Spatial Filtering of Small Ensemble-Based Estimations of Background Error Variances at Convective Scale. ∗
´ etrier ´ Benjamin Men , Thibaut Montmerle, Lo¨ık Berre and Yann Michel ´ eo-France/CNRS) ´ CNRM-GAME (Met Data assimilation schemes skills are partly determined by the accuracy of the background error covariance modeling, that should ensure a maximum flowdependency. A first step would be to use ensemble-based variances instead ´ eo´ of climatological homogeneous ones, as performed operationally at Met France at global scale (+ flow-dependent wavelet correlations). At convective scale however, only ensemble of small sizes are computationally affordable.
Thus, it is necessary to filter out the sampling noise arising in variances estimation [4]. For this, we merge the variance estimation theory with the optimal linear filtering theory to get a general criterion involving statistics of known quantities, that can be approximated within a local or global ergodicity assumption. For an homogeneous filtering, once the filtering function shape is chosen, this criterion provides a quasi-optimal filtering length-scale.
Merging theories towards an optimal filtering criterion for variances Background error variance estimation from N forecasts 1 e= Unbiased estimator : v N −1
N X
2 xk − hxi
k =1
Sampling noise e e e∗ v = v + ξ Raw variance True variance Sampling noise Variance of the sampling noise 2 2 N −3 ∗2 e − e var e ξ = (N−1)(NN2−3N+3) E Ξ E v (N−1)(N 2−3N+3) e is an estimation of the where E[·] denotes the expectation and Ξ 4th order centered moment : N X 4 1 e= xk − hxi Ξ N k =1
Optimal linear filtering theory b b e + f = v e∗ + v = F v ξ Filtered Filter Raw Filter True Filtering signal gain signal offset signal error ( −1 ∗ e ,v e Cov v e F = Cov v 2 b ∗ E kξk is minimized for e − FE v e f=E v Assumptions on the noise ∗ ∗ e e e e ξ = v − v has a zero expectation and is not correlated with v . Properties at optimality ( ( ∗2 b◦v e =E v e eT = 0 E v E b ξv T Orthogonality ⇒ e−v b) + var(b b =0 var(e ξ) = var(v ξ) E b ξv 2 ∗2 b e −E v Optimal filter ⇒ var b ξ =E v
Variance filtering optimality criterion 2 3 2 2 N −3N +6N−6 N e = 0 e−v b) − E v b + (N−1)(N e b var(v E v ◦ v − E Ξ] Merging both theories, we can prove that at optimality : 2 −3N+3) 2 (N−1)(N −3N+3) e e), filtered variance (v b) and raw 4th order centered moment (Ξ). This optimality criterion involves statistics of known quantities only : raw variance (v → Under local or global ergodicity assumption, expectation and variance can be replaced by spatial average and spatial variance.
→ For an homogeneous and isotropic filtering, optimization of the filtering lenght-scale by the dichotomy method is always and quickly converging.
Variances filtering : maps and scores
Conclusions and perspectives
Two large ensembles of Vars (90 members) have been set up for the global model ARPEGE (providing coupling fields) and the limited-area convective-scale model AROME [2]. After ensemble spin-up, 4 assimilations of the AROME ensemble have been kept for the experiments.
→ Homogeneous filtering of variances is far more sensitive to the filtering function length-scale than to its shape (not shown). → The filtrering function shape can be chosen to ensure the positivity of filtered variances, which is not the case for usual spectral and wavelet filters [1, 3, 4]. → The new criterion provides a quasi-optimal filtering length-scale in a homogeneous filtering framework. → This algorithm has several advantages: • It does not assume that the prior distribution is Gaussian. • It is always convergent in a few iterations (< 10). • It works with any homogeneous filtering method (spectral or recursive filters), and is easy to implement. • It is efficient whatever the ensemble size.
The ensemble has been split into two subsets : a small one of N members used to compute noisy variances that are filtered then, and the complementary set of 90-N members used to compute an independent reference, against which scores are computed to evaluate filtering algorithms. Small ensembles of 6 or 12 members have been considered.
→ An straightforward transposition of this algorithm has been successfully tested for local correlation Hessian tensors defined in [5] (not shown). → Tests are in progress to assess the impact of filtered variances in a deterministic 3D-Var and in an ensemble-variational hybrid system. Background error variances of unbalanced specific humidity at level 50 (∼ 945 hPa) computed with a 84-member (left) or a 6-member ensemble (center and right). The variances are homogeneously filtered on the right panel, with a Gaussian kernel.
References [1] M. Bonavita, L. Isaksen, and E. Holm. On the use of EDA background error variances in the ECMWF 4D-Var. Quarterly Journal of the Royal Meteorological Society, 2012. ´ etrier, ´ [2] B. Men T. Montmerle, L. Berre, and Y. Michel. Estimation and diagnosis of heterogeneous flow-dependent background error covariances at convective scale using either large or small ensembles. Quarterly Journal of the Royal Meteorological Society, submitted, 2013. [3] O. Pannekoucke, L. Raynaud, and M. Farge. A wavelet-based filtering of ensemble background-error variances. Quarterly Journal of the Royal Meteorological Society, accepted, 2013. [4] L. Raynaud, L. Berre, and G. Desroziers. Objective filtering of ensemble-based background-error variances. Quarterly Journal of the Royal Meteorological Society, 135(642):1177–1199, 2009.
RMSE of background error variances of unbalanced specific humidity, mesured against an independent reference (90-N members), for four different configurations. The grey-shaded area shows an estimation of the reference uncertainty. Variances are filtered with a Gaussian kernel.
∗
Corresponding author :
[email protected]
[5] A. T. Weaver and I. Mirouze. On the diffusion equation and its application to isotropic and anisotropic correlation modelling in variational assimilation. Quarterly Journal of the Royal Meteorological Society, 139:242–260, 2012.
