Accounting for observation uncertainty and bias due to unresolved scales with the Schmidt-Kalman filter

References

• Aravéquia, J. A., Szunyogh, I., Fertig, E. J., Kalnay, E., Kuhl, D. and co-authors. 2011. Evaluation of a strategy for the assimilation of satellite radiance observations with the local ensemble transform Kalman filter. Mon. Weather Rev. 139, 1932–1951. doi:10.1175/2010MWR3515.1
   Google Scholar
• Asher, R. B., Herring, K. D. and Ryles, J. C. 1976. Bias, variance, and estimation error in reduced order filters. Automatica 12, 589–600. doi:10.1016/0005-1098(76)90040-6
   Google Scholar
• Asher, R. B. and Reeves, R. M. 1975. Performance evaluation of suboptimal filters. IEEE Trans. Aerosp. Electron. Syst. AES-11, 400–405. doi:10.1109/TAES.1975.308092
   Google Scholar
• Brown, R. G. and Hwang, P. Y. 2012. Introduction to Random Signals and Applied Kalman Filtering. John Wiley & Sons, Hoboken, NJ, p. 192, Chapter 5.
   Google Scholar
• Brown, R. and Sage, A. 1971. Analysis of modeling and bias errors in discrete-time state estimation. IEEE Trans. Aerosp. Electron. Syst. AES-7, 340–354. doi:10.1109/TAES.1971.310375
   Google Scholar
• Cordoba, M., Dance, S. L., Kelly, G. A., Nichols, N. K. and Waller, J. A. 2017. Diagnosing atmospheric motion vector observation errors for an operational high-resolution data assimilation system. Q. Meteorol. Soc. 143, 333–341. doi:10.1002/qj.2925
   Google Scholar
• Daley, R. 1993. Estimating observation error statistics for atmospheric data assimilation. Ann. Geophysicae 11, 634–647.
   Google Scholar
• Dee, D. P. 2005. Bias and data assimilation. Q. J. R. Meteorol. Soc. 131, 3323–3343. doi:10.1256/qj.05.137
   Google Scholar
• Dee, D. 2004. Variational bias correction of radiance data in the ECMWF system. In: Proceedings of the ECMWF Workshop on Assimilation of High Spectral Resolution Sounders in NWP, Reading, UK, ECMWF.
   Google Scholar
• Derber, J. C. and Wu, W.-S. 1998. The use of TOVS cloud-cleared radiances in the NCEP SSI analysis system. Mon. Wea. Rev. 126, 2287–2299. doi:10.1175/1520-0493(1998)126<2287:TUOTCC>2.0.CO;2
   Google Scholar
• Desroziers, G., Berre, L., Chapnik, B. and Poli, P. 2005. Diagnosis of observation, background and analysis-error statistics in observation space. Q. J. R. Meteorol. Soc. 131, 3385–3396. doi:10.1256/qj.05.108
   Google Scholar
• Eyre, J. 2016. Observation bias correction schemes in data assimilation systems: a theoretical study of some of their properties. Q. J. R. Meteorol. Soc. 142, 2284–2291. doi:10.1002/qj.2819
   Google Scholar
• Fertig, E., Baek, S.-J., Hunt, B., Ott, E., Szunyogh, I. and co-authors. 2009. Observation bias correction with an ensemble Kalman filter. Tellus A: Dyn. Meteorol. Oceanogr. 61, 210–226. doi:10.1111/j.1600-0870.2008.00378.x
   Google Scholar
• Fielding, M. and Stiller, O. 2019. Characterizing the representativity error of cloud profiling observations for data assimilation. J. Geophys. Res. Atmos. 124, 4086–4103. doi:10.1029/2018JD029949
   Google Scholar
• Friedland, B. 1969. Treatment of bias in recursive filtering. IEEE Trans. Automat. Contr. 14, 359–367. doi:10.1109/TAC.1969.1099223
   Google Scholar
• Gelb, A. 1974. Applied Optimal Estimation. MIT Press, Cambridge, MA, and London, UK, Chapter 8, pp. 305–306.
   Google Scholar
• Grooms, I., Lee, Y. and Majda, A. J. 2014. Ensemble Kalman filters for dynamical systems with unresolved turbulence. Comput. Phys. 273, 435–452. doi:10.1016/j.jcp.2014.05.037
   Google Scholar
• Hodyss, D. and Satterfield, E. 2016. Mathematical concepts of data assimilation. In: Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications (eds. S. K. Park and L. Xu) Vol. 3. Springer, Berlin, pp. 177–194.
   Google Scholar
• Ignagni, M. 1981. An alternate derivation and extension of Friendland’s two-stage Kalman estimator. IEEE Trans. Automat. Contr. 26, 746–750. doi:10.1109/TAC.1981.1102697
   Google Scholar
• Janjić, T., Bormann, N., Bocquet, M., Carton, J., Cohn, S. and co-authors. 2018. On the representation error in data assimilation. Q. J. R. Meteorol. Soc. 144, 1257–1278. doi:10.1002/qj.3130
   Google Scholar
• Janjić, T. and Cohn, S. E. 2006. Treatment of observation error due to unresolved scales in atmospheric data assimilation. Mon. Weather Rev. 134, 2900–2915. doi:10.1175/MWR3229.1
   Google Scholar
• Jazwinski, A. H. 1970. Stochastic Processes and Filtering Theory. Academic Press, New York, NY, Chapter 7, p. 212.
   Google Scholar
• Kalman, R. E. 1960. A new approach to linear filtering and prediction problems. Journal of Basic Engineering 82, 35–45. doi:10.1115/1.3662552
   Google Scholar
• Karspeck, A. R. 2016. An ensemble approach for the estimation of observational error illustrated for a nominal 1° global ocean model. Mon. Wea. Rev. 144, 1713–1728. doi:10.1175/MWR-D-14-00336.1
   Google Scholar
• Lea, D., Drecourt, J.-P., Haines, K. and Martin, M. 2008. Ocean altimeter assimilation with observational-and model-bias correction. Q. J. R. Meteorol. Soc. 134, 1761–1774. doi:10.1002/qj.320
   Google Scholar
• Liu, Z.-Q. and Rabier, F. 2002. The interaction between model resolution, observation resolution and observation density in data assimilation: A one-dimensional study. Q J. R Meteorol. Soc. 128, 1367–1386. doi:10.1256/003590002320373337
   Google Scholar
• Ménard, R. 2010. Bias estimation. In: Data Assimilation: making Sense of Observations (eds. W. Lahoz, B. Khattatov and R. Menard). Springer, Berlin, pp. 113–135.
   Google Scholar
• Miyoshi, T., Sato, Y. and Kadowaki, T. 2010. Ensemble Kalman filter and 4D-Var intercomparison with the Japanese operational global analysis and prediction system. Mon. Weather Rev. 138, 2846–2866. doi:10.1175/2010MWR3209.1
   Google Scholar
• Moodey, A. J. 2013. Instability and regularization for data assimilation. PhD thesis, University of Reading.
   Google Scholar
• Nichols, N. 2010. Mathematical concepts of data assimilation. In: Data Assimilation: making Sense of Observations (eds. W. Lahoz, B. Khattatov and R. Menard). Springer, Berlin, pp. 13–39.
   Google Scholar
• Oke, P. R. and Sakov, P. 2008. Representation error of oceanic observations for data assimilation. J. Atmos. Oceanic Technol. 25, 1004–1017. doi:10.1175/2007JTECHO558.1
   Google Scholar
• Satterfield, E., Hodyss, D., Kuhl, D. D. and Bishop, C. H. 2017. Investigating the use of ensemble variance to predict observation error of representation. Mon. Weather Rev. 145, 653–667. doi:10.1175/MWR-D-16-0299.1
   Google Scholar
• Schmidt, S. F. 1966. Application of state-space methods to navigation problems. In: Advances in Control Systems (ed. C. T. Leondes) Vol. 3. Elsevier, Amsterdam, pp. 293–340.
   Google Scholar
• Schutgens, N. A., Gryspeerdt, E., Weigum, N., Tsyro, S., Goto, D. and co-authors. 2016. Will a perfect model agree with perfect observations? The impact of spatial sampling. Atmos. Chem. Phys. 16, 6335–6353. doi:10.5194/acp-16-6335-2016
   Google Scholar
• Simon, D. 2006. Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches. Wiley-Blackwell, Hoboken, NJ, pp. 309–312, Chapter 10.
   Google Scholar
• Stewart, L. M., Dance, S. L., Nichols, N. K., Eyre, J. R. and Cameron, J. 2014. Estimating interchannel observation-error correlations for IASI radiance data in the Met Office system. Q. J. R. Meteorol. Soc. 140, 1236–1244. doi:10.1002/qj.2211
   Google Scholar
• Todling, R. and Cohn, S. E. 1994. Suboptimal schemes for atmospheric data assimilation based on the Kalman filter. Mon. Wea. Rev. 122, 2530–2557. doi:10.1175/1520-0493(1994)122<2530:SSFADA>2.0.CO;2
   Google Scholar
• Waller, J. A., Ballard, S. P., Dance, S. L., Kelly, G., Nichols, N. K. and co-authors. 2016a. Diagnosing horizontal and inter-channel observation error correlations for SEVIRI observations using observation-minus-background and observation-minus-analysis statistics. Remote Sens. 8, 581. doi:10.3390/rs8070581
   Google Scholar
• Waller, J. A., Dance, S. L., Lawless, A. S., Nichols, N. K. and Eyre, J. 2014. Representativity error for temperature and humidity using the Met Office high-resolution model. Q. J. R. Meteorol. Soc. 140, 1189–1197. doi:10.1002/qj.2207
   Google Scholar
• Waller, J. A., Simonin, D., Dance, S. L., Nichols, N. K. and Ballard, S. P. 2016b. Diagnosing observation error correlations for Doppler radar radial winds in the Met Office UKV model using observation-minus-background and observation-minus-analysis statistics. Mon. Wea. Rev. 144, 3533–3551. doi:10.1175/MWR-D-15-0340.1
   Google Scholar
• Zhu, Y., Derber, J., Collard, A., Dee, D., Treadon, R. and co-authors. 2014. Enhanced radiance bias correction in the National Centers for Environmental Prediction’s gridpoint statistical interpolation data assimilation system. Q. J. R. Meteorol. Soc. 140, 1479–1492. doi:10.1002/qj.2233
   Google Scholar