2.1.  

Forecast error reduction, observation impact, and FSO

In the KMA UM 4DVAR system, the 27-hour forecast error ($e$) with respect to the true state ${\mathbf{x}}_t$ which is measured by the total energy norm (Lorenc and Marriott, 2014) is expressed as

where ${\mathbf{x}}^f$ is the 27-h forecast; $M_{\mathit{\text{domain}}}$ is the mass of the atmosphere in the model domain; $N^2$ is the square of the buoyancy frequency; $r^2$ is the square of the earth radius; $g$, $\theta$, $c$, $\rho$, $L$, $C_p$, and $\epsilon$ are the gravity, potential temperature, speed of sound, air density, specific latent heat of condensation, specific heat capacity, and weighting factor on the moisture term, respectively; $u^{\prime }$, $v^{\prime }$, ${\theta }^{\prime }$, $P^{\prime }$, and $q^{\prime }$ denote the forecast errors that correspond to the zonal wind, meridional wind, potential temperature, pressure, and specific humidity, respectively; $\varphi$, $\lambda$, and r denote the latitude, longitude, and earth radius, respectively; and C is a diagonal matrix that denotes the dry total energy norm with $\epsilon$ = 0. The non-zero diagonal components in $\mathbf{C}$ correspond to all grid points from the surface to 150 hPa. Because ${\mathbf{x}}_t$ is unknown, the analysis ${\mathbf{x}}_a$ of the KMA UM 4DVAR system was substituted for ${\mathbf{x}}_t$. The KMA UM 4DVAR has a 6-h assimilation window (T − 3, T + 3) starting from 03, 09, 15, and 21 UTC. The 3-h forecast from T − 3 is used as the analysis at T + 0. The adjoint model to calculate the observation impact is integrated back to T − 3, thus the 27-h forecast error is used instead of 24-h forecast error in this study.

The nonlinear forecast error reduction (FER; $\delta e$) is defined as the difference between the error of the forecast integrated from the analysis (${\mathbf{x}}_a^f$) and the error of the forecast integrated from the background (${\mathbf{x}}_b^f$) (Jung et al., 2013; Kim and Kim, 2014):

The ${\mathbf{x}}_a$ is determined from the optimal linear analysis equation as

where ${\mathbf{x}}_b$ is background, $\mathbf{K}={\left({\mathbf{B}}^{-1}+{\mathbf{H}}^T{\mathbf{R}}^{-1}\mathbf{H}\right)}^{-1}{\mathbf{H}}^T{\mathbf{R}}^{-1}$ denotes the Kalman gain where H is the linear observation operator, y is observation, and d represents the innovation vector. By applying the linearised model (the perturbation forecast (PF) model (M) in the KMA UM 4DVAR system) to Equation (3) and using the relationships ${\mathbf{x}}_a^f\approx \mathbf{M}{\mathbf{x}}_a$ and ${\mathbf{x}}_b^f\approx \mathbf{M}{\mathbf{x}}_b$, $\delta e$ is approximated in the observation space as where ${\mathbf{M}}^T$ is the adjoint of the PF (APF) model. The approximated FER ($\delta e$) in the observation space is called the observation impact.

Then FSO, the gradient of $\delta e$ with respect to $\mathbf{y}$, is represented as

The observation impact corresponding to the ith observation type is expressed using FSO as

In the KMA UM, the APF model ${\mathbf{M}}^T$ is linearised along an average trajectory between two forecast trajectories initialised at the analysis and the background (Lorenc and Marriott, 2014), and the adjoint of the Kalman gain is designed as an iterative linear system which is an algorithm for estimating the minimum of a cost function, similar to the algorithm by Cardinali (2009). Lorenc and Marriott (2014) showed that the linear and nonlinear perturbation growths based on an average trajectory between two forecast trajectories initialised at the analysis and the background are closer compared to those based on either analysis or background forecast trajectory. Furthermore, Lorenc and Marriott (2014) demonstrated that correlations between nonlinear FERs for various runs are higher for the FERs based on the average trajectory than that based on either analysis or background forecast trajectory. Lorenc and Marriott (2014) mentioned that these results are caused by the fact that the PF model of the 4DVAR system of the United Kingdom Met Office (UKMO) and the KMA does not use the Taylor expansion that assumes infinitesimal perturbation, but uses the linearised function of the analysis increment due to the batch of observations that allows finite perturbations.

2.2.  

Forecast sensitivity to error covariance parameters and error covariance impact

In a DA system, the analysis is obtained by combining background and observations considering the associated error covariances $\mathbf{B}$ and $\mathbf{R}$. $\mathbf{B}$ and $\mathbf{R}$ can be adjusted using parametric variables (i.e. proper weighting) in parametric space (Desroziers et al., 2009):

where i, P, $s^b$, and $s_i^o$ are the observation subset, the total number of observation subset, the background error covariance parameter, and the observation error covariance parameter, respectively; $s^b$ and $s_i^o$ should be positive for all i.

Daescu and Todling (2010) derived the forecast sensitivity to error covariance parameters in parametric space. The forecast sensitivity to error covariance parameters has been globally calculated in the Naval Research Laboratory (NRL) atmospheric variational DA system (Daescu and Langland, 2013) and regionally calculated in the Advanced Research version of the Weather Research and Forecasting (WRF)-ARW system (Jung et al., 2013; Kim et al., 2017).

Following Daescu and Todling (2010) and Daescu and Langland (2013), the forecast sensitivity to error covariance parameters is represented as:

where $\frac{\delta e}{\delta s^b}$ is the forecast sensitivity to background error covariance parameter ($s^b$– sensitivity), $\frac{\delta e}{\delta s_i^o}$ is the forecast sensitivity to observation error covariance parameter ($s_i^o$– sensitivity), and $h$ is the nonlinear observation operator. The intrinsic property between $s^b$– sensitivity and $s_i^o$–sensitivity is ${\sum }_{i=1}^P\frac{\delta e}{\delta s_i^o}+\frac{\delta e}{\delta s^b}=0$. The approximated FER in the parametric space (i.e. error covariance impact) is expressed as where $\delta s^b$ and $\delta s_i^o$ are the adjustment parameters of the B and Rs, respectively.

Different from Daescu and Todling (2010) and Daescu and Langland (2013) in which the $\delta e$ and forecast sensitivity to error covariance parameters (i.e. Equation (8a) and (8b)) are calculated based on an analysis trajectory, those in this study are calculated based on an average trajectory between two forecast trajectories initialised at the analysis and the background, as in Equation (4). Although the forecast sensitivity to error covariance parameters is more directly associated with the analysis trajectory, the average trajectory between two forecast trajectories initialised at the analysis and the background is assumed to be a substitute for the analysis trajectory in this study. The average trajectory is used to reduce the computational cost and storage space for data backup in the operational KMA UM 4DVAR system. By using the FSO calculated routinely in the operational KMA UM 4DVAR system, additional adjoint integrations to evaluate the FSO based on the analysis trajectory are not necessary. Because the methodology suggested in this study aims at application to the operational KMA NWP system, the computational cost and storage space for adjoint integration are important issues. In addition, the difference between the analysis trajectory and the average trajectory may not be large for short-term forecasts in the UKMO and KMA UM 4DVAR system as shown in Lorenc and Marriott (2014).

2.3.  

Error covariance adjustment parameters

Using matrix representation, the error covariance impact in Equation (9) can be expressed as

where m denotes the number of realisations of Equation (9) from specific analyses and n denotes the total number of error covariance adjustment parameters (i.e. $\delta s^b$ and $\delta s_i^o$). Note that n is the sum of 1 and the total number of observation subset P in Equation (8b).

If the first matrix on the right-hand side of Equation (10) has a full rank or is over-determined, then the error covariance adjustment parameters (i.e. the vector on the right-hand side of Equation (10)) can be calculated using multiple linear regression. The estimated error covariance adjustment parameters are used to calculate the error covariance parameters as:

Then, Equation (11a) and (11b), along with Equation (7a) and (7b), are used to obtain $\mathbf{B}$ and $\mathbf{R}$.

However, $\delta s^b$ and $\delta s_i^o$ cannot be uniquely determined by Equation (10) because the first matrix on the right-hand side of Equation (10) is rank deficit by the relationship ${\sum }_{i=1}^P\frac{\delta e}{\delta s_i^o}+\frac{\delta e}{\delta s^b}=0$. To calculate $\delta s_i^o$ in Equation (10), an appropriate value needs to be assigned to $\delta s^b$, which converts the first matrix on the right-hand side of Equation (10) to full rank. After assigning the pre-determined $\delta s^b$, Equation (10) becomes

Because the first matrix on the right-hand side of Equation (12) has full rank, the observation error covariance adjustment parameters $\delta s_i^o$ (i.e. the vector on the right-hand side of Equation (12)) can be calculated using multiple linear regression.

2.4.  

Experimental framework

The KMA UM version 7.7 with 4DVAR DA system version 27.2 (Courtier et al., 1994; Clayton, 2004) was used for this study. The model domain covers 769 × 1024 horizontal grid points with a resolution of approximately 25 km; 70 vertical eta-height hybrid layers exist from the surface to 80 km. The eta-height hybrid layers follow terrain near the surface but evolve to constant height surfaces as the layers go up. To make the minimisation more affordable, the KMA UM 4DVAR system uses a simplification operator, which truncates the forecast trajectory to a reduced resolution and dynamically balances the truncated trajectories (Lorenc and Payne, 2007). The model domain of the 4DVAR DA system covers 217 × 288 horizontal grid points with a resolution of approximately 80 km.

The physical parameterisations used in the KMA UM include Edwards–Slingo radiation (Edwards and Slingo, 1996), mixed-phase precipitation (Wilson and Ballard, 1999), the UKMO surface exchange scheme (Essery et al., 2001), the non-local boundary layer (Lock et al., 2000), the new gravity wave drag scheme (Webster et al., 2003), and the mass flux convection scheme (Kershaw and Gregory, 1997; Gregory et al., 1997). The same physical parameterisations are employed in the PF and APF models of the DA system, with the exception that a fixed boundary layer scheme is used instead of the non-local boundary layer scheme and that the simplified moisture physics is used instead of the mixed-phase precipitation and mass flux convection schemes. That is, the DA system uses simple physics parameterisations instead of complex physics parameterisations in the KMA UM to achieve numerical stability. The FSO tool is version 27.2 developed by the UKMO (Lorenc and Marriott, 2014). The assimilated observations include all operational observations from the KMA (Table 1).

The error covariance parameters were estimated for July and August 2012. Then, the adjusted observation error variance using the error covariance parameters was applied to the operational KMA UM 4DVAR system for August 2012 to verify the suitability of the adjusted observation error variance for operational forecasts. The analyses and forecasts using the current operational observation error variances and the analyses and forecasts with the adjusted observation error variances were compared to verify the effect of the newly adjusted observation error variances. The experiment using the operational (adjusted) observation error variances is called the CTL (ADJ_COV) experiment. For both ADJ_COV and CTL experiments, the operational B matrix of the KMA UM 4DVAR system was used and not inflated.

As in most operational DA system, the B and R of the KMA UM 4DVAR system are error variances with only block diagonal and diagonal elements, respectively. Therefore, the background and observation error covariances are the background and observation error variances in this study.

3.1.  

Characteristics of forecast sensitivity to error covariance parameters

Figure 1 shows the time-averaged forecast sensitivity to error covariance parameters and standard deviation in August 2012. The error bar in Fig. 1 was calculated based on the first-order auto regression, as discussed in Kim and Kim (2014). On the vertical axis, the B matrix indicates $s^b$– sensitivity and others correspond to $s_i^o$– sensitivity for each type of observation. The $s^b$– sensitivity is a linear combination of all FSOs (Equation (8a)) and can be projected onto the observation space (Daescu and Todling, 2010; Daescu and Langland, 2013; Jung et al., 2013). The $s^b$– sensitivity is negative, which implies that $\delta s^b$ should be positive and B be inflated to have negative $\delta e$ (i.e. reduction of the forecast error), as indicated in Equation (9). All average $s_i^o$– sensitivity values are positive except for the sensitivity associated with SONDE_q and SURFACE_q, which implies that all Rs except for SONDE_q and SURFACE_q need to be deflated to reduce the forecast error. The $s_i^o$– sensitivity of ATOVS is largest, followed by that of IASI, Geo_AMV, SONDE winds (SONDE_uv), and aircraft winds (AIRCRAFT_uv). A large $s_i^o$– sensitivity indicates that the variation in $\delta e$ may be large when the associated error covariance is adjusted. The $s_i^o$– sensitivities associated with SONDE and surface-specific humidity (i.e. SONDE_q and SURFACE_q) are relatively small, which may be attributable to the dry energy norm used to calculate $\delta e$.

3.2.  

Determining error covariance adjustment parameters

The $\delta s^b$ needs to be determined to solve $\delta s_i^o$ in Equation (12). As shown in Fig. 1, the negative $s^b$– sensitivity implies that B should be inflated to reduce the forecast error, similar to the results of Daescu and Todling (2010) and Jung et al. (2013). To inflate B, $\delta s^b$ should be positive from Equation (7a) and (11a). The magnitude of inflation may depend on the DA system. Then the question is how much inflation is appropriate in the KMA UM 4DVAR system. The B of the hybrid ensemble/4DVAR DA system of the KMA is a linear combination of the statistical error covariance and the ensemble-based error covariance. Clayton et al. (2013) showed that a combination of 100% weighted statistical B and 30% weighted ensemble-based B resulted in the better analysis in a hybrid ensemble/4DVAR DA system at the UKMO which is a similar system with that in the KMA. Although theoretically, the sum of statistical and ensemble-based B needs to be 100%, operational performance of the UM was improved by adding 30% weighted ensemble-based B to the 100% weighted statistical B. Therefore, the 30% inflation of the statistical B may be appropriate in the KMA UM 4DVAR system to lead better forecasts. The validity of specifying $\delta s^b$ as 0.3 is discussed in detail at the end of this subsection.

Once $\delta s^b$ was pre-determined as 0.3, $\delta s_i^o$ in Equation (12) were calculated using multiple linear regression of Chatterjee and Hadi (1986). The $s^b$– sensitivity, $s_i^o$– sensitivity, and error covariance impact $\delta e$ are in the parametric space and should be known to calculate $\delta s_i^o$. Because the error covariance impact cannot be diagnosed without $\delta s^b$ and $\delta s_i^o$, the observation impact in Equation (4) was used as a substitute for the error covariance impact. The similarity between the observation impact and error covariance impact is discussed in detail in the following Section 3.4. The $s^b$– sensitivity (Equation (8a)), $s_i^o$– sensitivity (Equation (8b)), and observation impact (Equation (4)) for July and August 2012 were used to solve Equation (12) in a preconditioned conjugate gradient algorithm of the multiple linear regression method. The $s^b$– sensitivity and $s_i^o$– sensitivity greater than three times of standard deviation from the time-averaged values were considered outliers and not included in solving Equation (12). By solving Equation (12) by the aforementioned method, $\delta s_i^o$ that correspond to $\delta s^b$ are obtained.

Table 2 shows $\delta s_i^o$ obtained by solving Equation (12) based on data for July and August 2012. Because $\delta s_i^o$ are independent variables of the multiple linear regression equation, as shown in Wilks (2006), they can be used to adjust the error covariance of August 2012. Because the $s_i^o$–sensitivity of ATOVS was largest (Fig. 1), the reduction of the R of ATOVS used in the KMA UM 4DVAR system by 72.14% (Table 2) may have large reduction of forecast error. The R of ATOVS for July and August, which was diagnosed by one analysis-forecast iteration using the method suggested by Desroziers et al. (2005), indicates that the current R of ATOVS should be deflated by 62.2%, which supports that the 72.14% reduction in the ATOVS R may be appropriate. ATOVS observations include AMSU-A, AMSU-B, and HIRS (Table 1); the Rs of AMSU-A, AMSU-B, and HIRS used in the KMA UM 4DVAR system are shown in Table 3. In addition, Lupu et al. (2015) suggested the average 60% deflation of the R for 33 IASI channels assimilated in the ECMWF 4DVAR system. The Rs of AMSU-A and IASI used in the KMA UM 4DVAR system are similar to those in the UKMO 4DVAR system, but are 25–50% greater than those of the ECMWF 4DVAR system. Considering the overinflation of the Rs of AMSU-A and IASI in the KMA UM 4DVAR system, the 72.14% reduction seems appropriate for the ATOVS R. The overinflation of observation error variances in most operational centres is associated with the compensation for the incorrect assumption of uncorrelated observation errors.

To check the validity of 30% inflation of the statistical B, dependence of $\delta s_i^o$ on the choice of $\delta s^b$ was investigated. The $\delta s^b$ was varied to 0.1, 0.3, 0.5, and the corresponding $\delta s_i^o$ was calculated. The $\delta s_i^o$ of ATOVS corresponding to 0.1, 0.3, 0.5 $\delta s^b$ are −0.8714, −0.7214, and −0.4714, respectively. As $\delta s^b$ increases, the absolute value of $\delta s_i^o$ for most observation types decrease approaching 0, which results in additional reduction in forecast error (not shown). Figure 2 shows the $\delta s_i^o$ of each observation type corresponding to 0.1, 0.3, 0.5, 0.7, and 0.9 $\delta s^b$. When $\delta s^b$ is 0.1, $\delta s_i^o$ of Geo_AMV, AIRCRAFT_uv, SONDE_q, SURFACE_t are less than −1. When $\delta s^b$ is 0.9, $\delta s_i^o$ of SSMIS is above 1. When $\delta s^b$ is 0.3, 0.5, 0.7, $\delta s_i^o$ of all observation types show physically meaningful values between −1 and 1. Because the $s_i^o$– sensitivity of SURFACE_q and SONDE_q are negative (Fig. 1), the $\delta s_i^o$ of SURFACE_q and SONDE_q need to be positive. Only 0.3 $\delta s^b$ provides positive $\delta s_i^o$ for SURFACE_q and SONDE_q, which is consistent with the sensitivities shown in Fig. 1. Therefore, 0.3 $\delta s^b$ is the most appropriate choice among the several values tested.

3.3.  

Time series of FER

Figure 3a shows the time series of the nonlinear FER, the approximated FER in the observation space (i.e. observation impact), and the approximated FER in the parametric space (i.e. error covariance impact) for July and August 2012. The time series shows generally negative values because the forecast error integrated from the analysis is smaller than the forecast error integrated from the background due to the DA effect (Langland and Baker, 2004; Gelaro et al., 2007; Jung et al., 2013; Lorenc and Marriott, 2014). Because the approximated FERs that correspond to 06, 12, 18 UTC 6, 00 UTC 7, 12 UTC 21, 12 UTC 22, 06 UTC 30 July 2012, 06, 12, 18 UTC 6, 00 UTC 7, and 12 UTC 17 August 2012 were zero or positive due to numerical instability, the approximated FERs for those times were not used for determining $\delta s_i^o$. The numerical instability is originated from problems in the computational stability of the adjoint model’s time-integration scheme and temporal resolution of the adjoint model, which may be associated with the static instability of the linearisation trajectory, small grid length near the poles, and steep orography as mentioned in Joo et al. (2012).

The error covariance impact in July and August 2012 was estimated using the $\delta s^b$ and $\delta s_i^o$ in Table 2. The correlation coefficient between the error covariance impact and the observation impact of July and August 2012 is 0.98 (Fig. 3a), which is quite reasonable because the observation impact is used as a substitute for the error covariance impact to calculate the $\delta s_i^o$. Once the 16 $\delta s_i^o$ of July and August 2012 listed in Table 2 were calculated, the $\delta s_i^o$ were applied to calculate the error covariance impact of August 2012 by the inner product with $s^b$– sensitivity and $s_i^o$– sensitivity of August 2012, and 0.3 $\delta s^b$. The resulting error covariance impact is similar to the observation impact of August 2012 (Fig. 3b). Therefore, the number of predictors used in the multiple linear regression is appropriate for statistically estimating the error covariance impact of August 2012, as indicated by Wilks (2006). In an additional experiment, the error covariance impact calculated by the error covariance adjustment parameters in July 2012 and the forecast sensitivity to error covariance parameters in August 2012 were similar to the observation impact in August 2012, which implies that the number of predictors was appropriate for statistically forecasting the error covariance impact in August 2012 (not shown) although some of the error covariance adjustment parameters in July 2012 are not physically meaningful due to small number of data during one month period for the multiple linear regression.

3.4.  

Cycling experiment and verification

To verify the impact of the adjusted observation error variance using $\delta s_i^o$, the adjusted observation error variance was applied to the operational KMA UM 4DVAR system for NWP in August 2012. Because the $s_i^o$– sensitivity of ATOVS is largest (Fig. 1), the ATOVS R was adjusted by applying the $\delta s_i^o$ in Table 2 to Equation (11b). Then, 24-h forecasts were generated from the updated analyses ($\mathbf{x}{\left(\stackrel{⌢}{\sigma }\right)}_a$) using the deflated ATOVS R in the cycling DA system for August 2012 (ADJ_COV). These forecasts were compared with the operational (i.e. control) forecasts generated from the analyses ($\mathbf{x}{(\sigma )}_a$) using the operational ATOVS R in the cycling DA system (CTL) for the same period. For both ADJ_COV and CTL experiments, the operational B matrix of the KMA UM 4DVAR system was used and not inflated. Thus, the difference between ADJ_COV and CTL is caused by the reduced R of ATOVS. The green shading in Fig. 3b represents the additional error covariance impact by applying the $\delta s_i^o$ of ATOVS in ADJ_COV compared to CTL, which suggests that additional approximated FER is expected when using a new analysis generated by the adjusted R of ATOVS for forecasts. The approximated FERs corresponding to 06, 12, 18 UTC 5, 00 UTC 6, 12 UTC 17, and 12 UTC 21 August 2012 were zero or positive due to numerical instability (Fig. 3b).

The updated analysis began at 03 UTC 24 July 2012 for spin-up. The residual within the assimilation window (i.e. O-A; the difference between observations and analysis trajectory within the assimilation window) of CTL and ADJ_COV were compared for the corresponding time range (i.e. 0–5 h from T–3 that is 03 UTC in the KMA UM) (Fig. 4). Similarly, the residual in the 24–29 h forecast range (i.e. O-F: the difference between observations and forecast trajectory) of CTL and ADJ_COV were compared (Fig. 4). The O-A (O-F) was verified within a 5-h period because the observation window of the KMA UM 4DVAR is from the T–3 (forecast time) to 5 h after the T–3 (forecast time). The observations used for the verification were quality-controlled over the globe and assimilated in the KMA UM 4DVAR system.

The ratio between the time-integrated O-A of ADJ_COV and that of CTL in August 2012 was calculated as

where ${||·||}_2$ denotes the l₂-norm and 124 (including 6 analysis times that showed the numerical instability in the approximated FER calculation) denotes the number of analysis times of August 2012. A ratio smaller (greater) than 1 implies that the O-A of ADJ_COV is smaller (greater) than that of CTL. That is, a ratio smaller (greater) than 1 implies that the analysis or short-term forecasts (i.e. 0–5 h) from the analysis of ADJ_COV is more (less) similar to the observations compared with the analysis of CTL. Figure 5 shows the ratio in Equation (13) for August 2012, stratified by each observation type and variable. Compared with the CTL analysis, the ADJ_COV analysis was closer to the observations for all data types and variables, but especially ATOVS (AMSU-A, AMSU-B, and HIRS) observations. The time-integrated O-A of ADJ_COV was reduced by 10.62% (3.83% when excluded ATOVS) compared with that of CTL.

The ratio between the time-integrated O-F of ADJ_COV and that of CTL in August 2012 is calculated as

where $\mathbf{x}{(·)}^{fa}$ denotes the 24–29 h forecast integrated from the analysis in August 2012. To minimise the interpolation error in the observation operator developed on the grid-based KMA UM, the forecast wind was decomposed into latitudinal and longitudinal wind. Figure 6 shows the ratio between the time-integrated O-F of ADJ_COV and that of CTL in August 2012, stratified by each observation type and variable. For other observation types and variables, with the exception of PRFL_v, MTSAT_v, MSG_v, and GPSRO, the O-F was reduced in ADJ_COV compared to CTL. By adjusting the R of ATOVS, the O-F of ADJ_COV is reduced compared with the satellite observations, SONDE observations (TEMP, PRFL, and PILOT), and surface observations (SYNOP, METAR, BUOY, and TCBOGUS). The reduction of the O-F of ADJ_COV is the greatest for TCBOGUS, which implies that the forecast of ADJ_COV is much closer to the synthetic observations near the centres of tropical cyclones than that of CTL. Figure 7 is a time series of the average ratio between the O-F of ADJ_COV and that of CTL for all observation types during August 2012. With the exception of 20 analysis times, the O-F for all other analysis times (84%) was reduced for ADJ_COV. The time-integrated O-F was reduced by 5.4% for ADJ_COV compared with CTL.

Vertical profiles of the root mean square (RMS) of the average O-F of ADJ_COV and CTL verified by SONDE observations are shown in Fig. 8. The O-F of ADJ_COV verified by SONDE observational variables (i.e. zonal wind, meridional wind, temperature, and specific humidity) were smaller than that of CTL. The differences between the O-F of ADJ_COV and CTL increased from the surface towards the mid-troposphere but decreased near the upper troposphere. The small differences in the O-F of ADJ_COV and CTL in the upper troposphere may reflect the small number of reliable SONDE observations in the upper troposphere. Therefore, the O-F of ADJ_COV and CTL should be verified with another reliable observation type in the upper troposphere.

Figure 9 shows the RMS of the average O-F of ADJ_COV and CTL compared with AMSU-A observations for the Meteorological Operational Satellite-A (Metop-A) and the National Oceanic and Atmospheric Administration (NOAA) 19 satellite. The reduction rate for ADJ_COV is calculated as

The channel numbers 3, 7 (8), 15 for Metop-A (NOAA 19) were excluded in Fig. 9 because they are not used in the KMA UM. The O-F of ADJ_COV was smaller than that of CTL for most channels, with the exception of channel 13 of Metop-A AMSU-A. An average of 2.34% (3.53%) of the O-F verified by Metop-A (NOAA 19) AMSU-A was reduced for ADJ_COV compared with CTL (Fig. 9). The O-F for channels 11–14 (with the exception of channel 13 of Metop-A AMSU-A), which are sensitive in the upper troposphere, decreased for both satellites. In addition, the O-F of ADJ_COV verified by the hyperspectral sensor of IASI was reduced compared with that of CTL, showing 21.7% reduction rate for ADJ_COV (Fig. 10). Ozone and window channels of IASI were not used. In addition, the O-F for ADJ_COV in the upper troposphere was reduced similarly with IASI when verified by the AIRS sensor (not shown).