Weight structure of the Local Ensemble Transform Kalman Filter: A case with an intermediate atmospheric general circulation model

Abstract

The Local Ensemble Transform Kalman Filter (LETKF) computes analysis by using a weighted average of the first‐guess ensemble with surrounding observations within a localization cut‐off radius. Since overlapped observations are assimilated at neighbouring grid points, the LETKF results in spatially smooth weights. This study explores the spatial structure of the weights with the intermediate atmospheric model SPEEDY (Simplified Parameterizations, Primitive Equation Dynamics). Based on the characteristics of the weight structure, we also aim to improve the weight interpolation (WI) method, which we use to compute the weights at coarser reference points and interpolate the weights into higher‐resolution model grid points. The results show that larger localization and sparser observations result in spatially smoother weights. WI is less detrimental when weight patterns are spatially smoother. An advanced WI method with observation‐density‐dependent reference points results in better forecasts than those with uniformly distributed reference points. This improvement may be due to the spatially inhomogeneous localization function realized by the WI method with observation‐density‐dependent reference points. The spatial distribution of the optimal localization scales shows that larger (smaller) localization is beneficial in sparsely (densely) observed regions. The WI method is computationally more efficient with larger ensembles since the additional computational cost for the WI is lower than that for the LETKF.

Implementing Hybrid Background Error Covariance into the LETKF with Attenuation-Based Localization: Experiments with a Simplified AGCM

“…As seen in Figures 5b and f Ocean (Figure 11b), which was consistent with the findings of Kotsuki et al (2020) that the optimal localization scale was larger in sparsely observed regions compared to densely observed regions. By contrast, increasing only Le degraded the RMSEs over the tropics and in the Northern Hemisphere (Figure 11b), perhaps because the optimal localization scale for densely observed regions is smaller than that for sparsely observed regions (Kotsuki et al 2020). In addition, error covariance tends to be smaller over the Increasing only Lc had detrimental effects on RMSEs, especially in sparsely observed regions of the southern Pacific Ocean (Figure 11c).…”

“…For the original LETKF implementation in SPEEDY based on Hunt et al (2007), the computational costs increased by O (m 3 ) due to the eigenvalue decomposition of (𝐏 ̃𝑎) −1 = 𝐈 + (𝐇𝐙 𝑏 ) 𝑇 𝐑 −1 𝐇𝐙 𝑏 (black lines of Figure 13), which was also observed for previous SPEEDY-LETKF experiments (cf. Table 2 of Kotsuki et al 2020). Figure 13 is consistent with our expectation that for large (m + c), the cost of the Optimal Eigen-Decomposition (OED) ETKF formulation would be proportional to O ((m + c) 2 ) and not…”

“…These experimental settings were determined according to previous SPEEDY-LETKF experiments (Greybush et al 2011;Miyoshi et al 2014;Miyoshi 2016, 2019;Kotsuki et al 2020). The number of assimilated observations at an observing station is 7 We performed a series of observing system simulation experiments using SPEEDY.…”

Revisiting Online and Offline Data Assimilation Comparison for Paleoclimate Reconstruction: An Idealized OSSE Study

“…SPEEDY is a hydrostatic model with a primitive equation dynamics and simplified parameterizations that include essential processes such as convection, large‐scale condensation, longwave and shortwave radiation, and boundary layer turbulence. SPEEDY has been used to study atmospheric predictability (e.g., Abid et al., 2015; Bahaga et al., 2015; Ehsan et al., 2013) and has been used as a testbed for DA studies first by Miyoshi (2005) and a number of follow‐on studies (e.g., Amezcua et al., 2014; Greybush et al., 2011; Hatfield et al., 2018; Kalnay et al., 2007; Kondo & Miyoshi, 2016, 2019; Kotsuki et al., 2020; Li et al., 2009; Miyoshi, 2011; Miyoshi et al., 2014). We choose SPEEDY for its ability to represent realistic physics with a low computational cost.…”

Empirical determination of the covariance of forecast errors: An empirical justification and reformulation of hybrid covariance models

“…It is important to note that, although the ensemble member 1 corresponds to the true state at the time we are initialising the ensemble, this is no longer valid after the first DA cycle. Using this initial ensemble, we used the LETKF to assimilate globally distributed radiosonde observations (Kotsuki et al ., 2020). However, to assimilate these observations using the LETKF, it is important to bear in mind that the implementation of ensemble Kalman filters in realistic frameworks (i.e., use of relative small ensemble members ) is characterised by two problems: forecast error covariance matrices (i) are rank deficient and (ii) contain spurious long‐distance correlations.…”