On the representation error in data assimilation

Representation, representativity, representativeness error, forward interpolation error, forward model error, observation operator error, aggregation error and sampling error are all terms used to refer to components of observation error in the context of data assimilation. This paper is an attempt to consolidate the terminology that has been used in the earth sciences literature and was suggested at a European Space Agency workshop held in Reading in April 2014. We review the state-of-the-art, and through examples, motivate the terminology. In addition to a theoretical framework, examples from application areas of satellite data assimilation, ocean reanalysis and atmospheric chemistry data assimilation are provided. Diagnosing representation error statistics as well as their use in state-of-the-art data assimilation systems is discussed within a consistent framework.

[1]  A. Jazwinski.  Stochastic Processes and Filtering Theory , 1970 .

[2]  P. Bauer,et al.  Satellite cloud and precipitation assimilation at operational NWP centres , 2011 .

[3]  Dominik Jacques,et al.  The Impacts of Representing the Correlation of Errors in Radar Data Assimilation. Part II: Model Output as Background Estimates , 2015 .

[4]  Claire E. Bulgin,et al.  Sampling uncertainty in gridded sea surface temperature products and Advanced Very High Resolution Radiometer (AVHRR) Global Area Coverage (GAC) data , 2016 .

[5]  D. P. DEE,et al.  Bias and data assimilation , 2005 .

[6]  Richard Ménard,et al.  Error covariance estimation methods based on analysis residuals: theoretical foundation and convergence properties derived from simplified observation networks , 2016 .

[7]  Lin Wu,et al.  Optimal representation of source‐sink fluxes for mesoscale carbon dioxide inversion with synthetic data , 2011 .

[8]  Graeme Kelly,et al.  Understanding Atmospheric Motion Vector Vertical Representativity Using a Simulation Study and First-Guess Departure Statistics , 2015 .

[9]  Neill E. Bowler.  Accounting for the effect of observation errors on verification of MOGREPS , 2008 .

[10]  C. Bishop,et al.  Resilience of Hybrid Ensemble/3DVAR Analysis Schemes to Model Error and Ensemble Covariance Error , 2004 .

[11]  Peter R. Oke,et al.  Representation Error of Oceanic Observations for Data Assimilation , 2008 .

[12]  Marc Berenguer,et al.  A Study of the Error Covariance Matrix of Radar Rainfall Estimates in Stratiform Rain , 2008 .

[13]  Henk Eskes,et al.  Representativeness errors in comparing chemistry transport and chemistry climate models with satellite UV-Vis tropospheric column retrievals , 2015 .

[14]  Anthony Hollingsworth,et al.  The statistical structure of short-range forecast errors as determined from radiosonde data , 1986 .

[15]  Svetlana N. Losa,et al.  Weak constraint parameter estimation for a simple ocean ecosystem model: what can we learn about the model and data? , 2004 .

[16]  Marc Berenguer,et al.  A Study of the Error Covariance Matrix of Radar Rainfall Estimates in Stratiform Rain. Part II: Scale Dependence , 2009 .

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

[18]  J. Carton,et al.  A Reanalysis of Ocean Climate Using Simple Ocean Data Assimilation (SODA) , 2008 .

[19]  Elizabeth A. Satterfield,et al.  The Treatment, Estimation, and Issues with Representation Error Modelling , 2017 .

[20]  Qiang Wang,et al.  Finite element ocean circulation model based on triangular prismatic elements, with application in studying the effect of topography representation , 2008 .

[21]  F. Rabier,et al.  The potential of high‐density observations for numerical weather prediction: A study with simulated observations , 2003 .

[22]  D. Dee.  On-line Estimation of Error Covariance Parameters for Atmospheric Data Assimilation , 1995 .

[23]  Sarah L. Dance,et al.  Diagnosing Horizontal and Inter-Channel Observation Error Correlations for SEVIRI Observations Using Observation-Minus-Background and Observation-Minus-Analysis Statistics , 2016, Remote. Sens..

[24]  Nancy Nichols,et al.  Estimating interchannel observation‐error correlations for IASI radiance data in the Met Office system † , 2014 .

[25]  Andrew Collard,et al.  Selection of IASI channels for use in numerical weather prediction , 2007 .

[26]  Jens Schröter,et al.  Assimilating NOAA SST data into BSH operational circulation model for the North and Baltic Seas: Part 2. Sensitivity of the forecast's skill to the prior model error statistics , 2014 .

[27]  Nancy Nichols,et al.  Diagnosing Observation Error Correlations for Doppler Radar Radial Winds in the Met Office UKV Model Using Observation-Minus-Background and Observation-Minus-Analysis Statistics , 2016 .

[28]  Clive D Rodgers,et al.  Microwindow selection for high-spectral-resolution sounders. , 2002, Applied optics.

[29]  Kayo Ide,et al.  Coastal ocean data assimilation using a multi-scale three-dimensional variational scheme , 2015, Ocean Dynamics.

[30]  Peter M. Lyster,et al.  Assimilation of Stratospheric Chemical Tracer Observations Using a Kalman Filter. Part I: Formulation , 2000 .

[31]  T. Hamill.  Interpretation of Rank Histograms for Verifying Ensemble Forecasts , 2001 .

[32]  Mark Buehner,et al.  Interchannel error correlation associated with AIRS radiance observations : Inference and impact in data assimilation , 2007 .

[33]  S. Cohn,et al.  An Introduction to Estimation Theory , 1997 .

[34]  Hendrik Elbern,et al.  Chemical state estimation for the middle atmosphere by four‐dimensional variational data assimilation: A posteriori validation of error statistics in observation space , 2010 .

[35]  Niels Bormann,et al.  Atmospheric Motion Vectors from Model Simulations. Part I: Methods and Characterization as Single-Level Estimates of Wind , 2014 .

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

[37]  C.,et al.  Analysis methods for numerical weather prediction , 2022 .

[38]  M. Joly,et al.  Objective classification of air quality monitoring sites over Europe , 2012 .

[39]  Preprint Mps,et al.  Representativity error for temperature and humidity using the Met Office high resolution model , 2012 .

[40]  Alicia R. Karspeck,et al.  An Ensemble Approach for the Estimation of Observational Error Illustrated for a Nominal 1° Global Ocean Model , 2016 .

[41]  R. S. Seaman,et al.  Absolute and Differential Accuracy of Analyses Achievable with Specified Observational Network Characteristics , 1977 .

[42]  E. Kalnay,et al.  Simultaneous estimation of covariance inflation and observation errors within an ensemble Kalman filter , 2009 .

[43]  Thomas Kaminski,et al.  On aggregation errors in atmospheric transport inversions , 2001 .

[44]  Herschel L. Mitchell,et al.  Discretization error and signal/error correlation in atmospheric data assimilation : (II). The effect of unresolved scales , 1997 .

[45]  Marc Bocquet,et al.  Accounting for representativeness errors in the inversion of atmospheric constituent emissions: application to the retrieval of regional carbon monoxide fluxes , 2012 .

[46]  Nancy Nichols,et al.  Data assimilation with correlated observation errors: experiments with a 1-D shallow water model , 2013 .

[47]  S. B. Healy,et al.  Use of discrete Fourier transforms in the 1D‐Var retrieval problem , 2005 .

[48]  Stephen E. Cohn,et al.  An Introduction to Estimation Theory (gtSpecial IssueltData Assimilation in Meteology and Oceanography: Theory and Practice) , 1997 .

[49]  R. Daley.  Estimating observation error statistics for atmospheric data assimilation , 1993 .

[50]  W. Briggs.  Statistical Methods in the Atmospheric Sciences , 2007 .

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

[52]  Leo Dr. Morbitzer,et al.  Die wichtigsten polymerphysikalischen Aspekte des Polycarbonats , 1988 .

[53]  Anu Dudhia,et al.  Improving the selection of IASI channels for use in numerical weather prediction , 2014 .

[54]  Michael Schulz,et al.  Will a perfect model agree with perfect observations? The impact of spatial sampling , 2016 .

[55]  Niels Bormann,et al.  Estimates of spatial and interchannel observation‐error characteristics for current sounder radiances for numerical weather prediction. II: Application to AIRS and IASI data , 2010 .

[56]  Carl Wunsch,et al.  Spatial Mapping of Time-Variable Errors in Jason-1 and TOPEX/Poseidon Sea Surface Height Measurements , 2007 .

[57]  Nancy Nichols,et al.  Observation error correlations in IASI radiance data , 2009 .

[58]  Peter Jan van Leeuwen.  Representation errors and retrievals in linear and nonlinear data assimilation , 2015 .

[59]  N. Bormann,et al.  Enhancing the impact of IASI observations through an updated observation‐error covariance matrix , 2016 .

[60]  M. Matricardi,et al.  Fast radiative transfer model for simulation of infrared atmospheric sounding interferometer radiances. , 1999, Applied optics.

[61]  Stephen E. Cohn,et al.  Treatment of Observation Error due to Unresolved Scales in Atmospheric Data Assimilation , 2006 .

[62]  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 .

[63]  Tijana Janjic Pfander.  Error due to Unresolved Scales in Estimation Problems for Atmospheric Data Assimilation. , 2001 .

[64]  G. Ueno,et al.  Bayesian estimation of the observation‐error covariance matrix in ensemble‐based filters , 2016 .

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

[66]  M. Matricardi.  Technical Note: An assessment of the accuracy of the RTTOV fast radiative transfer model using IASI data , 2009 .

[67]  Nancy Nichols,et al.  Theoretical insight into diagnosing observation error correlations using observation‐minus‐background and observation‐minus‐analysis statistics , 2016 .

[68]  A. Mcnally,et al.  A cloud detection algorithm for high‐spectral‐resolution infrared sounders , 2003 .

[69]  Arthur Gelb,et al.  Applied Optimal Estimation , 1974 .

[70]  P. L. Traon,et al.  High Resolution 3-D temperature and salinity fields derived from in situ and satellite observations , 2012 .

[71]  Gérald Desroziers,et al.  Diagnosis and adaptive tuning of observation‐error parameters in a variational assimilation , 2001 .

[72]  A. Lorenc.  A Global Three-Dimensional Multivariate Statistical Interpolation Scheme , 1981 .

[73]  Hendrik Elbern,et al.  Emission rate and chemical state estimation by 4-dimensional variational inversion , 2007 .

[74]  Preprint Mps,et al.  Conditioning of incremental variational data assimilation, with application to the Met Office system , 2010 .

[75]  Niels Bormann,et al.  Evaluation and assimilation of ATMS data in the ECMWF system , 2013 .

[76]  M. Kitchen,et al.  Representativeness errors in comparisons between radar and gauge measurements of rainfall , 1992 .

[77]  Ricardo Todling.  A Complementary Note to 'A Lag-1 Smoother Approach to System-Error Estimation': The Intrinsic Limitations of Residual Diagnostics , 2015 .

[78]  Nancy Nichols,et al.  Conditioning and preconditioning of the variational data assimilation problem , 2011 .

[79]  Nancy Nichols,et al.  Correlated observation errors in data assimilation , 2008 .

[80]  Herschel L. Mitchell,et al.  Discretization error and signal/error correlation in atmospheric data assimilation : (I). All scales resolved , 1997 .

[81]  Peter Bauer,et al.  Observation errors in all‐sky data assimilation , 2011 .

[82]  Amos S. Lawless,et al.  Accounting for model error in strong‐constraint 4D‐Var data assimilation , 2017 .

[83]  I. Zawadzki,et al.  On Radar-Raingage Comparison , 1975 .

[84]  Nancy Nichols,et al.  Diagnosing atmospheric motion vector observation errors for an operational high‐resolution data assimilation system , 2017 .

[85]  Florence Rabier,et al.  The interaction between model resolution, observation resolution and observation density in data assimilation: A one‐dimensional study , 2002 .

[86]  Florence Rabier,et al.  Properties and first application of an error‐statistics tuning method in variational assimilation , 2004 .

[87]  Scott E. Hannon,et al.  The Gastropod Fast Radiative Transfer Model for Advanced Infrared Sounders and Characterization of Its Errors for Radiance Assimilation , 2003 .

[88]  Roger Saunders,et al.  AMV errors: a new approach in NWP , 2008 .

[89]  Nick Rayner,et al.  EN4: Quality controlled ocean temperature and salinity profiles and monthly objective analyses with uncertainty estimates , 2013 .

[90]  C. Bulgin,et al.  Independent uncertainty estimates for coefficient based sea surface temperature retrieval from the Along-Track Scanning Radiometer instruments , 2016 .

[91]  W. Krajewski,et al.  On the estimation of radar rainfall error variance , 1999 .

[92]  Daniel J. Jacob,et al.  Balancing aggregation and smoothing errors in inverse models , 2015 .

[93]  Sarah L. Dance,et al.  Estimating correlated observation error statistics using an ensemble transform Kalman filter , 2014 .

[94]  Yves Candau,et al.  Regional scale ozone data assimilation using an ensemble Kalman filter and the CHIMERE chemical transport model , 2013 .

[95]  Niels Bormann,et al.  Estimates of spatial and interchannel observation‐error characteristics for current sounder radiances for numerical weather prediction. I: Methods and application to ATOVS data , 2010 .

[96]  Neill E. Bowler,et al.  On the diagnosis of model error statistics using weak‐constraint data assimilation , 2017 .

[97]  J. R. Eyre,et al.  The optimal density of atmospheric sounder observations in the Met Office NWP system , 2007 .

[98]  J. Eyre,et al.  Accounting for correlated error in the assimilation of high‐resolution sounder data , 2014 .

[99]  Witold F. Krajewski,et al.  Estimation of radar-rainfall error spatial correlation , 2009 .

[100]  John Nicol,et al.  The impact of radar and raingauge sampling errors when calibrating a weather radar , 2007 .

[101]  A. Mcnally,et al.  The assimilation of Infrared Atmospheric Sounding Interferometer radiances at ECMWF , 2009 .

[102]  Dick Dee,et al.  Adaptive bias correction for satellite data in a numerical weather prediction system , 2007 .

[103]  Craig H. Bishop,et al.  The benefits of correlated observation errors for small scales , 2015 .

[104]  Andrew C. Lorenc,et al.  Analysis methods for numerical weather prediction , 1986 .

[105]  Herschel L. Mitchell,et al.  Discretization error and signal/error correlation in atmospheric data assimilation , 1997 .

[106]  Jens Schröter,et al.  A finite-element ocean model: principles and evaluation , 2004 .

[107]  Andrew J. Majda,et al.  Ensemble Kalman filters for dynamical systems with unresolved turbulence , 2014, J. Comput. Phys..

[108]  Takemasa Miyoshi,et al.  Estimating and including observation-error correlations in data assimilation , 2013 .

[109]  R L Tao,et al.  The Broad Reach of Helminthology , 2001, Science.

[110]  J. Eyre,et al.  Assimilation of IASI at the Met Office and assessment of its impact through observing system experiments , 2009 .

[111]  Dominik Jacques,et al.  The Impacts of Representing the Correlation of Errors in Radar Data Assimilation. Part I: Experiments with Simulated Background and Observation Estimates , 2014 .

[112]  Philippe Courtier,et al.  Unified Notation for Data Assimilation : Operational, Sequential and Variational , 1997 .

[113]  O. Talagrand,et al.  Diagnosis and tuning of observational error in a quasi‐operational data assimilation setting , 2006 .

[114]  Dick Dee,et al.  Maximum-Likelihood Estimation of Forecast and Observation Error Covariance Parameters. Part I: Methodology , 1999 .

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

[116]  G. Kivman,et al.  An entropy approach to tuning weights and smoothing in the generalized inversion , 2001 .

[117]  Paul Poli,et al.  Diagnosis of observation, background and analysis‐error statistics in observation space , 2005 .

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

[119]  W. Paul Menzel,et al.  Intersatellite calibration of NOAA HIRS CO2 channels for climate studies , 2013 .

[120]  Lin Wu,et al.  Bayesian design of control space for optimal assimilation of observations. Part I: Consistent multiscale formalism , 2011 .

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

[122]  Preprint Mps.  The Error of Representation: Basic Understanding , 2014 .

[123]  P. Courtier,et al.  The ECMWF implementation of three‐dimensional variational assimilation (3D‐Var). I: Formulation , 1998 .

[124]  O. Talagrand,et al.  Impact of observational error on the validation of ensemble prediction systems , 2008 .

[125]  Dacian N. Daescu,et al.  Adjoint sensitivity of the model forecast to data assimilation system error covariance parameters , 2010 .