The Error Covariance Matrix Inflation in Ensemble Kalman Filter

The estimation accuracy of ensemble forecast errors is crucial to the assimilation results for all ensemble-based schemes. The ensemble Kalman filter (EnKF) is a widely used scheme in land surface data assimilation, without using the adjoint of a dynamical model. In EnKF, the forecast error covariance matrix is estimated as the sampling covariance matrix of the ensemble forecast states. However, past researches on EnKF have found that it can generally lead to an underestimate of the forecast error covariance matrix, due to the limited ensemble size, as well as the poor initial perturbations and model error. This can eventually result in filter divergence. Therefore, using inflation to further adjust the forecast error covariance matrix becomes increasingly important. In this chapter, a new structure of the forecast error covariance matrix is proposed to mitigate the problems with limited ensemble size and model error. An adaptive procedure equipped with a second-order least squares method is applied to estimate the inflation factors of forecast and observational error covariance matrices. The proposed method is tested on the well-known atmosphere-like Lorenz-96 model with spatially correlated observational systems. The experiment results show that the new structure of the forecast error covariance matrix and the adaptive estimation procedure lead to improvement of the analysis states

Soil Moisture Assimilation Using a Modified Ensemble Transform Kalman Filter Based on Station Observations in the Hai River Basin

Assimilating observations to a land surface model can further improve soil moisture estimation accuracy. However, assimilation results largely rely on forecast error and generally cannot maintain a water budget balance. In this study, shallow soil moisture observations are assimilated into Common Land Model (CoLM) to estimate the soil moisture in different layers. A proposed forecast error inflation and water balance constraint are adopted in the Ensemble Transform Kalman Filter to reduce the analysis error and water budget residuals. The assimilation results indicate that the analysis error is reduced and the water imbalance is mitigated with this approach

Optimization of terrestrial ecosystem model parameters using atmospheric CO2 concentration data with the Global Carbon Assimilation System (GCAS)

Author Posting. © American Geophysical Union, 2017. This article is posted here by permission of American Geophysical Union for personal use, not for redistribution. The definitive version was published in Journal of Geophysical Research: Biogeosciences 122 (2017): 3218–3237, doi:10.1002/2016JG003716.The Global Carbon Assimilation System that assimilates ground-based atmospheric CO2 data is used to estimate several key parameters in a terrestrial ecosystem model for the purpose of improving carbon cycle simulation. The optimized parameters are the leaf maximum carboxylation rate at 25°C (V25 max), the temperature sensitivity of ecosystem respiration (Q10), and the soil carbon pool size. The optimization is performed at the global scale at 1° resolution for the period from 2002 to 2008. The results indicate that vegetation from tropical zones has lower V25 max values than vegetation in temperate regions. Relatively high values of Q10 are derived over high/midlatitude regions. Both V25 max and Q10 exhibit pronounced seasonal variations at middle-high latitudes. The maxima in V25 max occur during growing seasons, while the minima appear during nongrowing seasons. Q10 values decrease with increasing temperature. The seasonal variabilities of V25 max and Q10 are larger at higher latitudes. Optimized V25 max and Q10 show little seasonal variabilities at tropical regions. The seasonal variabilities of V25 max are consistent with the variabilities of LAI for evergreen conifers and broadleaf evergreen forests. Variations in leaf nitrogen and leaf chlorophyll contents may partly explain the variations in V25 max. The spatial distribution of the total soil carbon pool size after optimization is compared favorably with the gridded Global Soil Data Set for Earth System. The results also suggest that atmospheric CO2 data are a source of information that can be tapped to gain spatially and temporally meaningful information for key ecosystem parameters that are representative at the regional and global scales.National Key R&D Program of China Grant Number: 2016YFA0600204; National Natural Science Foundation of China Grant Number: 415713382018-06-2

Ergodic theorems for stress release processes

This paper is devoted to a class of piecewise determined Markov processes, stress release processes. The models describe the regime of stress built up linearly by tectonic force and released incidentally by earthquakes. A structure of the processes is given, and Harris ergodicity and geometrical ergodicity of the processes are studied.piecewise determined Markov processes stress release processes

A method for constructing skillful seasonal forecasts using slow modes of climate variability

A methodology for constructing skillful statistical seasonal forecasts of climate fields is described and applied to predict the Southern Hemisphere summer mean sea level pressure anomalies for the period 1993--2004. The method employs a recently developed variance decomposition approach, which allows a separation of the predictable and unpredictable components of climate variation. The proposed forecast scheme is based on finding predictors for the amplitude time series of the dominant slow (or predictable) modes of interannual climate variation

A method for estimating and assessing modes of interannual variability in coupled climate models

The seasonal mean of a climate variable consists of: slow-external; slow-internal; and intraseasonal components. Using an analysis of variance-based method, the interannual variability of the seasonal mean from an ensemble of coupled atmosphere-ocean general circulation model (CGCM) realisations is separable into these three components. Eigenvalue decomposition is applied to the covariance matrices to obtain, for each component, the dominant modes of variability (eigenvectors) and their associated variance (eigenvalues) for the climate variable. Here, a method is described that assesses the modes of interannual variability in CGCMs against those obtained from reanalysis data based on observations. A metric is defined based on the pattern correlation between the observed and modelled modes of variability, and the ratio of their associated variances. This metric is applied to monthly mean southern hemisphere 500 hPa geopotential height from the second half of the 20th century. It is shown that CGCMs have clear differences in the slow-component of modes of interannual variability, related to external forcings and/or slowly-varying internal variability. References C. S. Frederiksen and X. Zheng. Coherent structures of interannual variability of the atmospheric circulation: the role of intraseasonal variability. Frontiers in Turbulence and Coherent Structures, World Scientific Lecture Notes in Complex Systems, Vol. 6, Eds Jim Denier and Jorgen Frederiksen, World Scientific Publications, 87–120, 2007. doi:10.1142/6320 C. E. Leith. The standard error of time-average estimates of climatic means. J. Appl. Meteor., 12:1066–1069, 1973. doi:10.1175/1520-0450(1973)012<1066:TSEOTA>2.0.CO;2 X. Zheng and C. S. Frederiksen. Variability of seasonal-mean fields arising from intraseasonal variability: part 1, methodology. Clim. Dynam., 23:177–191, 2004. doi:10.1007/s00382-004-0428-7 C. S. Frederiksen and X. Zheng. Variability of seasonal-mean fields arising from intraseasonal variability. Part 3: Application to SH winter and summer circulations. Clim. Dynam., 28:849–866, 2007. doi:10.1007/s00382-006-0214-9 S. Grainger, C. S. Frederiksen and X. Zheng. A method for evaluating the modes of variability in general circulation models. ANZIAM J., 50:C399–C412, 2008. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/1431 G. A. Meehl, C. Covey, K. E. Taylor, T. Delworth, R. J. Stouffer, M. Latif, B. McAvaney and J. F. B. Mitchell. The WCRP CMIP3 multimodel dataset: A new era in climate change research. Bull. Amer. Meteor. Soc., 88:1383–1394, 2007. doi:10.1175/BAMS-88-9-1383 K. E. Taylor, R. J. Stouffer and G. A. Meehl. An overview of CMIP5 and the experiment design. Bull. Amer. Meteor. Soc. 93:485–498, 2012. doi:10.1175/BAMS-D-11-00094.1 X. Zheng, M. Sugi and C. S. Frederiksen. Interannual variability and predictability in an ensemble of climate simulations with the MRI-JMA AGCM. J. Meteor. Soc. Jap., 82:1–18, 2004. doi:10.2151/jmsj.82.1 H. von Storch and F. W. Zwiers. Statistical Analysis in Climate Research. Cambridge University Press, 484pp, 1999. doi:10.1017/cbo9780511612336 S. Grainger, C. S. Frederiksen and X. Zheng. Modes of interannual variability of Southern Hemisphere atmoshperic circulation in CMIP3 models: assessment and projections. Clim. Dynam. 41:479–500, 2013. doi:10.1007/s00382-012-1659-7 S. Grainger, C. S. Frederiksen and X. Zheng. Estimating components of covariance between two climate variables using model ensembles. ANZIAM J., 52:C318–C332, 2011. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/3928 G. P. Compo, J. S. Whitaker, P. D. Sardeshmukh, N. Matsui, R. J. Allan, X. Yin, B. E. Gleason, R. S. Vose, G. Rutledge, P. Bessemoulin, S. Bronnimann, M. Brunet, R. I. Crouthamel, A. N. Grant, P. Y. Groisman, P. D. Jones, M. C. Kruk, A. C. Kruger, G. J. Marshall, M. Maugeri, H. Y. Mok, O. Nordli, T. F. Ross, R. M. Trigo, X. L. Wang, S. D. Woodruff and S. J. Worley. The Twentieth Century Reanalysis Project. Quart. J. Roy. Meteor. Soc. 137:1–28, 2011. doi:10.1002/qj.776 N. A. Rayner, D. E. Parker, E. B. Horton, C. K. Folland, L. V. Alexander, D. P. Rowell, E. C. Kent and A. Kaplan. Global analyses of sea surface temperature, sea ice, and night marine air temperature since the late nineteenth century. J. Geophys. Res., 108(D14):4407, 2003. doi:10.1029/2002JD00267

Estimating components of covariance between two climate variables using model ensembles

The seasonal mean of a climate variable is considered to consist of: (a)~slow-external; (b)~slow-internal; and (c)~intraseasonal components. Using an Analysis of Variance-based method, the interannual variability of the seasonal mean from an ensemble of coupled atmosphere-ocean general circulation model realisations is separable into these components. Here, we propose a method for analysing the covariability of these components between pairs of climate variables. In particular, the method allows for an estimate of the covariability of the projected time series of the modes of variability of one climate variable with the time series of another. To illustrate this, the relationship between time series of the modes of variability of 500\,hPa geopotential height and sea surface temperature is examined for an ensemble of coupled general circulation model realisations. The method is applicable to other atmospheric climate variables and datasets. References C. S. Frederiksen and X. Zheng. Coherent structures of interannual variability of the atmospheric circulation: the role of intraseasonal variability. Frontiers in Turbulence and Coherent Structures, World Scientific Lecture Notes in Complex Systems, Vol. 6, Eds Jim Denier and Jorgen Frederiksen, World Scientific Publications, 87--120, 2007. C. E. Leith. The standard error of time-average estimates of climatic means. J. Appl. Meteor., 12:1066--1069, 1973. doi:10.1175/1520-0450(1973)012<1066:TSEOTA>2.0.CO;2 X. Zheng and C. S. Frederiksen. Variability of seasonal-mean fields arising from intraseasonal variability. Part 1, methodology. Climate Dynamics, 23:177--191, 2004. doi:10.1007/s00382-004-0428-7 X. Zheng and C. S. Frederiksen. Validating interannual variability in an ensemble of AGCM simulations. J. Climate, 12:2386--2396, 1999. doi:10.1175/1520-0442(1999)012<2386:VIVIAE>2.0.CO;2 C. S. Frederiksen and X. Zheng. Variability of seasonal-mean fields arising from intraseasonal variability. part 2, application to nh winter circulations. Climate Dynamics, 23:193--206, 2004. doi:10.1007/s00382-004-0429-6 C. S. Frederiksen and X. Zheng. Variability of seasonal-mean fields arising from intraseasonal variability. Part 3: Application to SH winter and summer circulations. Climate Dynamics, 28:849--866, 2007. doi:10.1007/s00382-006-0214-9 S. Grainger, C. S. Frederiksen and X. Zheng. A method for evaluating the modes of variability in general circulation models. ANZIAM J., 50:C399--C412, 2008. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/1431 X. Zheng, M. Sugi and C. S. Frederiksen. Interannual variability and predictability in an ensemble of climate simulations with the MRI-JMA AGCM. J. Meteor. Soc. Jap., 82:1--18, 2004. doi:10.2151/jmsj.82.1 S. Grainger, C. S. Frederiksen, X. Zheng, D. Fereday, C. K. Folland, E. K. Jin, J. L. Kinter, J. R. Knight, S. Schubert and J. Syktus. Modes of variability of Southern Hemisphere atmospheric circulation estimated by AGCMs. Climate Dynamics, 36:473--490, 2011. doi:10.1007/s00382-009-0720-7 H. von Storch and F. W. Zwiers. Statistical Analysis in Climate Research. Cambridge University Press, 484pp, 1999. G. A. Meehl, C. Covey, T. Delworth, M. Latif, B. McAvaney, J. F. B. Mitchell, R. J. Stouffer and K. E. Taylor. The WCRP CMIP3 multimodel dataset: A new era in climate change research. Bull. Amer. Meteor. Soc., 88:1383--1394, 2007. doi:10.1175/BAMS-88-9-1383 E. Kalnay, M. Kanamitsu, R. Kistler, W. Collins, D. Deaven, L. Gandin, M. Iredell, S. Saha, G. White, J. Woollen, Y. Zhu, A. Leetmaa, R. Reynolds, M. Chelliah, W. Ebisuzaki, W. Higgins, J. Janowiak, K. C. Mo, C. Ropelewski, J. Wang, R. Jenne and D. Joseph. The NCEP/NCAR 40-year reanalysis project. Bull. Amer. Meteor. Soc., 77:437--471, 1996. doi:10.1175/1520-0477(1996)077<0437:TNYRP>2.0.CO;2 N. A. Rayner, D. E. Parker, E. B. Horton, C. K. Folland, L. V. Alexander, D. P. Rowell, E. C. Kent and A. Kaplan. Global analyses of sea surface temperature, sea ice, and night marine air temperature since the late nineteenth century. J. Geophys. Res., 108(D14):4407, 2003. doi:10.1029/2002JD002670 J. M. Arblaster and G. A. Meehl. Contributions of external forcings to Southern Annular Mode trends. J. Climate, 19:2896--2905, 2006. doi:10.1175/JCLI3774.

Estimating the potential predictability of Western Australian surface temperature using monthly data

The seasonal mean of a climate variable is considered to be a statistical random variable with two components: a slow component related to slowly varying (time scales of a season or more) forcings from external and internal atmospheric sources, and an intraseasonal component related to forcings from weather variability with time scales less than a season. Here, an extension of a previous Analysis of Variance method is proposed which deals with climate data in all seasons when estimating the intraseasonal variability. By removing this from the total variability, an estimate for the slow component, and hence the long range predictability, of the seasonal mean is made. The method is applied to monthly surface temperature data for Western Australia from 1951--2000. References R. J. B. Fawcett, D. A. Jones and G. S. Beard. A verification of publicly issued seasonal forecasts issued by the Australian Bureau of Meteorology: 1998--2003. Aust. Meteor. Mag., 54:1--13, 2005. C. S. Frederiksen and X. Zheng. Coherent Structures of Interannual Variability of the Atmospheric Circulation: The Role of Intraseasonal Variability. Frontiers in Turbulence and Coherent Structures, World Scientific Lecture Notes in Complex Systems, Vol. 6, Eds Jim Denier and Jorgen Frederiksen, World Scientific Publications, 87--120, 2007. S. Grainger, C. S. Frederiksen and X. Zheng. Estimating the potential predictability of Australian surface maximum and minimum temperature. Climate Dynamics, Accepted, 2008. D. A. Jones and B. C. Trewin. The spatial structure of monthly temperature anomalies over Australia. Aust. Meteor. Mag., 49:261--276, 2000. E. N. Lorenz. On the existence of extended range predictability. J. Appl. Meteor., 12:543--546, doi:10.1175/1520-0450(1973)012<0543:OTEOER>2.0.CO;2, 1973. I. G. Watterson. The diurnal cycle of surface air temperature in simulated present and doubled CO2 climates. Climate Dynamics, 13:533--545, doi:10.1007/s003820050181, 1997. X. Zheng and C. S. Frederiksen. Variability of seasonal-mean fields arising from intraseasonal variability. Part 1, methodology. Climate Dynamics, 23:177--191, doi::{10.1007/s00382-004-0428-7}, 2004. X. Zheng, H. Nakamura and J. A. Renwick. Potential predictability of seasonal means based on monthly time series of meteorological variables. J. Climate, 13:2591--2604, doi:10.1175/1520-0442(2000)013<2591:PPOSMB>2.0.CO;2, 2000. X. Zheng, M. Sugi and C. S. Frederiksen. Interannual variability and predictability in an ensemble of climate simulations with the MRI-JMA AGCM. J. Meteor. Soc. Jap., 82:1--18, doi:10.2151/jmsj.82.1, 2004

A method for evaluating the modes of variability in general circulation models

The seasonal mean of an atmospheric climate variable is considered to be a statistical random variable with two components: a slow component related to slowly varying forcing from external and internal atmospheric sources (time scale of a season or more), and an intraseasonal component related to forcing from weather variability with time scale less than a season. Here, a method is proposed to compare the modes of variability obtained from eigenvalue decomposition of the slow and intraseasonal covariance matrices estimated from reanalysis data with modes of variability estimated from a set of coupled general circulation models. As an example, the method is applied to the Southern Hemisphere summer 500hPa geopotential height for the period 1951--2000. The method is applicable to many other atmospheric climate variables and datasets. References G. P. Cressman. An operational objective analysis system. Mon. Wea. Rev., 87:367--374, doi:10.1175/1520-0493(1959)087<0367:AOOAS>2.0.CO;2, 1959. C. S. Frederiksen and X. Zheng. Variability of seasonal-mean fields arising from intraseasonal variability. part 2, application to nh winter circulations. Climate Dynamics, 23:193--206, doi:10.1007/s00382-004-0429-6, 2004. C. S. Frederiksen and X. Zheng. Coherent Structures of Interannual Variability of the Atmospheric Circulation: The Role of Intraseasonal Variability. Frontiers in Turbulence and Coherent Structures, World Scientific Lecture Notes in Complex Systems, Vol. 6, Eds Jim Denier and Jorgen Frederiksen, World Scientific Publications, 87--120, 2007. C. S. Frederiksen and X. Zheng. Variability of seasonal-mean fields arising from intraseasonal variability: Part 3: Application to SH winter and summer circulations. Climate Dynamics, 28:849--866, doi:10.1007/s00382-006-0214-9, 2007. N. Higham. Computing the nearest correlation matrix---a problem from finance. I.M.A. J. Numerical Analysis, 22:329--343, doi:10.1093/imanum/22.3.329, 2002. E. Kalnay et al. The NCEP/NCAR 40-year reanalysis project. Bull. Amer. Meteor. Soc., 77:437--471, doi:10.1175/1520-0477(1996)077<0437:TNYRP>2.0.CO;2, 1996. G. A. Meehl et al. The WCRP CMIP3 multimodel dataset: A new era in climate change research. Bull. Amer. Meteor. Soc., 88:1383--1394, doi:10.1175/BAMS-88-9-1383, 2007. H. von Storch and F. W. Zwiers. Statistical Analysis in Climate Research. Cambridge University Press, 484pp, 1999. K. E. Taylor. Summarizing multiple aspects of model performance in a single diagram. J. Geophys. Res., 106:7183--7192, http://www.agu.org/pubs/crossref/2001/2000JD900719.shtml, 2001. X. Zheng and C. S. Frederiksen. Variability of seasonal-mean fields arising from intraseasonal variability. Part 1, methodology. Climate Dynamics, 23:177--191, doi:10.1007/s00382-004-0428-7, 2004. X. Zheng, M. Sugi and C. S. Frederiksen. Interannual variability and predictability in an ensemble of climate simulations with the MRI-JMA AGCM. J. Met. Soc. Jap., 82:1--18, doi:10.2151/jmsj.82.1, 2004

Estimating the intermonth covariance between rainfall and the atmospheric circulation

The seasonal mean of a climate variable consists of a slow and intraseasonal component. Existing methods for deriving coupled patterns between intraseasonal components assume stationarity and first order autoregressive processes. This does not hold for a variable such as rainfall where the daily data consists of dichotomous (on/off) events. It is possible to formulate a more general method for such two-state climate variables but it requires an estimate of the intermonth covariance. We use a stochastic two-state first-order Markov chain model fitted to daily Australian rainfall data to provide an estimate of the intermonth covariance with daily 500hPa atmospheric geopotential height anomalies. We show that the estimate of the intermonth covariance is much smaller than the within-month covariance between rainfall and the 500hPa height intraseasonal component. References C. S. Frederiksen and X. Zheng. Coherent Structures of Interannual Variability of the Atmospheric Circulation: The Role of Intraseasonal Variability. Frontiers in Turbulence and Coherent Structures, World Scientific Lecture Notes in Complex Systems, Vol. 6, Eds Jim Denier and Jorgen Frederiksen, World Scientific Publications, 87--120, 2007. X. Zheng and C. S. Frederiksen. Variability of seasonal-mean fields arising from intraseasonal variability. Part 1, methodology. Climate Dynamics, 23:177--191, 2004. doi:10.1007/s00382-004-0428-7 C. S. Frederiksen and X. Zheng. A Method for constructing skilful seasonal forecasts using slow modes of climate variability. ANZIAM J., 48:C89--C103, 2007. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/114 X. Zheng and C. S. Frederiksen. A study of predictable patterns for seasonal forecasting of New Zealand rainfall. J. Climate, 19:3320--3333, 2006. doi:10.1175/JCLI3798.1 X. Zheng and C. S. Frederiksen. Statistical Prediction of Seasonal Mean Southern Hemisphere 500hPa Geopotential Heights. J. Climate, 20:2791--2809, 2006. doi:10.1175/JCLI4180.1 C. S. Frederiksen and X. Zheng. A Method for extracting coupled patterns of predictable and chaotic components in pairs of climate variables. ANZIAM J., 46:C276--C289, 2005. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/959 C. S. Frederiksen, S. Grainger and X. Zheng. A Method for estimating the potential long-range predictability of precipitation over Western Australia. ANZIAM J., 2008. 50:C569--C583, http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/1411 R. W. Katz and X. Zheng. Mixture Model For Overdispersion of Precipitation. J. Climate, 12:2528--2537, 1999. doi:10.1175/1520-0442(1999)012<2528:MMFOOP>2.0.CO;2 D. S. Wilks. Statistical Methods in the Atmospheric Sciences. (second edition). Academic Press, 627pp, 2006. W. Feller. An Introduction to Probability Theory and Its Applications. Vol. 2. John Wiley and Sons, 626pp, 1966. D. A. Jones and G. Weymouth. An Australian monthly rainfall data set. Technical Report No. 70, Bur. Met. Australia, 1997. E. Kalnay, M. Kanamitsu, R. Kistler, W. Collins, D. Deaven, L. Gandin, M. Iredell, S. Saha, G. White, J. Woollen, Y. Zhu, M. Chelliah, W. Ebisuzaki, W. Higgins, J. Janowiak, K. C. Mo, C. Ropelewski, J. Wang, A. Leetmaa, R. Reynolds, Roy Jenne, and D. Joseph. The NCEP/NCAR 40-year reanalysis project. Bull. Amer. Meteor. Soc., 77:437--471, 1996. doi:10.1175/1520-0477(1996)077<0437:TNYRP>2.0.CO;