1. Weak constraint 4D-Var at ECMWF
How to deal with model error in data assimilation
Patrick Laloyaux - Earth System Assimilation Section
Acknowledgement: Massimo Bonavita, Jacky Goddard, Elias Holm, Simon Lang, Mike Fisher, Yannick Tremolet
In in Sept 2011
Out in June 2013
In in Nov 2016

2. 4D-Var formulations Bias-blind data assimilation
designed to correct the background with observations when errors are random with zero means
model is assumed to be perfect
Biases in reality
Observations (satellite miscalibrations)
Observation operator (approximations in radiative transfer calculations)
Model trajectory (inaccurate surface forcing, simplified representations of physics)

3. Biases in observations and observation operators
Departure between observations and model depends on the scan position of the satellite instrument
AMSUA channel 7
Obs-FG bias
Limb
Nadir
Limb
Departure between observations and model depends on the spectroscopy used in the radiative transfer model
Obs-FG bias
Channel number (HIRS)

4. 4D-Var formulations Bias-aware data assimilation (VarBC)
estimate simultaneously the initial condition and parameters of the observation bias model
the bias model copes with instrument miscalibration or systematic errors in the observation operator
the model is a strong constraint

5. Biases in model trajectory
Estimating the systematic error in the model over a short period of time is not straightforward (disentangle model error and IC error)
Timeseries of estimated temperature model error after 12 hours [fc12h – an]
Pressure hPa
Nov 2016
Aug 2017

6. 4D-Var formulations
Bias-aware data assimilation (VarBC + Weak constraint)
Introduce additional controls to target an unbiased analysis and to move the assimilation away from a perfect-model trajectory
The model error represents the systematic error which develops in the model over the assimilation window
The model error covariance matrix Q constrains the model error field
Size on the control vector increased by ~25%

7. Implemented in operations (22 Nov 2016, CY43R1)
Weak constraint 4D-Var has been implemented in operations
constant model forcing applied every hour
 only active above 40 hPa
Estimate the model error covariance matrix (Q)
run the ensemble forecasting system (ENS) with SPPT (51 members with the same initial condition for 20 days)
 differences after 12 hours are used to compute Q

8. Impact of weak constraint 4D-Var
Comparison between strong and weak constraint 4D-VAR.
Verification against GPS-RO (good for highlighting improvements in the stratosphere)

9. Impact of weak constraint 4D-Var
Comparison between strong and weak constraint 4D-VAR in the stratosphere. Verification against radiosondes.

10. Monitoring of the weak constraint 4D-Var in operations
Timeseries of estimated temperature model error after 12 hours [fc12h – an]
Pressure hPa
Timeseries of temperature model error estimated by the weak constraint 4D-Var
Pressure hPa
Nov 2016
Aug 2017
Negative (positive) model errors are corrected by positive (negative) forcings, especially in the upper stratosphere

11. Weak constraint 4D-Var in the troposphere
Model error variability for vorticity at 850hPa (September 2016)
Spurious day-to-day variability near airports
T399
T64
Q T399
Q T64
Aircraft observations are fitted in the 4D-Var by changing the model error
current model error covariance matrix contains small scales (T399)
new model error covariance matrix contains longer scales (T64)
localisation techniques and better set of ensemble forecasts to build Q

12. Summary and conclusions
4D-Var designed to correct the background with observations when errors are random with zero means
VarBC to deal with systematic errors in observations and observation operators
Weak constraint to deal with systematic errors in the model
Weak constraint 4D-Var performs well in the stratosphere but more work is required to disentangle background and model errors in the troposphere
Q matrix with longer correlation lengthscales and proper localisation is required
Within the OOPS framework, ECMWF is interested to use the weak constraint
4D-Var parallelization (subwindows)