Hunt et al. (2007) introduced the LETKF as a computationally efficient EnKF by combining the local ensemble Kalman filter (Ott et al., 2004) and ETKF (Bishop et al., 2001). Let Xt be a matrix composed of m ensemble state vectors. The ensemble mean vector and perturbation matrix of Xt are given by x‾t (∈Rn) and Zt≡xt(1)-x‾t,…,xt(m)-x‾t (∈Rn×m) respectively, where n is the system size. The subscript t indicates the time, and the superscript (i) denotes the ith ensemble member. The analysis equations of the LETKF are specified by 1Xt,LETKFa=x‾tb⋅1+ZtbTt,LETKF,2Tt,LETKF=P̃taYtbTRt-1yto-HtXtb‾⋅1+m-1P̃ta1/2, where 1 is a row vector whose elements are all 1 (∈Rm); T is the ensemble transform matrix (∈Rm×m); P̃ is the error covariance matrix in the ensemble space (∈Rm×m); Y≡HZ is the ensemble perturbation matrix in the observation space (∈Rp×m); R is the observation error covariance matrix (∈Rp×p); y is the observation vector (∈Rp); H is the linear observation operator matrix (∈Rp×n); and H is the observation operator that may be nonlinear. Here, p is the number of observations. The superscripts “o”, “b”, and “a” denote the observation, background (prior), and analysis (posterior), respectively. The matrix TLETKF denotes the ensemble transform matrix of the LETKF. The first and second terms on the right-hand side of Eq. (2) correspond to the updates of ensemble mean and perturbation, respectively. The ETKF computes the analysis error covariance matrix in the ensemble space P̃ta by 3P̃ta=m-1βI+YtbTRt-1Ytb-1, where β is a multiplicative inflation factor. Equations (1) and (2) are derived from the Kalman filter equations, given by 4x‾ta=x‾tb+Ktyto-HtXtb‾,5Kt=PtbHtTHtPtbHtT+Rt-1=PtaHtTRt-1,6Pta=I-KtHtPtb⟺Pta-1=Ptb-1+HtTRt-1Ht, where K is the Kalman gain (∈Rn×p), and P is the error covariance matrix in the model space (∈Rn×n). Derivations of Eqs. (5) and (6) are detailed in Appendix A. The EnKF approximates the error covariance matrix by P≈1m-1ZZT. Hunt et al. (2007) provide more details on deriving Eqs. (1) and (2) from the Kalman filter equations. For nonlinear observation operators, the following approximation is used: 7Y≡HZ≈HX-HX‾⋅1. The LETKF provides P̃ta and P̃ta1/2 of Eq. (2) by solving the eigenvalue decomposition of P̃ta-1=UΛUT, where U is a square matrix (∈Rm×m) composed of eigenvectors, and Λ is a diagonal matrix (∈Rm×m) composed of the corresponding eigenvalues. The eigenvalue decomposition leads to P̃ta=UΛ-1UT and P̃ta1/2=UΛ-1/2UT.

The localization is practically important for mitigating sampling errors in the ensemble-based error covariance with a limited ensemble size (Houtekamer and Zhang, 2016). With localization, the LETKF computes the transform matrix TLETKF at every model grid point independently by assimilating a subset of observations within the localization cut-off radius. The LETKF employs localization by inflating the observation error variance so that observations distant from the analysis model grid point have fewer impacts (Hunt et al., 2007; Miyoshi and Yamane, 2007).

For local analysis schemes, a spatially smooth transition of the transform matrix TLETKF is essential to prevent abrupt changes in the analyses of neighboring grid points. The LETKF realizes a smooth transition of the transform matrix by using the symmetric square root of P̃ta (Hunt et al., 2007). The symmetric square root matrix minimizes the mean-square distance between identity matrix I and m-1P̃ta1/2; therefore, the analysis ensemble perturbations can be closer to the background ensemble perturbation.

The EnKF generally underestimates the error variance, mainly because of model errors, nonlinear dynamics, and limited ensemble size. Therefore, a covariance inflation technique is used to inflate the underestimated error variance (Houtekamer and Mitchell, 1998). Among several kinds of covariance inflations methods, the present study considers multiplicative inflation (Anderson and Anderson, 1999) and relaxation to the prior scheme (Zhang et al., 2004), as implemented by Whitaker and Hamill (2012). In multiplicative inflation, the ensemble-based covariance is multiplied by a factor β (Pb→βPb). This multiplicative inflation is employed in Eq. (3) in the LETKF. In this study, the LETKF experiments use the approach of Miyoshi (2011), which adaptively estimates a spatially varying inflation factor on the basis of the innovation statistics of Desroziers et al. (2005). In realistic problems, covariance relaxation methods are often used to inflate the posterior perturbation (e.g., Terasaki et al., 2019; Kotsuki et al., 2019b). This study utilizes the relaxation to prior spread (RTPS; Whitaker and Hamill, 2012), given by 8Zt(k)a←1-α+ασt(k)bσt(k)aZt(k)a, where α is the RTPS parameter, and σ is the ensemble spread. The subscript (k) denotes the kth model variable or the kth component of the state vector. Although this study only uses multiplicative inflation for the LETKF experiments, the posterior error perturbation is inflated by RTPS for LPF and LPFGM.

Here, we describe the LPF in the form of the ensemble transform matrix of the LETKF (Reich, 2013; Penny and Miyoshi, 2016; Potthast et al., 2019). The PF is a direct Monte Carlo realization of Bayes' theorem, given by 9πxt|y1:to=πyto|xtπxt|y1:t-1oπyto|y1:t-1o, where πxt|y1:to is the probability of state x given all observations yo up to time t. The PF approximates the prior probability density function (PDF), which appears in the numerator of Eq. (9), using an ensemble forecast 10πLPFxt|y1:t-1o≈∑i=1mwtb(i)δxt-xtb(i), where δ is the Dirac's delta function, and wb(i) is the prior weight of the ith particle (i.e., ensemble member) at the previous analysis time. If particles are resampled at the previous analysis time, all particles have the same weight wb(i)=1/mi=1,…,m. The sum of the weights ∑i=1mwta,b(i) is always 1. The denominator on the right-hand side of Eq. (9) can be estimated by 11πyto|y1:t-1o=∫πyto|xtπxt|y1:t-1odxt≈∑i=1mwtb(i)π(yto|xtb(i)).. This study assumes a Gaussian likelihood function, given by 12πyto|xt∝exp⁡-12yto-HtxtTRt-1yto-Htxt. This assumption means the observation error distribution is assumed to be Gaussian. Then, the posterior can be expressed by 13πLPFxt|y1:to≈∑i=1mwta(i)δxt-xtb(i),14wta(i)=wtbiπLPFy1:to|xtbi/∑k=1mwtb(k)πLPFy1:to|xtb(k)=wtb(i)qt(i)/∑k=1mwtb(k)qt(k),15qt(i)=exp⁡-12yto-Htxtb(i)TRt-1yto-Htxtb(i), where q is the likelihood. Equation (14) results in posterior weights that satisfy ∑i=1mwta(i)=1.

To mitigate weight collapse, the local PF (LPF) solves the PF equations by assimilating local observations surrounding the analysis grid point. This study uses Penny and Miyoshi's (2016) approach that computes the analysis at every model grid point independently, with the observation error covariance R being inflated by the inverse of a localization function, as in the LETKF (Penny and Miyoshi, 2016).

Sampling importance resampling (SIR) is a technique for applying the PF for high-dimensional dynamics with a limited amount of particles. The resampling process rearranges the particles to effectively represent the densest areas of the posterior PDF. After resampling, each particle has equal weights, and the posterior PDF is given by 16πLPFxt|y1:to≈1m∑i=1mδxt-xta(i). The resampling process can be expressed as equally valid to the ensemble transform matrix of the ETKF (Reich, 2013; Penny and Miyoshi, 2016), given by 17Xt,LPFa=XtbTt,LPF=x‾tb⋅1+ZtbTt,LPF=x‾tb⋅1+ZtbTt,LPF, where TLPF denotes the ensemble transform matrix of the LPF. Here, we applied x‾tb⋅1Tt,LPF=x‾tb⋅1 based on the following necessary condition of the ensemble transform matrix for the LPF: 18∑i=1mTt,LPF(i,j)=1j=1,…,m. In addition to Eq. (18), the ideal resampling matrix satisfies the following two conditions:

19∑j=1mTt,LPF(i,j)=m⋅wta(i) for i=1,…m⇔x‾tb=∑i=1mxtb(i)⋅wta(i),

The resampling matrix significantly affects the filter performance (Farchi and Bocquet, 2018). The present study constructs resampling matrices based on the Monte Carlo approach. This resampling method uses m random numbers r(i) (i=1,…,m) drawn from uniform distribution U ([0,1]) and accumulated weights wt,acc: 20wt,acc(0)=0,wt,acc(i)=wt,acc(i-1)+wta(i)i=1,…,m. By definition, wt,acc(m) is 1. After sorting r to be the ascending order rsorted, we generate the resampling matrix using Algorithm 1. This procedure is similar to the resampling of Potthast et al. (2019), but it employs an additional treatment so that the resampling matrix is close to the identity matrix (line 10 of Algorithm 1; Fig. 1b and c).

Classical resampling can produce the same particles, resulting in the loss of diversity of posterior particles. Therefore, additional treatments are required to generate slightly different posterior particles. Potthast et al. (2019) proposed adding Gaussian random noise to Tt,LPF to prevent the generation of the same particles. As an alternative, the present study uses the Monte Carlo approach that repeats Algorithm 1 many times with different random numbers and takes the average of the generated resampling matrices (Fig. 1). Using different random numbers r, the resampling matrices differ (Fig. 1b and c), even if the same weight is used (Fig. 1a). By averaging the resampling matrices generated by using different random numbers, we get matrices that have higher weights in the diagonal components and lower weights in the off-diagonal components (Fig. 1d and e). The Monte Carlo approach makes the resampling matrix closer to the identity matrix, which is beneficial for a smooth transition in space. In addition, this stochastic approach approximates Eq. (19) using the Monte Carlo approach. The generated transform matrix with 200 samples (Fig. 1c) is close to that with 10 000 samples (Fig. 1d) in the case of 40 particles. Therefore, the transform matrices are generated by averaging 200 sampled matrices in subsequent LPF experiments. Hereafter, we call this resampling method “MC resampling”, which is used in the following experiments. The number of required samples for MC resampling is briefly investigated in Appendix B.

The effective particle (or ensemble) size Neff (Kong et al., 1994) is useful to measure the diversity of particles in LPF: 21Nt,eff=1/∑i=1mwta(i)2∈1,m. If Neff is sufficiently large (Neff≈m), no resampling is needed. This study considers a tunable parameter N0 as a criterion for resampling: 22Tt,LPF←I if Nt,eff>N0Tt,LPF else Without resampling (Nt,eff>N0), the posterior weight is succeeded for the subsequent forecast as follows: 23wt+1b←1-τ⋅wta+τ/m, where τ is the tunable forgetting factor τ∈[0:1]. Here, τ=1 means the subsequent prior weight wt+1b has the same weights 1/m, whereas the posterior weight wta is completely succeeded to wt+1b when τ=0. This weight succession (Eq. 23) can be interpreted as a temporal localization that reduces the impact of observations temporally distant from the assimilation time. The weight succession in local PF is not trivial, because the weight (or likelihood) of particles would move with the dynamical flow. Similar discussions can be found for the advection of localization functions (e.g., Ota et al., 2013; Kotsuki et al., 2019a). This study assumes no flow motion of the weights and simply uses Eq. (23) at each grid point independently.

Although the LETKF applies inflation to the prior error covariance (i.e., Pb→βPb), this inflation is suboptimal to the LPF, partly because the weight collapse occurs more easily if multiplicative inflation is applied to the prior perturbation (see Eq. 15). Therefore, the LPF usually applies inflation to the posterior particles (e.g., Penny and Miyoshi, 2016; Farchi and Bocquet, 2018). The present study uses the RTPS to inflate posterior perturbations for the LPF. We do not use the RTPS for the LETKF experiments, since the Miyoshi (2011) adaptive multiplicative inflation is known to outperform the manually tuned RTPS for idealized twin experiments with SPEEDY, based on the authors' preliminary experiments and Yoichiro Ota (personal communication, 2021).

Preliminary experiments showed that the RTPS outperformed the relaxation to prior perturbation (RTPP; Zhang et al., 2004) for the LPF. Having some noise in the transform matrix is important for the LPF to maintain the diversity of posterior particles. The RTPP makes the transform matrix closer to the identity matrix, resulting in less diverged posterior particles. Therefore, the RTPS would be a more suitable relaxation method than the RTPP for LPF.

For the LPF to avoid weight collapse, assimilating too many independent observations in the local area is not desirable (van Leeuwen et al., 2019). To solve this problem, hybrid algorithms of EnKF and PF have been explored to efficiently assimilate massive observation data. The Gaussian mixture extension of the LPF is one such hybrid algorithm (Hoteit et al., 2008; Stordal et al., 2011; WP22). In the Gaussian mixture extension, the prior PDF is approximated by a combination of Gaussian distributions centered at the values of the particles, given by: 24πLPFGMxt|y1:t-1o≈∑i=1mwtb(i)Nxtb(i),P^tb, where Nxtb(i),P^tb is the Gaussian kernel, with mean xtb(i) and covariance P^tb. Here, hat indicates matrices for the Gaussian kernels (e.g., prior error covariance P^b, observation error covariance R^, and Kalman gain K^). The covariance of the Gaussian kernel uses the sampled covariance matrix Ptb, such that 25P^tb=γPtb≈γm-1ZtbZtbT, where γ (>0) is a tunable parameter that regulates the amplitude and width of the Gaussian kernel (i.e., uncertainty of particles). For example, larger γ reduces amplitude and increases the width of the Gaussian kernel. Because the kernel is supposed to have a Gaussian distribution, increasing the Gaussian kernel's width results in a decrease in amplitude. The LPFGM results in the same analysis of the LPF when γ is 0 (γ→0).

The Gaussian mixture performs a two-step update to obtain the posterior particles. The first update moves the center of the Gaussian kernel with observations. Since kernels are Gaussian, we can use the Kalman filter for the first update: 26xta(i)=xtb(i)+K^tyto-Hxtb(i)(i=1,…,m)⇔Xt,GMa=Xtb+K^tyto⋅1-HtXtb, where K^=P^aHTR-1, as in Eq. (5). In the LETKF, the Kalman gain K is computed by K=PaHTR-1≈ZbP^aZbTHTR-1. Since the Gaussian kernel uses the ensemble-based error covariance (Eq. 25), we can apply the exact algorithms of the LETKF to compute P^a by replacing β of Eq. (3) with γ (i.e., P^a=m-1γI+YtbTRt-1Ytb-1). Here, the same gain matrix K^ is applied to update each particle independently, while the gain matrix K^ is based on the forecast error covariance estimated from the entire ensemble. Consequently, Eq. (26) is equivalent to 27Xt,GMa=x‾tb⋅1+ZtbTt,GM, where 28St=P^taYtbTRt-1yto⋅1-HtXtb, and 29Tt,GM=St+I. Here, Tt,GM denotes the ensemble transform matrix of the Gaussian mixture.

The second update resamples the particles based on the likelihood of the posterior kernel, given by 30qt(i)=exp⁡-12yto-Htxtb(i)TR^t-1yto-Htxtb(i), where 31R^t=Rt+HtP^tbHtT. Hoteit et al. (2008) and Stordal et al. (2011) used Eq. (30) for computing the likelihood of posterior kernels. Alternatively, WP22 suggested that using Eq. (15) instead of Eq. (30) is a reasonable approximation in the case of a smaller variance of P^tb compared to the observation departure yto-Htxtb(i). This study follows the WP22's approximation, because computing the inverse of Eq. (31) is computationally much more expensive than computing the inverse of diagonal R. Using WP22's approximation, the solution of the LPFGM is given by 32Xt,LPFGMa=Xt,GMaTt,LPF=x‾tb⋅1+ZtbTt,GMTt,LPF=x‾tb⋅1+ZtbTt,GMTt,LPF. Here, we used x‾tb⋅1Tt,LPF=x‾tb⋅1 (cf. Eq. 18). Consequently, the transform matrix of the LPFGM (TLPFGM) is given by 33Tt,LPFGM=Tt,GMTt,LPF. Namely, the LPFGM can be described as the ensemble transform matrix form. Representing the LPFGM with only one transform matrix TLPFGM is practically beneficial if one aims to reduce computational costs by the weight interpolation for the LETKF, in which the transform matrices at higher-resolution model grid points are interpolated by transform matrices at coarser model grid points (Yang et al., 2009; Kotsuki et al., 2020). The weight interpolation is also useful for ensuring a spatially smooth transition of the transform matrix for LPFs (Potthast et al., 2019).

The two-step update of the LPFGM may appear to use the same observations twice, but this is not true. To understand the principles here, we consider a simple scalar example with H=1.0, with illustrations in Fig. 2. Let πyto be an observation PDF (Fig. 2, top row), and the prior and posterior PDFs of the LPF are given by πLPFxt≈1m∑i=1mδxt-xtb(i) and πLPFxt|yto ,respectively (Fig. 2a, second and third rows). The LPF employs resampling by approximating the posterior PDF as a combination of prior particles, such that πLPFxt|yto≈∑i=1mwta(i)δxt-xtb(i) (Fig. 2a, bottom row). Next, we focus on the fifth particle of the LPFGM (Fig. 2b). Prior and posterior PDFs of the fifth particle are given by πxt(5)=Nxtb(5),P^tb and πxt(5)|yto=Nxtb(5),P^tb⋅Nyto,Rt, respectively (Fig. 2b, second and third rows). Since the Gaussian kernels are assumed for the prior particles, the center of the posterior kernel moves such that πxt(5)|yto∝Nxta(5),P^ta, where xta(5) and P^ta can be computed by the Kalman filter (from the blue circle to the red circle in Fig. 2b). Since the LPFGM moves all particles, the posterior PDF of the LPFGM is given by πLPFGMxt|yto≈1m∑i=1mπxt(i)|yto (Fig. 2c, third row). These movements correspond to the first update of the LPFGM. In contrast to the LPF, the posterior PDF of the LPFGM is approximated by a combination of the posterior Gaussian kernels (red circles of Fig. 2c, third row). The LPFGM employs the resampling based on the moved particles such that πLPFGMxt|yto≈∑i=1mwta(i)Nxta(i),P^ta (Fig. 2c, bottom row). This resampling corresponds to the second update of the LPFGM. As seen in this example, the two-step update of the LPFGM does not use the same observations twice.

We implemented the LPFGM based on the available LETKF code from Miyoshi (2005) and from follow-on studies (Miyoshi and Yamane, 2007; Kondo and Miyoshi, 2019; Kotsuki et al., 2020; https://github.com/takemasa-miyoshi/letkf, last access: 16 November 2022). Figure 3 compares the workflows of the LETKF, LPF, and LPFGM. All DA methods execute the same first four steps (steps A–D). After step D, the LETKF involves four additional steps (steps E–H) to compute TLETKF. The LPF computes the weights of particles (step J), followed by the generation of the transform matrix TLPF (step K). The LPFGM first executes steps E–G, as in the LETKF, and then executes step I to compute TGM, followed by the LPF algorithms to compute TLPF (steps J and K). Finally, the LPFGM multiplies TGM and TLPF to compute TLPFGM (step L). At the end of the process, the transform matrix is applied to the prior perturbation matrix Zb to produce the analysis ensemble (step M) followed by the RTPS (step N).

If the LETKF code is available, the LPF in the transform matrix form can be developed by coding two steps, J and K. The LPFGM can also be developed easily by coding two more steps, I and L, if the LETKF and LPF codes are available.

Here, we compare the computational complexities of the LETKF, LPF, and LPFGM algorithms (Table 1). The total cost (CT) of a DA cycle is identical to the overhead of the assimilation system (CH) plus n times the average local analysis cost (CL) and m times the cost of a one-member model forecast (CM): 34CTn,p,m=CHn,p,m+n⋅CLpL,m+m⋅CM, where pL is the number of local observations within the localization cut-off radius. We assume that the overhead and model costs are equivalent among the three methods. In addition, the total computational cost of DA usually depends on the local analysis cost (CL): 35CLLETKF=Om3+m2pL,36CLLPF=Om2NMC,37CLLPFGM=Om3+m2pL+m2NMC, where NMC is the number of times the resampling matrices are generated by Algorithm 1. The number of local observations pL is usually much greater than the ensemble size m and NMC. In this case, O(m2pL) is dominant for the LETKF and the LPFGM.

Here, the computational cost of LPF is more expensive than that with a simpler resampling algorithm, such as the stochastic uniform resampling (Om; Penny and Miyoshi 2016; Farchi and Bocquet, 2018), due to the relatively complex Algorithm 1 (Om2NMC). We could not use such simpler approaches in this study, since they do not yield ideal resampling matrices that satisfy the two conditions (Eq. 19 and spatially smooth transition of Tt,LPF). The computational cost Om2NMC of LPF is still much smaller than the LETKF if NMC≪pL.

This study used the intermediate global atmospheric general circulation model SPEEDY (Molteni, 2003) to compare the LETKF, LPF, and LPFGM. The SPEEDY model is a computationally inexpensive hydrostatic model with fundamental physical parameterization schemes such as surface flux, radiation, convection, cloud, and condensation. The SPEEDY model has 96×48 grid points in the horizontal plane (T30 ∼3.75∘ × 3.75∘) and seven sigma-coordinate seven vertical layers. The SPEEDY model consists of five prognostic variables: temperature (T), specific humidity (Q), zonal wind (U), meridional wind (V) at seven layers, and surface pressure (Ps). The SPEEDY model coupled with the LETKF (SPEEDY-LETKF) has been widely used in DA studies (Miyoshi, 2005, and many follow-on studies, e.g., Miyoshi, 2011; Kondo et al., 2013; Kondo and Miyoshi, 2016, 2019; Kotsuki and Bishop, 2022). We implemented the LPF and LPFGM based on the existing SPEEDY-LETKF code, following the procedures described in Sect. 2.4.

The experimental settings follow the previous SPEEDY-LETKF experiments (Miyoshi, 2005, and follow-on studies, e.g., Kotsuki et al., 2020). A series of idealized and identical experiments (also known as observing system simulation experiments) were conducted without model errors. We first performed a spin-up run for one year, initialized by the standard atmosphere during rest, followed by the nature run started at 00:00 UTC on 1 January of the second year. We assumed diagonal observation error covariance R (i.e., uncorrelated observation error). Gaussian noise was added to the nature run to produce observation data at 6 h intervals. The standard deviations of the Gaussian noise are 1.0 K for T, 1.0 m s-1 for U and V, 0.1 g kg-1 for Q, and 1 hPa for surface pressure. This study considered two observing networks (Fig. 4a and b): the regularly distributed network (hereafter REG2) and the radiosonde-like inhomogeneous network (hereafter RAOB). We observe T, U, and V at all seven layers, whereas Q was observed at the first–fourth layers. The ensemble size is 40, and their initial conditions were taken from an independent single deterministic SPEEDY forecast with sufficient spin-up simulation.

Table 2 summarizes the settings of the LETKF, LPF, and LPFGM experiments. The observation error variance was inflated for the localization by using the Gaussian-based function given by 38l=exp⁡-12dh/ρh2+dv/ρv2 if dh<210/3ρh and dv<210/3ρv0 else where l is the localization function, and its inverse l-1 is multiplied and used to inflate R. dh and dv are horizontal and vertical distances (km and log(Pa)) between the observation and analysis grid point. ρh and ρv are tunable horizontal and vertical localization scales (km and log(Pa)). Subscripts “h” and “v” represent horizontal and vertical respectively. This study set the vertical localization scale ρv to be 0.1 log(Pa) following Greybush et al. (2011) and Kondo et al. (2013). The horizontal localization scales of the LETKF were tuned manually before the experiments to minimize the first-guess root mean square error (RMSE) of the fourth layer (∼500 hPa) temperature of the SPEEDY-LETKF, since this is among the most important variables for medium-range NWP.

The LETKF experiments used the adaptive multiplicative inflation method of Miyoshi (2011), in which the inflation factor β was estimated adaptively. On the basis of sensitivity experiments for γ, we chose γ=1.5. Sensitivity to γ is discussed in Sect. 4.3.3.

We first performed a one-year (from January to December) SPEEDY-LETKF experiment over the second year following the one-year spin up. We then performed LETKF, LPF, and LPFGM experiments from January to April in the third year, initialized by the first-guess ensemble of the LETKF at 00:00 UTC on 1 January of the third year. The results from the last three months, i.e., February to April, were used for verification. We assessed the 6 h forecast or background RMSE for T at the fourth model level (∼500 hPa). While this study mainly discusses the RMSE and ensemble spread for T, similar results are observed for other variables. Here, this study will concentrate on the fourth-level T as an important variable for NWPs, since the major goal of this study is to investigate the stabilities of LPF and LPFGM compared with the LETKF. Humidity verification or accounting for nonlinear observation operators and non-Gaussian observation errors are still crucial studies to investigate the advantages of the LPF and LPFGM with respect to the LETKF.

We first compare the LETKF, LPF, and LPFGM with REG2, where the manually tuned horizontal localization scale for the LETKF is 600 km. First, sensitivities to the horizontal localization scale (ρh) and RTPS parameter (α) are investigated for the LPF. Figure 5 indicates the time-mean background RMSEs and ensemble spreads for LETKF and LPF. The LPF requires large inflation (α≥1.0) to avoid filter divergence. With large inflation (α≥1.0), the LPF experiment with ρh=600 km resulted in the smallest RMSEs among the three LPF experiments. The best-performing localization scale of LPF was found to be similar to that of the LETKF. The four LPF experiments exhibited larger RMSEs than the LETKF, as demonstrated by the previous studies with a low-dimensional L96 model (Poterjoy, 2016; Penny and Miyoshi, 2016; Farichi and Bouquet, 2018). The LPF experiment with ρh=700 km shows filter divergence when α=1.0. The ensemble spreads of LPF are smaller than the RMSEs when α=1.0. This under-dispersive ensemble is a typical behavior of LPF (e.g., Poterjoy and Anderson, 2016).

Second, we compare the time series of the background RMSEs, ensemble spreads, and effective particle sizes Neff (Fig. 6). The RMSE and ensemble spread are consistent in the LETKF. However, the LPF generally shows smaller ensemble spreads than the RMSEs. Since the beginning of the experiments on 1 January, the three LPF experiments showed rapid increases in the ensemble spread (Fig. 6a) and a rapid decrease in the effective particle size (Fig. 6b) within two weeks. After the rapid changes, the ensemble spreads and effective particle sizes were stabilized. The three LPF experiments increased RMSEs until the beginning of March. The LPF with ρh=500 km showed filter divergence in April. The LPF with ρh=400 km and ρh=600 km also showed increasing trends in RMSE until the end of April, while their ensemble spreads and effective particle sizes were stable.

Next, sensitivities to the RTPS parameter (α), resampling frequency (N0), and forgetting factor (τ) are investigated for the LPFGM. The best performing localization scale of LPF was identical to that of the LETKF (Fig. 5). Therefore, we choose the same 600 km localization scale for the LPFGM. Figure 7 compares the time-mean background RMSEs and ensemble spreads for LETKF and LPFGM. In the LPFGM experiments, we investigate six experimental settings for τ=0.0, 0.1, 0.2, 0.3, 0.5, and 1.0. With τ=0.0 (Fig. 7a), the LPFGM shows RMSEs similar to those of LETKF, with best-performing parameters for α and N0. Regulating the resampling frequency (N0) would be needed for LPFGM, because the LPFGM shows filter divergence when resampling is employed at all assimilation steps, excluding the case of α=1.0 (magenta line in Fig. 7a). The LPFGM employs two ensemble updates: the Gaussian mixture and resampling. Owing to the first update by the Gaussian mixture, the LPFGM would not require resampling at all assimilation steps. With τ=0.0, the best-performing relaxation parameter α is approximately 0.6–0.9, excluding the case employing resampling at all assimilation steps.

Increasing τ leads to the LPFGM being less sensitive to the resampling parameter N0 and RTPS parameter α (Fig. 7b–f), implying that the LPFGM is more stable when the LPFGM forgets weights (τ≥0.1). Additionally, the LPFGM requires less inflation (i.e., smaller α) when the forgetting factor τ becomes larger. Owing to the first update by the Gaussian mixture, succeeding weights completely may be unnecessary for the LPFGM. In addition, the method of succeeding weights in the local PF is not trivial, because the weights would move with the dynamical flow. Because the results of the LPFGM experiments without weight succession (τ=1.0) were superior to those with weight succession (τ=0.0), the remainder of this paper focuses on the LPFGM experiments without weight succession (τ=0.0).

The time series of background RMSEs and the effective particle sizes for LPFGM experiments show that the LPFGM with every-time resampling (N0=40) exhibits large RMSEs and the smallest effective particle sizes (Fig. 8, magenta). In contrast, the LPFGM experiments with infrequent resampling (N0=10, 5, and 2) show small RMSEs similar to that of the LETKF and maintain larger effective particle sizes than the LPF (Fig. 8, red, blue, and green lines).

Finally, we compare the spatial patterns of the RMSEs at a fourth model level for the LETKF, LPF, and LPFGM, with best-performing parameter settings (Fig. 9). The LETKF and LPFGM show larger RMSEs in the tropics and polar regions (Fig. 9a and c), possibly because of uncertainties from convective dynamics in the tropics and sparser observations in the polar regions. The LPF shows a larger RMSE than the LETKF globally (Fig. 9b and d). In contrast, slight improvements were observed globally in the LPFGM relative to the LETKF, as indicated by generally warm colors in Fig. 9e, especially around the North Pole.

Here, we compare the LETKF, LPF, and LPFGM with the realistic observing network RAOB. With this spatially inhomogeneous observing network, all LPF experiments showed filter divergence, even with a broad range of the localization and RTPS parameters. Since there are fewer assimilated observations in RAOB than in REG2, the weight collapse would not be the primary cause of the LPF filter divergence. Because the LPF creates posterior particles by linearly combining prior particles, it is preferable for the LPF to have observations within the range of the prior particles. Filter divergence must be prevented by maintaining synchronization between the LPF and the observations. The LPF would require more observations for the synchronization than the LETKF, which was also shown in the authors' initial tests with L96. In addition, the LPFGM was unstable with weight succession (τ=0.0). Therefore, this section focuses on the comparison of the LETKF and LPFGM without weight succession (τ=1.0) only.

First, sensitivities to the RTPS parameter (α) and resampling frequency (N0) are investigated for the LPFGM. Figure 10 compares the time-mean background RMSEs and ensemble spreads for the LETKF and LPFGM. The best-performing horizontal localization scale for the LETKF is 1100 km, and the same localization scale is used for the LPFGM. The LPFGM with every-time resampling (N0=40) shows the largest RMSEs (Fig. 10a, magenta). By contrast, some LPFGM experiments show smaller RMSEs than the LETKF (Fig. 10b). In this experimental setting, the resampling parameter of N0=2.0 shows the most stable performance. With N0=2.0, the LPFGM outperforms the LETKF slightly for α=0.5–0.7 (green line of Fig. 10b).

The time series of the background RMSEs and effective particle sizes show that the LPFGM with every-time resampling (N0=40; Fig. 11a, magenta) and relatively frequent resampling (N0=10; Fig. 11a, red) caused filter divergences. The times when the filters diverged seem to correspond to the times when the effective particle sizes decreased (Fig. 11b, magenta and red). This reduction in effective particle size is a typical sign of filter divergence for PF (Snyder et al., 2008). One may speculate that the assumption of R^t≈Rt in the LPFGM (Sect. 2.3) would be a reason for the filter divergence. However, adopting the appropriate norm (R^t=Rt+HtP^tbHtT) for the LPFGM did not eliminate the filter divergence (not shown). An effective solution for avoiding the filter divergence with every-time resampling is still unknown, but a possible reason is the method of creating a transform matrix using the Monte Carlo method that approximates two ideal conditions (Eq. 19 and a spatially smooth transition). For deeper understanding, using other resampling techniques such as the optimal transport (see Sect. 4.3.2) would be useful. In contrast, the LPFGM experiments with infrequent resampling (N0=5 and 2) show stable RMSEs and maintain a larger effective particle size. These stable LPFGM experiments maintain the amplitude of the ensemble spreads similarly to that of the RMSEs.

We further explore regions where the LPFGM outperforms the LETKF. Figure 12 compares the space-based patterns of the background RMSEs for T (K) at the fourth model level for LETKF and LPFGM. Errors in sparsely recognized regions – such as the South Pacific Ocean, Indian Ocean, and polar regions – tend to be larger (Fig. 12a and b). The LPFGM outperforms the LETKF in such sparsely observed regions (Fig. 12c). The regions showing improvements correspond to the regions where the ensemble spreads and first-guess non-Gaussianity are larger (Fig. 13). Here, we measure the non-Gaussianity by means of Kullback–Leibler divergence (Kullback and Leibler, 1951).

The first update of the LPFGM (the Gaussian mixture) is similar to the LETKF update, as discussed. Therefore, the improvements of the LPFGM with respect to the LETKF would be owing to the resampling step. Previous studies demonstrated that the LPF outperformed the LETKF for non-Gaussian DA with non-Gaussian observation and prior PDFs (Poterjoy and Anderson, 2016; Poterjoy, 2016; Penny and Miyoshi, 2016). Therefore, the improvements of the LPFGM would come from the consideration of non-Gaussian prior PDF in sparsely observed regions.

Finally, we investigate the region where the difference in RMSE between the LETKF and LPFGM is large (120–60∘ W and 70–30∘ S; indicated by black dashed rectangles in Figs. 12 and 13). Figure 14 compares the time series of RMSE, ensemble spread, effective particle size, and Kullback–Leibler divergence averaged over the region for the LETKF and LPFGM. Here, we conducted an additional LPFGM experiment that employs no resampling (i.e., N0=0) to investigate the importance of resampling. Vertical red lines in Fig. 14 represent the cases when the LETKF has large RMSEs greater than 0.8 K. Figure 14a indicates that the LPFGM tends to mitigate large RMSEs, which is in contrast to the LETKF. Since no significant increase is seen in the Kullback–Leibler divergence at the four cases (Fig. 14c), the large RMSEs of LETKF would not be caused by analyses with highly non-Gaussian first-guess ensembles. For the last three cases, the LPFGM has significantly smaller RMSEs than the LETKF. The improvement of the LPFGM would be partially led by larger ensemble spread than the LETKF (Fig. 14b). However, the LPFGM with resampling (N0=2) outperforms the LPFGM without resampling (N0=0), indicating that resampling improves the RMSE. The LETKF and LPFGM exhibit reduced effective particle size at the three later cases (Fig. 14d). Because of the smaller effective particle size, fewer ensembles predict states that are closer to observations, and the LPFGM employed more resampling in the region. In addition to the particle shift, the LPFGM would reduce RMSE further by the resampling.

The LPFGM occasionally resulted in larger RMSE than the LETKF (e.g., beginning of March in Fig. 14a). However, the LPFGM has the potential to outperform the LETKF overall by reducing the RMSE in sparsely observed regions. While operational NWP systems assimilate massive observations, the LPFGM would be useful when spatially and temporally sparse observations are used – e.g., in the twentieth-century reanalysis projects (e.g.,  Compo et al., 2011; Laloyaux et al., 2018) and paleoclimate reconstructions (Acevedo et al., 2017; Okazaki and Yoshimura, 2017).

Finally, we discuss other issues for potential further improvement of the LPF and LPFGM in future studies.