Improved analysis‐error covariance matrix for high‐dimensional variational inversions: application to source estimation using a 3D atmospheric transport model

Variational methods are widely used to solve geophysical inverse problems. Although gradient-based minimization algorithms are available for high-dimensional problems (dimension >106), they do not provide an estimate of the errors in the optimal solution. In this study, we assess the performance of several numerical methods to approximate the analysis-error covariance matrix, assuming reasonably linear models. The evaluation is performed for a CO2 flux estimation problem using synthetic remote-sensing observations of CO2 columns. A low-dimensional experiment is considered in order to compare the analysis error approximations to a full-rank finite-difference inverse Hessian estimate, followed by a realistic high-dimensional application. Two stochastic approaches, a Monte-Carlo simulation and a method based on random gradients of the cost function, produced analysis error variances with a relative error 120%), a new preconditioner that efficiently accumulates information on the diagonal of the inverse Hessian dramatically improves the results (relative error <50%). Furthermore, performing several cycles of the BFGS algorithm using the same gradient and vector pairs enhances its performance (relative error <30%) and is necessary to obtain convergence. Leveraging those findings, we proposed a BFGS hybrid approach which combines the new preconditioner with several BFGS cycles using information from a few (3–5) Monte-Carlo simulations. Its performance is comparable to the stochastic approximations for the low-dimensional case, while good scalability is obtained for the high-dimensional experiment. Potential applications of these new BFGS methods range from characterizing the information content of high-dimensional inverse problems to improving the convergence rate of current minimization algorithms.

[1]  William C. Davidon,et al.  Variable Metric Method for Minimization , 1959, SIAM J. Optim..

[2]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..

[3]  J. J. Moré,et al.  Quasi-Newton Methods, Motivation and Theory , 1974 .

[4]  Albert Tarantola,et al.  Inverse problem theory - and methods for model parameter estimation , 2004 .

[5]  Tatsuya Yokota,et al.  Global Concentrations of CO2 and CH4 Retrieved from GOSAT: First Preliminary Results , 2009 .

[6]  Dylan B. A. Jones,et al.  Construction of non-diagonal background error covariance matrices for global chemical data assimilation , 2010 .

[7]  F. L. Dimet,et al.  On deterministic error analysis in variational data assimilation , 2005 .

[8]  Serge Gratton,et al.  A reduced and limited‐memory preconditioned approach for the 4D‐Var data‐assimilation problem , 2011 .

[9]  Rebecca Castano,et al.  Corrigendum to "The ACOS CO 2 retrieval algorithm – Part 1: Description and validation against synthetic observations" published in Atmos. Meas. Tech., 5, 99–121, 2012 , 2012 .

[10]  L. Berre,et al.  Filtering of Background Error Variances and Correlations by Local Spatial Averaging: A Review , 2010 .

[11]  Susan S. Kulawik,et al.  Inverse modeling of CO 2 sources and sinks using satellite observations of CO 2 from TES and surface flask measurements , 2011 .

[12]  Claude Lemaréchal,et al.  Some numerical experiments with variable-storage quasi-Newton algorithms , 1989, Math. Program..

[13]  J. Dennis,et al.  Sizing and least-change secant methods , 1993 .

[14]  D. Nychka.  Data Assimilation” , 2006 .

[15]  Philippe Courtier,et al.  Four‐Dimensional Assimilation In the Presence of Baroclinic Instability , 1992 .

[16]  Mark Buehner,et al.  Evaluation of a Spatial/Spectral Covariance Localization Approach for Atmospheric Data Assimilation , 2012 .

[17]  J. Nocedal.  Updating Quasi-Newton Matrices With Limited Storage , 1980 .

[18]  Trissevgeni Stavrakou,et al.  Inversion of CO and NO x emissions using the adjoint of the IMAGES model , 2004 .

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

[20]  I. Yu. Gejadze,et al.  Computation of the analysis error covariance in variational data assimilation problems with nonlinear dynamics , 2011, J. Comput. Phys..

[21]  Didier Auroux,et al.  Limited-Memory BFGS Diagonal Preconditioners for a Data Assimilation Problem in Meteorology , 2000 .

[22]  Serge Gratton,et al.  On A Class of Limited Memory Preconditioners For Large Scale Linear Systems With Multiple Right-Hand Sides , 2011, SIAM J. Optim..

[23]  Ian G. Enting,et al.  Inverse problems in atmospheric constituent transport , 2002 .

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

[25]  D. Daescu.  On the Sensitivity Equations of Four-Dimensional Variational (4D-Var) Data Assimilation , 2008 .

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

[27]  R. Bannister.  A review of forecast error covariance statistics in atmospheric variational data assimilation. II: Modelling the forecast error covariance statistics , 2008 .

[28]  J. Seinfeld,et al.  Development of the adjoint of GEOS-Chem , 2006 .

[29]  Ionel M. Navon,et al.  Second-Order Information in Data Assimilation* , 2002 .

[30]  Philippe Ciais,et al.  What eddy‐covariance measurements tell us about prior land flux errors in CO2‐flux inversion schemes , 2012 .

[31]  Wah June Leong,et al.  A class of diagonal preconditioners for limited memory BFGS method , 2013, Optim. Methods Softw..

[32]  J. Renaud.  Numerical Optimization, Theoretical and Practical Aspects— , 2006, IEEE Transactions on Automatic Control.

[33]  Serge Gratton,et al.  Limited‐memory preconditioners, with application to incremental four‐dimensional variational data assimilation , 2008 .

[34]  Jean Charles Gilbert,et al.  Numerical Optimization: Theoretical and Practical Aspects , 2003 .

[35]  Pauli Heikkinen,et al.  Inferring regional sources and sinks of atmospheric CO 2 from GOSAT XCO 2 data , 2013 .

[36]  François-Marie Bréon,et al.  Contribution of the Orbiting Carbon Observatory to the estimation of CO2 sources and sinks: Theoretical study in a variational data assimilation framework , 2007 .

[37]  Dylan B. A. Jones,et al.  Modeling global atmospheric CO 2 with improved emission inventories and CO 2 production from the oxidation of other carbon species , 2010 .

[38]  Gérald Desroziers,et al.  Use of randomization to diagnose the impact of observations on analyses and forecasts , 2005 .

[39]  Dimitris Menemenlis,et al.  Carbon monitoring system flux estimation and attribution: impact of ACOS-GOSAT XCO2 sampling on the inference of terrestrial biospheric sources and sinks , 2014 .

[40]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[41]  L. Nazareth.  A Relationship between the BFGS and Conjugate Gradient Algorithms and Its Implications for New Algorithms , 1979 .

[42]  Ross N. Bannister,et al.  A review of forecast error covariance statistics in atmospheric variational data assimilation. I: Characteristics and measurements of forecast error covariances , 2008 .