes in its properties; the cause and implication of such changes are strongly debated. Here the authors examine the influences of two particular attributors of the ocean09恪癮tmospheric system. The roles of stochastic forcing (SF) in the atmosphere and decadal changes in the temperature of subsurface water entrained into the mixed layer (Te) in modulating ENSO are compared to one another using coupled ocean09恪癮tmosphere models of the tropical Pacific climate system. Two types of coupled models are used. One is an intermediate coupled model (ICM) and another is a hybrid coupled model (HCM), both of which consist of the same intermediate ocean model (IOM) with an empirical parameterization for Te, constructed via singular value decomposition (SVD) analysis of the IOM simulated historical data. The differences in the ICM and HCM are in the atmospheric component: the one in the ICM is an empirical feedback model for wind stress (0367), and that in the HCM is an atmospheric general circulation model (AGCM; ECHAM4.5). The deterministic component of atmospheric 0367 variability, representing its signal response (0367Sig) to an external SST forcing, is constructed statistically by an SVD analysis from a 24-member ensemble mean of the ECHAM4.5 AGCM simulations forced by observed SST; the SF component (0367SF) is explicitly estimated from the ECHAM4.5 AGCM ensemble and HCM simulations. Different SF representations are specified in the atmosphere: the SF effect can be either absent or present explicitly in the ICM, or implicitly in the HCM where the ECHAM4.5 AGCM is used as a source for SF. Decadal changes in the ocean thermal structure observed in the late 1970s are incorporated into the coupled systems through the Te parameterizations for the two subperiods before (196309恪79) and after (198009恪96) the climate shift (T6309恪79e and T8009恪96e), respectively. The ICM and HCM simulations well reproduce interannual variability associated with El Ni01±o in the tropical Pacific. Model sensitivity experiments are performed using these two types of coupled models with different realizations of SF in the atmosphere and specifications of decadal Te changes in the ocean. It is demonstrated that the properties of ENSO are modulated differently by these two factors. The decadal Te changes in the ocean can be responsible for a systematic shift in the phase propagation of ENSO, while the SF in the atmosphere can contribute to the amplitude and period modulation in a random way. The relevance to the observed decadal ENSO variability in the late 1970s is discussed.
    Zhang R. H., F. Zheng, J. Zhu, and Z. G. Wang, 2013: A successful real-time forecast of the 2010-11 La Niña event. Sci. Rep., 3,1108, doi: 10.1038/srep01108.10.1038/srep0110823346365e49859a1-e895-4f7b-90b5-1d2e90766fe7c2be70db9eed1eb05a7cafcc13853e40http%3A%2F%2Flabs.europepmc.org%2Fabstract%2FPMC%2FPMC3552287refpaperuri:(2824948dabf5f33f50859bbc916a124d)http://labs.europepmc.org/abstract/PMC/PMC3552287ABSTRACT During 2010-11, a La Ni&ntilde;a condition prevailed in the tropical Pacific. An intermediate coupled model (ICM) is used to demonstrate a real-time forecast of sea surface temperature (SST) evolution during the event. One of the ICM's unique features is an empirical parameterization of the temperature of subsurface water entrained into the mixed layer (T(e)). This model provided a good prediction, particularly of the "double dip" evolution of SST in 2011 that followed the La Ni&ntilde;a event peak in October 2010. Thermocline feedback, explicitly represented by the relationship between T(e) and sea level in the ICM, is a crucial factor affecting the second cooling in 2011. Large negative T(e) anomalies were observed to persist in the central equatorial domain during 2010-11, inducing a cold SST anomaly to the east during July-August 2011 and leading to the development of a La Ni&ntilde;a condition thereafter.
    Zhang R. H., C. Gao, X. B. Kang, H. Zhi, Z. G. Wang, and L. C. Feng, 2015: ENSO modulations due to interannual variability of freshwater forcing and ocean biology-induced heating in the tropical Pacific. Sci. Rep., 5,18506, doi: 10.1038/srep18506.10.1038/srep18506aa3f00b5e001cf336c6533289fd037dchttp%3A%2F%2Feuropepmc.org%2Farticles%2FPMC4683514%2Fhttp://europepmc.org/articles/PMC4683514/Recent studies have identified clear climate feedbacks associated with interannual variations in freshwater forcing (FWF) and ocean biology-induced heating (OBH) in the tropical Pacific. The interrelationships among the related anomaly fields are analyzed using hybrid coupled model (HCM) simulations to illustrate their combined roles in modulating the El Niño-Southern Oscillation (ENSO). The HCM-based supporting experiments are performed to isolate the related feedbacks, with interannually varying FWF and OBH being represented individually or collectively, which allows their effects to be examined in a clear way. It is demonstrated that the interannual freshwater forcing enhances ENSO variability and slightly prolongs the simulated ENSO period, while the interannual OBH reduces ENSO variability and slightly shortens the ENSO period, with their feedback effects tending to counteract each other.
    Zhang S., X. Zou, and J. E. Ahlquist, 2001: Examination of numerical results from tangent linear and adjoint of discontinuous nonlinear models. Mon. Wea. Rev., 129( 11), 2791- 2804.e87e3e83516cc49bbdcaf06283cdb8abhttp%3A%2F%2Fadsabs.harvard.edu%2Fcgi-bin%2Fnph-data_query%3Fbibcode%3D2001MWRv..129.2791Z%26db_key%3DPHY%26link_type%3DABSTRACT/s?wd=paperuri%3A%28b3ee69db961290912f9d787a4266667e%29&filter=sc_long_sign&tn=SE_xueshusource_2kduw22v&sc_vurl=http%3A%2F%2Fadsabs.harvard.edu%2Fcgi-bin%2Fnph-data_query%3Fbibcode%3D2001MWRv..129.2791Z%26db_key%3DPHY%26link_type%3DABSTRACT&ie=utf-8&sc_us=1812729966643523739
    Zhang S., M. J. Harrison, A. T. Wittenberg, A. Rosati, J. L. Anderson, and V. Balaji, 2005c: Initialization of an ENSO forecast system using a parallelized ensemble filter. Mon. Wea. Rev., 133( 11), 3176- 3201.10.1175/MWR3024.1a9816a2e-275b-43b1-be61-d9750ecb72cb676d6bd510b545ed4583758b6f9219achttp%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F2005MWRv..133.3176Zrefpaperuri:(656e75ebeaef2434aa84d777646ce1d1)http://adsabs.harvard.edu/abs/2005MWRv..133.3176ZNot Available
    Zhang S., M. J. Harrison, A. Rosati, and A. Wittenberg, 2007: System design and evaluation of coupled ensemble data assimilation for global oceanic climate studies. Mon. Wea. Rev., 135( 10), 3541- 3564.23f1211717e3690331b5400ab00e1685http%3A%2F%2Ficesjms.oxfordjournals.org%2Fexternal-ref%3Faccess_num%3D10.1175%2FMWR3466.1%26link_type%3DDOIhttp://icesjms.oxfordjournals.org/external-ref?access_num=10.1175/MWR3466.1&amp;link_type=DOI
    Zhang S., Y. S. Chang, X. Yang, and A. Rosati, 2014: Balanced and coherent climate estimation by combining data with a biased coupled model. J.Climate, 27( 3), 1302- 1314.10.1175/JCLI-D-13-00260.1d54561989da58886ff753ffa5f4cb751http%3A%2F%2Fonlinelibrary.wiley.com%2Fresolve%2Freference%2FXREF%3Fid%3D10.1175%2FJCLI-D-13-00260.1http://onlinelibrary.wiley.com/resolve/reference/XREF?id=10.1175/JCLI-D-13-00260.1Abstract Given a biased coupled model and the atmospheric and oceanic observing system, maintaining a balanced and coherent climate estimation is of critical importance for producing accurate climate analysis and prediction initialization. However, because of limitations of the observing system (e.g., most of the oceanic measurements are only available for the upper ocean), directly evaluating climate estimation with real observations is difficult. With two coupled models that are biased with respect to each other, a biased twin experiment is designed to simulate the problem. To do that, the atmospheric and oceanic observations drawn from one model based on the modern climate observing system are assimilated into the other. The model that produces observations serves as the truth and the degree by which an assimilation recovers the truth steadily and coherently is an assessment of the impact of the data constraint scheme on climate estimation. Given the assimilation model bias of warmer atmosphere and colder ocean, where the atmospheric-only (oceanic only) data constraint produces an overcooling (overwarming) ocean through the atmosphere-搊cean interaction, the constraints with both atmospheric and oceanic data create a balanced and coherent ocean estimate as the observational model. Moreover, the consistent atmosphere-搊cean constraint produces the most accurate estimate for North Atlantic Deep Water (NADW), whereas NADW is too strong (weak) if the system is only constrained by atmospheric (oceanic) data. These twin experiment results provide insights that consistent data constraints of multiple components are very important when a coupled model is combined with the climate observing system for climate estimation and prediction initialization.
    Zhang X. F., S. Q. Zhang, Z. Y. Liu, X. R. Wu, and G. J. Han, 2015a: Parameter optimization in an intermediate coupled climate model with biased physics. J.Climate, 28( 3), 1227- 1247.6126e262-008a-4520-a8c4-f965ee992fde416fd8107351a205a2a3e1de44f26062http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F2015jcli...28.1227z/s?wd=paperuri%3A%28d3be56f5b29ef06073f40fea96eb4198%29&filter=sc_long_sign&tn=SE_xueshusource_2kduw22v&sc_vurl=http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F2015jcli...28.1227z&ie=utf-8&sc_us=9433316375189316790
    Zhang X. F., G. J. Han, D. Li, X. R. Wu, W. Li, and P. C. Chu, 2015b: Variational estimation of wave-affected parameters in a two-equation turbulence model. J. Atmos. Oceanic Technol., 32( 3), 528- 546.10.1175/JTECH-D-14-00087.111ea80ba-f077-4cf5-9483-fb19664f43cef993fb0aa7b05a8437b0ab168cdc150ehttp%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F2015JAtOT..32..528Zhttp://adsabs.harvard.edu/abs/2015JAtOT..32..528ZNot Available
    Zheng F., J. Zhu, R. H. Zhang, and G. Q. Zhou, 2006: Ensemble hindcasts of SST anomalies in the tropical Pacific using an intermediate coupled model. Geophys. Res. Lett., 33( 19), L19604.10.1029/2006GL026994a643ac8e307005ceb7e558583fe8a59dhttp%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1029%2F2006GL026994%2Fabstracthttp://onlinelibrary.wiley.com/doi/10.1029/2006GL026994/abstract&nbsp; &nbsp; Ensemble hindcasts of sea surface temperature (SST) anomalies in the tropical Pacific are studied using an intermediate coupled model (ICM), in which an ensemble Kalman filter (EnKF) data assimilation system is implemented to provide the initial ensemble. A linear, first-order Markov stochastic model is adopted to represent model errors. Parameters in the stochastic model are estimated by comparing observation-minus-forecast values over 30 years. Twelve-month, 120 ensemble hindcasts are performed over the period 1995-2004, each with 100 ensemble members. This ensemble technique provides a simple method of extending the standard ICM forecasts to the probabilistic domain. The results show that the prediction skill of the ensemble mean is better than that of one single deterministic forecast using the same ICM. For the probabilistic perspective, those ensemble forecasts have their ensembles following observed SST anomaly variations well.
    Zheng F., J. Zhu, H. Wang, and R. H. Zhang, 2009: Ensemble hindcasts of ENSO events over the past 120 years using a large number of ensembles. Adv. Atmos. Sci.,26, 359-372, doi: 10.1007/s00376-009-0359-7.10.1007/s00376-009-0359-746b4cb1872e3da23cf3785e06025ff11http%3A%2F%2Fwww.cnki.com.cn%2FArticle%2FCJFDTotal-DQJZ200902020.htmhttp://d.wanfangdata.com.cn/Periodical_dqkxjz-e200902020.aspxBased on an intermediate coupled model (ICM), a probabilistic ensemble prediction system (EPS) has been developed. The ensemble Kalman filter (EnKF) data assimilation approach is used for generating the initial ensemble conditions, and a linear, first-order Markov-Chain SST anomaly error model is embedded into the EPS to provide model-error perturbations. In this study, we perform ENSO retrospective forecasts over the 120 year period 1886--2005 using the EPS with 100 ensemble members and with initial conditions obtained by only assimilating historic SST anomaly observations. By examining the retrospective ensemble forecasts and available observations, the verification results show that the skill of the ensemble mean of the EPS is greater than that of a single deterministic forecast using the same ICM, with a distinct improvement of both the correlation and root mean square (RMS) error between the ensemble-mean hindcast and the deterministic scheme over the 12-month prediction period. The RMS error of the ensemble mean is almost 0.2oC smaller than that of the deterministic forecast at a lead time of 12 months. The probabilistic skill of the EPS is also high with the predicted ensemble following the SST observations well, and the areas under the relative operating characteristic (ROC) curves for three different ENSO states (warm events, cold events, and neutral events) are all above 0.55 out to 12 months lead time. However, both deterministic and probabilistic prediction skills of the EPS show an interdecadal variation. For the deterministic skill, there is high skill in the late 19th century and in the middle-late 20th century (which includes some artificial skill due to the model training period), and low skill during the period from 1906 to 1961. For probabilistic skill, for the three different ENSO states, there is still a similar interdecadal variation of ENSO probabilistic predictability during the period 1886--2005. There is high skill in the late 19th century from 1886 to 1905, and a decline to a minimum of skill around 1910--50s, beyond which skill rebounds and increases with time until the 2000s.
    Zhu J., G. Q. Zhou, C. X. Yan, W. W. Fu, and X. B. You, 2006: A three-dimensional variational ocean data assimilation system: scheme and preliminary results. Science in China Series D: Earth Sciences, 49( 11), 1212- 1222.10.1007/s11430-006-1212-97b0e0f30e237011aca3c68bec7409f6bhttp%3A%2F%2Flink.springer.com%2F10.1007%2Fs11430-006-1212-9http://d.wanfangdata.com.cn/Periodical_zgkx-ed200611009.aspxA new 3DVAR-based Ocean Variational Analysis System (OVALS) is developed. OVALS is capable of assimilating in situ sea water temperature and salinity observations and satellite altimetry data. As a component of OVALS, a new variational scheme is proposed to assimilate the sea surface height data. This scheme considers both the vertical correlation of background errors and the nonlinear temperature-salinity relationship which is derived from the generalization of the linear balance constraints to the nonlinear in the 3DVAR. By this scheme, the model temperature and salinity fields are directly adjusted from the altimetry data. Additionally, OVALS can assimilate the temperature and salinity profiles from the ARGO floats which have been implemented in recent years and some temperature and salinity data such as from expendable bathythermograph, moored ocean buoys, etc. A 21-year assimilation experiment is carried out by using OVALS and the Tropical Pacific circulation model. The results show that the assimilation system may effectively improve the estimations of temperature and salinity by assimilating all kinds of observations. Moreover, the root mean square errors of temperature and salinity in the upper depth less than 420 m reach 0.63 and 0.34 psu.
    Zhu J. S., A. Kumar, H. Wang, and B. H. Huang, 2015: Sea surface temperature predictions in NCEP CFSv2 using a simple ocean initialization scheme. Mon. Wea. Rev., 143, 3176- 3191.10.1175/MWR-D-14-00297.1529a13e523e8d440aae9ab8f94e555d4http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F2015MWRv..143.3176Zhttp://adsabs.harvard.edu/abs/2015MWRv..143.3176ZAbstract In contrast to operational climate predictions based on sophisticated ocean data assimilation schemes at the National Centers for Environmental Predictions (NCEP), this study applied a simple ocean initialization scheme to the NCEP latest seasonal prediction model - Climate Forecast System version 2 (CFSv2). In the scheme, sea surface temperature (SST) was the only observed information applied to derive ocean initial states. The physical basis for the method is that, through air-sea coupling, SST is capable of reproducing some observed features of ocean evolutions by forcing the atmospheric winds. SST predictions based on the scheme are compared against hindcasts from the National (lately North American) Multimodel Ensemble (NMME) project. It was found that due to substantial biases in the tropical eastern Pacific in the ocean initial conditions produced by SST assimilation, ENSO SST predictions were not as good as those with sophisticated initialization schemes, e.g., hindcasts in the NMME project. However, in other basins, SST predictions based on simple ocean initialization procedure were not worse (sometimes even better) than those with sophisticated initialization schemes. These comparisons indicate that it was helpful that subsurface ocean information be assimilated to improve the tropical Pacific SST predictions, while SST-based ocean assimilation was an effective way to enhance SST prediction capability in other ocean basins. By examining multimodel ensembles with the simple scheme-based hindcasts either included or excluded in NMME, it is also suggested that including the hindcast would generally benefit multimodel ensemble forecasts. In addition, possible ways to further improve ENSO SST predictions with the simple initialization scheme are also discussed.
    Zou X., I. M. Navon, M. Berger, K. H. Phua, T. Schlick, and F. X. Le Dimet, 1993: Numerical experience with limited-memory quasi-Newton and truncated Newton methods. SIAM Journal on Optimization, 3( 3), 582- 608.10.1137/080302992559443b519a5b0c2c34b5f0e64b270http%3A%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D1230158http://www.ams.org/mathscinet-getitem?mr=1230158Computational experience with several limited-memory quasi-Newton and truncated Newton methods for unconstrained nonlinear optimization is described. Comparative tests were conducted on a well-known test library [J. J. Mor&eacute;, B. S. Gaxbow, and K. E. Hillstrom, ACM Trans. Math. Software, 7 (1981), pp. 17&#150;41], on several synthetic problems allowing control of the clustering of eigenvalues in the Hessian spectrum, and on some large-scale problems in oceanography and meteorology. The results indicate that among the tested limited-memory quasi-Newton methods, the L-BFGS method [D. C. Liu and J. Nocedal, Math. Programming, 45 (1989), pp. 503&#150;528] has the best overall performance for the problems examined. The numerical performance of two truncated Newton methods, differing in the inner-loop solution for the search vector, is competitive with that of L-BFGS. &copy;1993 Society for Industrial and Applied Mathematics
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Manuscript received: 01 December 2015
Manuscript revised: 28 January 2016
Manuscript accepted: 19 February 2016
通讯作者: 陈斌, bchen63@163.com
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Testing a Four-Dimensional Variational Data Assimilation Method Using an Improved Intermediate Coupled Model for ENSO Analysis and Prediction

  • 1. Key Laboratory of Ocean Circulation and Waves, Institute of Oceanology, Chinese Academy of Sciences, Qingdao 266071
  • 2. University of Chinese Academy of Sciences, Beijing 100029
  • 3. Key Laboratory of Marine Environmental Information Technology, State Oceanic Administration, National Marine Data and Information Service, Tianjin 300000
  • 4. Laboratory for Ocean and Climate Dynamics, Qingdao National Laboratory for Marine Science and Technology, Qingdao 266237

Abstract: A four-dimensional variational (4D-Var) data assimilation method is implemented in an improved intermediate coupled model (ICM) of the tropical Pacific. A twin experiment is designed to evaluate the impact of the 4D-Var data assimilation algorithm on ENSO analysis and prediction based on the ICM. The model error is assumed to arise only from the parameter uncertainty. The "observation" of the SST anomaly, which is sampled from a "truth" model simulation that takes default parameter values and has Gaussian noise added, is directly assimilated into the assimilation model with its parameters set erroneously. Results show that 4D-Var effectively reduces the error of ENSO analysis and therefore improves the prediction skill of ENSO events compared with the non-assimilation case. These results provide a promising way for the ICM to achieve better real-time ENSO prediction.

1. Introduction
  • ENSO is the strongest interannual phenomenon in the tropical Pacific, directly inducing climate anomalies worldwide. Thus, accurately and effectively predicting ENSO events is of great significance to society. In recent decades, great advancements have been made in understanding ENSO and developing models for its real-time prediction (e.g., Bjerknes, 1969; Wyrtki, 1975; McCreary, 1983; Cane et al., 1986; Zhang et al., 2013). At present, various types of air-sea coupled models have been developed, including intermediate coupled models (ICMs; e.g., Zebiak and Cane, 1987; Balmaseda et al., 1994; Zhang et al., 2003), hybrid coupled models (e.g., Neelin, 1990; Barnett et al., 1993; Zhang and Gao, 2015), and fully coupled general circulation models (e.g., Philander et al., 1992; Rosati et al., 1997). Currently, these coupled models enable us to make six-month to one-year real-time ENSO predictions in advance with reasonable success.

    However, certain challenges still exit in the real-time prediction of ENSO events. For example, model biases cause ENSO simulations to depart far away from observations, making model state estimation and prediction inaccurate (Zhang et al., 2005a). In addition, high-quality ocean observations are very scarce, which results in uncertainties in ocean state estimation (Wang et al., 2000). Accordingly, difficulties emerge in providing accurate initial ocean conditions for ENSO prediction. Thus, it is essential to find a way to make model solutions coherent with observations by producing optimal initial conditions for predictions. To this end, data assimilation is an effective way to provide optimal initializations for ENSO analysis and prediction. However, a related issue is how to effectively use limited observations in data assimilation (Mu et al., 2015). To achieve this, observing system experiments need to be performed to identify target observations with sensitive domains where data assimilation can be used effectively to improve prediction. Thus, an effective data assimilation system necessarily includes not only highly-quality data and a good data assimilation scheme, but also a method for these two aspects to be combined in a smart way, ultimately providing much more realistic initial analysis fields for predictions (Mu et al., 2015).

    Various data assimilation algorithms have been used to initialize ENSO prediction. Originally, the nudging method, which directly forces the model solution to approximate observations, was used to force the modeled SST to reflect observed values to initialize ENSO prediction (e.g., Chen et al., 1995; Kirtman and Zebiak, 1997; Kumar et al., 2014; Zhu et al., 2015). Later, ensemble Kalman filters (Evensen, 1994) were introduced into ocean modeling, providing probabilistic forecasts of ENSO (e.g., Houtekamer and Mitchell, 1998; Zheng et al., 2006, 2009; Zhang et al., 2007; Tang et al., 2014; Wu et al., 2014). As an important branch of data assimilation, variational (three- and four-dimensional) methods (3D-Var and 4D-Var, respectively) are also widely used in ENSO analysis and prediction. For example, the 4D-Var data assimilation method pursues the analysis solutions by minimizing the distance between the model trajectory and observation time series [i.e., the so-called cost function (e.g., Tang and Hsieh, 2001; Zhang et al., 2001; Han et al., 2006, 2015; Peng and Xie, 2006; Zhang et al., 2015b)]. Compared with economic 3D-Var analysis (Derber and Rosati, 1989; Zhu et al., 2006), the 4D-Var data assimilation method is more dynamically and mathematically consistent (e.g., Dommenget and Stammer, 2004; Sugiura et al., 2008). For instance, (Weaver et al., 2003) assimilated in situ temperature data into an OGCM by the 3D-Var and 4D-Var methods, and demonstrated that 4D-Var is more effective than 3D-Var in producing a consistent ocean state between model solutions and observations. Additionally, the 4D-Var method has been applied to ENSO prediction using various models and has achieved some success (Kleeman et al., 1995; Galanti et al., 2003; Dommenget and Stammer, 2004). The main difficulty in 4D-Var-based initialization of ENSO prediction is that the method requires the development of an adjoint model to compute the gradient of the cost function with respect to the control variables, which is very complicated and time-consuming.

    Our goal in this study is to implement the 4D-Var method to an improved intermediate coupled model (ICM) that was developed for ENSO studies (e.g., Zhang et al., 2003; Zhang et al., 2005b). The ICM used is a simplified coupled ocean-atmosphere model with two statistical submodels for the temperature of subsurface water entrained into the mixed layer (T e) and wind stress (τ). T e is optimally calculated in terms of sea level (SL) anomalies using an EOF analysis technique. Wind stress anomalies are estimated based on an SVD analysis between SST anomalies and τ anomalies (Zhang et al., 2003, 2015). Therefore, the wind anomalies are represented as a response to SST, and the subsurface thermal effect on SST is parameterized by the ocean dynamical field. Although the ICM has been used for realistic predictions of ENSO (Zhang et al., 2013), it has not yet applied the 4D-Var method to initialize the real-time prediction. Since the 4D-Var method is more dynamically and mathematically consistent in offering an initial ocean state for improving forecast accuracy, we specifically address the following question in this paper: Can the ENSO forecast skill generated by the ICM be increased by using the 4D-Var method?

    Herein, we provide a detailed description of the incorporation of the 4D-Var data assimilation method into the aforementioned ICM, including the development of the associated tangent linear model and adjoint model. Based on the successful implementation of the 4D-Var data assimilation formulation into the ICM, we then report the preliminary results of a series of sensitivity experiments. Previously, (Zheng et al., 2009) incorporated an ensemble Kalman filter method into the ICM and achieved improved ENSO prediction.

    The paper is organized as follows: Section 2 describes the ICM and 4D-Var data assimilation method. The experimental setup is introduced in section 3, and the assimilation impacts are analyzed in section 4. Finally, a conclusion and discussion are presented in section 5.

2. Methodology
  • In this section, we briefly describe the ICM that has been routinely used to make ENSO predictions [see a summary of the model ENSO forecasts at the International Research Institute for Climate and Society(IRI) website: http://iri.columbia.edu/climate/ENSO/currentinfo/SST_table.html];the real-time prediction results are posted on the IRI website every month, now referred to as the IOCAS (Institute of Oceanology/ Chinese Academy of Sciences) ICM. Then, the 4D-Var data assimilation procedure is described, including its tangent linear model and adjoint model and the corresponding minimization processes.

  • The ICM consists of a dynamic ocean model, an SST anomaly model, and two statistical anomaly models for Te and τ. The atmosphere component is a simple empirical statistical model for the τ anomaly, which depicts the response of τ to an SST field. It is constructed by the SVD method, based on historical data of the SST and τ; symbolically, the relation between these two anomalous fields is expressed as τ=ατ Fτ ( SST inter), in which Fτ is the relationship between τ and SST inter derived using statistical methods from historical data, and ατ is a scalar parameter indicating the strength of wind forcing. The combined SVD method is used to obtain the covariance between the SST and zonal and meridional τ fields. The seasonality of interannual τ variability is taken into account with 12 τ models constructed for each month. The τ field is then used to drive the ocean model.

    The ocean component of the ICM includes a dynamical ocean model, an SST anomaly model, and a statistical model of Te. The dynamical ocean model was developed by (Keenlyside and Kleeman, 2002), based on the (McCreary, 1981) baroclinic model. It includes linear and nonlinear parts. In the vertical direction, the modal decomposition approach is adopted to solve the linear part, which retains the first 10 baroclinic modes, whereas the higher 11 to 30 modes are represented only in the two surface layers. The nonlinear part is highly simplified and represented as the residual term in the momentum equation and is used to make a correction to the linear solutions that are ignored by the linear assumption that can be broken down in the equatorial region. It is worth noting that by introducing the horizontal stratification variation and partial nonlinear effects, the dynamic ocean model can simulate features of the actual equatorial current system well, such as the equatorial undercurrent and surface current and their seasonal variability (Keenlyside and Kleeman, 2002).

    The SST anomaly model, which is embedded in the dynamical ocean model, describes the evolution of interannual temperature anomalies over the surface mixed layer. The time tendency of the SST anomaly is determined by its horizontal advection and diffusion terms, vertical advection and diffusion terms and thermal dissipation. The diagnostic analysis of the SST budget demonstrates that the vertical advection and diffusion terms (which are related to the T e anomaly) are important in determining the variation in the SST anomaly. The SST anomaly model is equipped with a parameterization for T e that is diagnosed by the sea level anomaly (SL inter) field based on an EOF. The relationship between the T e anomaly (T' e) and SL inter can be written as T'e TeF Te (SL inter), in which F Te is the relationship between T' e and SL inter derived using statistical methods from historical data, and α Te is a scalar parameter introduced as the subsurface thermal forcing strength.

    For each time step, the integration of the ICM can be sequentially implemented as follows (Zhang et al., 2005a): First, the SST anomaly equation is integrated to update the SST anomaly, which is used to calculate the τ anomaly based on the τ model. Second, the obtained τ anomaly field is used as the forcing to drive the dynamic ocean to update the SL, current fields in the mixed layer, and vertical velocity at the bottom of the mixed layer. Third, the T e anomaly is calculated using the updated SL anomaly based on the T e model, which is then used to simulate the vertical thermal effect in the SST anomaly equation. Repeating these processes can provide interannual variations of the oceanic and atmospheric wind fields. Further details regarding the ICM can be found in the study by Zhang and Gao (2015).

  • The 4D-Var method achieves the analysis solution of initial fields through minimizing the distance between the model trajectory and observation, which is constrained strictly by the model dynamical equations (Klinker et al., 2000).

    In general, the governing equations of the ICM can be symbolically expressed as follows (Kalnay, 2003): \begin{equation} \begin{array}{rcl} \dfrac{\partial{X}}{\partial t}&=&F({X}) ,\\[3mm] {X}|_{t_0}&=&{X}_0 , \end{array} (1)\end{equation} where t is time and t0 is the initial time; X is the vector of control variables, which includes SST, SL and horizontal ocean current velocities (U and V) in the ICM; X0 is the initial value of X; and F is the nonlinear forward operator.

    For the 4D-Var algorithm, the cost function can be formulated as (Kalnay, 2003) \begin{eqnarray} J({X}_0)&=&\dfrac{1}{2}[{X}(t_0)-{X}_{b}]^{T}{B}^{-1}[{X}(t_0)-{X}_{b}]+\nonumber\\ &&\dfrac{1}{2}\sum_{i=1}^N\{{H}[{X}(t_i)]-{Y}_{o}(t_i)\}^{T}{R}^{-1}\{{H}[{X}(t_i)]-{Y}_{o}(t_i)\} (2)\end{eqnarray} where the superscript "T" represents the transpose of a matrix and subscripts "b" and "o" represent the background field and observation, respectively; N indicates the number of integrations in the minimization time window; Yo represents the observation; and B, R and H represent the background error covariance matrix, the observation error covariance matrix and the observation operator, respectively. In this study, B and R are simply set as the identity matrix multiplied by the standard deviation of the observational error.

    An optimization algorithm is needed to obtain the optimal solutions. The input arguments of an optimization algorithm include the initial guess and the number of control variables, the cost function and the gradient of the cost function with respect to the control variable. The computation of the gradient of the cost function involves the backward integration of the adjoint model. Mathematically, if we consider the adjoint model as an operator, the adjoint model is the transpose of the tangent linear model that is the linearization of the nonlinear forward model. Whether an optimization algorithm can correctly yield an analysis solution depends on the accuracy of the gradient. Thus, it is necessary to examine the accuracy of the gradient computed by the adjoint model. At this point, the tangent linear model is an effective tool to perform the abovementioned verification. In this section, we simply introduce the tangent linear model and the adjoint model of the ICM, as well as the optimization algorithm used in this study.

    2.2.1. The tangent linear model

    The tangent linear model results from the linearization of the original nonlinear model. The model is not directly involved in the 4D-Var data assimilation procedure, but it is helpful for developing the adjoint model and testing whether the adjoint model is correct.

    The tangent linear model of the ICM can be expressed as (Kalnay, 2003) \begin{equation} \begin{array}{rcl} \dfrac{\partial{X}'}{\partial t}&=&\dfrac{\partial F({X})}{\partial{X}}{X}'={M}({X}){X}' ,\\[3mm] {X}'|_{t_0}&=&{X}'_0 , \end{array} (3)\end{equation} where X' is a small perturbation vector of X and M(X)=∂ F(X)/∂X is the tangent linear operator of F, which is a first-order approximation.

    To verify whether the established tangent linear model of the ICM is correct, one can use a formula based on the first-order approximation as follows (Navon et al., 1992): \begin{equation} {RV}=\dfrac{\|F({X}+\delta{X}')-F({X})\|}{\delta\|{M}({X},{X}')\|}={\bf 1}+O(\delta) , (4)\end{equation} where ||・|| is the L2-norm; δ is a small value ranging from 0 to 1, and O(δ) is the high-order small perturbation. RV is the ratio of the differences between the ICM variable tendencies caused by a small perturbation in δ to the perturbation calculated by the tangent linear model, which ideally should approach 1. The test results (double precision) of the tangent linear model in association with the 4D-Var method based on the ICM are shown in Table 1. As δ gradually decreases by one order of magnitude from 10-1 to 10-5, the value of RV consistently approaches 1. It should also be noted that when δ is too small, e.g., with a decrease by one order of magnitude from 10-6 to 10-10, the value of RV conversely becomes slightly larger, which is the result of a truncation error. Thus, it is evident that the established tangent linear model of the ICM is correct.

    2.2.2. The adjoint model

    Generally, the adjoint model is an efficient solution for evaluating the gradient of the cost function with respect to high-dimensional control variables in the 4D-Var data assimilation method. The model is the transpose of the tangent linear model, i.e., it features the reverse of the temporal and spatial integration and other characteristics.

    The equations of the adjoint model of the ICM can be written as follows (Kalnay, 2003): \begin{equation} \begin{array}{rcl} -\dfrac{\partial{X}^*}{\partial t}=\left(\dfrac{\partial F({X})}{\partial {X}}\right)^{T}{X}^\ast={M}^{T}{X}^\ast={M}^\ast{X}^\ast ,\\ {X}^\ast|_{t=N}&=&0 , \end{array} (5)\end{equation} where X* is the adjoint of X and M*=(∂ F(X)/∂X) T=M T is the adjoint of M, which is the tangent linear model of the ICM. The gradient of the cost function is obtained by a backward integration of the adjoint model.

    Based on the relationship between the tangent linear model and the adjoint model, one can verify the accuracy of the adjoint model using the following formula (Navon et al., 1992): \begin{equation} \langle{MX}_0,{MX}_0\rangle=\langle{M}^\ast{MX}_0,{X}_0\rangle , (6)\end{equation}

    where \(\langle,\rangle\) represents the inner product between the two vectors. For the LHS of Eq. (6), the tangent linear model is integrated forward using the initial condition X0 to obtain MX0, which is then used to compute its own inner product. For the RHS of Eq. (6), the adjoint model is integrated from the initial condition MX0 to obtain M*MX0, which is used to compute the inner product with the initial condition X0. Then, how one inner equals the other can be checked with a given precision.

    Following the above-described approach, we perform a set of sensitivity experiments to demonstrate how the accuracy of the adjoint model of the ICM is affected by the length of the assimilation time window in the 4D-Var data assimilation process. Table 2 presents the test results for the experiments, obtained using different assimilation time windows (days). The results show that at least the first 10 valid digits of \(\langle{MX_0},MX_0\rangle\) are equal to those of \(\langle{M}^\ast{MX_0},X_0\rangle\) when using different assimilation time windows, indicating that the adjoint model is accurate. Additionally, as the length of the assimilation window becomes longer from 4 days to 28 days, the equal valid digits become shorter from 12 to 10; this is because the nonlinearity becomes stronger as the length of the assimilation window becomes longer. Note that the experimental settings in the tangent linear model and adjoint model must remain the same as in the original nonlinear model, including the resolution, time step, physical processes and simplified dynamics.

    2.2.3. The minimization procedure

    After the adjoint model of the ICM is properly constructed, a minimization algorithm is used to find the 4D-Var analysis solution. First, the ICM model is integrated forward from an initial guess of X0 to obtain the cost function J. Second, the ICM is integrated backward with the adjoint model to obtain the gradient of J with respect to X0. Third, the Limited-Memory BFGS (L-BFGS) algorithm (Liu and Nocedal, 1989) is used to minimize the cost function to obtain the analysis solution of X0 (Zou et al., 1993). The L-BFGS algorithm is an improved version of the BFGS algorithm, which is a quasi-Newton algorithm. The L-BFGS requires four input arguments: an initial guess for the value of X0, the dimension of X0, the cost function J, and the gradient of the cost function with respect to X0.

    An example of the convergence of the cost function with respect to the iteration number is shown in Fig. 1. The figure shows that the cost function rapidly reaches equilibrium after four iterations. Thus, the 4D-Var based on the ICM is efficient and reliable. To save on computational cost, we set the maximum value of the iteration number to 20, which is sufficient to satisfy the convergence of the cost function. At this point, the 4D-Var data assimilation based on the ICM has been established.

3. Assimilation experiments
  • To partly reflect reality, we design a biased twin experiment (Zhang et al., 2014; Zhang et al., 2015a; Wu et al., 2016) to test the 4D-Var method, and report the results in this section. The twin experiment includes the observing network, model error and the assimilation schemes. Note that the model settings are the same for all data assimilation and prediction experiments, which prevents the "initial shock" that can result from the inconsistency of the coupled model physics and initial conditions in the transition from the assimilation phase to the prediction phase using the ICM (Keenlyside et al., 2005).

  • We assume that only the SST anomaly is observed and sampled once a day from the "truth" model that takes the default values of model parameters. The observed position of the SST anomaly is assumed to be the same as that of the model grids. To simulate the observational error, Gaussian noise with a mean and standard deviation of 0 and 0.2 is added to the sampled "truth" daily SST anomaly.

    For the assimilation model, we assume that the model error only arises from parameter perturbations to roughly mimic the model error in the real situation. The default values of three model parameters are modified: the coupling coefficient between the SST anomaly and τ anomaly, ατ, which is varied from 1.03 to 1.03× 1.01; the vertical diffusivity coefficient, Kv, which is varied from 1.0× 10-3 to 1.0× 10-3× 0.95; and the thermal damping coefficient, Λ, which is varied from 1/(100× 86400) to 1/(100× 86400)× 1.01. The modifications of these parameters cause the trajectories of the assimilation model to depart from those of the "truth" model. However, the basic ENSO features simulated by the assimilation model, such as the spatiotemporal structure and the amplitude, remain unchanged. Starting from the same initial conditions, these two simulations are respectively conducted for 200 model years. Note that each model calendar month is assumed to have 30 days in this study. Figure 2a shows the time series of the Niño3.4 indices for the "truth" model and the assimilation model in the first 100-year simulations. It is clear that both model simulations can simulate the prominent ENSO features. Note that the stochastic forcing of the atmospheric wind field is not included in the ICM; thus the ENSO events depicted by the ICM are quite regular (Zhang et al., 2008; Zhang and Gao, 2015). In addition, even though the two model simulations start from the same initial conditions, the simulated Niño3.4 indices gradually depart from each other. To detect the significant periods of ENSO events produced by the two simulations, we perform a power spectrum analysis of the Niño3.4 indices with the total 200-year outputs (Fig. 2b). The results indicate that both model simulations have a 2-7-year period that passes the 95% confidence level. The "truth" model has a dominant period of 3.81 years, whereas the assimilation model has a dominant period of 3.92 years.

    Figure 1.  Variation in the cost function with respect to the iteration number. Here, the cost function is defined as the sum of the background error term and observational error term, which decreases rapidly and converges to a constant value.

    Figure 2.  (a) Time series of the Niño3.4 indices for the "truth" model (blue) and the assimilation model (red) in the first 100-year simulations. (b) Power spectrum analysis of the Niño3.4 indices during the first 200-year simulations for the "truth" model (blue) and the assimilation model (red), with the 95% confidence level indicated by the dashed curve.

    Figure 3.  Schematic diagram illustrating the process of the 4D-Var data assimilation. The "truth" model and the assimilation model are integrated from the same initial condition (restart0), and the models gradually diverge from each other. Then, the "truth" model is integrated for 20 years from restart1 to sample "observations", which are assimilated into the assimilation model with 4D-Var to obtain the optimal initial condition for ENSO prediction.

  • In this study, three experiments are conducted to evaluate the 4D-Var data assimilation method based on the ICM: Expt. 1 is the control experiment of the "truth" model, used to produce the "observation" field; Expt. 2 is the 4D-Var assimilation experiment, which provides the optimal initial conditions by assimilating the "observation" of the SST anomaly; and Expt. 3 is the non-assimilation experiment of the assimilation model. The simulation period comprises 20 years from model time 2080/01/01 to 2099/12/30.

    Figure 3 is a schematic diagram illustrating the experimental configuration for Expt. 2. From the initial condition (restart0) at model time 2000/01/01 (which is represented as 1 January 2000, in the model time), both the "truth" model and the assimilation model are integrated for 80 years. It is clear that the two simulations diverge from each other (see Fig. 2a). Then, the "truth" model is further integrated for 20 years from restart1 at model time 2080/01/01 to generate the "observations". The assimilation model is then integrated forward by assimilating "observations" from restart2 at model time 2080/01/01. In this 4D-Var data assimilation process, the "observed" SST anomaly is assimilated into the assimilation model at the first step of every day. The length of the minimization time window in this 4D-Var is determined by trial and error. In this study, considering the nonlinearity effect and the computational efficiency, the minimization time window is set to 15 days in length. For example, starting from restart2 at model time 2080/01/01, the assimilation model is subject to assimilating daily "observations" of the SST anomaly within a 15-day window to obtain the optimal initial condition at model time 2080/01/01. The assimilation model is then integrated with the optimal initial condition until model time 2080/01/16 to enter the next data assimilation cycle. Thus, each month has two data assimilation cycles.

    Figure 4.  Time series of RMSEs for prior anomalies of (a) SST (units: $^\circ$C), (b) zonal $\tau$ (units: dyn cm$^-2$) and (c) SL (units: cm) over the full tropical Pacific region (30$^\circ$N-30$^\circ$S, 124$^\circ$E-78$^\circ$W). Here, the RMSE is calculated on the first day of each month in the first 10-year simulations for the assimilation (red) and non-assimilation (blue) experiments.

    Figure 5.  Spatial distributions of RMSEs for prior anomalies of (a, f) SST (units: $^\circ$C), (b, g) zonal $\tau$ (units: dyn cm$^-2$), (c, h) meridional $\tau$ (units: dyn cm$^-2$), (d, i) SL (units: cm), and (e, j) $T_e$ (units: $^\circ$C). Here, the RMSE is calculated from results obtained for the first 20-year simulations for the assimilation (left panels) and non-assimilation (right panels) experiments.

    Figure 6.  Longitude-time sections of SST anomalies along the equator for the (a) "truth" value, (b) assimilation experiment and (c) non-assimilation experiment during the first 12-year simulations (model time period of 2080/01/01 to 2091/12/30). Contour interval: 0.5$^\circ$C.

    Figure 7.  As in Fig. 6 but for zonal wind anomalies. Contour interval: 0.1 dyn cm$^-2$. 1 dyn cm$^-2$ = 0.1 N m$^-2$

    Figure 8.  As in Fig. 6 but for $T_e$ anomalies. Contour interval: 1$^\circ$C.

    Figure 9.  (a) Time series of the Niño3.4 indices (units: $^\circ$C) for the "truth" value (green), assimilation experiment (red) and non-assimilation experiment (blue) during the first 10-year simulations. (b) Time series of the absolute errors of the Niño3.4 indices (unit: $^\circ$) for the assimilation (red) and non-assimilation (blue) experiment during the first 10-year simulations.

    The key measure for assessing assimilation quality is the prior RMSE, which is defined as \begin{equation} \label{eq6} {RMSE}=\sqrt{\dfrac{1}{G}\sum_{i=1}^G({X}_i-{X}_{{truth}_i})^2} , (7)\end{equation} where X is the control vector; X truth is the corresponding "truth" vector obtained from Expt. 1; i is the grid index; and G is the total number of model grids. When the RMSE falls to a value that changes only slightly, the assimilation method is considered to have a converged solution. The assimilation period is chosen to have 20 model years. The results show that the spin-up period of the state estimation is approximately 2 years.

4. Assessing the impacts of data assimilation
  • The principles of the twin experiment are introduced in section 3, where the assimilation model is assimilated with the "observation" field to retrieve the analysis solution. To assess the success of this 4D-Var data assimilation method based on the ICM, the focus will be on ENSO phenomena when performing the twin experiment. In this section, the effect of assimilation on ENSO analysis is demonstrated first, and that on ENSO prediction is then discussed.

  • In this section, we first check the time series of the RMSEs of several key variables. Figure 4 shows the time series of RMSEs for prior anomalies of SST, zonal τ and SL over the full tropical Pacific region (30°N-30°S, 124°E-78°W) for Expt. 2 and Expt. 3. Being directly assimilated, the RMSE of the SST anomaly (Fig. 4a) is rapidly reduced. Because the τ anomaly is directly diagnosed from the SST anomaly using its SVD model, it is the first beneficiary of the assimilation of the SST anomaly. The RMSE of the zonal wind anomaly (Fig. 4b) is rapidly reduced, similar to the RMSE of the SST anomaly. In contrast, the SL anomaly is indirectly affected by the SST anomaly assimilation, causing most RMSEs of the SL anomaly produced in Expt. 2 to be smaller than those produced in Expt. 3 (Fig. 4c). All the RMSEs indicate that assimilating the "observations" of the SST anomaly into the ICM by 4D-Var can improve the model state estimate, thus being able to provide optimal initial conditions.

    Secondly, we examine the spatial RMSEs for Expt. 2 and Expt. 3. Figure 5 plots the spatial distributions of RMSEs for prior anomalies of SST, zonal and meridional τ, SL and Te for Expt. 2 and Expt. 3. The RMSE here for each grid is calculated as follows: \begin{equation} \label{eq7} {RMSE}_{i,j}=\sqrt{\dfrac{1}{N}\sum_{t=1}^N({X}_{i,j,t}-{X}_{{truth}_{i,j,t}})^2} , (8)\end{equation} where X represents the vector of the anomaly variables, including SST, zonal and meridional τ, SL and Te; X truth is the corresponding "truth" value of X; i and j represent the (i,j) grid; t is the time index; and N is the total number of analysis times. The RMSEs of all variables for Expt. 2 (Figs. 5a-e) are much smaller than those for Expt. 3 (Figs. 5f-j), but the spatial patterns are quite similar. For the SST anomaly, both maximum RMSEs [0.15°C for Expt. 2 (Fig. 5a) and 0.5°C for Expt. 3 (Fig. 5f)] are centered in the eastern and central equatorial Pacific. For the zonal τ anomaly, the maximum RMSEs are located in the central equatorial Pacific, with values of 0.027 dyn cm-2 and 0.16 dyn cm-2 for Expt. 2 (Fig. 5b) and Expt. 3 (Fig. 5g), respectively. Similar results are obtained for the meridional τ anomaly (Figs. 5c and h), SL anomaly (Figs. 5d and i) and Te anomaly (Figs. 5e and j). Generally speaking, the differences in the RMSEs of the SST and τ (zonal and meridional components) anomalies between Expt. 2 and Expt. 3 are much larger than those of the Te and SL anomalies. The reason is the fact that only the "observations" of the SST anomaly are assimilated in Expt. 2 and τ anomalies are directly calculated from the SST anomaly field using the τ model. Thus, the assimilation process has a direct effect on the SST anomaly field and thereby on the τ anomaly field. In contrast, the SL and Te fields are indirectly impacted by the SST anomaly assimilation through the model physical processes. In general, the RMSEs produced by Expt. 2 are slightly smaller than those produced by Expt. 3. These results demonstrate that the 4D-Var method can effectively reduce the error in the initial conditions, thereby leading to more accurate state estimations for ENSO events.

    Thirdly, we check the temporal evolution of the SST and τ anomalies. Figure 6 shows the longitude-time sections of the SST anomalies along the equator during the first 12-year simulations (model time from 2080/01/01 to 2091/12/30) for the "truth" fields, Expt. 2 and Expt. 3. It can be seen that the ENSO period, spatial structure and phase transition are well represented in the ICM. Excluding the spin-up period of 4D-Var, Expt. 2 (Fig. 6b) can retain nearly the same variability of the SST anomaly as in the "truth" simulation (Fig. 6a). For Expt. 3 (Fig. 6c), the biases arise from the initial conditions and the three model parameter perturbations cause the modeled SST anomaly to differ greatly from the "truth" field. For example, the amplitude of the modeled SST anomaly exhibits some bias, especially in the eastern Pacific. Additionally, the phase transition time of the SST anomaly also differs from the "truth" value. Similar to Fig. 6, Fig. 7 illustrates the longitude-time sections of zonal τ anomalies along the equator. The spatiotemporal structure and amplitude of the zonal τ anomaly produced by Expt. 2 (Fig. 7b) are much more consistent with the "truth" field than those produced by Expt. 3 (Fig. 7c).

    The ocean subsurface fields play an important role in the development of the ENSO events. To capture the ENSO events, it is necessary to adequately depict the Te field. Figure 8 shows the longitude-time sections of Te anomalies along the equator for the "truth" fields, Expt. 2 and Expt. 3. Through the model adjustment achieved by assimilating "observations" of the SST anomaly, the spatiotemporal evolution of Te produced by Expt. 2 (Fig. 7b) is in good agreement with the "truth" field (Fig. 7a) compared with that produced by Expt. 3 (Fig. 7c).

    Finally, we check the analysis quality of ENSO produced by 4D-Var by taking the Niño3.4 index as the key parameter of ENSO. Figure 9a shows the time series of the Niño3.4 indices during the first 10-year simulations for the "truth" value, Expt. 2 and Expt. 3. It can be seen that Expt. 2 (red dashed) can keep tracking the "truth" value (green dotted) very well, whereas Expt. 3 (blue dashed) shows some deviation. For clarity, the time series of the absolute errors of the Niño3.4 indices in Expt. 2 and Expt. 3 are presented in Fig. 9b. It is evident that the absolute error produced by Expt. 2 is much smaller than that produced by Expt. 3. Furthermore, the absolute error produced by Expt. 3 gradually becomes larger (even reaching approximately 1.2°C) due to the existence of model error. This finding again demonstrates that the 4D-var data assimilation can recover ENSO conditions well. Thus, the high level of agreement between the assimilation results and the "truth" value can provide a better initialization for ENSO prediction.

  • In general, a better prediction of ENSO events is a strict test for model simulation and analysis through data assimilation. Therefore, improved prediction accuracy is an important indicator for assessing the quality of the 4D-Var data assimilation approach. Based on state estimation with a 2-year spin-up period, we perform an array of 1-year forecast experiments starting from the analysis solutions on the first day in each month between the model time 2082/01/01 and 2099/12/01. Thus, there are 18× 12=216 forecast experiments in total, which are used to perform the statistical analysis. The prediction results with and without data assimilation are compared below.

    Figure 10.  Time series of the Niño3.4 indices for the "truth" value (green) and predictions made at 12-month lead times using initial conditions with (red) and without (blue) data assimilation during the model time period 2083/01 to 2099/12.

    Figure 10 presents the time series of the Niño3.4 indices for the "truth" value and predictions made at 12-month lead times using initial conditions with and without data assimilation during the model time period 2083/01-2099/12. The Niño3.4 indices in the assimilation case are very close to the "truth" value, whereas those in the non-assimilation case depart to a certain extent from the "truth" value. The correlation coefficient between the "truth" and the predicted Niño3.4 index in the assimilation case is 0.99, whereas that between the "truth" and the predicted Niño3.4 index in the non-assimilation case is 0.84. The RMSEs of the predicted Niño3.4 index for the assimilation and non-assimilation cases are 0.05 and 0.66 in the 1-year lead time. The results are likely idealized to a certain extent because they are evaluated in a twin experiment, but these experiments provide us with important information about the way the 4D-Var data assimilation approach can effectively improve the model state estimation and prediction of ENSO events using the ICM.

5. Conclusion and discussion
  • Data assimilation is an effective way to improve the accuracy of model simulations and analyses for weather and climate through an optimal combination of model solutions and observations. In particular, the advanced 4D-Var data assimilation method is more dynamically and mathematically consistent in constraining numerical models with observations to achieve the optimal initialization for ENSO analysis and prediction. In this study, we implement the 4D-Var method based on an improved ICM that has been routinely used for real-time ENSO prediction. The construction of the 4D-Var assimilation system includes the tangent linear model and adjoint model of the ICM and a minimization procedure. Strict testing justifies the accuracy of the adjoint model and the effectiveness of the 4D-Var in constraining dynamical models with observations.

    The impacts of the optimal initialization produced by 4D-Var on ENSO analysis and prediction are evaluated through a biased twin experiment. In this study, only "observations" of the SST anomaly are assimilated into the model to optimize the initial conditions. Results show that, compared with the non-assimilation case, the assimilation results are more consistent with the "truth" value, and the RMSEs of the anomalies for the SST, τ, SL and Te fields are much smaller (especially for the SST and τ fields). Additionally, the prediction accuracy is improved by optimizing the initial conditions. The results obtained in this study provide some insight into the way in which ENSO prediction can be improved with the 4D-Var algorithm.

    The work performed in this study is a first step towards improving real-time ENSO analysis and prediction by applying the 4D-Var algorithm in the ICM. Further modeling studies using the 4D-Var are underway. As noted above, the ICM has been successfully used for real-time ENSO prediction, whose result, now named IOCAS ICM, is collected and posted every month at IRI/Columbia University, a multi-model product for real-time ENSO monitoring and prediction (see the IRI website). In this application, however, no sophisticated data assimilation is applied in the ICM; instead, a simple initialization method is currently taken for the model forecast, as follows: The observed interannual SST anomalies are the only field used in the prediction initialization (Zhang et al., 2013). In real-time practice, experimental predictions are typically conducted near the middle of each month, when the monthly mean SST fields from the previous month and the weekly mean SST fields from the first week of the current month are available from NOAA's Environment Modeling Center (Reynolds et al., 2002), which can be obtained online from the IRI data library. Then, the observed SST anomalies are used to derive interannual τ inter fields using the empirical τ model. The derived τ inter fields are taken to force the ocean model to produce an initial ocean state for the first day of each month, from which predictions are made. Additionally, as part of the initialization procedure, the observed SST anomalies are directly inserted into the ICM when making predictions. Based on results from this paper, the 4D-Var method will be incorporated in the ICM for real-time ENSO predictions.

    Additionally, even without data assimilation, the forecasts using the ICM show a fairly high level of skill (Fig. 10, blue line) because the ENSO events simulated are so regular. This is attributed to the fact that stochastic atmospheric wind forcing is not included in the ICM (Zhang et al., 2008). In a more realistic global coupled climate model, however, the forecast skill of Niño 3.4 SST initialized by the SST-nudging scheme is very limited (Kumar et al., 2014; Zhu et al., 2015). In the future, we plan to assess the impact of the 4D-Var data assimilation in a more realistic way by including stochastic atmospheric forcing in the ICM, whose effects on ENSO simulations were evaluated by (Zhang et al., 2008).

    Furthermore, the 4D-Var method can also be used to optimize the model parameters, as demonstrated by the ensemble Kalman filter (Wu et al., 2012, 2016). For example, the performance of the ICM is sensitive to ατ and α Te (Zhang et al., 2005a; 2008); we plan to use 4D-Var to optimally determine these two parameters to further improve the ENSO prediction skill. In addition, the oceanic subsurface state has a considerable effect on SST in the tropical Pacific; thus, in addition to assimilating the observed SST field, observed subsurface thermal fields need to be assimilated into the ICM. In addition to the assimilation of oceanic fields, that of atmospheric data can also be considered. Note that during the 4D-Var assimilation process (the forward and backward time integrations of the model and its adjoint model), τ anomalies are internally determined using its anomaly model from the corresponding SST anomalies. Thus, the ICM with the 4D-Var has already taken into account the coupling between the ocean and atmosphere. So, the observed τ anomaly field can be introduced into the 4D-Var assimilation processes in a fairly straightforward way (that is, the coupled data assimilation). Taking all these together, it can ultimately be expected that real-time ENSO forecasting using the ICM can be improved through optimal initialization and parameter optimization using the 4D-Var data assimilation method.

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