Motivated by the spurious variance loss encountered during covariance propagation in atmospheric and other large-scale data assimilation systems, we consider the problem for state dynamics governed by the continuity and related hyperbolic partial differential equations. This loss of variance has been attributed to reduced-rank representations of the covariance matrix, as in ensemble methods for example, or else to the use of dissipative numerical methods. Through a combination of analytical work and numerical experiments, we demonstrate that significant variance loss, as well as gain, typically occurs during covariance propagation, even at full rank. The cause of this unusual behavior is a discontinuous change in the continuum covariance dynamics as correlation lengths become small, for instance in the vicinity of sharp gradients in the velocity field. This discontinuity in the covariance dynamics arises from hyperbolicity: the diagonal of the kernel of the covariance operator is a characteristic surface for advective dynamics. Our numerical experiments demonstrate that standard numerical methods for evolving the state are not adequate for propagating the covariance, because they do not capture the discontinuity in the continuum covariance dynamics as correlations lengths tend to zero. Our analytical and numerical results show that this leads to significant, spurious variance loss in certain regions and gain in others. The results suggest that developing local covariance propagation methods designed specifically to capture covariance evolution near the diagonal may prove a useful alternative to current methods of covariance propagation.


  1. covariance propagation
  2. variance loss
  3. data assimilation
  4. advective systems

MSC codes

  1. 35L65
  2. 47B38
  3. 65M22
  4. 86A10
  5. 65M75
  6. 65M32
  7. 65M80

Get full access to this article

View all available purchase options and get full access to this article.


J. L. Anderson and S. L. Anderson, A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts, Mon. Weather Rev., 127 (1999), pp. 2741–2758, https://doi.org/10.1175/1520-0493(1999)127%3C2741:AMCIOT%3E2.0.CO;2.
J. Barkmeijer, M. Van Gijzen, and F. Bouttier, Singular vectors and estimates of the analysis-error covariance metric, Q. J. Roy. Meteor. Soc., 124 (1998), pp. 1695–1713, https://doi.org/10.1002/qj.49712454916.
J. Berner, U. Achatz, L. Batté, L. Bengtsson, A. D. L. Cámara, H. M. Christensen, M. Colangeli, D. R. B. Coleman, D. Crommelin, S. I. Dolaptchiev, C. L. E. Franzke, P. Friederichs, P. Imkeller, H. Järvinen, S. Juricke, V. Kitsios, F. Lott, V. Lucarini, S. Mahajan, T. N. Palmer, C. Penland, M. Sakradzija, J.-S. von Storch, A. Weisheimer, M. Weniger, P. D. Williams, and J.-I. Yano, Stochastic parameterization: Toward a new view of weather and climate models, Bull. Amer. Meteorol. Soc., 98 (2017), pp. 565–588, https://doi.org/10.1175/BAMS-D-15-00268.1.
G. Beylkin and J. Keiser, An adaptive pseudo-wavelet approach for solving nonlinear partial differential equations, in Multiscale Wavelet Methods for Partial Differential Equations, Wavelet Anal. Appl. 6, Academic Press, San Diego, CA, 1997, pp. 137–197.
R. Buizza and T. Palmer, The singular-vector structure of the atmospheric global circulation, J. Atmos. Sci., 52 (1995), pp. 1434–1456, https://doi.org/10.1175/1520-0469(1995)052%3C1434:TSVSOT%3E2.0.CO;2.
S. E. Cohn, Dynamics of short-term univariate forecast error covariances, Mon. Weather Rev., 121 (1993), pp. 3123–3149, https://doi.org/10.1175/1520-0493(1993)121%3C3123:DOSTUF%3E2.0.CO;2.
S. E. Cohn, The principle of energetic consistency in data assimilation, in Data Assimilation, Springer, Berlin, Heidelberg, 2010, pp. 137–216, https://doi.org/10.1007/978-3-540-74703-1_7.
R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II: Partial Differential Equations, Interscience Publishers, New York, 1962.
P. Courtier and O. Talagrand, Variational assimilation of meteorological observations with the adjoint vorticity equation, II: Numerical results, Q. J. Roy. Meteor. Soc, 113 (1987), pp. 1329–1347, https://doi.org/10.1002/qj.49711347813.
J. Crank and P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Math. Proc. Cambridge Philos. Soc., 43 (1947), pp. 50–67, https://doi.org/10.1017/S0305004100023197.
R. Daley, Atmospheric Data Analysis, Cambridge University Press, Cambridge, UK, 1991.
M. Ehrendorfer and J. Tribbia, Optimal prediction of forecast error covariances through singular vectors, J. Atmos. Sci, 54 (1997), pp. 286–313, https://doi.org/10.1175/1520-0493(1999)127%3C2741:AMCIOT%3E2.0.CO;2.
G. Evensen, Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics, J. Geophys. Res., 99 (1994), pp. 10143–10162, https://doi.org/10.1029/94JC00572.
R. Furrer and T. Bengtsson, Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants, J. Multivariate Anal., 98 (2007), pp. 227–255, https://doi.org/10.1016/j.jmva.2006.08.003.
G. Gaspari and S. E. Cohn, Construction of correlation functions in two and three dimensions, Q. J. Roy. Meteor. Soc., 125 (1999), pp. 723–757, https://doi.org/10.1002/qj.49712555417.
G. Gaspari, S. E. Cohn, J. Guo, and S. Pawson, Construction and application of covariance functions with variable length-fields, Q. J. Roy. Meteor. Soc., 132 (2006), pp. 1815–1838, https://doi.org/10.1256/qj.05.08.
J. Holton and G. J. Hakim, An Introduction to Dynamic Meteorology, 5th ed., Academic Press, New York, 2013.
P. L. Houtekamer and H. L. Mitchell, An adaptive ensemble Kalman filter, Mon. Weather Rev., 128 (2000), pp. 416–433, https://doi.org/10.1175/1520-0493(2000)128%3C0416:AAEKF%3E2.0.CO;2.
P. L. Houtekamer and H. L. Mitchell, Ensemble Kalman filtering, Q. J. Roy. Meteor. Soc., 131 (2005), pp. 3269–3289, https://doi.org/10.1256/qj.05.135.
J. K. Hunter and B. Nachtergaele, Applied Analysis, World Scientific, River Edge, NJ, 2001.
A. H. Jazwinski, Stochastic Processes and Filter Theory, Academic Press, New York, 1970.
R. Kalman, A new approach to linear filtering and prediction problems, Trans. ASME Ser. D. J. Basic Engrg., 82 (1960), pp. 35–45, https://doi.org/10.1115/1.3662552.
E. Kalnay, Atmospheric Modeling, Data Assimilation and Predictability, Cambridge University Press, Cambridge, UK, 2003.
P. Lax and B. Wendroff, Systems of conservation laws, Comm. Pure Appl. Math., 13 (1960), pp. 217–237, https://doi.org/10.1002/cpa.3160130205.
C. E. Leith, Theoretical skill of Monte Carlo forecasts, Mon. Weather Rev., 102 (1974), pp. 409–418.
E. Lorenz, A study of the predictability of a 28-variable atmospheric model, Tellus, 17 (1965), pp. 321–333, https://doi.org/10.3402/tellusa.v17i3.9076.
P. M. Lyster, S. E. Cohn, B. Zhang, L.-P. Chang, R. Ménard, K. Olson, and R. Renka, A Lagrangian trajectory filter for constituent data assimilation, Q. J. Roy. Meteor. Soc., 130 (2004), pp. 2315–2334, https://doi.org/10.1256/qj.02.234.
P. S. Maybeck, Stochastic Models, Estimation and Control, Vol. 2, Academic Press, London, 1982.
R. Ménard, S. E. Cohn, L. P. Chang, and P. M. Lyster, Assimilation of stratospheric chemical tracer observations using a Kalman filter, Part I: Formulation, Mon. Weather Rev., 128 (2000), pp. 2654–2671, https://doi.org/10.1175/1520-0493(2000)128.
R. Ménard, S. Skachko, and O. Pannekoucke, Numerical discretization causing error variance loss and the need for inflation, Q. J. Roy. Meteor. Soc., 147 (2021), pp. 3498>–3520, https://doi.org/10.1002/qj.4139.
H. L. Mitchell and P. L. Houtekamer, An adaptive ensemble Kalman filter, Mon. Weather. Rev., 128 (2000), pp. 416–433, https://doi.org/10.1175/1520-0493(2000)128.
O. Pannekoucke, M. Bocquet, and R. Ménard, Parametric covariance dynamics for the nonlinear diffusive Burgers equation, Nonlinear Processes Geophys., 25 (2018), pp. 481–495, https://doi.org/10.5194/npg-25-481-2018.
O. Pannekoucke, R. Ménard, M. E. Aabaribaoune, and M. Plu, A methodology to obtain model-error covariances due to the discretization scheme from the parametric Kalman filter perspective, Nonlinear Processes Geophys., 28 (2021), pp. 1–22, https://doi.org/10.5194/npg-28-1-2021.
O. Pannekoucke, S. Ricci, S. Barthelemy, R. Ménard, and O. Thual, Parametric Kalman Filter for chemical transport models, Tellus A, 68 (2016), 31547–31561, https://doi.org/10.3402/tellusa.v68.31547.
N. A. Phillips, The spatial statistics of random geostrophic modes and first-guess errors, Tellus A, 38A (1986), pp. 314–332, https://doi.org/10.1111/j.1600-0870.1986.tb00418.x.
M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, New York, 1972.
M. Shearer and R. Levy, Partial Differential Equations: An Introduction to Theory and Applications, Princeton University Press, Princeton, NJ, 2015.
O. Talagrand and P. Courtier, Variational assimilation of meteorological observations with the adjoint vorticity equation. I: Theory, Q. J. Roy. Meteor. Soc., 113 (1987), pp. 1311–1328, https://doi.org/10.1002/qj.49711347812.
D. Zwillinger, ed., CRC Standard Mathematical Tables and Formulae, 30th ed., CRC Press, Boca Raton, FL, 1996.

Information & Authors


Published In

cover image SIAM/ASA Journal on Uncertainty Quantification
SIAM/ASA Journal on Uncertainty Quantification
Pages: 886 - 914
ISSN (online): 2166-2525


Submitted: 26 August 2021
Accepted: 24 February 2022
Published online: 5 August 2022


  1. covariance propagation
  2. variance loss
  3. data assimilation
  4. advective systems

MSC codes

  1. 35L65
  2. 47B38
  3. 65M22
  4. 86A10
  5. 65M75
  6. 65M32
  7. 65M80



Shay Gilpin Contact the author
Department of Applied Mathematics, University of Colorado Boulder, Boulder, CO 80303-0429 USA ([email protected]).
Tomoko Matsuo
Ann and H. J. Smead Department of Aerospace Engineering Sciences, University of Colorado Boulder, Boulder, CO, Department of Applied Mathematics, University of Colorado Boulder, Boulder, CO 80303-0429 USA ([email protected]).
Stephen E. Cohn
Global Modeling and Assimilation Office, NASA Goddard Space Flight Center, Greenbelt, MD 20771 USA ([email protected]).

Funding Information

This work was supported by the National Science Foundation (NSF) Graduate Research Fellowship Program under grant DGE-1650115. The work of the second author was supported by the NSF CAREER Program under grant AGS-1848544. The work of the third author was supported by the National Aeronautics and Space Administration (NASA) under Modeling, Analysis and Prediction (MAP) program Core funding to the Global Modeling and Assimilation Office (GMAO) at Goddard Space Flight Center (GSFC). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the NSF or of NASA.

Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.







Copy the content Link

Share with email

Email a colleague

Share on social media