Study of conservation laws with the Local Ensemble Transform Kalman Filter

Numerical discretization schemes have a long history of incorporating the most important conservation properties of the continuous system in order to improve the prediction of the nonlinear flow. The question arises whether data assimilation algorithms should follow a similar approach. To address this issue, we explore the conservation properties during data assimilation using perfect model experiments with a 2D shallow-water model preserving important properties of the true nonlinear flow. The data assimilation scheme used here is the Local Ensemble Transform Kalman Filter with varying observed variables, inflation, localization radius and thinning interval. It is found that, during the assimilation, the total energy of the analysis ensemble mean converges with time towards the nature run value. However, enstrophy, divergence and the energy spectra are strongly affected by the data assimilation settings. Having in mind that the conservation of both the kinetic energy and enstrophy by the momentum advection schemes in the case of non-divergent flow prevents a systematic and unrealistic energy cascade towards the high wave numbers, we test the effects on the prediction depending on the type of error in the initial condition. During the assimilation, we assess the downward nonlinear energy cascade through a scalar, domain-averaged noise measure. We show that the accumulated noise during assimilation and the error of analysis are good indicators of the quality of the prediction.

[1]  P. Houtekamer,et al.  Ensemble size, balance, and model-error representation in an ensemble Kalman filter , 2002 .

[2]  Ratko Vasic,et al.  A Class of Conservative Fourth-Order Advection Schemes and Impact of Enhanced Formal Accuracy on Extended-Range Forecasts , 2011 .

[3]  Gary D. Egbert,et al.  Estimates of M2 Tidal Energy Dissipation from TOPEX/Poseidon Altimeter Data , 2001 .

[4]  R. Anthes.  The dynamics and energetics of mature tropical cyclones , 1974 .

[5]  Tijana Janjić,et al.  Assimilation of Mode-S EHS Aircraft Observations in COSMO-KENDA , 2016 .

[6]  A. Chorin,et al.  Implicit particle filtering for models with partial noise, and an application to geomagnetic data assimilation , 2011, 1109.3664.

[7]  P. Leeuwen,et al.  Nonlinear data assimilation in geosciences: an extremely efficient particle filter , 2010 .

[8]  Colin J. Cotter,et al.  Probabilistic Forecasting and Bayesian Data Assimilation , 2015 .

[9]  Rick Salmon.  Poisson-Bracket Approach to the Construction of Energy- and Potential-Enstrophy- Conserving Algorithms for the Shallow-Water Equations , 2004 .

[10]  Peter R. Oke,et al.  Impacts of localisation in the EnKF and EnOI: experiments with a small model , 2007 .

[11]  R. Daley.  Atmospheric Data Analysis , 1991 .

[12]  P. Houtekamer,et al.  Data Assimilation Using an Ensemble Kalman Filter Technique , 1998 .

[13]  Pierre Brasseur,et al.  A demonstration of ensemble-based assimilation methods with a layered OGCM from the perspective of operational ocean forecasting systems , 2003 .

[14]  Takemasa Miyoshi,et al.  Local Ensemble Transform Kalman Filtering with an AGCM at a T159/L48 Resolution , 2007 .

[15]  Roland Potthast,et al.  Kilometre‐scale ensemble data assimilation for the COSMO model (KENDA) , 2016 .

[16]  Georg A. Gottwald,et al.  Controlling balance in an ensemble Kalman filter , 2014 .

[17]  T. Janjić,et al.  Testing particle filters on simple convective‐scale models Part l: A stochastic cloud model , 2016 .

[18]  J. Kepert.  Covariance localisation and balance in an Ensemble Kalman Filter , 2009 .

[19]  Lars Nerger,et al.  Domain localization in ensemble based Kalman filter algorithms , 2011 .

[20]  S. Cohn,et al.  Assessing the Effects of Data Selection with the DAO Physical-Space Statistical Analysis System* , 1998 .

[21]  Jens Schröter,et al.  Impact of combining GRACE and GOCE gravity data on ocean circulation estimates , 2011 .

[22]  E. Lorenz.  The predictability of a flow which possesses many scales of motion , 1969 .

[23]  Martin Verlaan,et al.  Conservation of Mass and Preservation of Positivity with Ensemble-Type Kalman Filter Algorithms , 2014 .

[24]  Geir Evensen,et al.  Assimilation of SLA and SST data into an OGCM for the Indian Ocean , 2002 .

[25]  A. Arakawa,et al.  A Potential Enstrophy and Energy Conserving Scheme for the Shallow Water Equations , 1981 .

[26]  E. Kalnay,et al.  Balance and Ensemble Kalman Filter Localization Techniques , 2011 .

[27]  Jens Schröter,et al.  Sequential assimilation of multi-mission dynamical topography into a global finite-element ocean model , 2008 .

[28]  Matthias Sommer,et al.  A conservative scheme for the shallow‐water system on a staggered geodesic grid based on a Nambu representation , 2009 .

[29]  H. Ngodock,et al.  The Maintenance of Conservative Physical Laws within Data Assimilation Systems , 2003 .

[30]  Jens Schröter,et al.  Assimilation of geodetic dynamic ocean topography with ensemble basedKalman filter , 2012 .

[31]  Lawrence L. Takacs,et al.  Data Assimilation Using Incremental Analysis Updates , 1996 .

[32]  Jeffrey L. Anderson,et al.  A Monte Carlo Implementation of the Nonlinear Filtering Problem to Produce Ensemble Assimilations and Forecasts , 1999 .

[33]  Louis J. Wicker,et al.  Wind and Temperature Retrievals in the 17 May 1981 Arcadia, Oklahoma, Supercell: Ensemble Kalman Filter Experiments , 2004 .

[34]  Peter Lynch,et al.  Initialization of the HIRLAM Model Using a Digital Filter , 1992 .

[35]  S. Cohn,et al.  Ooce Note Series on Global Modeling and Data Assimilation Construction of Correlation Functions in Two and Three Dimensions and Convolution Covariance Functions , 2022 .

[36]  Nils Gustafsson,et al.  Dynamical response of equatorial waves in variational data assimilation , 2004 .

[37]  Thierry Penduff,et al.  A four-year eddy-permitting assimilation of sea-surface temperature and altimetric data in the South Atlantic Ocean , 2002 .

[38]  George C. Craig,et al.  Testing particle filters on simple convective‐scale models. Part 2: A modified shallow‐water model , 2016 .

[39]  Jean-Michel Brankart,et al.  Implementation of a multivariate data assimilation scheme for isopycnic coordinate ocean models: Application to a 1993–1996 hindcast of the North Atlantic Ocean circulation , 2003 .

[40]  Julius Chang,et al.  General circulation models of the atmosphere , 1977 .

[41]  A. M. Clayton,et al.  On the Relationship between Incremental Analysis Updating and Incremental Digital Filtering , 2004 .

[42]  Eugenia Kalnay,et al.  Atmospheric Modeling, Data Assimilation and Predictability , 2002 .

[43]  P. Houtekamer,et al.  A Sequential Ensemble Kalman Filter for Atmospheric Data Assimilation , 2001 .

[44]  Georg A. Gottwald,et al.  Controlling Overestimation of Error Covariance in Ensemble Kalman Filters with Sparse Observations: A Variance-Limiting Kalman Filter , 2011, 1108.5801.

[45]  Trevor Davies,et al.  An Overview of Numerical Methods for the Next Generation U.K. NWP and Climate Model , 1997 .

[46]  C. Snyder,et al.  Assimilation of Simulated Doppler Radar Observations with an Ensemble Kalman Filter , 2003 .

[47]  Richard Asselin,et al.  Frequency Filter for Time Integrations , 1972 .

[48]  B. Khattatov,et al.  Data assimilation : making sense of observations , 2010 .

[49]  G. Zängl,et al.  The ICON (ICOsahedral Non‐hydrostatic) modelling framework of DWD and MPI‐M: Description of the non‐hydrostatic dynamical core , 2015 .

[50]  R. Handel,et al.  Can local particle filters beat the curse of dimensionality , 2013, 1301.6585.

[51]  G. S. Ketefian,et al.  A mass, energy, vorticity, and potential enstrophy conserving lateral fluid-land boundary scheme for the shallow water equations , 2008, J. Comput. Phys..

[52]  Zavisa Janjic,et al.  Nonlinear Advection Schemes and Energy Cascade on Semi-Staggered Grids , 1984 .

[53]  Istvan Szunyogh,et al.  Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter , 2005, physics/0511236.