SOSTENIBILIDAD MENTAL
FUENTE: EL AUTOR.
The km-scale ensemble data assimilation (KENDA) system (Schraff et al., 2016) imple- ments the LETKF described above in COSMO-DE as a 4 dimensional LETKF (see Section 3.1.4). During the COSMO forward integration, observation operators are applied when-
Figure 3.4.: Schematic comparing the full-resolution model grid (squares and crosses) with the sparse analysis grid (crosses). In this case, the sparse analysis grid consists of every third grid point in zonal and meridional direction of the full-resolution grid. The analysis grid uses approximately 11% as many grid points as the full-resolution grid. This figure is adapted from Yang et al. (2009).
ever observations are available, i.e. a model equivalent is generated for each observation within the forward integration between two analysis time steps. Thus, KENDA weights the ensemble members according to their trajectory over the first guess window. In the standard mode, KENDA uses 40 ensemble members contributing to the calculation of the weights in Equation (3.35). An additional deterministic run is updated via
wbdet = ˜Pa(Yb)TR−1(y− ybdet), (3.53) where yb
det = H(xbdet). KENDA implements various features relevant for convective scale
data assimilation summarized in the following.
3.3.1. Sparse grid analysis
The KENDA suite allows to compute the analysis weights on a coarsened grid (Yang et al., 2009). In ensemble space, the weighting coefficients are calculated from similar data sets when localization regions overlap sufficiently. In this case, the spatial variability of the weights is small and Yang et al. (2009) suggest to choose a limited number of grid points for the analysis, i.e. to perform the analysis on a sparse grid (Figure 3.4). After the analysis step, the information from the coarse analysis grid is transferred back to the high resolution grid by interpolation. The interpolation of the weights is performed in ensemble space before calculating the analysis ensemble in model space, i.e the background maintains the full high resolution information and the analysis increments thus may still capture small- scale features. Yang et al. (2009) demonstrate a significant decrease of computational
costs while accuracy is hardly affected. Furthermore, the coarse grid analysis might even be more robust against unwanted imbalances, e.g. leading to gravity waves. In this study, every third grid point in zonal and meridional direction of the full-resolution model grid is used in the analysis. Therefore, the analysis grid uses approximately 11% as many grid points as the full-resolution grid.
3.3.2. Inflation and relaxation
In practice, it is commonly found that the ensemble Kalman filter as described previously may diverge from the observations, i.e. the observations are not within the range of the ensemble. One possible reason is that the LETKF formulas do not account for model error. Even in perfect model environments, the background error tends to become too small making the filter too confident about the background (Hunt et al., 2007). In case the ensemble does not provide sufficient spread, the filter fails to gain information from the observation. One possible solution is to increase the ensemble spread artificially. The most common method is multiplicative inflation (Anderson and Anderson, 1999). In the framework described above, multiplicative inflation can be included easily by replacing Equation (3.34) with
˜
Pa = (N− 1)I/ρ + (Yb)TR−1Yb−1, (3.54)
for an inflation factor ρ. Common choices are ρ ∈ [1.03, 1.1].
Relaxation approaches represent an alternative to increase ensemble spread. The meth- ods of Zhang et al. (2004) - relaxation to prior perturbations (RTPP), and Whitaker and Hamill (2012) - relaxation to prior spread (RTPS), are implemented in KENDA and discussed in Harnisch and Keil (2015). In RTPP, the analysis ensemble perturbations x0a
i = xai−xafor i = 1, . . . , N are relaxed towards the first guess perturbations x0bi = xbi −xb
at each analysis point:
x0ai ← (1 − α)x0ai + αx0bi . (3.55)
The parameter α controls the influence of the background spread. Zhang et al. (2004) recommend the heuristic value α = 0.75. The second method, RTPS, relaxes the analysis ensemble spread σa towards the prior ensemble spread σb:
σa ← (1 − α)σa+ ασb, where (3.56) σ(a,b) = r (n− 1)−1X x0(a,b) i 2 . (3.57)
Whitaker and Hamill (2012) recommend α = 0.95 in order to give a high weight to the prior spread. Equation (3.56) can be rewritten as
σa ← (1 − α)σa+ ασb · x0ai (3.58) ⇔ x0ai σa ← xi0a(σa− ασa+ ασb) (3.59) ⇔ x0ai ← x0ai ασ b− σa σa + 1 (3.60)
Thus, RTPS is a purely multiplicative inflation, whereas RTPP is partly multiplicative and partly additive.
3.3.3. Cycling
In the Kalman filter framework, a cycling between model runs and assimilation steps is needed as shown in Equation (3.15) and illustrated in Figure 3.1. At DWD, the NWP model COSMO-DE and the LETKF implemented by KENDA are coupled via shell scripts, called basic cycling (BACY) environment. COSMO and KENDA communicate via their output files within BACY. COSMO is reinitialized after each assimilation step. To avoid a dry start of COSMO, i.e. a start where all hydrometeor contents are set to zero, the hydrometeor contents qc, qi, qr , qs, qg (cloud drops, cloud ice, rain, snow and graupel) are passed between both modules.
Initial conditions and boundary data are produced within BACY via the downscaling module int2lm (Sch¨attler and Blahak, 2015) applied to model fields of the global model ICON (Z¨angl et al., 2015), which as well runs an LETKF. This means that each ensemble member of KENDA is provided with initial and boundary conditions from a different ICON ensemble member.