In this study, we implement the IAU (Bloom et al., 1996) based on existing ocean data assimilation systems (Balmaseda et al., 2015; Martin et al., 2015). The procedure for one assimilation cycle is as follows: (i) conduct model integration up to the middle of an assimilation window; (ii) assimilate observations within the window and save the analysis increments in temperature, salinity, and horizontal velocity; and (iii) conduct model integration over the assimilation window adding the increments equally distributed to each time step. The IAU reduces noise from high-frequency gravity waves associated with initial shocks, but the computational cost of the model integration is 1.5 times that of the standard method in which the analyses performed at the beginning of the window are used for the model initial conditions. Following Miyazawa et al. (2012), all analysis variables (SSH, temperature, salinity, and horizontal velocities) are used for initial conditions in the standard method. Although there are various IAU methods, the SSH increments are not included in most of the existing ocean data assimilation systems (Table 2 of Martin et al., 2015), mainly because the SSH increments tend to cause initial shocks. Even without the SSH increments, the SSH would be modified properly in response to the temperature and salinity increments. Therefore, we adopt the analysis increments of temperature, salinity, and horizontal velocity except for SSH.

Following previous studies (Kunii and Miyoshi, 2012; Penny et al., 2013; Torn et al., 2006), atmospheric and lateral boundary conditions are artificially perturbed for each ensemble member. Atmospheric forcing of the ith ensemble member at a time t, w(i)(t), is given by 1wit=wt+αwt+δti-1n∑i=1nwt+δti, where wt is atmospheric forcing at a time t, α(=0.2) is an arbitrary constant, wt+δti is atmospheric forcing at the same time as wt but in a different year, and n (=100) is the ensemble size. Here, the year in wt+δti is changed every month. As is clear from Eq. (1), the ensemble mean of the atmospheric forcing wit is equivalent to wt.

Lateral boundary conditions for each ensemble member are obtained from a monthly mean global ocean reanalysis dataset for different years. Namely, the ensemble mean of the lateral boundary condition corresponds to a monthly climatology. These perturbed atmospheric and lateral boundary conditions play a role equivalent to additive inflation (Houtekamer and Zhang, 2016).

Three covariance inflation methods (MULT, RTPP, and RTPS) are adopted in this study. MULT inflates forecast error covariance Pf by a factor of ρ(>1): 2Pinff=ρPorigf, where the subscripts inf and orig denote inflated and original (i.e., before inflation), respectively. Both RTPP and RTPS restore the analysis ensemble perturbation towards the forecast ensemble perturbations maintaining the analysis ensemble mean, as represented by 3Xinfa=αRTPPXf+1-αRTPPXorigaand4Xinfa(i)=αRTPSσf(i)+(1-αRTPS)σoriga(i)σoriga(i)Xoriga(i). Here, X[=(x1-x‾,…,xi-x‾,…,xn-x‾)] is the ensemble perturbation matrix whose ith column consists of the perturbations of the ith ensemble member, where xi and x‾ are the state vector of the ith ensemble member and ensemble mean; the superscripts a and f denote analysis and forecast; and αRTPP and αRTPS are the relaxation parameters in the RTPP and RTPS, respectively. σ(i) is the ensemble spread of the ith variable of state vector x, as represented by 5σ(i)=n-1-1X(i)X(i)T. In the RTPP and RTPS, the relaxation parameters are generally defined between 0 and 1, where αRTPP=0 and αRTPS=0 correspond to no inflation, and αRTPP=1 and αRTPS=1 correspond to the inflated analysis ensemble spread being equivalent to the forecast ensemble spread. RTPP and RTPS are thought to have side effects in maintaining the dynamic balance (Houtekamer and Zhang, 2016; Ying and Zhang, 2015).

The σ-coordinate regional ocean model used in this study is based on version 1.0 of the Stony Brook Parallel Ocean Model (sbPOM; Jordi and Wang, 2012) and constructed for the northwestern Pacific region (15–50∘ N, 117–180∘ E) with a horizontal resolution of 0.25∘ and 50 σ-layers (Table 2). The bottom topography is derived from ETOPO1, a 1 arcmin global relief model of Earth's surface (Amante and Eakins, 2009). We apply a Gaussian filter with e-folding scales of 200 km to the topography to reduce the pressure gradient errors in σ-coordinate models caused by steep bottom slopes (Mellor et al., 1994) and fulfill the condition Hi+1-Hi/Hi+1+Hi<0.2, where Hi and Hi+1 are bottom topographies at adjacent grids. Monthly (seasonal) temperature and salinity climatologies from the World Ocean Atlas 2018 (WOA18; Locarnini et al., 2019; Zweng et al., 2019) with a horizontal resolution of 1∘ and 57 (102) layers are used for an initial condition over depths shallower (deeper) than 1500 m. Lateral boundary conditions for temperature, salinity, and horizontal velocity are obtained from version 3.7.2 of Simple Ocean Data Assimilation (SODA; Carton et al., 2018) with a horizontal resolution of 0.5∘ and 50 layers. Here, to satisfy volume conservation, flow relaxation (Guo et al., 2003) is applied to the horizontal velocity at the lateral boundary. The Japanese 55-year Reanalysis (JRA-55; Kobayashi et al., 2015) with horizontal and temporal resolutions of 1.25∘ and 6 h, respectively, is adopted for the atmospheric boundary conditions, including air temperature and specific humidity at 2 m, wind velocity at 10 m, shortwave radiation, total cloud fraction, sea level pressure, and precipitation. We also use river discharge from the Japan Aerospace Exploration Agency (JAXA)'s land surface and river simulation system, Today's Earth Global (TE-Global; https://www.eorc.jaxa.jp/water/, last access: 11 November 2022), with horizontal and temporal resolutions of 0.25∘ and 3 h, respectively. The atmospheric and lateral boundary conditions are perturbed as described in Sect. 2.2, except for the rainfall and river discharge.

The model is driven by wind stresses as well as heat and freshwater fluxes using bulk formulae in which bulk coefficients are estimated from the Coupled Ocean–Atmosphere Response Experiment (COARE), version 3.5, bulk algorithm (Brodeau et al., 2017; Edson et al., 2013). The horizontal diffusivity coefficient is calculated by a Smagorinsky type formulation with a coefficient of 0.1 (Smagorinsky et al., 1965) and is assumed to be one-fifth of the horizontal viscosity coefficient. The vertical diffusivity coefficient is estimated by the Level 2.5 version of Nakanishi and Niino (2009). The model is spun up from 1 January 2011 to 6 July 2015 using the initial condition with no motion. During the spin-up period, simulated temperatures and salinity are nudged towards the monthly and seasonal climatologies from WOA18 with a 90 d timescale to damp northward overshooting of the Kuroshio. We have confirmed that the perturbed boundary conditions substantially increase the ensemble spread even with the nudging (not shown).

We implement the three-dimensional local ensemble transform Kalman filter (3D-LETKF; Hunt et al., 2007; Miyoshi and Yamane, 2007) with 100 ensemble members to assimilate the following observations on a 1 d assimilation interval (Table 3): satellite SSTs from Himawari-8 and GCOM-W, SSS from the SMOS (http://www.esa.int/Applications/Observing_the_Earth/SMOS, last access: 11 November 2022) and Soil Moisture Active Passive (SMAP) version 4.3 (Meissner et al., 2018), SSH consisting of satellite SSH anomalies from the Copernicus Marine Environment Monitoring Service (CMEMS; http://marine.copernicus.eu/, last access: 11 November 2022) and MDOT estimated from simulated SSH averaged in 2012–2014, and in situ temperature and salinity from the Global Temperature and Salinity Profile Programme (GTSPP; Sun et al., 2010) and Advanced automatic QC (AQC) Argo Data version 1.2a (https://www.jamstec.go.jp/argo_research/dataset/aqc/index_dataset.html, last access: 11 November 2022). We exclude satellite SSS within 100 km of the coasts, SSH for bottom topography shallower than 200 m, in situ temperature and salinity duplicated between the GTSPP and AQC Argo datasets, and observations without the best quality flags or whose differences from the forecasts are larger than the values in the gross error check in Table 3. Following Miyazawa et al. (2012) and Penny et al. (2013), the localization scales based on a Gaussian function are chosen to be 300 km and 100 m in the horizontal and vertical directions, respectively. An observational error covariance matrix is assumed to be diagonal using the observation errors in Table 3.

We conduct sensitivity experiments combining the IAU and covariance inflation methods (no inflation (NO INFL), RTPP, RTPS, and MULT) to investigate their impacts on the geostrophic balance and accuracy. We set the relaxation parameters in the RTPP and RTPS experiments to αRTPP=αRTPS=0.9 without the IAU and to αRTPP=αRTPS=0.5,0.6,…,1.2 with the IAU, and we set the inflation parameter to ρ=1.052 (inflating the forecast ensemble spread by 5 %) in the MULT experiments, regardless of the application of the IAU. In this study, we do not explore all values of the relaxation and inflation parameters because of the limitations of computational resources. Hereafter, we refer to the RTPP experiments implemented with and without the IAU as the RTPP+IAU and RTPP experiments, respectively, and we refer to the RTPP+IAU experiment with a relaxation parameter of 0.5 as the RTPP05+IAU experiment. Kotsuki et al. (2017) indicated that the RTPP and RTPS do not consider the model error explicitly and that the optimal relaxation parameter may be larger than 1.0. Therefore, we perform experiments with a relaxation parameter value >1. To clarify the effects of the IAU and covariance inflation methods, the NO INFL experiment is defined as a control experiment in this study (see Eqs. 9 and 10).

We integrate the LETKF-based ocean data assimilation system from 7 July 2015 at the start date of the Himawari-8 observations to 31 December 2016, applying the SSS nudging with 90 d timescale to damp a surface freshening drift, as in the model spin-up described in Sect. 3.1. Furthermore, we conduct 11 d ensemble forecast experiments initialized on the first day of each month in 2016 by the forecasts from the NO INFL, NO INFL+IAU, RTPP09, RTPP09+IAU, RTPS09, and RTPS09+IAU experiments, with the SSS nudging applied with a 90 d timescale. We estimate Δ NBE from the ensemble analysis increments on days 1 and 16 of each month, the RMSDs from the daily averaged ensemble mean analyses and forecasts, and the ensemble spread from the daily mean ensemble analyses. As described in Sect. 2.4.2, the statistical analyses are applied to IRs of area-averaged Δ NBE and analysis RMSDs in all analysis experiments. The results of the RTPP11+IAU and RTPP12+IAU experiments are not shown because numerical instability developed.

We first compare the geostrophic balance for the various sensitivity experiments using spatiotemporally averaged Δ NBE over the whole system domain for 2016 (Fig. 1). The NO INFL+IAU experiment has the best geostrophic balance with significant improvement relative to the NO INFL experiment; thus, the IAU plays a role in enhancing the balance, probably because the IAU reduces noise of the high-frequency gravity waves associated with initial shocks. This result is consistent with Yan et al. (2014), who demonstrated that the IAU reduces the spurious oscillation of vertical velocity in twin experiments using a relatively idealized EnKF-based ocean data assimilation system. In contrast, as the RTPP09 and RTPS09 experiments show significantly larger Δ NBE than the NO INFL experiment, RTPP and RTPS contribute to breaking the balance. The MULT+IAU and MULT experiments give such a large Δ NBE (2.11×10-10 and 5.22×10-10 s-2, respectively) that the MULT breaks the balance considerably even if the IAU is applied.

The RTPP+IAU and RTPS+IAU experiments provide significant improvement when the relaxation parameters are αRTPP≤0.9 and αRTPS≤0.8, respectively. The RTPP11+IAU experiment becomes numerically unstable in December 2015, and the RTPS11+IAU experiment also significantly degrades the balance; thus, relaxation parameters larger than 1.0 do not appear to be appropriate for the EnKF-based ocean data assimilation system. The combinations of the IAU and RTPP/RTPS, in which the relaxation parameters are set to αRTPP≤0.9 and αRTPS≤0.8, appear to maintain geostrophic balance, likely because the IAU counteracts the RTPP/RTPS by improving the balance.

To investigate spatial characteristics of the geostrophic balance, Δ NBE is temporally averaged over the whole year 2016 (Fig. 2). Here, the RTPP09+IAU and RTPS11+IAU experiments are shown from the RTPP+IAU and RTPS+IAU experiments because they have the best accuracy, as seen in Sect. 4.2. The NO INFL, RTPP09, and RTPS09 experiments produce less-balanced fields in the midlatitude region, especially around the KE (Fig. 2a, c, e). In the RTPS11+IAU experiment, the balance is also lost in higher-latitude regions (Fig. 2f). In the MULT and MULT+IAU experiments, there are almost no balanced regions with Δ NBE smaller than 1.5×10-10 s-2 (not shown). The NO INFL+IAU and RTPP09+IAU experiments show substantial improvement around the KE region, although a relatively large Δ NBE remains along the KE in the RTPP09+IAU experiment (Fig. 2b, d). Thus, in general, although the balance in the analysis field is not maintained around the KE region, this imbalance is substantially reduced in the NO INFL+IAU and RTPP09+IAU experiments.