We thank two anonymous reviewers for their most constructive suggestions and comments that have significantly improved our work. This publication is based on work supported by funds from the KAUST GCR Academic Excellence Alliance program.
Anderson, J. L., , and S. L. Anderson, 1999: A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts. Mon. Wea. Rev., 127, 2741–2758.
Bishop, C. H., , B. J. Etherton, , and S. J. Majumdar, 2001: Adaptive sampling with the ensemble transform Kalman filter. Part I: Theoretical aspects. Mon. Wea. Rev., 129, 420–436.
Burgers, G., , P. J. van Leeuwen, , and G. Evensen, 1998: On the analysis scheme in the ensemble Kalman filter. Mon. Wea. Rev., 126, 1719–1724.
Cohn, S., , and R. Todling, 1996: Approximate data assimilation schemes for stable and unstable dynamics. J. Meteor. Soc. Japan, 74, 63–75.
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), 10 143–10 162.
Evensen, G., , and P. J. van Leeuwen, 1996: Assimilation of Geosat altimeter data for the Aghulas Current using the ensemble Kalman filter with a quasigeostrophic model. Mon. Wea. Rev., 124, 85–96.
Hamill, T. M., , J. S. Whitaker, , and C. Snyder, 2001: Distance-dependent filtering of background error covariance estimates in an ensemble Kalman filter. Mon. Wea. Rev., 129, 2776–2790.
Hamill, T. M., , J. S. Whitaker, , J. L. Anderson, , and C. Snyder, 2009: Comments on “Sigma-point Kalman filter data assimilation methods for strongly nonlinear systems.” J. Atmos. Sci., 66, 3498–3500.
Hoteit, I., , and D. T. Pham, 2004: An adaptively reduced-order extended Kalman filter for data assimilation in the tropical Pacific. J. Mar. Syst., 45, 173–188.
Hoteit, I., , D. T. Pham, , and J. Blum, 2001: A semi-evolutive partially local filter for data assimilation. Mar. Pollut. Bull., 43, 164–174.
Hoteit, I., , D. T. Pham, , and J. Blum, 2002: A simplified reduced order Kalman filtering and application to altimetric data assimilation in tropical Pacific. J. Mar. Syst., 36, 101–127.
Hoteit, I., , D. T. Pham, , G. Triantafyllou, , and G. Korres, 2008: A new approximate solution of the optimal nonlinear filter for data assimilation in meteorology and oceanography. Mon. Wea. Rev., 136, 317–334.
Houtekamer, P. L., , and H. L. Mitchell, 1998: Data assimilation using an ensemble Kalman filter technique. Mon. Wea. Rev., 126, 796–811.
Lorenz, E. N., , and K. A. Emanuel, 1998: Optimal sites for supplementary weather observations: Simulation with a small model. J. Atmos. Sci., 55, 399–414.
Luo, X., , I. M. Moroz, , and I. Hoteit, 2010: Scaled unscented transform Gaussian sum filter: Theory and application. Physica D, 239, 684–701.
Pham, D. T., , J. Verron, , and M. C. Roubaud, 1998: A singular evolutive extended Kalman filter for data assimilation in oceanography. J. Mar. Syst., 16, 323–340.
Tippett, M. K., , J. L. Anderson, , C. H. Bishop, , T. M. Hamill, , and J. S. Whitaker, 2003: Ensemble square root filters. Mon. Wea. Rev., 131, 1485–1490.
Verlaan, M., , and A. W. Heemink, 1997: Tidal flow forecasting using reduced rank square root filters. Stoch. Hydrol. Hydraul., 11, 349–368.
Wang, D., , and X. Cai, 2008: Robust data assimilation in hydrological modeling—A comparison of Kalman and H-infinity filters. Adv. Water Resour., 31, 455–472.
Whitaker, J. S., , and T. M. Hamill, 2002: Ensemble data assimilation without perturbed observations. Mon. Wea. Rev., 130, 1913–1924.
The deduction will be similar in case that ui and vi are correlated colored noise. Readers are referred to, for example, Simon (2006, chapter 7) for the details.
If, in contrast, the observation is very unreliable, then one may choose a negative value for γ such that the background has relatively more weight in the update. In this work we confine ourselves to the scenario γ ≥ 0.