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Steven Fletcher
CIRA Fellow

Dr. Steven J. Fletcher FRMetS

Senior Research Scientist/Scholar

Mailing Address:
Dr. Steven J. Fletcher
Cooperative Institute for Research in the Atmosphere
Colorado State University
1375 Campus Delivery
Fort Collins, CO 80523-1375
  • Office Location:
    CIRA Room 112
  • 970-491-8376
About Me:

Dr. Steven J. Fletcher FRMetS is a leading research scientist in the field of non-Gaussian based data assimilation, with over 20 years’ experience in data assimilation.  He has derived many different versions of lognormal and mixed Gaussian-lognormal forms of variational, ensemble, representer, and Kalman filter based data assimilation systems.  In the October of 2017 Dr. Fletcher was elected as a Fellow of the Royal Meteorological Society for his over 15 years of contributions to nonlinear and non-Gaussian based data assimilation research.

In 2010 Dr. Fletcher proved that full field strong and weak constraint could be expressed as a 4-dimensional Bayes problem:

Fletcher, S. J., 2010: Mixed lognormal-Gaussian four-dimensional data assimilation. Tellus, 62A, 266—287.

In 2014 Drs Fletcher and Jones were able to derive lognormal and mixed Gaussian-lognormal based 3D and 4D incremental VAR systems that were was the first ever non-additive incremental based 3D and 4D VAR systems:

Fletcher, S. J. and A. S. Jones, 2014: Multiplicative and Additive Incremental Variational Data Assimilation for Mixed Lognormal-Gaussian Errors. Mon. Wea. Rev., 142, 2521–2544,

In 2022 Dr. Fletcher published the proof of the first ever nonlinear lognormal and mixed Gaussian-lognormal based Kalman filter.  This approach had eluded scientist for over 60 years, but was achievable due to understanding which descriptive statistic that the lognormal distribution is defined by: its median:

Fletcher, S. J., M. Zupanski, M. R. Goodliff, A. J. Kliewer, A. S. Jones, J. M. Forsythe, T.-C. Wu, M. J. Hossen and S. Van Loon, 2023: Lognormal and Mixed Gaussian-lognormal based Kalman Filters. Mon. Wea. Rev., 151, 761-774.

To help consolidate the information to understand data assimilation and the different components involved Dr. Fletcher has written three textbooks; two associated with data assimilation, and one centered on the numerical modeling techniques associated with semi-Lagrangian advection:

Fletcher, S. J., 2022: Data Assimilation for the Geosciences: From Theory to Applications (2nd Edition), Elsevier

Fletcher, s. J., 2019: Semi-Lagrangian Methods and their Applications in Geoscience. Elsevier.

Fletcher, S. J., 2017: Data Assimilation for the Geosciences: From Theory to Applications, Elsevier

Dr Fletcher has received several National Science Foundation grants, as well as NOAA and DoD grants to explore non-Gaussian based data assimilation development.  More recently Dr. Fletcher has been working with series of postdoctoral fellows to develop approaches to determine when the dynamical system’s error structure is going to change from Gaussian to non-Gaussian with machine learning techniques.

Dr. Fletcher is very active at the American Geophysical Union, having convened a session in the Nonlinear Geophysics (NG) section on Advances in Data Assimilation, Predictability, and Uncertainty Quantification since 2010.  In 2013 he became the Fall Meeting Program Committee representative for the NG section and is now the second most senior member of the committee after the chairperson.

He has supervised three postdocs on non-Gaussian data assimilation and machine learning: Drs. Anton Kliewer, Michael Goodliff, and Jakir Hossen.  His current postdoc is Dr. Senne Van Loon who is investigating the reverse lognormal distribution based variational and ensemble data assimilation algorithms, as well as the bounded Johnson distribution as a possible new alternative.

Dr. Fletcher received his B.Sc.(HONS) in Mathematics and Statistics from the University of Reading in the United Kingdom (1998), his M.Sc. in Numerical Solutions to Differential Equations again from the University of Reading (1999), (Advisor: Prof. Nancy Nichols) and his Ph.D. in Data Assimilation also from the Mathematics department at the University of Reading (2004), (Advisors: Prof. Nancy Nichols and Prof. Ian Roulstone, Met Office/U. Surrey).  His areas of interest are numerical weather prediction and modeling, data assimilation, control theory, calculus of variation, control variable transforms and preconditioning for large scale nonlinear inverse problems and non-Gaussian probability models for data assimilation, as well as techniques to use machine learning algorithms to optimize data assimilation systems.

From 2004 to 2006, he worked as a post doctoral fellow at the Cooperative Institute for Research in the Atmosphere (CIRA) under a National Science Foundation grant into the use of control theory in Numerical weather prediction. During this time he published the first ever non-Gaussian three-dimensional variational data assimilation system for lognormally distributed observational errors. Besides this he also published the first paper to combine Gaussian and lognormal errors together to allow for the simultaneous assimilation of different observational error types. He has also worked with the Maximum Likelihood Ensemble Filter (MLEF) with Dr. Milija Zupanski and Prof. Mike Navon of Florida State University exploring the size of the ensemble and the dynamics in the shallow water equations.

From 2006 to 2010 Dr. Fletcher was a research scientist at the Center for Geosciences/Atmospheric Research (CG/AR) in CIRA. During this time he developed a Bayesian based probability model for 4DVAR to enable any distribution to be used. In particular he derived the 4DVAR equivalent for the mixed lognormal-Gaussian distribution.  As part of the review process for the Fletcher (2010) Tellus paper Dr. Fletcher derived and coded into the Lorenz 1963 model, the equations for a mixed distribution constant model error term.  Also during this time Dr. Fletcher worked on detecting important dynamical properties of the MLEF with respect to Lyapunov and Bred vectors. He has also worked with the DA group at the National Center for Atmospheric Research on their preconditioner for the Weather, Research and Forecasting model’s 4D VAR system for the US Air Force.

During 2008 to 2011,  Dr. Fletcher worked on techniques associated with how to assimilate both binary MODIS 500m snow cover observations with coarse AMSR-E 25km snow water equivalence into the snow evolution model SnowModel.  There were four different schemes developed from simple nudging to a rule based iterative direct insertion technique as well as combination of the two to optimize the data available.  This work was tested over two different spacial domains; the first in south-eastern Colorado and the second over a region of the tri-state are of Colorado, Wyoming and Nebraska, during the winter season of 2006-2007.  This was work done in collaboration with Drs. Glen Liston, Christopher Hiemstra and Steven Miller.

Between 2012 and 2014 Dr. Fletcher worked with the data assimilation scientist at the Naval Research Laboratory in Monterey, CA, on the Navy’s numerical weather prediction system investigating updating the background error covariance model as well as non-Gaussian possibilities inside of the observation space based DA system.  Also during this period Dr. Fletcher was awarded an NSF grant to investigate the non-Gaussian effects inside of Gaussian data assimilation systems.

Between 2013 and 2015 Dr. Fletcher supervised Dr. Anton Kliewer as a postdoctoral fellow on the first of Dr. Fletcher’s NSF awards, to look at detecting non-Gaussian behavior of moisture variables.  The award also lead to the first ever mixed-Gaussian-lognormal temperature and mixing ratio microwave brightness temperature retrieval system.  This work also lead to discovering a sensitivity to the initial conditions of the Newton-Rhapson solver which is some form of Newton Fractal.

In 2017 Dr. Fletcher was successful in obtaining a second NSF award to investigate the impacts of non-Gaussian behavior on data assimilation systems.  As part of this award a new postdoc was hired at CIRA, Dr. Michael Goodliff who oversaw machine learning techniques to detect changes in probability density distributions with different dynamical systems.

In 2020 Dr. Fletcher received a third NSF award to look at how machine learning could help detect changes in the distributions to inform the data assimilation which formulation should be used to minimize the analysis error.  As part of this award there have been two postdoc: Dr. Jakir Hossen, and the current postdoc Dr. Senne Van Loon.  This award has lead to the development of the reverse-lognormal based data assimilation methods, as well as the nonlinear Gaussian, and nonlinear lognormal based Kalman filter.  Recently there has been work looking at the Johnson distributions as possible dynamical form of error modeling.

My Work/Projects:

NSF: Establishing links between atmospheric dynamics and non-Gaussian distribution and quantifying their effects on numerical weather prediction

July 2017 - June 2020.

This NSF award is looking at the impacts of non-Gaussian dynamics have on data assimilation systems, and different machine learning techniques to detect a change of probability density functions due to a change in the dynamics.

NRL: Regional, Seasonal and Large Dynamical Scale Based Covariance and Humidity Control Variable Transform implementation in NAVDAS-AR.

March 2015 - March 2020

This project examines the updating of the background error covariance matrix in the US Navy operational numerical weather prediction.  As part of this work a NMC based method will be coded to develop a method to update the covariance matrix when there are operational upgrades to the 4DVAR system.  The second part of this work will be to introduce a normalized version of the relative humidity to couple the moisture field increment with the virtual temperature and pressure increments.