The field of view is susceptible to deformation as the scan angle increases when the satellite sensor performs a cross-track scanning to both sides of scan lines, which leads to unavoidable observation biases relative to the nadir FOV. The optical pathlength is longer as the scan angle increases, also leading to scan angle biases. In addition, although the fast radiative transfer models are become increasingly sophisticated and accurate, they can not only simulate the brightness temperature at the nadir, but also simulate the brightness temperature at other scan angles. However, atmospheric inhomogeneity increases with the increase of scan angle. If the model cannot fully simulate atmospheric inhomogeneity, it may also lead to bias in the simulated brightness temperature that depend on the scan angle (Zou et al., 2011). For each channel, the global or regional mean bias of every scan position (angle) relative to the central nadir position is calculated as: 1Sθ=R‾(θ)-R‾(θ=0) Where R‾ refers to the mean radiance at different scan angles, θ represents the scan position (scan angle) and S denotes the scan bias. A significant improvement made by Harris and Kelly (2001) to this scheme is incorporating the dependency of biases on latitude band. The Earth is divided into 18 latitude bands with 10° interval and the correction coefficient is computed respectively. After a long period of lager sample size statistics, it has been shown that the scan biases of HIRAS-II do not exhibit a pronounced dependence on latitude band as found in microwave instruments. Therefore, the influence of latitude is not considered for the scan bias correction in this study.

The O-B biases generally exhibit variations associated with the properties of air-mass (and the surface) due to the inaccuracies in the radiance calibration of satellite instruments, the error from the fast radiative transfer models and NWP systems. The air-mass bias correction scheme primarily involves establishing a multivariate linear regression equation that relates the air-mass predictors xi(i=1,2,…,n) to the air-mass biases, the air-mass biases rj for each channel j can be calculated: 2rj=∑i=1najixi+cj Where aji and cj are calculated by least square fitting using a large number of samples. 3aji=∑k=1n〈Dj,xk〉X,Xki-1 Here, 〈…,…〉 represents covariance, k is the sample number, X is the vector of xk, and Dj denotes the O-B bias in channel j.

The success of air-mass bias correction depends on the selection of air-mass predictors. The commonly used air-mass predictors in the ECMWF and NCEP-GSI assimilation systems are shown in Table 1. Here, p represents the air-mass predictors in the ECMWF assimilation system and p′ represents the air-mass predictors in GSI. The predictor p1, p2 , p5 and p6 reflect the mode background-errors at different layers and various dependencies within the forward model. The predictor p3 represents the systematic error of near-surface channels. In addition, it can compensate somewhat for the different emissivity characteristics of different surface types for these channels (Harris and Kelly, 2001). The water vapor is one of the main input components in the fast radiative transfer mode, the predictor p4 can reflect the model simulated error to some extent. The predictors p8–p10 are employed to correct residual biases after scan bias correction (Dee and Uppala, 2009). The Parameter thickness is calculated as 4Predthickness=kth×∑iN-1tv(i)×ln⁡P(i)P(i+1) Here, kth=gasconstant/gravity(gasconstant= 287.0 J K⁻¹ kg⁻¹, gravity = 9.81 N kg⁻¹), N is atmosphere levels, P is atmospheric pressure, tv is the parameter characterizing atmospheric temperature and humidity and calculated as 5tv(i)=Ti+T(i+1)2×1.0+0.608×qi+q(i+1)2 where, T and q represent RTM level temperatures and moistures, respectively.

In the NECP-GSI assimilation system, the predictor p0′ represents a global constant offset. The predictor p1′ is a function of the satellite scan angle θ and is primarily used to correct residual scan biases. The predictor p2′ is only used for the correction of clear-sky microwave instrument radiance over the ocean to eliminate residual cloud interference. For non-microwave instruments, the value of the predictor is set to 0. p3′ and p4′ is the predictor of “temperature lapse rate”, τ and T represents the vertical variation rate of transmittance and temperature, respectively. The predictor p4′ represents the convolution of τ and T, and p3′ is the square of the former. The predictor p3′ and p4′ reflect instrument and RTM errors. When the frequency of channel is shifted or the spectral settings in the RTM are inaccurate, the calculated transmittance profile will move up/down in the atmosphere (if the atmosphere is not isothermal). If the transmittance is moved up slightly, the weight function of the channel is shifted upwards and the brightness temperature should be decreased if the temperature decreases with height. Conversely, the brightness temperature will increase in the case of a temperature inversion. (Zhu et al., 2014).

Harris and Kelly (2001) as well as Liu et al. (2007) used four predictors (model surface skin temperature, model total column water vapor, thickness of 1000–300 hPa and thickness of 200–50 hPa) as the optimal combination to the TOVS's air-mass bias correction. Auligné et al. (2007) employed a predictor combination of thickness of 1000–300 hPa, thickness of 200–50 hPa, thickness of 50–5 hPa and thickness of 10–1 hPa to correct the air-mass biases in ATOVS. The predictor combinations used in the above-mentioned studies are primarily designed for microwave instruments. The predictors used in the ECMWF for all infrared hyperspectral instruments operating on polar-orbiting platforms are summarized in Table 2 (Auligné et al., 2007; Collard and McNally, 2009; Eresmaa et al., 2017). In order to evaluate the optimal combination for FY-3E/HIRAS-II, several typical channels (737.5, 900, 1040, 1279.375, 1476.25 and 1809.375 cm⁻¹) are taken as examples in our study. The importance of predictors p1–p6 for FY-3E/HIRAS-II is evaluated based on the diagnostic scheme proposed by Auligné et al. (2007). Since we use a two-step bias correction scheme, angle-related predictors are not involved in the assessment. This diagnostic scheme evaluates the bias correction effect based on the ability of each predictor to reduce the root mean square error of radiance biases. The results are shown in Fig. 1. Figure 1a and b show the spectral positions and the weighting functions (WF) of each selected channel. The wavenumber of HIRAS-II channel No. 141, 401, 626, 1008, 1323 and 1855 are 737.5, 900, 1040, 1279.375, 1476.25 and 1809.375 cm⁻¹, respectively. The peak heights of weighting function are 535.2, 1070.9, 29.1, 852.8, 300 and 500.2 hPa in sequence. The six channels correspond to the long-wave CO₂ absorption band, the long-wave window channel, the O₃ absorption band, and the water vapor absorption bands at three different heights in turn. Figure 1c indicates the importance assessment of different predictors in different channels, where the horizontal axis indicates the channel No. and the vertical axis is the diagnostic coefficients. A higher coefficient indicates a stronger correlation. The different colored bars represent different predictors. It can be seen from Fig. 1c that the O₃ channel 626 and the water vapor channel 1323 with higher height of weight functions have a higher correlation in predictors thickness of 10–1 and 50–5 hPa. However, other four channels commonly used for data assimilation systems are strongly correlated with the model background surface skin temperature, model total column water vapor, thickness of 1000–300 hPa, and thickness of 200–50 hPa. Data assimilation systems generally do not assimilate strong O₃ absorption channels at present and the water vapor content in upper atmosphere is scarce, so a predictor combination including model surface skin temperature, model total column water vapor, thickness of 1000–300 hPa and thickness of 200–50 hPa is selected to correct the air-mass biases for HIRAS-II in this research. Furthermore, to assess the effectiveness of our bias correction scheme a comparison is made in this study between the correction results of the NCEP-GSI predictor combination (p0′–p4′) and the proposed method.

As we discussed earlier, model surface skin temperature, model total column water vapor, thickness of 1000–300 hPa and thickness of 200–50 hPa are selected as predictors for bias correction. The multiple linear regression model requires a linear relationship between continuous independent variables and dependent variables. Therefore, we examine the correlation between HIRAS-II O-B bias and predictors p1–p6 based on samples from 1 to 31 January 2023. Figure 2a–f show the distribution of a typical HIRAS-II channel 1855 (1809.375 cm⁻¹, the peak height of weight function is 500 hPa) O-B bias with predictors p1–p6, respectively. In order to obtain a correct analysis of the relationship between the O-B bias and predictors p1–p6, the O-B biases eliminate the influence of scan bias. The horizontal coordinate of each subgraph is the value of each predictor, the vertical coordinate is the O-B bias and the color represents the number of samples. The black solid line in figure shows the first-order polynomial fit of the O-B bias and predictors, the correlation coefficients are given in the subtitles of each figure. From Fig. 2a–f, there is a certain linear correlation between the O-B bias and predictors p1–p4, with the highest correlation coefficient of 0.49, and there is no obvious linear relationship between the predictors p5–p6.

The variation of mean biases with the scan angles before and after scan-angle correction for the six typical channels is illustrated in Fig. 3. The dot-dash line represents the bias distribution before correction and the dashed and solid lines represent the bias distribution after correction for scan angle and air-mass, respectively. The vertical axis is the mean O-B biases and the horizontal axis is the positions of the field of regards (FORs) for each scan line (i.e., the scan angle) with the nadir between FOR14 and FOR15. It is can be seen from Fig. 3 that the biases change with the scan angles for all channels before correction. The bias increases as the scan angle increases, especially for the lower tropospheric water vapor channel No. 1008 with value up to 1.5 K caused by the larger scan angle with respect to nadir. Additionally, there is an asymmetrical bias distribution on both sides of the nadir in all channels, particularly for the lower height channels 141 and 1008. This is primarily caused by the non-90° inclination angle of the polar orbit satellite, leading to inconsistent latitude of the field of view on both sides of the scan line. The phenomenon resulted in the HIRAS-II observation being higher on one side and a lower on the other side, which in turn causes an asymmetric distribution of O-B bias. After the scan-angle bias correction, the mentioned biases from the limb FOR relative to nadir and the asymmetrical biases on both sides have been eliminated and the mean bias of each scan position is basically consistent. However, there are still some residual biases (dotted line) in the properties of air mass caused by inaccurate simulation (some channels reaching 0.5–1 K). It can be clearly seen from the black solid line in Fig. 3 that the mean biases of all scan positions in all channels approach 0 K after the air-mass correction.

Based on the O-B biases of each HIRAS-II channel after scan-angle bias correction from 1 to 14 January 2023 and corresponding air-mass predictors (model surface skin temperature, model total column water vapor, thickness of 1000–300 hPa and thickness of 200–50 hPa) data, the air-mass correction coefficients are fitted using Eq. (3). Then the coefficients are applied for air-mass bias correction (referred to as EXP-2). Figure 4a to f and g to l show the scatterplots of observed brightness temperature (O) and background simulated brightness temperature (B) for the six representative channels before and after the air-mass bias correction. The dashed line represents the y=x contour and the color shade represent the density of the radiance data number. The values in each subplot are the mean value and standard deviation of O-B. From Fig. 4a, b, and d, it is evident that significant negative biases are exhibited in channels 141, 401, and 1008 (with a lower height of weight function) when the scene temperature is high before the EXP-2 correction, while water vapor channels with a higher height of weight function (Fig. 4e–f) show a relatively warm bias compared to the background simulation (with mean biases ranging from 0.2 to 0.4 K). The scatter of these channels are all concentrated and evenly distributed near the y=x contour after air-mass correction, with mean bias close to 0 K and a significantly reduced standard deviation.

The bias correction scheme proposed in this paper has been compared with the air-mass bias correction scheme of NCEP-GSI to further evaluate its correction effectiveness. The air-mass predictors used in GSI for infrared instruments are p0′, p1′, p3′, and p4′ listed in Table 1 (hereafter referred to as EXP-GSI). Figure 5 illustrates the histogram of O-B biases before and after correction using two different schemes. The x axis corresponds to the O-B bias and y axis is the probability density function (PDF) of O-B. The black curves in the figure represents the observation residuals O-B before correction and the red and blue curves are the O-B residuals after correction using the EXP-2 and EXP-GSI schemes, respectively. The EXP-GSI scheme still displays notable positive biases in most channels after correction, especially in the channels with a height of weight function below 500 hPa. Considering that ninety percent of atmospheric moisture is confined below 500 hPa, the omission of water vapor as a predictor in the EXP-GSI scheme could potentially explain the suboptimal correction results. In contrast, the biases in all channels of the EXP-2 scheme are distributed close to a Gaussian distribution centered at zero, with the most significant correction effect in channel 1008. However, the corrected bias distribution of window channel 401 still exhibits a long tail on the left side, which may be attributed to the incomplete removal of cloud contamination scene during quality control.

In order to examine whether the distribution of O-B biases after correction satisfies a normal distribution with a mean of 0, Table 4 presents the values before and after correction for these two schemes. Mean represents the mean value, STD represents the standard deviation, Kurtosis value indicates the steepness of the sample distribution and Skewness is the asymmetry of the sample distribution. Kurtosis of 3 and skewness of 0 indicate a normal distribution. It can be seen that the kurtosis and skewness values in all channels after EXP-GSI correction have not changed significantly, while both the observation residuals and standard deviations in all channels show a significant reduction after correction by EXP-2. The O-B bias of all other channels expect for channel 141 with higher kurtosis 5.4 are close to a normal distribution with kurtosis value 3 and skewness value 0. This indicates a significant improvement in the correction effect for EXP-2.

It is important to verify the potential impact of different BC methods on HIRAS-II radiance data assimilation. The analyzed fields will deviate from the true state if the model first guess or observation contain systematic errors. This study uses WRFDA V4.4 to validate the effects of different bias correction schemes on NWP. The WRFDA model has been developed by the National Center for Atmospheric Research (NCAR). It provides different methods of data assimilation and can assimilate a wide range of observations. The WRF three-dimensional variational data assimilation system (WRF-3DVar) minimizes the so-called variational cost function J(x) as Eq. (6) (Barker et al., 2004). 6Jx=12x-xbTB-1x-xb+12(y-Hx)TR-1(y-Hx) Where x is the atmospheric state vectors, xb is the first guess (background), B is the background error covariance, y is the observation vector, R is the observation error covariance and H stands for the observation operator by using RTTOV v12.3.

A new HIRAS-II data assimilation module is created in WRFDA by adding the reading and QC (The QC steps as described in Sect. 3, but the MR method (Eyre and Menzel, 1989) was used for cloud detection) interfaces. A total of 485 channels are selected for data assimilation, with specific positions in the spectrum shown as black dots in Fig. 1a. Four parallel experiments are designed to assess the effects of different bias correction on analysis. These experiments differed in predictors and their detailed configurations are shown in Table 3. The initial and boundary conditions for all experiments are provided by the NECP GFS 6 h forecast data. The simulation domain of all experiments is approximately from 0 to 60° N and from 70 to 150° E, with a 9 km grid spacing on 889 × 828 horizontal grids and 60 vertical levels up to 1 hPa. The bias correction coefficients used for experiments are fitted by the predictors and HIRAS-II O-B biases obtained from 17 to 31 July 2023, and the fitted coefficients are used in 1-month assimilation experiments from 1 to 31 August 2023.

This study assesses the effect of different bias correction schemes on NWP based on 1-month assimilation experiments from 1 to 31 August 2023. The mean O-B bias and mean O-A (observation minus analysis) bias at 485 assimilated channels during the 1-month experiments period after QC are plotted in Fig. 6a and b, respectively. The horizontal coordinate is the ordinal numbers of the assimilation channels, arranged from longwave to shortwave. The vertical coordinate is the mean bias. The black curves, blue curves and red curves in figures represent the results of experiment NO BC, experiment EXP-GSI and experiment EXP-2, respectively. As shown in Fig. 6a, the majority of the HIRAS channels without bias correction exhibit significant negative biases up to -2.5 K. The O₃ absorption bands from channel NO. 161 to NO. 182 with an average bias of 5 K due to the fixed default O₃ profiles used in RTTOV. Although the EXP-GSI scheme can reduce the absolute deviation of most channels, it still shows a certain positive deviation. It is consistent with the results shown in Fig. 5. After EXP-2 bias correction, the average biases of almost all channels are around 0 and only channel NO. 183 to NO. 223 and channel NO. 452 to NO. 472 still have a small negative bias (maximum not exceeding -0.32 K). This indicates that EXP-2 scheme is effective in the bias correction for HIRAS-II assimilated channels. From Fig. 6b, there is obvious difference in O-A biases with and without bias correction. The mean O-A bias of the experiment without bias correction (NO BC) shows large deviations in the O₃ absorption band (channel NO. 161 to NO. 182), the near-surface water vapor absorption band (channel NO. 189 to NO. 297) and the shortwave CO₂ absorption band (channel NO. 452 to NO. 485). There is a significant improvement in O-A for all channels relative to O-B after bias correction (both for EXP- GSI and EXP-2). The mean O-A of EXP-2 (red curve) is essentially 0 for all channels except the O₃ absorption band and the shortwave CO₂ absorption band. It shows that the EXP-2 scheme is effective in improving the analysis.

In order to specifically analyze the improvement of the analysis by different schemes, three data assimilation experiments at 00:00 UTC 7 August 2023 were selected for analysis. Figure 7 shows the spatial distribution of HIRAS-II O-B biases. The colored dots in figures represent the values of the O-B bias, with colder colors being negative bias and warmer colors being positive bias. The shading indicates the observed FY-4A AGRI brightness temperature of the window channel 13 at a central wavelength of 12 µm. Figure 7a–c show the distribution of O-B biases passed QC in 900 cm⁻¹ channel (1000 hPa) for the NOBC, EXP-GSI and EXP-2, respectively. Figure 7d–f are as above but for 1476 cm⁻¹ channel (300 hPa). It can be seen that the O-B biases without BC have either cold or warm bias (Fig. 7a and d). The most O-B biases after EXP-GSI BC are slightly warmer (Fig. 7b and e), especially in the region from 0 to 30° N. The overall O-B biases after EXP-2 BC is near 0 (Fig. 7c and f) and the correction effect is significant.

To further validate DA results, the analysis is verified against the ERA5 0.25° × 0.25° data for each experiment. Figure 8a and c show the vertical RMSE (Root Mean Square Error) profiles of temperature and water vapor, respectively. The black solid line represents experiment NOBC, the blue solid line represents experiment EXP-GSI, the red solid line represents experiment EXP-2 and the black dotted line represents experiment EXP-CONV. Compared with experiment EXP-COVN, experiments EXP-2, EXP-GSI and EXP-NOBC assimilated a large amount of HIRAS-II data covering from the ground to the upper atmosphere. Therefore, it can be seen from Fig. 8a that the temperature analysis field accuracy is effectively improved (closest to ERA5), including experiment EXP-NOBC. The RMSE of temperature analysis fields from NOBC, EXP-2 and EXP-GSI significantly decrease relative to EXP-CONV in all atmosphere (Fig. 8a), the RMSE of water vapor analysis fields slightly decrease relative to EXP-CONV below 800 hPa (Fig. 8c). The RMSE profiles from EXP-2 and EXP-GSI normalized by the RMSE from the experiment NOBC are displayed in Fig. 8b and d to validate the influence of different bias correction schemes. The horizontal axis represents the RMSE percentage reduction relative to NOBC. A value less than 100 % indicates a decrease in RMSE after bias correction and values greater than 100 % means the RMSE increase reversely. Figure 8b shows that EXP-2 correction scheme effectively improves the temperature analysis fields accuracy in the upper and near-surface levels. EXP-2 scheme shows a better improvement than EXP-GSI schemes, especially in the upper levels. For the water vapor analysis fields (Fig. 8d), all BC schemes showed less significant improvements relative to NOBC compared to the temperature analysis fields, with changes in RMSE within 0.5 %. It is still the EXP-2 scheme reducing the largest RMSE at 400 to 800 hPa atmosphere. This may be due to the warmer O-B biases after the EXP-GSI correction.