Ensemble Data Assimilation on a Non-Conservative Adaptive Mesh
- 1. Ensemble Data Assimilation on a Non- Conservative Adaptive Mesh Colin Guider Alberto Carrassi and Ali Aydogdu C.K.R.T. Jones
- 2. Outline 1. Motivation – a dynamical/thermodynamical sea ice model (neXtSIM) 2. Challenges in developing a suitable ensemble-based data assimilation method 3. Development of algorithm and results for 1- dimensional example
- 3. The neXtSIM Model Variables Center variables • Ice thickness ℎ and snow thickness ℎ 𝑠 • Ice concentration 𝐴 • Ice damage 𝑑 • Internal stress tensor 𝜎 Nodal variables • Sea ice velocity u Methodology Lagrangian model running on triangular, unstructured mesh When mesh becomes too distorted, a remeshing process occurs
- 4. Data Assimilation with neXtSIM Mesh-based Challenges Each ensemble member will have its own adaptive mesh Remeshing occurs independently for each ensemble member, and may occur at different times Ensemble meshes will have different sizes – how can we compute ensemble statistics? How do we implement an ensemble-based method (EnKF-like) method for neXtSIM? Observation-based Challenges Combination of satellite observations and in-situ observations How do we incorporate Eulerian and Lagrangian observations in same model? How will we assimilate different types of observations at different times?
- 5. At analysis time, project each ensemble mesh onto a fixed “supermesh” At most one vertex in each mesh box – fill in empty boxes by interpolating Perform the analysis on this filled-in supermesh The supermesh can be fixed in time or change at each analysis time Proposed Solution: Super-meshing
- 6. A 1-Dimensional Example We consider the dissipative form of Burgers’ equation 𝑢 𝑡 = 𝜀𝑢 𝑧𝑧 − 𝑢𝑢 𝑧 for 𝑧 ∈ 0,1 with periodic boundary conditions and initial condition 𝑢 0, 𝑧 = sin 2𝜋𝑧 + 1 2 sin 𝜋𝑧. We take the parameter 𝜖 = 0.005. Our goal is to estimate 𝑢 at various points in time and space, given noisy observations of 𝑢.
- 7. Super-meshing in the 1-D Case Each ensemble member will have its own adaptive mesh The adaptive mesh, 𝑧(𝑡), evolves in time We consider the mesh valid if 𝛿1 ≤ 𝑧𝑖+1 − 𝑧𝑖 < 𝛿2 for all 𝑖 Otherwise, a remeshing process occurs To left is example with 𝛿1 = 0.02 and 𝛿2 = 0.05
- 8. • Each ensemble member has its own adaptive mesh • Values of 𝑢 are defined at adaptive mesh points • Ensemble members evolve according to adaptive moving mesh equations • Evolution of 𝑢 and evolution of mesh are coupled • Remeshing occurs when mesh points become too close together, or too far apart Evolution of Ensemble Members
- 9. Projecting onto the supermesh The smaller mesh parameter, 𝛿1, is what allows us to define a constant- dimensional state space for our DA algorithm. Specifically, letting 𝑁 = 1 𝛿1 , we write
- 10. Defining the State Space We then define 𝑧𝑖 𝑡 = 𝑧 𝑡, 𝑗 𝑢𝑖 𝑡 = 𝑢(𝑡, 𝑧 𝑡, 𝑗 ) (mesh point) (physical variable) if 𝑧(𝑡, 𝑗) ∈ 𝐿𝑖. With this definition, we can define our state vector
- 11. Filling in the Ensemble Members For now, we only perform data assimilation on the physical variables, and not the mesh points themselves Ensemble members will generally have different meshes, so they will have different “active” cells We address this by “filling in” each ensemble mesh This allows us to carry out DA in the standard way on a state space of full dimension
- 12. Ensemble Kalman Filter (EnKF) with super-meshing (1D-case) Define to be the matrix of forecast ensemble members. We subtract the matrix whose columns contain the ensemble means to form the matrix of anomalies: We assume we have 𝑝 observations of 𝑢 at certain points in the interval 0,1 , and that we have an observation operator that maps the state space to the observation space. We can then define the matrix
- 13. Ensemble Kalman Filter (EnKF) with super-meshing (1D-case) We assume the observations have covariance R. We then compute the Kalman gain The analysis update to the forecast ensemble is performed using the stochastic EnKF where
- 14. We run the experiment from time 𝑡 = 0 to 𝑡 = 1, performing data assimilation time step 𝑡 𝑜𝑏𝑠 = 0.2 We use an ensemble of size 𝑁𝑒 = 20. Initially, all ensemble members have the same mesh. Initial ensemble values of 𝑢 centered at true initial value of 𝑢 with standard deviation 𝜎𝑒𝑛𝑠 = 2 Numerical Experiment
- 15. Forecast and Analysis Ensembles
- 16. Forecast and Analysis Means
- 17. Forecast and Analysis Means
- 18. RMSE and Spread
- 19. Run experiment with Lagrangian observations Test a more interesting one-dimensional model (Kuramoto-Sivashinsky experiment in progress) Further develop a “low-resolution” model, in which the state space is defined by larger remeshing parameter Eventually work up to two-dimensional models, with neXtSIM the ultimate goal Future Directions
- 20. Essential bibliography Bouillon, S. and P. Rampal: Presentation of the dynamical core of neXtSIM, a new sea ice model, Ocean Modelling, 91, 23–37, 2015. Bouillon, S. and P. Rampal: On producing sea ice deformation data sets from SAR-derived sea ice motion, The Cryosphere, 9, 663–673, 2015. Guider, C., M. Rabatel, A. Carrassi and C.K.R.T. Jones: Data assimilation on a non-conservative adaptive mesh. Geophysical Research Abstracts, vol. 19, EGU2017-706 Rampal, P., Bouillon, S., Ólason, E., and Morlighem, M.: neXtSIM: A new Lagrangian sea ice model, The Cryosphere, 10, 1055–1073, 2016. Huang, W. and R. Russell: Adaptive Moving Mesh Methods, Springer, 2011. Thank you !