Accounting for correlated AMSU-A observation errors in an ensemble Kalman filter 18 August 2014 The World Weather Open Science Conference, Montreal, Canada Peter Houtekamer, Mikhail Tsyrulnikov * and Herschel Mitchell * HydroMetCenter of Russia

Overview Motivation, Innovation statistics, Description of observation error statistics, Changes to the Ensemble Kalman Filter, Experimental results, Impact of observation bias, Outlook.

Motivation The EnKF at Environment Canada uses a Monte Carlo procedure to simulate the evolution of errors in a data-assimilation cycle. With a realistic sampling of all sources of error, the ensemble spread would be representative of the ensemble mean error (as is the case in OSSEs with the EnKF). Using current descriptions of observation error and accounting for known model uncertainty, we obtain an underdispersive ensemble. To increase spread, we roughly double it by adding an isotropic “model error” term. We do not think the actual uncertainty in the forecast model is this large. We think that it would be most productive, at this stage, to improve the description of the “observation error”.

AMSU-A observations Radiance observations, in particular AMSU-A radiances, are an important source of information for modern data-assimilation systems. In general, we try to improve quality by increasing the volume of the radiance observations. With the increased density, the error correlations between nearby observations become more important. Current assimilation schemes neglect error correlations and use a corresponding inflation of the observation error variances. Normally, to be internally consistent, this inflation should become larger with higher density. The method would appear ad hoc and we think it would be more productive to specify the best knowledge on observation error statistics.

Use of innovations to tune observation error variances To tune error statistics, it is common to use innovation amplitudes: Innovation = O – H (x) = true value - H(truth) + ε(O) – H(ε(x)), where x is the ensemble mean background (i.e forecast). The error variance of the innovations is given by: (O-H(x),O-H(x))= (ε(O),ε(O)) + (H(ε(x)),H(ε(x))) – 2 (ε(O),H(ε(x))) In the special case that the observation error ε(O) does not correlate with the background error H(ε(x)), we have: (O-H(x),O-H(x))= (ε(O),ε(O)) + (H(ε(x)),H(ε(x))) From which it follows that: (ε(O),ε(O)) < (O-H(x),O-H(x))

Innovation statistics for AMSU-A channels 4 – 12 (used in EnKF) Normally innovation amplitudes would be bigger than the uncertainty in the observations and the uncertainty in the background. In the EnKF, the specified observation standard deviation exceeds the innovation amplitude. We would like to clarify or correct this situation. Our initial hypothesis was that the specified observation errors are too large.

The argument for inflating observation error variances In operational data-assimilation systems, it is common to assume that all observations have independent errors. To some extent, the neglect of correlations can be compensated for by inflation of the observation error variance (Liu and Rabier 2003 for horizontal correlations). We found the optimal diagonal matrix to be: Thus, in the presence of horizontal observation error correlations, tuning for optimal analysis quality can lead to an overestimate of observation error variances. Due to both temporal and horizontal observation error correlations, it is not evident what the tuned observation error variances correspond to. In a simple 2-variable example with P=I, H=I and

Estimated horizontal error correlations for AMSU-A channel 8. Unconventional results (significant horizontal correlations for the observational error) are due to not assuming that observation and forecast error are independent (Gorin & Tsyrulnikov, MWR, 2011). The paper also estimates temporal and interchannel correlations for channels 6-8.

Short Term (ST) and Long Term (LT) components The horizontal correlation is the result of the addition of a slowly evolving component with long temporal (LT) scales and a more rapidly evolving component with short temporal (ST) scales. In our experiments, the correlations determined for channel 8 are used for all channels.

Evolution of observation error fields Observation error = independent error (ω)+ correlated error (z). The z-field for each AMSU-A channel is defined on a horizontal grid. The same z-field is used for different satellites. The z-field evolves in time and space using Markov sequence: z(k+1) = N z(k) – τ (k) Here N and τ form a stochastic model which respects desired spatial and temporal correlations. O = H(truth) + L z + ω Here L is a bi-linear interpolation operation and ω is a random (uncorrelated) error. Interchannel correlations can be dealt with (optional).

Example of an observation error field Each ensemble member uses a different observation error (z) field.

The Kalman filter equations In a Kalman filter, the gain matrix K is computed as: Since the z-field evolves in time and space, and enters the observation equation, it can be estimated by the analysis. In the augmented-space analysis, the z-field is appended to the regular model state x: The covariance matrix for this vector is:

Changes to the Kalman filter equations For the extended-state formulation, the gain K should be computed as: To permit some early experiments, with only relatively minor code changes, the cross-covariances of x and z have been neglected and the z-field is not updated by the analysis. This leads to: Hereis the covariance matrix of the uncorrelated error.

Experimental strategy We use a low-resolution (with a 400 by 200 uniform global grid) configuration of the Canadian operational EnKF. The model has 74 vertical levels with a model top near 2 hPa (as in operations). Qualitative results are similar to those of the operational system. A total of 288 ensemble members are used. Short (two week) assimilation cycles, starting December , are performed. The envisioned strategy was to gradually introduce correlated error components and have corresponding reductions of the observation error variances to make these more realistic.

Impact of using correlations on 6h verifcations Red : experiment using horizontal and temporal correlations (only the short-term component is used). Blue : reference experiment with diagonal covariance. Verification of the ensemble mean 6h forecast against the global radiosonde network. Typical results are fairly neutral. Attempts to reduce observation error variances typically show somewhat degraded verifications above 200 hPa (not shown).

Impact of correlations on spread After an assimilation cycle of two weeks, the cycle using horizontal and temporal correlations shows more spread in the trial field ensemble than the reference cycle. While it would seem reasonable that correlated observations contain less information, it is hard to make this point with a simple example.

Summary of experimental results After some 50 experiments, we have no configuration that beats the reference. We did not manage to reduce observation error variances. What is wrong? 1)With horizontal, temporal and inter-channel correlations and correlated and uncorrelated observation error variances, the parameter space is large and it is hard to tune for optimal results. 2)We needed to extrapolate estimated observation error descriptions for high and low-peaking channels. 3)We may need to move to the full extended-state-vector description.

Impact of bias correction Our center is testing a major change to the AMSU-A bias correction procedure. For a number of channels, the amplitude of the bias correction change is comparable to the total innovation amplitude. This suggests temporal error correlations play an important role. It may be impossible to reduce the assumed observation error variances for the EnKF.

Interpretation and outlook Observation bias is an observation error with a large spatial and long temporal component. Bias will have an impact on innovation amplitudes. That is, true observation errors are really large. They have not just been tuned to be large to account for horizontal error correlations. The (full) description of the EnKF with no neglected terms should be able to handle observation error with a correlated component with large spatial and long temporal components. Work will continue towards a comprehensive treatment of observation error in the EnKF.

Conclusions Thanks for your attention