The Local Ensemble Tangent Linear Model: an enabler for coupled model 4D‐Var

A leading Data Assimilation (DA) technique in meteorology is 4D-Var which relies on the Tangent Linear Model (TLM) of the nonlinear model and its adjoint. The difficulty of building and maintaining traditional TLMs and adjoints of coupled ocean–wave–atmosphere–etc. models is daunting. On the other hand, coupled model ensemble forecasts are readily available. Here, we show how an ensemble forecast can be used to construct an accurate Local Ensemble TLM (LETLM) and adjoint of the entire coupled system. The method features a local influence region containing all the variables that could possibly influence the time evolution of some target variable(s) near the centre of the region. We prove that high accuracy is guaranteed provided that (i) the ensemble perturbations are governed by linear dynamics, and (ii) the number of ensemble members exceeds the number of variables in the influence region. The approach is illustrated in a simple coupled model. This idealized coupled model has some realistic features including reasonable predictability limits in the upper atmosphere, lower atmosphere, upper ocean and lower ocean of 10, 96, 160 and 335 days, respectively. In addition, the length-scale of eddies in the ocean is about one fifth of those in the atmosphere. The easy manner in which the adjoint is obtained from the LETLM is also described and illustrated by demonstrating how the LETLM adjoint predicts the high sensitivity of oceanic boundary-layer evolution to changes in the atmosphere. Finally, the feasibility of LETLMs for 4D-Var is demonstrated. Specifically, a case is considered with a 5-day data assimilation window in which nonlinear terms play a significant role in the evolution of forecast error; it is shown that the posterior mode delivered by 4D-Var with an LETLM, its adjoint and ten outer loops approximately recovers the true state in spite of a spatially sparse observational network.

[1]  Sasa Gabersek,et al.  Effect of Two-Way Air–Sea Coupling in High and Low Wind Speed Regimes , 2010 .

[2]  Edward N. Lorenz,et al.  Designing Chaotic Models , 2005 .

[3]  R. Buizza,et al.  The forecast skill horizon , 2015 .

[4]  Craig H. Bishop,et al.  Adaptive Ensemble Covariance Localization in Ensemble 4D-VAR State Estimation , 2011 .

[5]  A. Simmons,et al.  The ECMWF operational implementation of four‐dimensional variational assimilation. I: Experimental results with simplified physics , 2007 .

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

[7]  Ronald M. Errico,et al.  Sensitivity Analysis Using an Adjoint of the PSU-NCAR Mesoseale Model , 1992 .

[8]  Liang Xu,et al.  Development of NAVDAS-AR: formulation and initial tests of the linear problem , 2005 .

[9]  Philippe Courtier,et al.  Sensitivity of forecast errors to initial conditions , 1996 .

[10]  A. Rosati,et al.  System Design and Evaluation of Coupled Ensemble Data Assimilation for Global Oceanic Climate Studies , 2007 .

[11]  Neill E. Bowler,et al.  Comparison of Hybrid-4DEnVar and Hybrid-4DVar Data Assimilation Methods for Global NWP , 2015 .

[12]  Y. Trémolet.  Incremental 4D-Var convergence study , 2007 .

[13]  Matthias Morzfeld,et al.  To what extent is your data assimilation scheme designed to find the posterior mean, the posterior mode or something else? , 2016 .

[14]  Keith Haines,et al.  Origin and impact of initialisation shocks in coupled atmosphere-ocean forecasts , 2015 .

[15]  A. Lorenc,et al.  Operational implementation of a hybrid ensemble/4D‐Var global data assimilation system at the Met Office , 2013 .

[16]  Ann Henderson-Sellers,et al.  A Climate Modelling Primer: McGuffie/A Climate Modelling Primer , 2005 .

[17]  Sean Milton,et al.  Coupled versus uncoupled hindcast simulations of the Madden‐Julian Oscillation in the Year of Tropical Convection , 2014 .

[18]  P. Courtier,et al.  A strategy for operational implementation of 4D‐Var, using an incremental approach , 1994 .

[19]  Jeffrey K. Uhlmann,et al.  Unscented filtering and nonlinear estimation , 2004, Proceedings of the IEEE.

[20]  Jeffrey L. Anderson.  A Local Least Squares Framework for Ensemble Filtering , 2003 .

[21]  Travis A. Smith,et al.  Ocean-wave coupled modeling in COAMPS-TC: A study of Hurricane Ivan (2004) , 2013 .

[22]  Jeffrey L. Anderson.  An Ensemble Adjustment Kalman Filter for Data Assimilation , 2001 .

[23]  Andrew C. Lorenc,et al.  4D‐Var and the butterfly effect: Statistical four‐dimensional data assimilation for a wide range of scales , 2007 .

[24]  Craig H. Bishop,et al.  Localized Ensemble-Based Tangent Linear Models and Their Use in Propagating Hybrid Error Covariance Models , 2016 .

[25]  Uang,et al.  The NCEP Climate Forecast System Reanalysis , 2010 .

[26]  Liang Xu,et al.  Development of NAVDAS-AR: non-linear formulation and outer loop tests , 2006 .

[27]  Craig H. Bishop,et al.  Comparison of Hybrid Ensemble/4DVar and 4DVar within the NAVDAS-AR Data Assimilation Framework , 2013 .

[28]  A. Lorenc,et al.  The Met Office global four‐dimensional variational data assimilation scheme , 2007 .

[29]  Monique Tanguay,et al.  Extension of 3DVAR to 4DVAR: Implementation of 4DVAR at the Meteorological Service of Canada , 2007 .

[30]  Kristian Mogensen,et al.  A coupled data assimilation system for climate reanalysis , 2016 .

[31]  Q. Xiao,et al.  An Ensemble-Based Four-Dimensional Variational Data Assimilation Scheme. Part I: Technical Formulation and Preliminary Test , 2008 .

[32]  Mark Buehner,et al.  Four-dimensional ensemble-variational data assimilation for global deterministic weather prediction , 2013 .

[33]  Adam A. Scaife,et al.  Global Seasonal forecast system version 5 (GloSea5): a high‐resolution seasonal forecast system , 2015 .

[34]  Amos S. Lawless,et al.  Exploring strategies for coupled 4D-Var data assimilation using an idealised atmosphere–ocean model , 2015 .