Comparing ensemble-variational assimilation methods for NWP: Can hybrid-4denvar match hybrid-4dvar? WWOSC, Montreal, August 2014. Andrew Lorenc, Neill Bowler, Adam Clayton, David Fairbairn and Stephen Pring Crown copyright Met Office
Outline of Talk Background Variational DA methods: Hybrid-4DVar. Adds flow-dependent ensemble covariances to traditional incremental 4DVar (using linear and adjoint models). Hybrid-4DEnVar. Use ensemble trajectories: no need to integrate linear & adjoint models. Results of initial trials comparing these. What we need to do to improve 4DEnVar. Crown copyright Met Office Andrew Lorenc 2
Background Powerful computers enable us to compute the evolution and growth of forecast errors. This is used within 4DVar to combine observations at different times in a short time-window. Ensemble Kalman filters can compute error evolution over longer periods, to estimate error covariances. Variational methods have some advantages over EnKF: ensemble-var methods try to keep these while adding Errors Of The Day (EOTD). Crown copyright Met Office Andrew Lorenc 3
Variational Methods Use operations on model fields to define the structure of background forecast errors. These, plus simpler descriptions of observation errors, are at the heart of an iterative algorithm to find the best analysis, using all observations. Ideal for dense but incomplete observations, from satellites or radars space & time gradients in an observed field can give information about unobserved fields; all scales are handled correctly. 4DVar has been the favourite DA method for operational NWP for the last decade (Rabier 2005). Crown copyright Met Office Andrew Lorenc 2014 4
Key weaknesses of 4DVar 1. Scientific: Background errors are modelled using a covariance which is usually assumed to be stationary, isotropic and homogeneous. Need to allow for Errors of The Day. 2. Technical: The minimisation requires repeated sequential runs of a (low resolution) linear model and its adjoint. Inefficient on massively parallel computers; difficult development when the forecast model is redesigned. The Met Office has addressed 1 in its hybrid 4DVar 3. (Clayton Scientific: et al. 2013). Does not naturally generate an analysis ensemble. In the work presented here we use a separate ensemble system. Our hybrid 4DEnVar developments (Lorenc et al. 2014) are attempting to also address 2. Crown copyright Met Office Andrew Lorenc 5
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u increments fitting a single u ob at 500hPa, at different times. 4D-Var at start of window at end of 6-hour window Hybrid 4D-Var Unfilled contours show T field. Clayton et al. 2013
Hybrid-4DVar Results Clayton et al. (2013) showed that hybrid-4dvar using 23 members directly, localized using alpha control variable, performed ~1% better than 4DVar. (It is now operational) However this increased the complexity of the system, and does not address foreseen scalability issues for 4DVar on future massively parallel computers. An advantage of the direct use of ensemble perturbations is that it can be extended to 4D ensemble trajectories, giving a simpler and more scalable algorithm. Crown copyright Met Office Andrew Lorenc 2013 11
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Hybrid 4DEnVar differences from hybrid-4dvar 4D trajectory is used from ensemble, rather than 3D states at beginning of window. 4D localisation fields and increment x c increment is constant in time, as in 3DVar FGAT No model integration inside minimisation, so costs like hybrid-3dvar No J c balance constraint, so additional initialisation is necessary. Crown copyright Met Office Andrew Lorenc 13
Comparison of hybrid-4denvar and hybrid-4dvar data assimilation methods for global NWP Andrew C Lorenc, Neill Bowler, Adam Clayton, David Fairbairn and Stephen Pring. Submitted to MWR Trials: Name DA Method Initialization 4DVar hybrid 4DVar J c 4DEnVar hybrid 4DEnVar 4DIAU 3DVar hybrid 3DVar IAU 3DEnVar hybrid 3DEnVar IAU 4DVar4DIAU hybrid 4DVar 4DIAU Trials for July 2013, based on lower res. operational global hybrid-4dvar (Clayton et al. 2013) NWP system: 640 481 70 deterministic model and 432 325 70 ensemble and PF & adjoint models in 4DVar. 44-member ensemble precalculated by MOGREPS-G (Bowler et al. 2008; Flowerdew and Bowler 2011). Crown copyright Met Office Andrew Lorenc 14
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Results of Trial 4DVar v 4DEnVar 3.138% Relative RMS error against observations for a sample of fields and forecast ranges. Hollow grey box is 2%, max is 10%. First / Second trial is better. #.###% is the average. Crown copyright Met Office Andrew Lorenc 16
The difference is due to the time-dimension 4DVar v 4DEnVar 3.138% 3DVar v 3DEnVar 0.007% 4DVar v 3DVar 3.506% 4DEnVar v 3DEnVar 0.474% Crown copyright Met Office Andrew Lorenc 17
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Much smaller differences due to the initialization 4DVar v 4DEnVar 3.138% 4DVar 4DIAU v 4DEnVar 2.594% 4DVar v 4DVar 4DIAU 0.531% Crown copyright Met Office Andrew Lorenc 19
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Single wind observation at start of 6 hour window, in jet Background trajectory 0 3 6 Ob is at at time 0. Crown copyright Met Office Andrew Lorenc 21
100% ensemble 1200km localization scale 4DEnVar 4DVar error Crown copyright Met Office Andrew Lorenc 22
50-50% hybrid 1200km localization scale 4DEnVar 4D-Var Crown copyright Met Office Andrew Lorenc 23
100% climatological B 4DEnVar 3DVar 4D-Var Crown copyright Met Office Andrew Lorenc 24
100% ensemble 500km localization scale 4DEnVar 4D-Var Crown copyright Met Office Andrew Lorenc 25
Relative Strong Constraint Errors We ran similar tests on a Hurricane Sandy case. Here the ensemble covariances dominated, making hybrid-4denvar perform better. Jet case Hurricane 1200km localization scale Sandy 4DEnVar 51% 57% En 4DVar 54% 69% Hybrid 4DEnVar 78% 66% Hybrid 4DVar 66% 75% When the ensemble covariances dominated the increments, and the horizontal localization was not too severe, 4DEnVar had better consistency with the strong constraint than 4DVar. Runs with smallest deviation from model constraint Crown copyright Met Office Andrew Lorenc 26
Conclusions from 4D analysis increment study 1. The main error in our hybrid-4denvar (v hybrid-4dvar) is that the climatological covariance is used as in 3DVar. 2. 3D localization not following the flow is not an important error for our 1200km localization scale and 6hour window, but does become important for a 500km scale. Crown copyright Met Office Andrew Lorenc 27
Improving 4DEnVar The maintenance and running costs of hybrid-4dvar are larger, so there is an incentive to improve hybrid-4denvar. We need to reduce the weight on climatological B relative to the ensemble covariance. We must first improve the ensemble covariances: a bigger ensemble; better ensemble generation; better filtering of ensemble covariance, e.g. localization. Encouraging progress has been made in all of these. Crown copyright Met Office Andrew Lorenc 28
Summary 4DEnVar retains the main advantages of 4DVar, and adds EOTD, in a simpler and more scalable system. Soon to be used for operational NWP in Canada (replacing 4DVar) and USA (replacing 3DVar). In the Met Office we added EOTD in hybrid-4dvar; hybrid-4denvar does not yet match this. R&D is underway to improve the filtered ensemble covariances, to improve 4DEnVar. Crown copyright Met Office Andrew Lorenc 2014 29
Questions and answers Clayton, A. M., Lorenc, A. C. and Barker, D. M. (2013), Operational implementation of a hybrid ensemble/4d-var global data assimilation system at the Met Office. Q.J.R. Meteorol. Soc., 139: 1445 1461. doi: 10.1002/qj.2054 Andrew Lorenc, Neill E. Bowler, Adam M. Clayton, David Fairbairn and Stephen R. Pring. 2014: "Comparison of hybrid-4denvar and hybrid-4dvar data assimilation methods for global NWP". Submitted to Mon. Wea. Rev. Crown copyright Met Office
Statistical, incremental 4DVar Hybrid PDF valid at t0 Statistical 4DVar approximates entire PDF by a 4D Gaussian defined by PF model. 4D analysis increment is a trajectory of the PF model. Lorenc & Payne 2007
Incremental 4D-Ensemble-Var Statistical 4D-Var approximates entire PDF by a Gaussian. 4D analysis is a (localised) linear combination of nonlinear trajectories. It is not itself a trajectory. Crown copyright Met Office Andrew Lorenc 32
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Spectral localisation smooths in space: σ b of pressure at level 21 Crown copyright Met Office Andrew Lorenc 34
Vertical crosscorrelation between q and divergence at an active point. Localisation (except parameter) retains plausible correlation between q and convergence below, divergence above. Crown copyright Met Office Andrew Lorenc 35
Trials of increased ensemble size and weight Modest improvement when increasing ensemble size Much larger improvement when ensemble weight is high 4DVar performs worse with high ensemble weight, 4DEnVar performs better Using ensemble modes from the wrong time brings a small benefit Crown copyright Met Office