doi:10.1007/s00376-016-6003-4

Anderson J. L., S. Anderson, 1999: A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts. Mon. Wea. Rev., 127, 2741- 2758.10.1175/1520-0493(1999)1272.0.CO;2

7055e7f275587995641eee2766ef174f

http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F1999MWRv..127.2741A

http://adsabs.harvard.edu/abs/1999MWRv..127.2741ANot Available

Barkmeijer J., 1992: Local error growth in a barotropic model. Tellus A, 44, 314- 323.10.1034/j.1600-0870.1992.t01-3-00003.x

8649328541d811287afd9fe2b6c14f1b

http%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1034%2Fj.1600-0870.1992.t01-3-00003.x%2Fabstract

http://onlinelibrary.wiley.com/doi/10.1034/j.1600-0870.1992.t01-3-00003.x/abstractABSTRACT The concept of local error growth is considered within the context of the non-viscous barotropic vorticity equation. Using the adjoint equation of the tangent linear barotropic vorticity equation, we determine error patterns in the initial state that result in the maximum linear error growth at some pre-chosen geographical area and prediction time. For some solutions of the nonlinear barotropic vorticity equation, we investigate to what extent error growth depends on the geographical position of the area considered. It appears that for short prediction times (<3days), the barotropic growth of kinetic energy of initial perturbations is exponential plus a substantial transient effect due to the non-orthogonality of the eigenmodes of the tangent linear operator with respect to the kinetic energy inner product. The latter mechanism initially causes very rapid perturbation growth, even for neutral flows.

Benettin G., L. Galgani, A. Giorgilli, and J. M. Strelcyn, 1980: Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: Theory. Meccanica, 15, 9- 20.10.1007/BF02128237

db41e98c-0da1-4c52-b2cf-23ee28e503d5

c375b5d107dc0d1180d8ef1a5a457fdc

http%3A%2F%2Flink.springer.com%2F10.1007%2Fbf02128237

refpaperuri:(3cc90f68f01107a05a62349e29a08786)

http://link.springer.com/10.1007/bf02128237Not Available

Bishop C. H., B. J. Etherton, and S. J. Majumdar, 2001: Adaptive sampling with the ensemble transform Kalman filter. Part I: theoretical aspects. Mon. Wea. Rev., 129, 420- 436.24610b9bb142a4812e0dd6e3b446b34c

http%3A%2F%2Fnsr.oxfordjournals.org%2Fexternal-ref%3Faccess_num%3D10.1175%2F1520-0493%282001%291292.0.CO%3B2%26link_type%3DDOI

http://nsr.oxfordjournals.org/external-ref?access_num=10.1175/1520-0493(2001)1292.0.CO;2&link_type=DOI

Bowler N. E., 2006: Comparison of error breeding, singular vectors, random perturbations and ensemble Kalman filter perturbation strategies on a simple model. Tellus A, 58, 538- 548.10.1111/j.1600-0870.2006.00197.x

5ce8a377-8e2f-4e3d-8104-0ece17c58e4d

d598fbc96ab13e338f9c45e1fc026536

http%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1111%2Fj.1600-0870.2006.00197.x%2Fcitedby

refpaperuri:(4a9b947e82663979db74e48166973349)

http://onlinelibrary.wiley.com/doi/10.1111/j.1600-0870.2006.00197.x/citedbyABSTRACT An experiment has been performed, using a simple chaotic model, to compare different ensemble perturbation strategies. The model used is a 300 variable Lorenz 95 model which displays many of the characteristics of atmospheric numerical weather prediction models. Twenty member ensembles were generated using five perturbation strategies, error breeding, singular vectors, random perturbations (RPs), the Ensemble Kalman Filter (EnKF) and the Ensemble Transform Kalman Filter (ETKF). Based on normal verification methods, such as rank histograms and spread of the perturbations, the RPs method performs as well as any other methodhis illustrates the limitations of using a simple model. Consideration of the quality of the background error information provided by the ensemble gives a better assessment of the ensemble skill. This measure indicates that the EnKF performs best, with the ETKF combined with RPs being the next most skillful. It was found that neither the ETKF, error breeding nor singular vectors provided useful background information on their own.Central to the success of the EnKF is the localization of the background error covariance which removes spurious long-range correlations within the ensemble. Computationally efficient versions of the EnKF (such as the ETKF) cannot accommodate covariance localization and their performance is seen to suffer. Applying the ETKF to a series of local domains has been tested, which allows covariance localization whilst remaining computational efficient, and this has been found to be nearly as effective as the EnKF with covariance localization.

Buizza R., P. L. Houtekamer, Z. Toth, G. Pellerin, M. Z, Wei, and Y. J. Zhu, 2005: A comparison of the ECMWF, MSC, and NCEP global ensemble prediction systems. Mon. Wea. Rev., 133, 1076- 1097.d6ae0c6ecff394896dd321deb3e191ab

http%3A%2F%2Fbioscience.oxfordjournals.org%2Fexternal-ref%3Faccess_num%3D10.1175%2FMWR2905.1%26link_type%3DDOI

http://bioscience.oxfordjournals.org/external-ref?access_num=10.1175/MWR2905.1&link_type=DOI

Carrassi A., A. Trevisan, and F. Uboldi, 2007: Adaptive observations and assimilation in the unstable subspace by breeding on the data assimilation system. Tellus A, 59( 2), 101- 113.10.1111/j.1600-0870.2006.00210.x

8187068327f4c0d19c1019f197958bf6

http%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1111%2Fj.1600-0870.2006.00210.x%2Fcitedby

http://onlinelibrary.wiley.com/doi/10.1111/j.1600-0870.2006.00210.x/citedbyResults of targeting and assimilation experiments in a quasi-geostrophic atmospheric model are presented and discussed. The basic idea is to assimilate observations in the unstable subspace (AUS) of the data-assimilation system. The unstable subspace is estimated by breeding on the data-assimilation system (BDAS). The analysis update has the same local structure as the observationally fozrced bred modes.

Cheung K. K. W., J. C. L. Chan, 1999: Ensemble forecasting of tropical cyclone motion using a barotropic model. Part I: perturbations of the environment. Mon. Wea. Rev., 127, 1229- 1243.

Descamps L., O. Talagrand, 2007: On some aspects of the definition of initial conditions for ensemble prediction. Mon. Wea. Rev., 135, 3260- 3272.10.1175/MWR3452.1

fe879b2caf3a0f47038475377af0a3a5

http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F2007MWRv..135.3260D

http://adsabs.harvard.edu/abs/2007MWRv..135.3260DFour methods for initialization of ensemble forecasts are systematically compared, namely the methods of singular vectors (SV) and bred modes (BM), as well as the ensemble Kalman filter (EnKF) and the ensemble transform Kalman filter (ETKF). The comparison is done on synthetic data with two models of the flow, namely, a low-order model introduced by Lorenz and a three-level quasigeostrophic atmospheric model. For the latter, both cases of a perfect and an imperfect model are considered. The performance of the various initialization methods is assessed in terms of the statistical reliability and resolution of the ensuing predictions. The relative performance of the four methods, which is statistically significant to a range of about 6 days, is in the order EnKF > ETKF > BM > SV. The difference between the former two methods and the latter two is on the whole more significant than the differences between EnKF and ETKF, or between BM and SV separately. The general conclusion is that, if the quality of ensemble predictions is assessed by the degree to which the predicted ensembles statistically sample the uncertainty on the future state of the flow, the best initial ensembles are those that best statistically sample the uncertainty on the present state of the flow.

Ding R. Q., J. P. Li, 2007: Nonlinear finite-time Lyapunov exponent and predictability. Physics Letters A, 364, 396- 400.10.1016/j.physleta.2006.11.094

c52a97486acf5c7f7ec380454abc48ff

http%3A%2F%2Fwww.sciencedirect.com%2Fscience%2Farticle%2Fpii%2FS037596010601944X

http://www.sciencedirect.com/science/article/pii/S037596010601944XABSTRACT In this Letter, we introduce a definition of the nonlinear finite-time Lyapunov exponent (FTLE), which is a nonlinear generalization to the existing local or finite-time Lyapunov exponents. With the nonlinear FTLE and its derivatives, the limit of dynamic predictability in large classes of chaotic systems can be efficiently and quantitatively determined.

Ding R. Q., J. P. Li, and K.-J. Ha, 2008: Trends and interdecadal changes of weather predictability during 1950s-1990s. J. Geophys. Res., 113,D24112, doi: 10.1029/2008JD010404.10.1029/2008JD010404

4e99cb4e725f7eaa1cf0c879ab0d4cf8

http%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1029%2F2008JD010404%2Fcitedby

http://onlinelibrary.wiley.com/doi/10.1029/2008JD010404/citedby[1] To study the atmospheric predictability from the view of nonlinear error growth dynamics, a new approach using the nonlinear local Lyapunov exponent (NLLE) is introduced by the authors recently. In this paper, the trends and interdecadal changes of weather predictability limit (WPL) during 1950s1990s are investigated by employing the NLLE approach. The results show that there exist significant trend changes for WPL over most of the globe. At three different pressure levels in the troposphere (850, 500, and 200 hPa), spatial distribution patterns of linear trend coefficients of WPL are similar. Significant decreasing trends in WPL could be found in the most regions of the northern midlatitudes and Africa, while significant increasing trends in WPL lie in the most regions of the tropical Pacific and southern mid-high latitudes. In the lower stratosphere (50 hPa), the WPL in the whole tropics all shows significant increasing trends, while it displays significant decreasing trends in the most regions of the Antarctic and northern mid-high latitudes. By examining the temporal variations of WPL in detail, we find that the interdecadal changes of WPL in most regions at different levels mainly happen in the 1970s, which is consistent with the significant climate shift occurring in the late 1970s. Trends and interdecadal changes of WPL are found to be well related to those of atmospheric persistence, which in turn are linked to the changes of atmospheric internal dynamics. Further analysis indicates that the changes of atmospheric static stability due to global warming might be one of main causes responsible for the trends and interdecadal changes of atmospheric persistence and predictability in the southern and northern mid-high latitudes. The increased sea surface temperature (SST) variability exerts a stronger external forcing on the tropical Pacific atmosphere that tends to enhance the persistence of tropical Pacific atmosphere. This process appears to be responsible for the increase of atmospheric predictability and persistence in the tropical Pacific since the late 1970s.

Ding R. Q., J. P. Li, and K.-H. Seo, 2010: Predictability of the Madden-Julian oscillation estimated using observational data. Mon. Wea. Rev., 138, 1004- 1013.10.1175/2009MWR3082.1

a63f25a7340ba12998736beda92a0aa8

http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F2010MWRv..138.1004D

http://adsabs.harvard.edu/abs/2010MWRv..138.1004DNot Available

Ding R. Q., J. P. Li, and K.-H. Seo, 2011: Estimate of the predictability of boreal summer and winter intraseasonal oscillations from observations. Mon. Wea. Rev., 139, 2421- 2438.10.1175/2011MWR3571.1

cfa82d458df50569729aae32846f7285

http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F2011MWRv..139.2421D

http://adsabs.harvard.edu/abs/2011MWRv..139.2421DNot Available

Duan W. S., Z. H. Huo, 2016: An approach to generating mutually independent initial perturbations for ensemble forecasts: orthogonal conditional nonlinear optimal perturbations. J. Atmos. Sci.,73, 997-1014, doi: 10.1175/JAS-D-15-0138.1.

Durran D. R., M. Gingrich, 2014: Atmospheric predictability: Why butterflies are not of practical importance.J. Atmos. Sci., 71, 2476- 2488.10.1175/JAS-D-14-0007.1

9622df5f1e1f43a0700833fd52981251

http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F2014JAtS...71.2476D

http://adsabs.harvard.edu/abs/2014JAtS...71.2476DNot Available

Epstein E. S., 1969: Stochastic dynamic prediction. Tellus, 21, 739- 759.10.3402/tellusa.v21i6.10143

46ad0ed3-01d9-4bec-a167-43471f2ad241

283fb6357090d5876a15eea2b10f272b

http://dx.doi.org/10.1111/j.2153-3490.1969.tb00483.x

http://dx.doi.org/10.1111/j.2153-3490.1969.tb00483.xThe question of analysis to obtain the initial stochastic statement of the atmospheric state is considered and one finds here too a promise of significant advantages over present deterministic methods. It is shown how the stochastic method can be used to assess the value of new or improved data by considering their influence on the decrease in the uncertainty of the forecast. Comparisons among physical-numerical models are also made more effectively by applying stochastic methods. Finally the implications of stochastic dynamic prediction on the question of predictability are briefly considered, with the conclusion that some earlier estimates have been too pessimistic.

Evensen G., 2003: The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean Dynamics, 53, 343- 367.10.1007/s10236-003-0036-9

fe06c7adae408eb1888b539a799f9cda

http%3A%2F%2Flink.springer.com%2F10.1007%2Fs10236-003-0036-9

http://link.springer.com/10.1007/s10236-003-0036-9The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

Evensen G., 2004: Sampling strategies and square root analysis schemes for the EnKF. Ocean Dynamics, 54, 539- 560.10.1007/s10236-004-0099-2

2cc4dfccac3482af78528fe442c005ba

http%3A%2F%2Flink.springer.com%2F10.1007%2Fs10236-004-0099-2

http://link.springer.com/10.1007/s10236-004-0099-2The purpose of this paper is to examine how different sampling strategies and implementations of the analysis scheme influence the quality of the results in the EnKF. It is shown that by selecting the initial ensemble, the model noise and the measurement perturbations wisely, it is possible to achieve a significant improvement in the EnKF results, using the same number of members in the ensemble. The results are also compared with a square root implementation of the EnKF analysis scheme where the analyzed ensemble is computed without the perturbation of measurements. It is shown that the measurement perturbations introduce sampling errors which can be reduced using improved sampling schemes in the standard EnKF or fully eliminated when the square root analysis algorithm is used. Further, a new computationally efficient square root algorithm is proposed which allows for the use of a low-rank representation of the measurement error covariance matrix. It is shown that this algorithm in fact solves the full problem at a low cost without introducing any new approximations.

Feng J., R. Q. Ding, D. Q. Liu, and J. P. Li, 2014: The application of nonlinear local Lyapunov vectors to ensemble predictions in the Lorenz systems.J. Atmos. Sci., 71( 10), 3554- 3567.10.1175/JAS-D-13-0270.1

fc4c34ac6a236575e4326e2c3b97a50c

http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F2014EGUGA..1616378F

http://adsabs.harvard.edu/abs/2014EGUGA..1616378FNonlinear local Lyapunov vectors (NLLVs) are developed to indicate orthogonal directions in phase space with different perturbation growth rates. In particular, the first few NLLVs are considered to be an appropriate orthogonal basis for the fast-growing subspace. In this paper, the NLLV method is used to generate initial perturbations and implement ensemble forecasts in simple nonlinear models (the Lorenz63 and Lorenz96 models) to explore the validity of the NLLV method. The performance of the NLLV method is compared comprehensively and systematically with other methods such as the bred vector (BV) and the random perturbation (Monte Carlo) methods. In experiments using the Lorenz63 model, the leading NLLV (LNLLV) captured a more precise direction, and with a faster growth rate, than any individual bred vector. It may be the larger projection on fastest-growing analysis errors that causes the improved performance of the new method. Regarding the Lorenz96 model, two practical measures, namely the spread/skill relationship and the Brier score, were used to assess the reliability and resolution of these ensemble schemes. Overall, the ensemble spread of NLLVs is more consistent with the errors of the ensemble mean, which indicates the better performance of NLLVs in simulating the evolution of analysis errors. In addition, the NLLVs perform significantly better than the BVs in terms of reliability and the random perturbations in resolution.

Fraedrich K., 1987: Estimating weather and climate predictability on attractors.J. Atmos. Sci., 44, 722- 728.10.1175/1520-0469(1987)044<0722:EWACPO>2.0.CO;2

0564d40859f80495b7f3f603fd36b4ce

http%3A%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D888056

http://www.ams.org/mathscinet-getitem?mr=888056Abstract Predictability is deduced from phase space trajectories of weather and climate variables which evolve on attractors (local surface pressure and a δ 18 O-record). Predictability can be defined by the divergence of initially close pieces of trajectories and estimated by the cumulative distance distributions of expanding pairs of points on the single variable trajectory. The e -folding expansion rates characterize predictability tune scales. As a first estimate one obtains a predictability time scale of about two weeks for the weather variable and 10–15 thousand years for the climate variable.

Gaspari G., S. E. Cohn, 1999: Construction of correlation functions in two and three dimensions.Quart. J. Roy. Meteor. Soc., 125, 723- 757.10.1002/qj.49712555417

6bd81eb7b6838de7499add8663f840e7

http%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1002%2Fqj.49712555417%2Ffull

http://onlinelibrary.wiley.com/doi/10.1002/qj.49712555417/fullAbstract This article focuses on the construction, directly in physical space, of simply parametrized covariance functions for data-assimilation applications. A self-contained, rigorous mathematical summary of relevant topics from correlation theory is provided as a foundation for this construction. Covariance and correlation functions are defined, and common notions of homogeneity and isotropy are clarified. Classical results are stated, and proven where instructive. Included are smoothness properties relevant to multivariate statistical-analysis algorithms where wind/wind and wind/mass correlation models are obtained by differentiating the correlation model of a mass variable. the Convolution Theorem is introduced as the primary tool used to construct classes of covariance and cross-covariance functions on three-dimensional Euclidean space R 3 . Among these are classes of compactly supported functions that restrict to covariance and cross-covariance functions on the unit sphere S 2 , and that vanish identically on subsets of positive measure on S 2 . It is shown that these covariance and cross-covariance functions on S 2 , referred to as being space-limited , cannot be obtained using truncated spectral expansions. Compactly supported and space-limited covariance functions determine sparse covariance matrices when evaluated on a grid, thereby easing computational burdens in atmospheric data-analysis algorithms. Convolution integrals leading to practical examples of compactly supported covariance and cross-covariance functions on R 3 are reduced and evaluated. More specifically, suppose that gi and gj are radially symmetric functions defined on R 3 such that gi (x) = 0 for |x| > di and gj (x) = 0 for |xv > dj , O < di,dj ≦, where |. | denotes Euclidean distance in R 3 . the parameters di and dj are ‘cut-off’ distances. Closed-form expressions are determined for classes of convolution cross-covariance functions Cij (x,y) := ( gi * gj )(x-y), i ≠ j , and convolution covariance functions Cii (x,y) := ( gi * gi )(x-y), vanishing for |x - y| > di + dj and |x - y| > 2 di , respectively, Additional covariance functions on R 3 are constructed using convolutions over the real numbers R , rather than R 3 . Families of compactly supported approximants to standard second- and third-order autoregressive functions are constructed as illustrative examples. Compactly supported covariance functions of the form C (x,y) := Co (|x - y|), x,y ∈ R 3 , where the functions Co ( r ) for r ∈ R are 5th-order piecewise rational functions, are also constructed. These functions are used to develop space-limited product covariance functions B (x, y) C (x, y), x, y ∈ S 2 , approximating given covariance functions B (x, y) supported on all of S 2 × S 2 .

Houtekamer P. L., H. L. Mitchell, 1998: Data assimilation using an ensemble Kalman filter technique. Mon. Wea. Rev., 126, 796- 811.10.1175/1520-0493(1998)126<0796:DAUAEK>2.0.CO;2

d3f8c1aa877d5eac5d73e53c0fdac200

http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F1998MWRv..126..796H

http://adsabs.harvard.edu/abs/1998MWRv..126..796HNot Available

Houtekamer P. L., H. L. Mitchell, 2001: A sequential ensemble Kalman filter for atmospheric data assimilation. Mon. Wea. Rev., 129, 123- 137.20195ce27846d743db66fe3982faf1b7

http%3A%2F%2Fbiomet.oxfordjournals.org%2Fexternal-ref%3Faccess_num%3D10.1175%2F1520-0493%282001%291292.0.CO%3B2%26link_type%3DDOI

http://biomet.oxfordjournals.org/external-ref?access_num=10.1175/1520-0493(2001)1292.0.CO;2&link_type=DOI

Kalnay E., 2002: Atmospheric Modeling,Data Assimilation and Predictability. Cambridge University Press, 221 pp.10.1198/tech.2005.s326

17d7fbb1594d0cade45315e4ab692760

http%3A%2F%2Fwww.jstor.org%2Fstable%2F25471085

http://www.jstor.org/stable/25471085CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Numerical weather prediction (NWP) now provides major guidance in our daily weather forecast. The accuracy of NWP models has improved steadily since the first successful experiment made by Charney, Fj!rtoft and von Neuman (1950). During the past 50 years, a large number of technical papers and reports have been devoted to NWP, but the number of textbooks dealing with the subject has been very small, the latest being the 1980 book by Haltiner & Williams, which was dedicated to descriptions of the atmospheric dynamics and numerical methods for atmospheric modeling. However, in the intervening years much impressive progress has been made in all aspects of NWP, including the success in model initialization and ensemble forecasts. Eugenia Kalnay recent book covers for the first time in the long history of NWP, not only methods for numerical modeling, but also the important related areas of data assimilation and predictability. It incorporates all aspects of environmental computer modeling including an historical overview of NWP, equations of motion and their approximations, a modern description of the methods to determine the initial conditions using weather observations and a clear discussion of chaos in dynamic systems and how these concepts can be

Lawrence A. R., M. Leutbecher, and T. N. Palmer, 2009: The characteristics of Hessian singular vectors using an advanced data assimilation scheme.Quart. J. Roy. Meteor. Soc., 135, 1117- 1132.10.1002/qj.447

eb65b6c81077bf94a8119707b7521a83

http%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1002%2Fqj.447%2Fcitedby

http://onlinelibrary.wiley.com/doi/10.1002/qj.447/citedbyNot Available

Leith C. E., 1974: Theoretical skill of Monte Carlo forecasts. Mon. Wea. Rev., 102, 409- 418.10.1175/1520-0493(1974)102<0409:TSOMCF>2.0.CO;2

86f6231bdfd8ee5b602d3893586fda21

http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F1974MWRv..102..409L

http://adsabs.harvard.edu/abs/1974MWRv..102..409LNot Available

Li J. P., J. F. Chou, 1997: Existence of the atmosphere attractor. Science in China Series D: Earth Sciences, 40( 3), 215- 220.10.1007/BF02878381

2190ea244556cc025d10cc855908ad41

http%3A%2F%2Flink.springer.com%2F10.1007%2FBF02878381

http://www.cnki.com.cn/Article/CJFDTotal-JDXG199702012.htm The global asymptotic behavior of solutions for the equations of large-scale atmospheric motion with the non-stationary external forcing is studied in the infinite-dimensional Hilbert space. Based on the properties of operators of the equations, some energy inequalities and the uniqueness theorem of solutions are obtained. On the assumption that external forces are bounded, the exsitence of the global absorbing set and the atmosphere attractor is proved, and the characteristics of the decay of effect of initial field and the adjustment to the external forcing are revealed. The physical sense of the results is discussed and some ideas about climatic numerical forecast are elucidated.

Li J. P., S. H. Wang, 2008: Some mathematical and numerical issues in geophysical fluid dynamics and climate dynamics. Communications in Computational Physics, 3, 759- 793.10.1007/978-1-4614-8963-4_5

9057e2d6-02ce-45d3-924c-e5444189700f

a48dc7c8a5503b67727eff1b53683535

http%3A%2F%2Fsourcedb.cas.cn%2Fsourcedb_iap_cas%2Fen%2Fpapers%2F200908%2Ft20090831_2459201.html

refpaperuri:(3061fd2c074e4ad9caba9a4235258fb0)

http://sourcedb.cas.cn/sourcedb_iap_cas/en/papers/200908/t20090831_2459201.htmlIn this article, we address both recent advances and open questions in some mathematical and computational issues in geophysical fluid dynamics (GFD) and climate dynamics. The main focus is on 1) the primitive equations (PEs) models and their related mathematical and computational issues, 2) climate variability, predictability and successive bifurcation, and 3) a new dynamical systems theory and its applications to GFD and climate dynamics.

Li J. P., R. Q. Ding, 2011: Temporal-spatial distribution of atmospheric predictability limit by local dynamical analogs.Mon. Wea. Rev., 139, 3265- 3283.10.1175/MWR-D-10-05020.1

94414680e18eb59458f83a2bb8f359ac

http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F2011mwrv..139.3265l

http://adsabs.harvard.edu/abs/2011mwrv..139.3265lAbstract To quantify the predictability limit of a chaotic system, the authors recently developed a method using the nonlinear local Lyapunov exponent (NLLE). The NLLE method provides a measure of local predictability limit of chaotic systems and is intended to supplement existing predictability methods. To apply the NLLE in studies of actual atmospheric predictability, an algorithm based on local dynamical analogs is devised to enable the estimation of the NLLE and its derivatives using experimental or observational data. Two examples are given to illustrate the effectiveness of the algorithm, involving the Lorenz63 three-variable model and the Lorenz96 forty-variable model; they reveal that the algorithm is applicable in estimating the NLLE of a chaotic system from its experimental time series. On this basis, the NLLE method is used to investigate temporalpatial distributions of predictability limits of the daily geopotential height and wind fields. The limit of atmospheric predictability varies widely with region, altitude, and season. The predictability limits of the daily geopotential height and wind fields are generally less than 3 weeks in the troposphere, whereas they are approximately 1 month in the lower stratosphere, revealing a potential predictability source for forecasting weather from the stratosphere. Further work is required to examine broader applications of the NLLE method in predictability studies of the atmosphere, ocean, and other systems.

Li J. P., R. Q. Ding, 2013: Temporal-spatial distribution of the predictability limit of monthly sea surface temperature in the global oceans. Int. J. Climatol., 33, 1936- 1947.10.1002/joc.3562

d591093830f02d95321208ab398e296a

http%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1002%2Fjoc.3562%2Fpdf

http://onlinelibrary.wiley.com/doi/10.1002/joc.3562/pdfNot Available

Li J. P., R. Q. Ding, and B. H. Chen, 2006: Review and Prospect on the Predictability Study of the Atmosphere. Review and Prospects of the Developments of Atmosphere Sciences in Early 21st Century. China Meteorology Press, 96- 104. (in Chinese)

Lorenz E. N., 1965: A study of the predictability of a 28-variable atmospheric model. Tellus, 17, 321- 333.10.1111/j.2153-3490.1965.tb01424.x

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http%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1111%2Fj.2153-3490.1965.tb01424.x%2Fabstract

http://onlinelibrary.wiley.com/doi/10.1111/j.2153-3490.1965.tb01424.x/abstractNot Available

Magnusson L., E. Källèn, and J. Nycander, 2008: Initial state perturbations in ensemble forecasting. Nonlinear Processes in Geophysics, 15, 751- 759.10.5194/npg-15-751-2008

c6dae7ce4c145a3213c8d22e0ca31101

http%3A%2F%2Fwww.oalib.com%2Fpaper%2F1377277

http://www.oalib.com/paper/1377277Due to the chaotic nature of atmospheric dynamics, numerical weather prediction systems are sensitive to errors in the initial conditions. To estimate the forecast uncertainty, forecast centres produce ensemble forecasts based on perturbed initial conditions. How to optimally perturb the initial conditions remains an open question and different methods are in use. One is the singular vector (SV) method, adapted by ECMWF, and another is the breeding vector (BV) method (previously used by NCEP). In this study we compare the two methods with a modified version of breeding vectors in a low-order dynamical system (Lorenz-63). We calculate the Empirical Orthogonal Functions (EOF) of the subspace spanned by the breeding vectors to obtain an orthogonal set of initial perturbations for the model. We will also use Normal Mode perturbations. Evaluating the results, we focus on the fastest growth of a perturbation. The results show a large improvement for the BV-EOF perturbations compared to the non-orthogonalised BV. The BV-EOF technique also shows a larger perturbation growth than the SVs of this system, except for short time-scales. The highest growth rate is found for the second BV-EOF for the long-time scale. The differences between orthogonal and non-orthogonal breeding vectors are also investigated using the ECMWF IFS-model. These results confirm the results from the Loernz-63 model regarding the dependency on orthogonalisation.

Magnusson L., J. Nycand er, and E. Källèn, 2009: Flow-dependent versus flow-independent initial perturbations for ensemble prediction. Tellus A, 61( 3), 194- 209.10.1111/j.1600-0870.2008.00385.x

218f02584b9ae2a70174ff58b538b89f

http%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1111%2Fj.1600-0870.2008.00385.x%2Fcitedby

http://onlinelibrary.wiley.com/doi/10.1111/j.1600-0870.2008.00385.x/citedbyEnsemble prediction relies on a faithful representation of initial uncertainties in a forecasting system. Early research on initial perturbation methods tested random perturbations by adding `white noise' to the analysis. Here, an alternative kind of random perturbations is introduced by using the difference between two randomly chosen atmospheric states (i.e. analyses). It yields perturbations (random field, RF, perturbations) in approximate flow balance.

Molteni F., R. Buizza, T. N. Palmer, and T. Petroliagis, 1996: The new ECMWF ensemble prediction system: methodology and validation.Quart. J. Roy. Meteor. Soc., 122, 73- 119.10.1002/qj.49712252905

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http%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1002%2Fqj.49712252905%2Ffull

refpaperuri:(d38acaf5dc0dec476fd404c53c2f7bb0)

http://onlinelibrary.wiley.com/doi/10.1002/qj.49712252905/fullThe ECMWF ensemble prediction system: methodology and validation. MOLTENI F. Quart. J. Roy. Meteor. Soc. 122, 73-119, 1996

Mu M., Z. Y. Zhang, 2006: Conditional nonlinear optimal perturbations of a two-dimensional quasigeostrophic model.J. Atmos. Sci., 63, 1587- 1604.10.1002/qj.256

bc158660b9c89cbf8721e7e8ca0260de

http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F2006JAtS...63.1587M

http://adsabs.harvard.edu/abs/2006JAtS...63.1587MCiteSeerX - Scientific documents that cite the following paper: Conditional nonlinear optimal perturbations of a two-dimensional quasigeostrophic model

Mu M., Z. N. Jiang, 2008: A new approach to the generation of initial perturbations for ensemble prediction: Conditional nonlinear optimal perturbation. Chinese Science Bulletin, 53, 2062- 2068.10.1007/s11434-008-0272-y

4b262054db4a6d12088d0a3f072c4570

http%3A%2F%2Flink.springer.com%2F10.1007%2Fs11434-008-0272-y

http://www.cnki.com.cn/Article/CJFDTotal-JXTW200813024.htm

Murphy A. H., 1973: A new vector partition of the probability score. J. Appl. Meteor., 12, 595- 600.98fc6802469e1f61885d283bfbac4f0c

http%3A%2F%2Fwww.emeraldinsight.com%2Fservlet%2Flinkout%3Fsuffix%3Db28%26dbid%3D16%26doi%3D10.1108%252FEJM-05-2012-0288%26key%3D10.1175%252F1520-0450%281973%290122.0.CO%253B2

http://xueshu.baidu.com/s?wd=paperuri%3A%28e7073650750802c7824e422f3b610106%29&filter=sc_long_sign&tn=SE_xueshusource_2kduw22v&sc_vurl=http%3A%2F%2Fwww.emeraldinsight.com%2Fservlet%2Flinkout%3Fsuffix%3Db28%26dbid%3D16%26doi%3D10.1108%252FEJM-05-2012-0288%26key%3D10.1175%252F1520-0450%281973%290122.0.CO%253B2&ie=utf-8&sc_us=10008998936017562578

Palatella L., A. Trevisan, 2015: Interaction of Lyapunov vectors in the formulation of the nonlinear extension of the Kalman filter. Physical Review E, 91, 042905.10.1103/PhysRevE.91.042905

c199aee48dac355c6321809562dd75bc

http%3A%2F%2Feuropepmc.org%2Fabstract%2FMED%2F25974560

http://med.wanfangdata.com.cn/Paper/Detail/PeriodicalPaper_PM25974560When applied to strongly nonlinear chaotic dynamics the extended Kalman filter (EKF) is prone to divergence due to the difficulty of correctly forecasting the forecast error probability density function. In operational forecasting applications ensemble Kalman filters circumvent this problem with empirical procedures such as covariance inflation. This paper presents an extension of the EKF that includes nonlinear terms in the evolution of the forecast error estimate. This is achieved starting from a particular square-root implementation of the EKF with assimilation confined in the unstable subspace (EKF-AUS), that is, the span of the Lyapunov vectors with non-negative exponents. When the error evolution is nonlinear, the space where it is confined is no more restricted to the unstable and neutral subspace causing filter divergence. The algorithm presented here, denominated EKF-AUS-NL, includes the nonlinear terms in the error dynamics: These result from the nonlinear interaction among the leading Lyapunov vectors and account for all directions where the error growth may take place. Numerical results show that with the nonlinear terms included, filter divergence can be avoided. We test the algorithm on the Lorenz96 model, showing very promising results.

Pe\na M., Z. Toth, 2014: Estimation of analysis and forecast error variances. Tellus A, 66, 21767.10.3402/tellusa.v66.21767

dcbd6bb39469335359a2cf0c8f32515f

http%3A%2F%2Fwww.researchgate.net%2Fpublication%2F269711012_Estimation_of_analysis_and_forecast_error_variances

http://www.researchgate.net/publication/269711012_Estimation_of_analysis_and_forecast_error_variancesAccurate estimates of error variances in numerical analyses and forecasts (i.e. difference between analysis or forecast fields and nature on the resolved scales) are critical for the evaluation of forecasting systems, the tuning of data assimilation (DA) systems and the proper initialisation of ensemble forecasts. Errors in observations and the difficulty in their estimation, the fact that estimates of analysis errors derived via DA schemes, are influenced by the same assumptions as those used to create the analysis fields themselves, and the presumed but unknown correlation between analysis and forecast errors make the problem difficult. In this paper, an approach is introduced for the unbiased estimation of analysis and forecast errors. The method is independent of any assumption or tuning parameter used in DA schemes. The method combines information from differences between forecast and analysis fields (erceived forecast errors) with prior knowledge regarding the time evolution of (1) forecast error variance and (2) correlation between errors in analyses and forecasts. The quality of the error estimates, given the validity of the prior relationships, depends on the sample size of independent measurements of perceived errors. In a simulated forecast environment, the method is demonstrated to reproduce the true analysis and forecast error within predicted error bounds. The method is then applied to forecasts from four leading numerical weather prediction centres to assess the performance of their corresponding DA and modelling systems. Error variance estimates are qualitatively consistent with earlier studies regarding the performance of the forecast systems compared. The estimated correlation between forecast and analysis errors is found to be a useful diagnostic of the performance of observing and DA systems. In case of significant model-related errors, a methodology to decompose initial value and model-related forecast errors is also proposed and successfully demonstrated. Keywords: uncertainty of analysis, forecast verification, estimation methods, data assimilation, ensemble forecasts (Published: 24 November 2014) Citation: Tellus A 2014, 66 , 21767, http://dx.doi.org/10.3402/tellusa.v66.21767

Stephenson D. B., C. A. S. Coelho, and I. T. Jolliffe, 2008: Two extra components in the Brier Score decomposition. Mon. Wea. Rev., 23, 752- 757.10.1175/2007WAF2006116.1

89e970acdb5dba5c19ac64d274147bc1

http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F2008WtFor..23..752S

http://adsabs.harvard.edu/abs/2008WtFor..23..752SAbstract The Brier score is widely used for the verification of probability forecasts. It also forms the basis of other frequently used probability scores such as the rank probability score. By conditioning (stratifying) on the issued forecast probabilities, the Brier score can be decomposed into the sum of three components: uncertainty, reliability, and resolution. This Brier score decomposition can provide useful information to the forecast provider about how the forecasts can be improved. Rather than stratify on all values of issued probability, it is common practice to calculate the Brier score components by first partitioning the issued probabilities into a small set of bins. This note shows that for such a procedure, an additional two within-bin components are needed in addition to the three traditional components of the Brier score. The two new components can be combined with the resolution component to make a generalized resolution component that is less sensitive to choice of bin width than is the traditional resolution component. The difference between the generalized resolution term and the conventional resolution term also quantifies how forecast skill is degraded when issuing categorized probabilities to users. The ideas are illustrated using an example of multimodel ensemble seasonal forecasts of equatorial sea surface temperatures.

Toth Z., E. Kalnay, 1993: Ensemble forecasting at NMC: The generation of perturbations. Bull. Amer. Meteor. Soc., 74, 2317- 2330.10.1175/1520-0477(1993)074<2317:EFANTG>2.0.CO;2

cbe63b780b83c7603b5d4d22f0d6e97f

http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F1993BAMS...74.2317T

http://adsabs.harvard.edu/abs/1993BAMS...74.2317TOn 7 December 1992, The National Meteorological Center (NMC) started operational ensemble forecasting. The ensemble forecast configuration implemented provides 14 independent forecasts every day verifying on days 1-10. In this paper we briefly review existing methods for creating perturbations for ensemble forecasting. We point out that a regular analysis cycle is a "breeding ground" for fast-growing modes. Based on this observation, we devise a simple and inexpensive method to generate growing modes of the atmosphere.The new method, "breeding of growing modes", or BGM, consists of one additional, perturbed short-range forecast, introduced on top of the regular analysis in an analysis cycle. The difference between the control and perturbed six-hour (first guess) forecast is scaled back to the size of the initial perturbation and then reintroduced onto the new atmospheric analysis. Thus, the perturbation evolves along with the time dependent analysis fields, ensuring that after a few days of cycling the perturbation field consists of a superposition of fast-growing modes corresponding to the contemporaneous atmosphere, akin to local Lyapunov vectors.The breeding cycle has been designed to model how the growing errors are "bred" and maintained in a conventional analysis cycle through the successive use of short-range forecasts. The bred modes should thus offer a good estimate of possible growing error fields in the analysis. Results from extensive experiments indicate that ensembles of just two BGM forecasts achieve better results than much larger random Monte Cado or lagged average forecast (LAF) ensembles. Therefore, the operational ensemble configuration at NMC is based on the BGM method to generate efficient initial perturbations.The only two methods explicitly designed to generate perturbations that contain fast-growing modes corresponding to the evolving atmosphere are the BGM and the method of Lorenz, which is based on the singular modes of the linear tangent model. This method has been adopted operationally at The European Centre for Medium-Range Forecasts (ECMWF) for ensemble forecasting. Both the BGM and the ECMWF methods seem promising, but since it has not yet been possible to compare in detail their operational performance we limit ourselves to pointing out some of their similarities and differences.

Toth Z., E. Kalnay, 1997: Ensemble forecasting at NCEP and the breeding method. Mon. Wea. Rev., 125, 3297- 3319.10.1175/1520-0493(1997)125<3297:EFANAT>2.0.CO;2

d77eecdcf56bec983b860925dfc56c2d

http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F1997MWRv..125.3297T

http://adsabs.harvard.edu/abs/1997MWRv..125.3297TPurpose: The aim of this study was to evaluate the effects of curing mode and viscosity on the biaxial flexural strength (FS) and modulus (FM) of dual resin cements. Methods: Eight experimental groups were created (n=12) according to the dual-cured resin cements (Nexus 2/Kerr Corp. and Variolink II/IvoclarVivadent), curing modes (dual or self-cure), and viscosities (low and high). Forty-eight cement discs of each product (0.5 mm thick by 6.0 mm diameter) were fabricated. Half specimens were light--activated for 40 seconds and half were allowed to self-cure. After 10 days, the biaxial flexure test was performed using a universal testing machine (1.27 mm/min, Instron 5844). Data were statistically analyzed by three-way ANOVA and Tukey's test (5%). Results: Light-activation increased FS and FM of resin cements at both viscosities in comparison with self-curing mode. The high viscosity version of light-activated resin cements exhibited higher FS than low viscosity versions. The viscosity of resin and the type of cement did not influence the FM. Light-activation of dual-polymerizing resin cements provided higher FS and FM for both resin cements and viscosities. Conclusion: The use of different resin cements with different viscosities may change the biomechanical behavior of these luting materials.

Trevisan A., R. Legnani, 1995: Transient error growth and local predictability: A study in the Lorenz system. Tellus A, 47, 103- 117.10.1034/j.1600-0870.1995.00006.x

aa6c4400de03ba3d68d42ff0f63180e3

http%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1034%2Fj.1600-0870.1995.00006.x%2Fcitedby

http://onlinelibrary.wiley.com/doi/10.1034/j.1600-0870.1995.00006.x/citedbyLorenz's three-variable convective model is used as a prototypical chaotic system in order to develop concepts related to finite time local predictability. Local predictability measures can be represented by global measures only if the instability properties of the attractor are homogeneous in phase space. More precisely, there are two sources of variability of predictability in chaotic attractors. The first depends on the direction of the initial error vector, and its dependence is limited to an initial transient period. If the attractor has homogeneous predictability properties, this is the only source of variability of error growth rate and, after the transient has elapsed, all initial perturbations grow at the same rate, given by the first (global) Lyapunov exponent. The second is related to the local instability properties in phase space. If the predictability properties of the attractor are not homogeneous, this additional source of variability affects both the transient and post-transient phases of error growth. After the transient phase all initial perturbations of a particular initial condition grow at the same rate, given in this case by the first local Lyapunov exponent. We consider various currently used indexes to quantify finite time local predictability. The probability distributions of the different indexes are examined during and after the transient phase. By comparing their statistics it is possible to discriminate the relative importance of the two sources of variability of predictability and to determine the most appropriate measure of predictability for a given forecast time. It is found that a necessary premise for choosing a relevant local predictability index for a specific system is the study of the characteristics of its transient. The consequences for the problem of forecasting forecast skill in operational models are discussed.

Trevisan A., F. Uboldi, 2004: Assimilation of standard and targeted observations within the unstable subspace of the observation-analysis-forecast cycle system.J. Atmos. Sci., 61, 103- 113.10.1175/1520-0469(2004)061<0103:AOSATO>2.0.CO;2

b93afd432c2099523d9e3e9221a7221e

http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F2004jats...61..103t

http://adsabs.harvard.edu/abs/2004jats...61..103tIn this paper it is shown that the flow-dependent instabilities that develop within an observation analysis forecast (OAF) cycle and that are responsible for the background error can be exploited in a very simple way to assimilate observations. The basic idea is that, in order to minimize the analysis and forecast errors, the analysis increment must be confined to the unstable subspace of the OAF cycle solution. The analysis solution here formally coincides with that of the classical three-dimensional variational solution with the background error covariance matrix estimated in the unstable subspace.The unstable directions of the OAF system solution are obtained by breeding initially random perturbations of the analysis but letting the perturbed trajectories undergo the same process as the control solution, including assimilation of all the available observations. The unstable vectors are then used both to target observations and for the assimilation design.The approach is demonstrated in an idealized environment using a simple model, simulated standard observations over land with a single adaptive observation over the ocean. In the application a simplified form is adopted of the analysis solution and a single unstable vector at each analysis time whose amplitude is determined by means of the adaptive observation. The remarkable reduction of the analysis and forecast error obtained by this simple method suggests that only a few accurately placed observations are sufficient to control the local instabilities that take place along the cycle.The stability of the system, with or without forcing by observations, is studied and the growth rate of the leading instability of the different control solutions is estimated. Whereas the model has more than one positive Lyapunov exponent, the solution of the OAF scheme that includes the adaptive observation is stable. It is suggested that a negative exponent can be considered a necessary condition for the convergence of a particular OAF solution to the truth, and that the estimate of the degree of stability of the control trajectory can be used as a simple criterion to evaluate the efficiency of data assimilation and observation strategies.The present findings are in line with previous quantative observability results with more realistic models and with recent studies that indicate a local low dimensionality of the unstable subspace.

Trevisan A., L. Palatella, 2011: Chaos and weather forecasting: the role of the unstable subspace in predictability and state estimation problems. International Journal of Bifurcation and Chaos, 21( 12), 3389- 3415.10.1142/S0218127411030635

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WOS:000300016000002

fc774617d6d0c74e56d2eb42e1e6bb35

http%3A%2F%2Fwww.worldscientific.com%2Fdoi%2Fabs%2F10.1142%2FS0218127411030635

http://www.worldscientific.com/doi/abs/10.1142/S0218127411030635In the first part of this paper, we review some important results on atmospheric predictability, from the pioneering work of Lorenz to recent results with operational forecasting models. Particular relevance is given to the connection between atmospheric predictability and the theory of Lyapunov exponents and vectors. In the second part, we briefly review the foundations of data assimilation methods and then we discuss recent results regarding the application of the tools typical of chaotic systems theory described in the first part to well established data assimilation algorithms, the Extended Kalman Filter (EKF) and Four Dimensional Variational Assimilation (4DVar). In particular, the Assimilation in the Unstable Space (AUS), specifically developed for application to chaotic systems, is described in detail.

Vannitsem S., C. Nicolis, 1997: Lyapunov vectors and error growth patterns in a T21L3 quasigeostrophic model.J. Atmos. Sci., 54, 347- 361.10.1175/1520-0469(1997)054<0347:LVAEGP>2.0.CO;2

c72165e612a8b150a2b386e2a5a169cd

http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F1997JAtS...54..347V

http://adsabs.harvard.edu/abs/1997JAtS...54..347VAbstract The authors report a systematic study on the short and intermediate time predictability properties of a quasigeostrophic T21L3 model in which emphasis is placed on the role of the Lyapunov vectors in the growth patterns of generic initial error fields. It is found that under scale-independent small-amplitude initial perturbations the evolution of the mean error is intimately related to the spectral distribution of the Lyapunov vectors. In the case of perturbations at a particular scale of motion the picture turns out to be more involved, particularly as far as mean error growth over all wavenumbers is concerned, and must appeal to coupling mechanisms between different scales. The role of the norm used for the measure of the mean error growth and the specific predictability properties at different vertical levels of the model are also analyzed.

Wang X., C. H. Bishop, 2003: A comparison of breeding and ensemble transform Kalman filter ensemble forecast schemes.J. Atmos. Sci., 60, 1140- 1158.10.1175/1520-0469(2003)0602.0.CO;2

e944df1a5d6726e5668650fe360597c4

http%3A%2F%2Fadsabs.harvard.edu%2Fabs%2F2003EAEJA.....8087W

http://adsabs.harvard.edu/abs/2003EAEJA.....8087WAbstract The ensemble transform Kalman filter (ETKF) ensemble forecast scheme is introduced and compared with both a simple and a masked breeding scheme. Instead of directly multiplying each forecast perturbation with a constant or regional rescaling factor as in the simple form of breeding and the masked breeding schemes, the ETKF transforms forecast perturbations into analysis perturbations by multiplying by a transformation matrix. This matrix is chosen to ensure that the ensemble-based analysis error covariance matrix would be equal to the true analysis error covariance if the covariance matrix of the raw forecast perturbations were equal to the true forecast error covariance matrix and the data assimilation scheme were optimal. For small ensembles (100), the computational expense of the ETKF ensemble generation is only slightly greater than that of the masked breeding scheme. Version 3 of the Community Climate Model (CCM3) developed at National Center for Atmospheric Research (NCAR) is used to test and compare these ensemble generation schemes. The NCEPCAR reanalysis data for the boreal summer in 2000 are used for the initialization of the control forecast and the verifications of the ensemble forecasts. The ETKF and masked breeding ensemble variances at the analysis time show reasonable correspondences between variance and observational density. Examination of eigenvalue spectra of ensemble covariance matrices demonstrates that while the ETKF maintains comparable amounts of variance in all orthogonal and uncorrelated directions spanning its ensemble perturbation subspace, both breeding techniques maintain variance in few directions. The growth of the linear combination of ensemble perturbations that maximizes energy growth is computed for each of the ensemble subspaces. The ETKF maximal amplification is found to significantly exceed that of the breeding techniques. The ETKF ensemble mean has lower root-mean-square errors than the mean of the breeding ensemble. New methods to measure the precision of the ensemble-estimated forecast error variance are presented. All of the methods indicate that the ETKF estimates of forecast error variance are considerably more accurate than those of the breeding techniques.

Wei M. Z., Z. Toth, R. Wobus, Y. J. Zhu, C. H. Bishop, and X. G. Wang, 2006: Ensemble transform Kalman filter-based ensemble perturbations in an operational global prediction system at NCEP. Tellus A, 58, 28- 44.10.1111/j.1600-0870.2006.00159.x

9327d5ee3404291fa0c4ad6c9a0f1523

http%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1111%2Fj.1600-0870.2006.00159.x%2Fcitedby

http://onlinelibrary.wiley.com/doi/10.1111/j.1600-0870.2006.00159.x/citedbyThe initial perturbations used for the operational global ensemble prediction system of the National Centers for Environmental Prediction are generated through the breeding method with a regional rescaling mechanism. Limitations of the system include the use of a climatologically fixed estimate of the analysis error variance and the lack of an orthogonalization in the breeding procedure. The Ensemble Transform Kalman Filter (ETKF) method is a natural extension of the concept of breeding and, as shown by Wang and Bishop, can be used to generate ensemble perturbations that can potentially ameliorate these shortcomings. In the present paper, a spherical simplex 10-member ETKF ensemble, using the actual distribution and error characteristics of real-time observations and an innovation-based inflation, is tested and compared with a 5-pair breeding ensemble in an operational environment.

Wei M. Z., Z. Toth, R. Wobus, and Y. J. Zhu, 2008: Initial perturbations based on the ensemble transform (ET) technique in the NCEP global operational forecast system. Tellus A, 60, 62- 79.10.1111/j.1600-0870.2007.00273.x

c630b34e2d19a77224c2d239c5c9c31d

http%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1111%2Fj.1600-0870.2007.00273.x%2Fcitedby

http://onlinelibrary.wiley.com/doi/10.1111/j.1600-0870.2007.00273.x/citedbyABSTRACT Since modern data assimilation (DA) involves the repetitive use of dynamical forecasts, errors in analyses share characteristics of those in short-range forecasts. Initial conditions for an ensemble prediction/forecast system (EPS or EFS) are expected to sample uncertainty in the analysis field. Ensemble forecasts with such initial conditions can therefore (a) be fed back to DA to reduce analysis uncertainty, as well as (b) sample forecast uncertainty related to initial conditions. Optimum performance of both DA and EFS requires a careful choice of initial ensemble perturbations.A can be improved with an EFS that represents the dynamically conditioned part of forecast error covariance as accurately as possible, while an EFS can be improved by initial perturbations reflecting analysis error variance. Initial perturbation generation schemes that dynamically cycle ensemble perturbations reminiscent to how forecast errors are cycled in DA schemes may offer consistency between DA and EFS, and good performance for both. In this paper, we introduce an EFS based on the initial perturbations that are generated by the Ensemble Transform (ET) and ET with rescaling (ETR) methods to achieve this goal. Both ET and ETR are generalizations of the breeding method (BM).he results from ensemble systems based on BM, ET, ETR and the Ensemble Transform Kalman Filter (ETKF) method are experimentally compared in the context of ensemble forecast performance. Initial perturbations are centred around a 3D-VAR analysis, with a variance equal to that of estimated analysis errors. Of the four methods, the ETR method performed best in most probabilistic scores and in terms of the forecast error explained by the perturbations. All methods display very high time consistency between the analysis and forecast perturbations. It is expected that DA performance can be improved by the use of forecast error covariance from a dynamically cycled ensemble either with a variational DA approach (coupled with an ETR generation scheme), or with an ETKF-type DA scheme.

Wolf A., J. B. Swift, H. L. Swinney, and J. A. Vastano, 1985: Determining Lyapunov exponents from a time series. Physica D, 16, 285- 317.10.1016/0167-2789(85)90011-9

baaa5d4fc1835d289fb01a91524965ba

http%3A%2F%2Fwww.sciencedirect.com%2Fscience%2Farticle%2Fpii%2F0167278985900119

http://www.sciencedirect.com/science/article/pii/0167278985900119We present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunov exponents is defined to be chaotic. Our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems: we monitor the long-term growth rate of small volume elements in an attractor. The method is tested on model systems with known Lyapunov spectra, and applied to data for the Belousov-Zhabotinskii reaction and Couette-Taylor flow.

Yoden S., M. Nomura, 1993: Finite-time Lyapunov stability analysis and its application to atmospheric predictability.J. Atmos. Sci., 50, 1531- 1543.10.1175/1520-0469(1993)0502.0.CO;2

acce085a99cc527702f019f7391a3174

http%3A%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D1220381

http://www.ams.org/mathscinet-getitem?mr=1220381Finite-time Lyapunov stability analysis is reviewed and applied to a low-order spectral model of barotropic flow in a midlatitude channel. The tangent linear equations of the model are used to investigate the growth of small perturbations superposed on a reference solution for a prescribed time interval. Three types of reference solutions of the model, stationary, periodic, and chaotic, are investigated to demonstrate usefulness of the analysis in the study of the atmospheric predictability problem.The finite-time Lyapunov exponents, which give the growth rate of small perturbations, depend upon the reference solution as well as the prescribed time interval. The finite-time Lyapunov vector corresponding to the largest Lyapunov exponent gives the streamfunction field of the fastest growing perturbation for the time interval. In the case of the chaotic reference solution, the streamfunction field has large amplitudes in limited areas for a small time interval. The areas of the large perturbation growth have some relation to the reference streamfunction field.A possible application of the finite-time Lyapunov exponents and vectors to the atmospheric predictability problem is discussed. These quantities might be used as several forecast measures of the time-dependent predictability in numerical weather predictions.

Zhang J., W. S. Duan, and X. F. Zhi, 2015: Using CMIP5 model outputs to investigate the initial errors that cause the "spring predictability barrier" for El Niño events. Science China Earth Sciences, 58( 6), 685- 696.10.1007/s11430-014-4994-1

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3f8214c469d75bc5ff9d90f96d0f82e1

http%3A%2F%2Flink.springer.com%2F10.1007%2Fs11430-014-4994-1

refpaperuri:(60b1f6a9262208ffa3e1e125bf94f4b1)

http://www.cnki.com.cn/Article/CJFDTotal-JDXG201505003.htmMost ocean-atmosphere coupled models have difficulty in predicting the El Ni-Southern Oscillation(ENSO) when starting from the boreal spring season. However, the cause of this spring predictability barrier(SPB) phenomenon remains elusive. We investigated the spatial characteristics of optimal initial errors that cause a significant SPB for El Ni events by using the monthly mean data of the pre-industrial(PI) control runs from several models in CMIP5 experiments. The results indicated that the SPB-related optimal initial errors often present an SST pattern with positive errors in the central-eastern equatorial Pacific, and a subsurface temperature pattern with positive errors in the upper layers of the eastern equatorial Pacific, and negative errors in the lower layers of the western equatorial Pacific. The SPB-related optimal initial errors exhibit a typical La Ni--like evolving mode, ultimately causing a large but negative prediction error of the Ni-3.4 SST anomalies for El Ni events. The negative prediction errors were found to originate from the lower layers of the western equatorial Pacific and then grow to be large in the eastern equatorial Pacific. It is therefore reasonable to suggest that the El Ni predictions may be most sensitive to the initial errors of temperature in the subsurface layers of the western equatorial Pacific and the Ni-3.4 region, thus possibly representing sensitive areas for adaptive observation. That is, if additional observations were to be preferentially deployed in these two regions, it might be possible to avoid large prediction errors for El Ni and generate a better forecast than one based on additional observations targeted elsewhere. Moreover, we also confirmed that the SPB-related optimal initial errors bear a strong resemblance to the optimal precursory disturbance for El Ni and La Ni events. This indicated that improvement of the observation network by additional observations in the identified sensitive areas would also be helpful in detecting the signals provided by the precursory disturbance, which may greatly improve the ENSO prediction skill.

Ziehmann C., L. A. Smith, and J. Kurths, 2000: Localized Lyapunov exponents and the prediction of predictability. Physics Letters A, 271, 237- 251.10.1016/S0375-9601(00)00336-4

aef304e3cb647d0a59ceaec1a98852ab

http%3A%2F%2Fwww.sciencedirect.com%2Fscience%2Farticle%2Fpii%2FS0375960100003364

http://www.sciencedirect.com/science/article/pii/S0375960100003364Every forecast should include an estimate of its likely accuracy, a current measure of predictability. Two distinct types of localized Lyapunov exponents based on infinitesimal uncertainty dynamics are investigated to reflect this predictability. Regions of high predictability within which any initial uncertainty will decrease are proven to exist in two common chaotic systems; potential implications of these regions are considered. The relevance of these results for finite size uncertainties is discussed and illustrated numerically.