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References

Master reference list for the Mathematical Reference, grouped by chapter. Per-chapter lists repeat the most load-bearing entries.

Foundational DA texts

  1. Evensen, G. (2009). Data Assimilation: The Ensemble Kalman Filter. 2nd ed., Springer.
  2. Carrassi, A., Bocquet, M., Bertino, L., & Evensen, G. (2018). Data assimilation in the geosciences: An overview of methods, issues, and perspectives. WIREs Climate Change, 9(5), e535.
  3. Asch, M., Bocquet, M., & Nodet, M. (2016). Data Assimilation: Methods, Algorithms, and Applications. SIAM.
  4. Evensen, G. (2003). The Ensemble Kalman Filter: theoretical formulation and practical implementation. Ocean Dynamics, 53, 343–367.

EnKF — stochastic formulation (chapters 4–5)

  1. Evensen, G. (1994). Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res., 99(C5), 10143–10162.
  2. Burgers, G., van Leeuwen, P. J., & Evensen, G. (1998). Analysis scheme in the ensemble Kalman filter. Mon. Wea. Rev., 126, 1719–1724.

ETKF and square-root variants (chapters 6–7)

  1. Bishop, C. H., Etherton, B. J., & Majumdar, S. J. (2001). Adaptive sampling with the ensemble transform Kalman filter. Part I: Theoretical aspects. Mon. Wea. Rev., 129, 420–436.
  2. Wang, X., Bishop, C. H., & Julier, S. J. (2004). Which is better, an ensemble of positive–negative pairs or a centered spherical simplex ensemble? Mon. Wea. Rev., 132, 1590–1605. (Square-root choices; the symmetric square root.)
  3. Livings, D. M., Dance, S. L., & Nichols, N. K. (2008). Unbiased ensemble square root filters. Physica D, 237(8), 1021–1028.
  4. Sakov, P. & Oke, P. R. (2008). Implications of the form of the ensemble transformation in the ensemble square root filters. Mon. Wea. Rev., 136, 1042–1053.
  5. Whitaker, J. S. & Hamill, T. M. (2002). Ensemble data assimilation without perturbed observations. Mon. Wea. Rev., 130, 1913–1924. (EnSRF / serial assimilation.)
  6. Nerger, L., Janjić, T., Schröter, J., & Hiller, W. (2012). A unification of ensemble square root Kalman filters. Mon. Wea. Rev., 140, 2335–2345. (ESTKF.)
  7. Tippett, M. K., Anderson, J. L., Bishop, C. H., Hamill, T. M., & Whitaker, J. S. (2003). Ensemble square root filters. Mon. Wea. Rev., 131, 1485–1490.

LETKF (chapter 8)

  1. Hunt, B. R., Kostelich, E. J., & Szunyogh, I. (2007). Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter. Physica D, 230, 112–126.
  2. Ott, E., Hunt, B. R., Szunyogh, I., Zimin, A. V., Kostelich, E. J., Corazza, M., Kalnay, E., Patil, D. J., & Yorke, J. A. (2004). A local ensemble Kalman filter for atmospheric data assimilation. Tellus A, 56(5), 415–428.

Ensemble Kalman processes — EKI / EKS / UKI (chapter 9)

  1. Iglesias, M. A., Law, K. J. H., & Stuart, A. M. (2013). Ensemble Kalman methods for inverse problems. Inverse Problems, 29(4), 045001.
  2. Garbuno-Inigo, A., Hoffmann, F., Li, W., & Stuart, A. M. (2020). Interacting Langevin diffusions: Gradient structure and ensemble Kalman sampler. SIAM J. Appl. Dyn. Syst., 19(1), 412–441.
  3. Huang, D. Z., Schneider, T., & Stuart, A. M. (2022). Iterated Kalman methodology for inverse problems. J. Comput. Phys., 463,
    1. (UKI.)
  4. Kovachki, N. B. & Stuart, A. M. (2019). Ensemble Kalman inversion: a derivative-free technique for machine learning tasks. Inverse Problems, 35(9), 095005.
  5. Iglesias, M. A. (2016). A regularizing iterative ensemble Kalman method for PDE-constrained inverse problems. Inverse Problems, 32(2), 025002. (Regularising Levenberg-Marquardt scheme.)

Smoothers (chapter 10)

  1. Evensen, G. & van Leeuwen, P. J. (2000). An ensemble Kalman smoother for nonlinear dynamics. Mon. Wea. Rev., 128, 1852–1867.
  2. Cosme, E., Verron, J., Brasseur, P., Blum, J., & Auroux, D. (2012). Smoothing problems in a Bayesian framework and their linear Gaussian solutions. Mon. Wea. Rev., 140, 683–695.
  3. Chen, Y. & Oliver, D. S. (2013). Levenberg–Marquardt forms of the iterative ensemble smoother for efficient history matching and uncertainty quantification. Comput. Geosci., 17, 689–703. (IES.)
  4. Bocquet, M. & Sakov, P. (2014). An iterative ensemble Kalman smoother. Q. J. R. Meteorol. Soc., 140(682), 1521–1535.
  5. Whitaker, J. S. & Compo, G. P. (2002). An ensemble Kalman smoother for reanalysis. Proc. Symp. on Observations, Data Assimilation and Probabilistic Prediction, AMS.
  6. Evensen, G., Raanes, P. N., Stordal, A. S., & Hove, J. (2019). Efficient implementation of an iterative ensemble smoother for data assimilation and reservoir characterization. Front. Appl. Math. Stat., 5, 47.

Localization (chapter 11)

  1. Gaspari, G. & Cohn, S. E. (1999). Construction of correlation functions in two and three dimensions. Q. J. R. Meteorol. Soc., 125, 723–757.
  2. Houtekamer, P. L. & Mitchell, H. L. (2001). A sequential ensemble Kalman filter for atmospheric data assimilation. Mon. Wea. Rev., 129, 123–137.
  3. Anderson, J. L. (2007). Exploring the need for localization in ensemble data assimilation using a hierarchical ensemble filter. Physica D, 230, 99–111.
  4. Thiebaux, H. J. & Pedder, M. A. (1987). Spatial Objective Analysis. Academic Press. (SOAR correlation model.)

Inflation (chapter 12)

  1. Anderson, J. L. & Anderson, S. L. (1999). A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts. Mon. Wea. Rev., 127, 2741–2758.
  2. Zhang, F., Snyder, C., & Sun, J. (2004). Impacts of initial estimate and observation availability on convective-scale data assimilation. Mon. Wea. Rev., 132, 1238–1253. (RTPP.)
  3. Whitaker, J. S. & Hamill, T. M. (2012). Evaluating methods to account for system errors in ensemble data assimilation. Mon. Wea. Rev., 140, 3078–3089. (RTPS.)
  4. Anderson, J. L. (2009). Spatially and temporally varying adaptive covariance inflation for ensemble filters. Tellus A, 61, 72–83.
  5. Hamill, T. M. & Whitaker, J. S. (2005). Accounting for the error due to unresolved scales in ensemble data assimilation: A comparison of different approaches. Mon. Wea. Rev., 133, 3132–3147. (Additive inflation.)
  6. Ledoit, O. & Wolf, M. (2004). A well-conditioned estimator for large-dimensional covariance matrices. J. Multivariate Anal., 88, 365–411.

Diagnostics (chapter 13)

  1. Desroziers, G., Berre, L., Chapnik, B., & Poli, P. (2005). Diagnosis of observation, background and analysis-error statistics in observation space. Q. J. R. Meteorol. Soc., 131(613), 3385–3396.
  2. Hersbach, H. (2000). Decomposition of the Continuous Ranked Probability Score for Ensemble Prediction Systems. Wea. Forecasting, 15(5), 559–570.
  3. Hamill, T. M. (2001). Interpretation of Rank Histograms for Verifying Ensemble Forecasts. Mon. Wea. Rev., 129(3), 550–560.
  4. Mehra, R. K. (1970). On the Identification of Variances and Adaptive Kalman Filtering. IEEE Trans. Automatic Control, 15(2), 175–184.
  5. Cardinali, C., Pezzulli, S., & Andersson, E. (2004). Influence-Matrix Diagnostic of a Data Assimilation System. Q. J. R. Meteorol. Soc., 130(603), 2767–2786.
  6. Gneiting, T. & Raftery, A. E. (2007). Strictly Proper Scoring Rules, Prediction, and Estimation. JASA, 102(477), 359–378.
  7. Fortin, V., Abaza, M., Anctil, F., & Turcotte, R. (2014). Why Should Ensemble Spread Match the RMSE of the Ensemble Mean? J. Hydrometeorol., 15(4), 1708–1713.
  8. Whitaker, J. S. & Loughe, A. F. (1998). The Relationship between Ensemble Spread and Ensemble Mean Skill. Mon. Wea. Rev., 126(12), 3292–3302.
  9. Liu, J. S. & Chen, R. (1998). Sequential Monte Carlo Methods for Dynamic Systems. JASA, 93(443), 1032–1044.

Differentiable & latent DA (chapters 14–15)

  1. Chen, Y., Sanz-Alonso, D., & Willett, R. (2023). Reduced-Order Autodifferentiable Ensemble Kalman Filters. Inverse Problems, 39(12), 124001. (ROAD-EnKF.)
  2. Griewank, A. & Walther, A. (2000). Algorithm 799: revolve — an implementation of checkpointing for the reverse or adjoint mode of computational differentiation. ACM TOMS, 26(1), 19–45.
  3. Peyron, M., Fillion, A., Gürol, S., Marchais, V., Gratton, S., Boudier, P., & Goret, G. (2021). Latent space data assimilation by using deep learning. Q. J. R. Meteorol. Soc., 147(740), 3759–3777.
  4. Amendola, M., Arcucci, R., Mottet, L., Casas, C. Q., Fan, S., Pain, C., Linden, P., & Guo, Y.-K. (2021). Data assimilation in the latent space of a convolutional autoencoder. ICCS 2021, LNCS 12746.

Adaptive step size (chapter 9)

  1. Iglesias, M. A. & Yang, Y. (2021). Adaptive regularisation for ensemble Kalman inversion. Inverse Problems, 37(2), 025008. (DataMisfitController.)

Ecosystem libraries

  • gaussx — structured linear operators; the Woodbury / matrix-determinant-lemma dispatch behind the gain and likelihood, the Gaspari-Cohn taper, and the classic inflation trio.
  • lineax — linear operators and solvers (AbstractLinearOperator for \(R\), \(Q\)).
  • equinox — module system; every filterax component is an eqx.Module PyTree.
  • optax — optimizers for the differentiable-training surface.
  • pipekit + pipekit-cycle — operator composition and the DA-cycle protocols of chapter 16.
  • vardax — variational DA sibling library (OI / 3DVar / 4DVar); filterax provides its ensemble-posterior bridge.