This study focuses on covariance localization in the Gridpoint Statistical Interpolation (GSI)-based 3DEnVar (Wang et al., 2008, 2013). In 3DEnVar with a pure ensemble-based BEC, the analysis increment δx is obtained by minimization of the cost function: 1Ja1,…,aK=12∑k=1KakTL-1ak+12Hδx-dTR-1Hδx-d,2δx=∑k=1Kak⋮ak∘xken, where ak (k=1,…,K) is the N-dimension control vector, L is the covariance localization (N×N matrix), R is the observation error covariance (M×M matrix), H is the linearized observation operator (M×NV matrix), and d is the M-dimension observation innovation vector (K: the ensemble size; N: the number of grid points; M: the number of assimilated observations; V: the number of analysis variables). xken is the NV-dimension kth ensemble perturbation vector (kth ensemble member subtracted by ensemble mean and normalized by K-1). In this formulation, the same localization length is applied to all analysis variables.

In applying SDL (Buehner and Shlyaeva, 2015), the analysis increment δx (NV-dimension vector) is obtained as: 3δx=∑k=1K∑w=1Wak,w⋮ak,w∘xk,wen, instead of Eq. (2), where ak is extended to the NW-dimension vector as ak=ak,1⋮ak,W (W: the number of scales in SDL), xken is separated to multiple scales as xken=∑w=1Wxk,wen, and L is extended to the NW×NW matrix as: 4L=L11/20⋱0LW1/2EL1T/20⋱0LWT/2, where Lw is the N×N localization matrix applied for wth scale of ensemble perturbations xk,wen, and E=I⋯I⋮⋱⋮I⋯I is the NW×NW matrix to combine each scale for localizing cross-scale covariances in SDL (“Cross” in Huang et al., 2021).

The calculation of the localization L is accomplished by the RF (Purser et al., 2003a) in the GSI-based 3DEnVar as shown in Fig. 1a, where the square root of the localization matrix Lw=Lw1/2LwT/2 is quasi-Gaussian and computed as: 5Lw1/2=FRFZFRFYFRFX. FRFX,FRFY, and FRFZ denote RFs in x-, y-, and z-directions, respectively (self-adjoint N×N matrices). These RFs should be applied recursively; for example, to obtain q1out⋮qNXout=FRFXq1in⋮qNXin, 6qimid=βqiin+∑j=1pαjqi-jmid is sequentially calculated from the smallest i, and after that, 7qiout=βqimid+∑j=1pαjqi+jout is sequentially calculated from the largest i, where qiin, qimid and qiout (i=1,…,NX) are NYNZ-dimension vectors (NX, NY, and NZ are the numbers of grid points in x-, y-, and z-directions, respectively, so N=NXNYNZ). qi-jmid (i-j≤0) and qi+jout (i+j≥NX) are zero. The coefficients β and αj (j=1,…,p; p is the order of RF) are set to make the filtering kernel of FRFX quasi-Gaussian as: 8Gpxp→∞→cGexp⁡-x2s2, where the coefficient cG is set to satisfy ∫-∞∞Gpx2dx=1. Since the resulting filtering kernel of FRFXFRFXT is the self-convolution of Gpx as: 9Gp∗Gpx≡∫-∞∞Gpx-x′Gpx′dx′p→∞→exp⁡-x22s2, s is the standard deviation of G∞∗G∞x, which is the same as the e-1/2-folding scale σ.

Since Eqs. (6) and (7) are calculated sequentially, RF in one-direction is efficiently parallelized only in the other direction; for example, FRFX is efficiently parallelized only for NYNZ and the parallelization for NX is impossible. Therefore, all-to-all communication to change the direction of parallelization, which degrades the parallelization efficiency with the large number of processors, is required to calculate FRFZFRFYFRFX (e.g., between FRFZ and FRFYFRFX). Note that L itself is also calculated in parallel for the ensemble size K considering the formulation in Eq. (1).

This study suggests to calculate the localization L with MGBF instead of RF. Although the original MGBF (Purser et al., 2022) superposes variables filtered in filter grids of multiple resolutions gt (t=1,…,T; the grid interval of gt+1 is twice coarser than gt), this study applies MGBF only for the coarsest filter grid gT for faster computation as shown in Fig. 1b, where Lw1/2 is computed as: 10Lw1/2=Dg0←g1FBFg1ZDg1←gTFBFgTYFBFgTXDgT←g1, where DgT←g1 (NgT×Ng1 matrix) is 2×2-points bilinear interpolations with doubling the coefficients to satisfy DgT←g1DgT←g1T=I, which is repeated from g1 (the finest filter grid) to gT (the coarsest filter grid), Dg1←gT (Ng1×NgT matrix) is linearly weighted biquadratic horizontal interpolations (down-sending) repeated from gT to g1, and Dg0←g1 (N×Ng1 matrix) is bilinear horizontal and vertical interpolations (mapping) from g1 to the analysis grid g0 (Ngt: the number of grid points in gt). The finest filter grid g1 is the same as the analysis grid g0 or coarser. Note that DgT←g1 is required only in SDL because DgT←g1EDgT←g1T=I in single-scale localization. FBFgtX, FBFgtY, and FBFgtZ denote isotropic line beta filters applied in each generation in x-, y-, and z-directions, respectively (self-adjoint Ngt×Ngt matrices); for example, the filtering kernel of FBFgtX is: 11Bp,tx=cB,t1-4X2pX≡xs22p+3≤12 where Bp,tx=0 in X>1/2, the coefficient cB,t is set to satisfy ∫-∞∞Bp,tx2dx=ωt (ωt: weight of gt where ∑t=1Tωt=ωT=1), and s is the standard deviation of the self-convolution of Bp,tx, which is the filtering kernel of FBFgtXFBFgtXT and can be shown to have the form: 12Bp,t∗Bp,tx=ωt1-X2p+1∑i=0pai,pXp-i1+X2iX≤1. If we generalize the definition of binomial coefficients: 13Ci,j=i!i-j!j!, then the coefficients can be expressed, 14ai,p=∑j=0min⁡i,p-i/2Cp,jCp-i,2jCi,jC2p,2j, where ⋅ is the floor function. In the particular case, p=2, these coefficients are a0,2=a1,2=a2,2=1. The filtering kernel of FBFgTXFBFgTXT obtained as the self-convolution of B2,Tx can be expanded as: 15B2,T∗B2,Tx=1-X51+5X+9X2+5X3+X4X≡xs14≤1, where s is the standard deviation of B2,T∗B2,Tx. Unlike RF, s is smaller than the e-1/2-folding scale σ in MGBF (here, s/σ∼0.92852).

In MGBF, not only FBFg1Z but also FBFgTX and FBFgTY are parallelized for NXNY because Eq. (11) is independently applied for each horizontal grid point only in the finite domain near the point. It indicates that communication between processors is limited to the exchange of halo grid points with spatially neighboring processors and all-to-all communication is not required in MGBF.

In this subsection, the filter responses of the RF- and MGBF-based localizations are compared with single pseudo-observation DA. Here, a single surface pressure observation was assimilated with -10 hPa innovation and 1 hPa observation error in the northern region of Hurricane Ian at 80° W and 31° N, where the first guess was the 1 h FV3LAM forecast at 16:00 UTC, 29 September.

Figure 3 shows analysis increments of sea-level pressure (SLP). Compared to the increments with the single-scale localization (Fig. 3a, b), the SDL created the larger scale flow-dependent increments both for the RF- and MGBF-based localizations (Fig. 3c, d) since the horizontal localization length in the larger-scale SDL was set to fourfold. The difference between the RF- and MGBF-based localizations was little compared to the difference between the single-scale localization and SDL.

To clarify the difference of the responses between the RF- and MGBF-based localizations in more detail, the meridional cross-section of the ratio of analysis increments with and without the localization (analysis increments in RF, MGBF00, MGBF04, and MGBF04σ divided by the increment without the localization), which are regarded as the filter responses of each experiment, are shown in Fig. 4. While the response of RF (cyan line) was almost the same as Gaussian, that of MGBF00 (brown line) was a little wider, and almost consistent with Eq. (15). The difference between MGBF00 (brown line) and MGBF04 (pink line) was hardly visible although it was slightly underestimated near the peak in MGBF04 due to the coarser filter grid. Compared to MGBF04 (pink line), the response of MGBF04σ (yellow line) was closer to Gaussian near the e-1/2-folding scale while it was smaller far from the observation.

In this subsection, the calculation time of the RF- and MGBF-based localizations and the qualities of the resulting analyses are compared. Figure 5 shows the computation times for localizations in analysis-forecast cycling experiments. The time for horizontal filtering in MGBF01–02 was smaller than that of MGBF00 because it was applied in the coarser filter grid g4; in MGBF02, it was about half of that in MGBF01 due to skipping the filter for g1. However, the total time for the localization in MGBF00–02 was larger than that in RF because the amount of the calculation and communication between processors in up-sending and down-sending were proportional to the number of grid points, which were large in MGBF00–02. On the other hand, the time for the localization in MGBF03–04, which applied a coarser g1 than MGBF00–02, was shorter than that in RF. In particular, the time for the localization in MGBF04, which applied vertical MGBF in the coarser vertical grid, was about 20 % of that in RF. In SDL, the total time for the localization was roughly twice that of single-scale localization both for the RF- and MGBF-based localizations, which means that the reduction of the computation time by the MGBF-based localization was also approximately twice in SDL. The reduction rate of the computation time by the MGBF-based localization was larger in the experiments with larger numbers of processors (not shown), which indicates that parallelization efficiency of the MGBF is higher than that of the RF including the all-to-all communication. Hereafter, only RF, MGBF04, MGBF04σ, RFSDL, MGBF04SDL, and MGBF04σSDL are focused to show the small difference of the analyses with computationally efficient MGBF from that with RF.

Despite the large reduction of the computation time, the difference of analysis increments of SLP between the RF- and MGBF-based localizations was small in both experiments with the single-scale localization and the SDL (Fig. 6). The relatively large difference near Hurricane Ian (Fig. 6c–f) is reasonable due to the large increment there (Fig. 6a, b). In the experiments with SDL, the difference is slightly larger in the maritime area (Fig. 6d, f) probably because the difference between RF and MGBF is more obvious in the large localization applied to the large-scale ensemble-based error covariance, which is also large in the maritime area. Note that the analysis increment was not spatially smoothed even in the MGBF-based localization with the coarse filter grid because the ensemble perturbations xk,wen in Eq. (3) was not affected by the MGBF. Moreover, the difference from RF (Fig. 6a) was slightly smaller in MGBF04σ (Fig. 6e) than that in MGBF04 (Fig. 6c), and the difference from RFSDL (Fig. 6b) was also slightly smaller in MGBF04σSDL (Fig. 6f) than that in MGBF04SDL (Fig. 6d) even though the computation times for MGBF04σ and MGBF04σSDL were almost the same as that for MGBF04 and MGBF04SDL, respectively (not shown).

The impact of the MGBF-based localization on the dynamical balance of the analysis was also small. Figure 7 shows the mean absolute pressure tendency of the forecast from the analysis at 16:00 UTC, 29 September. While it was smaller in the experiments with SDL than that with single-scale localization (consistent with Yokota et al., 2024b), the impact of the MGBF-based localization was relatively small; for example, the difference between RF (cyan line) and MGBF04 (pink line) was smaller than that between RF (cyan line) and RFSDL (blue line). However, this slight difference between the RF- and MGBF-based localizations was accumulated in the analysis-forecast cycle, and the pressure tendency of the forecast from the last analysis with the MGBF-based localization at 00:00 UTC, 30 September was larger than that with the RF-based localization (Fig. 8) probably because the MGBF was the compact-support filter and its filter response was limited to the finite region. Nevertheless, MGBF04σ (yellow line) showed a smaller deviation from RF (cyan line) than MGBF04 (pink line). Similarly, MGBF04σSDL (orange line) was closer to RFSDL (blue line) than MGBF04SDL (red line).

Figure 9 shows the first guess departure of assimilated in-situ temperature and horizontal wind observations in the whole analysis-forecast cycles. For temperature, the RMSE and cold bias in the experiments with SDL were smaller than those with single-scale localization (consistent with Yokota et al., 2024b), and the differences between the RF- and MGBF-based localizations were relatively small (Fig. 9a and b). For horizontal wind, on the other hand, the degradation of the RMSE by the MGBF-based localization (pink line in Fig. 9c) were not necessarily smaller than the improvement by the SDL (blue line in Fig. 9c) probably because the impact of SDL on horizontal wind was smaller than that on temperature. However, the difference from RF (cyan line) was smaller in MGBF04σ (yellow line) than that in MGBF04 (pink line), and the difference from RFSDL (blue line) was also smaller in MGBF04σSDL (orange line) than that in MGBF04SDL (red line).

Considering the results above, the quality of the analysis in RF and RFSDL was closer to that in MGBF04σ and MGBF04σSDL than that in MGBF04 and MGBF04SDL, respectively. It may indicate that the e-0.5-folding scale of the localization function σ is more sensitive to the quality of the analysis than the standard deviation s. Note that these differences of the analyses discussed here hardly affected the Hurricane Ian forecasts. In fact, the track forecasts and associated precipitation forecasts initiated with the last analyses with the MGBF-based localization at 00:00 UTC, 30 September were almost the same as those with the RF-based localization (Figs. 10 and 11a). The minimum SLP forecasts with the RF- and MGBF-based localizations were also almost the same and the differences were smaller than that with and without SDL (Fig. 11b).