RTTOV was first developed as a fast radiative transfer model at the European Centre for Medium-Range Weather Forecasts (ECMWF) in the early 1990s (Eyre, 1991; Eyre et al., 1993). The original fast radiative transfer model (RTTOV v3) was described by Eyre (1991). After the mid-1990s, development was continued by the Numerical Weather Prediction Satellite Application (https://nwp-saf.eumetsat.int/site/software/rttov/, last access: 15 October 2026) Facility (NWP SAF, Rayer, 1995), supported by the European Organisation for the Exploitation of Meteorological Satellites (EUMETSAT). Subsequently, starting with RTTOV v6 in March 2000, the NWP SAF made the technical documents for this fast radiative transfer model publicly available via its official website (Saunders et al., 2018). The current version, RTTOV v14 (Hocking et al., 2025), supports fast radiative transfer calculations for over 120 satellite sensors covering the visible, near-infrared, infrared, microwave, and ultraviolet.

In the RTTOV model, infrared spectral gas absorption is primarily caused by water vapor, ozone, and uniformly mixed gases with a nearly constant spatial and temporal distribution. The total channel transmittance is calculated as the product of the transmittances of these three components (as shown in Eq. 1). Equation (1) accounts for the absorption overlap among these gases, thereby reducing the adverse effect on the forward model's accuracy. 1τi,jtotal=τi,jmix⋅τi,jmix+wvτi,jmix⋅τi,jmix+wv+o3τi,jmix+wv

The relationship between channel transmittance and optical thickness is defined by Eq. (2). The optical depth for a given atmospheric layer is calculated using the iterative Eq. (3): 2di,j=-ln⁡τi,j3di,j-di,j-1=-ln⁡τi,jτi,j-1=∑k=1nai,j,kXk,j where Xk,j is the predictor, which is a function of zenith angle, temperature, humidity, and pressure. di,j denotes the optical thickness from atmospheric level j to the top of the atmosphere for channel i, andτi,j represents the transmittance from level j to the top of the atmosphere. ai,j,k is the transmittance coefficient, where i denotes the channel index, j denotes the pressure level index, and k denotes the predictor index. For selected atmospheric sample profiles, we use LBLRTM to compute an accurate channel-by-channel transmittance dataset. When the left-hand side of Eq. (3) and the transmittance predictors are known, Eq. (3) forms an overdetermined homogeneous linear system of equations: ai,j,k is the least-squares solution to this system.

The training dataset comprises 83 atmospheric profiles. While these profiles cover a wide range of extreme meteorological conditions (Chevallier et al., 2006), not every individual profile is utilized in the layer-by-layer training of the fast transmittance coefficients. Based on the instrument characteristics of the Hyperspectral Infrared Atmospheric Sounder, the spectral response function is constructed using the Hamming window function (Harris, 1987). Because the values of the two wings of the spectral response function (SRF) of interferometric instruments oscillate around zero, the shape of this function can lead to physically unrealistic negative transmittance values in strong absorption bands after convolution with the monochromatic spectrum (transmittance is close to zero). Additionally, for strong water vapor absorption channels, under conditions of abundant water vapor content, the total transmittance from a given level to the top of the atmosphere (TOA) becomes an extremely small quantity in the middle and lower atmosphere. Thus, division by small quantities (τi,jτi,j-1) arises when calculating the optical depth of this layer. This leads to abnormal values in the total transmittance, causing numerical instability in the least-squares solution for the system of Eq. (3). Threshold screening for transmittance achieves numerically stable and accurate atmospheric profile sampling and enhances the stability of the least-squares regression solution. However, the choice of thresholds significantly affects both the training profiles and the optimality of the least-squares solution. The threshold method sets distinct thresholds for different channels to select atmospheric profiles for the regression. When the level-to-space transmittance falls below this threshold, samples below that height (i.e., at higher pressure/lower altitude) are considered too strongly affected by computational errors and are therefore excluded. Since the magnitude of transmittance is relatively small below that height (i.e., at higher pressure/lower altitude), the transmittance from this layer to the surface layer is considered to make a negligible contribution to the total atmospheric transmittance.

Ordinary least squares (OLS) obtains the optimal solution to the objective function when the sum of squared deviations between actual and predicted values is minimized. The error function being minimized, for this optimal solution, is: 4S=∑m=1m=p(ym-y^m)2m=1,2,3,…,p where ym denotes the optical thickness of a single atmospheric layer for the mth sample, and y^m denotes the fitted values of the mth sample, m denotes the index of the training profiles involved in the statistical analysis from 1 to 83×6, and p denotes the total number of samples in the training dataset. S denotes the sum of squared deviations between actual and predicted values. When S is minimized, the optimal solution for y^m is obtained.

The ordinary least squares method assumes homoscedasticity, giving equal weight to all data involved in the calculation. However, in practice, some data are more important than others. To retain a sufficient number of samples involved in the calculation while reducing the contribution of non-critical data, a method of assigning different weights to the data is adopted.

Weighted least squares enables models to better fit datasets with heteroscedastic properties by assigning different weights to different data points. In this case, Eq. (4) becomes: 5S=∑m=1m=pWm(ym-y^m)2(m=1,2,3,…,p) where Wm represents the weight factor.

The weight factor Wj=τj⋅τj is selected, and the approach screens out non-physical negative transmittance from satellite radiances.  From the top of the atmosphere to the surface, as the integrated transmittance from a given layer to space decreases, the layer lies closer to the surface. For these near-surface layers, their contribution to the total atmospheric absorption decreases. This is because the overlying atmosphere has already absorbed a significant portion of the incoming radiation, leaving less energy to be absorbed by the lower layers. Consequently, for layers with small transmittance (i.e., near-surface layers with weak signal), their regression weight should be reduced accordingly. Based on this physical understanding, we select a function of transmittance as the basic form of the weight coefficient. After analyzing the characteristics of transmittance variation and comparing the effects of different weight functions (such as τj, τj⋅τj, τj2, τj, etc.) on the regression results, we ultimately adopt the integrated transmittance from the layer out to space, τj⋅τj as the weight factor in the weighted least squares method.

In the second weighting scheme, the weighting factors are applied to both the independent variables (predictors) and the dependent variable prior to the statistical regression, and the regression coefficients are subsequently derived using the weighted data. This entails weighting the left-hand side term dj-dj-1 and the right-hand side term Xk,j, before solving the system of equations, as shown in Eqs. (6) and (7): 6Xk,j-1′=WjXk,j-17(dj-dj-1)′=Wj(dj-dj-1)

The weight factor expression is Wj=τj⋅τj. Equation (3) is transformed into Eq. (8): 8dj-dj-1′=-ln⁡τj/τj-1′=∑k=1Kaj,kXk,j′

Equation (4) becomes: 9S=∑m=1pWm2(ym-y^m)2(m=1,2,3,……,p)

This study optimizes the transmittance coefficient for the HIRAS-II 6.7 µm water vapor absorption channel (Matricardi and Saunders, 1999; Matricardi, 2007) using two methods: atmospheric profile selection and a weighted least-squares algorithm. We evaluate the forward model by comparing its output with both line-by-line radiative transfer model (LBLRTM) simulations and actual HIRAS-II observations. LBLRTM v12.11 by Atmospheric and Environmental Research, Inc. (AER) is used to compute monochromatic transmittances from 650 to 2600 cm⁻¹ (3.8 to 15.4 µm) at a 0.001 cm⁻¹ spectral resolution for 83 profile training datasets. The calculations cover 101 pressure levels (from 1100 to 0.005 hPa) and six zenith angles, accounting for refractive effects. They include absorption by a uniform mixture of 26 gases (CO₂, N₂O, CO, CH₄, O₂, NO, SO₂, NO₂, NH₃, HNO₃, OH, HF, HCl, HBr, HI, ClO, OCS, H₂CO, HOCl, N₂, HCN, CH₃Cl, H₂O₂, C₂H₂, C₂H₆, PH₃), water vapor (line and continuum), and ozone. These high-resolution transmittances are then convolved with the HIRAS-II spectral response function using Eq. (12) to produce the “true” transmittance values.

We calculated fast transmittance coefficients for the HIRAS mid-wave channels within the RTTOV framework using the RTTOV v7 predictors (Saunders et al., 2018). To evaluate the impact of the threshold method and weighted least squares on forward model accuracy, we selected channels representing different atmospheric layers. These include the 6.7 µm absorption peak (channel 1372, central wavenumber 1506.875 cm⁻¹), mid-level (channel 1239, central wavenumber 1423.75 cm⁻¹; channel 1596, central wavenumber 1646.875 cm⁻¹) and lower-level (channel 1169, central wavenumber 1380 cm⁻¹; channel 1729, central wavenumber 1730 cm⁻¹), chosen from the left and right ends of the peak region. These channels were selected to analyze the effects of the threshold method and weighted least squares on forward model accuracy, respectively.

We analyze the impact of the threshold and weighted least-squares methods on forward model accuracy according to the experimental design in Table 1. Based on the 1976 U.S. standard atmospheric profile, we employ the RTTOV radiative transfer model to analyze mid-wave spectrum simulations of brightness temperature and calculate the height of the weighting function for selected channels (as shown in Fig. 1). The weighting function is calculated using Eq. (10): 10W=∂τ∂ln⁡P where W denotes the weighting function, τ represents the channel transmittance, and P indicates atmospheric pressure. This derivative with respect to log-pressure is computed using forward-difference discretization. As shown in Fig. 1, the weighting function peak indicates the atmospheric altitude sensed by the channel.

To comprehensively evaluate the influence of different parameterization strategies on the computational accuracy of fast transmittance and brightness temperature, we have designed four experimental schemes, incorporating the aforementioned typical channels identified through Jacobian weighting-function analysis. Specifically, LSM0 serves as the baseline scheme without any data processing. LSM1 selects a reasonable threshold of 10-4 and applies ordinary least squares (OLS) regression after thresholding. LSM2 adopts the threshold-based method, incorporating weighted least squares (WLS) for the regression. LSM3 applies weighting factors to both the independent variables and the dependent variable, prior to performing the statistical regression. These four schemes are systematically compared to quantitatively assess the impact of the respective parameterization approaches on the simulation accuracy of fast transmittance and brightness temperature.

This study uses 83 profile training datasets from Matricardi (2008b), which are widely adopted for training fast transmittance coefficients in radiative transfer models, including RTTOV, CRTM, and ARMS. These profiles provide global coverage of temperature, water vapor, and ozone across 101 pressure levels (1100 to 0.005 hPa). They represent a range of atmospheric conditions, including seasonal extremes. Temperature ranges from a minimum of 159.62 K to a maximum of 318.26 K. The spatial distribution of the profiles is shown in Fig. 2a, while the vertical distribution characteristics of temperature and humidity profiles are depicted in Fig. 2b and c, respectively.

The Hyperspectral Infrared Atmospheric Sounder-II (HIRAS-II) has a spectral coverage of 650–2550 cm⁻¹ (3.8 to 15.4 µm) using continuous spectral scanning with a spectral resolution of 0.625 cm⁻¹. Three bands cover the 15 µm CO₂ band (LW: 648.75–1169.375 cm⁻¹), the 6.7 µm H₂O band (MW: 1167.5–1921.25 cm⁻¹), and the 4.3 µm CO₂ band (SW: 1919.375–2551.25 cm⁻¹), totaling 3053 channels (http://www.nsmc.org.cn/nsmc/cn/instrument/HIRAS-2.html, last access: 15 October 2026). Satellite observation data were obtained from the Fengyun Satellite Remote Sensing Data Service Network (http://data.nsmc.org.cn/DataPortal/cn/home/index.html, last access: 15 October 2026). FY-3F HIRAS-II L1 observation data from August 2024 and February 2025 were selected to represent summer and winter, respectively.

Based on 83 atmospheric profiles, profiles were selected using both unconstrained and threshold methods to calculate transmittance coefficients. The results were validated against LBLRTM benchmarks. At the left wing of the 6.7 µm absorption band, features due to SO₂, N₂, and CH₄ are present. These gases are all contained within a uniform mixture, which increases the complexity of transmittance calculations. The change in transmittance caused by the threshold method is shown in Fig. 5. Channel No. 1169 (central wavenumber 1380 cm⁻¹) lies on the far wing of the 6.7 µm absorption line, representing water vapor absorption in the lower atmosphere, as shown in Fig. 5a. Profile 82 represents the maximum values of all level variables among the set of 83 profile samples. For this profile type, applying the threshold has a negligible effect on fast transmittance calculations. The impact of threshold magnitude is essentially the same as that seen for dry profiles, as shown by Profile 81. As the water vapor content decreases, the impact of the threshold on transmittance calculation accuracy increases, but it is mainly limited to altitudes below 600 hPa, which corresponds roughly to the channel detection height (peak height of the weighting function). Furthermore, the lower the water vapor content, the clearer the improvement in transmittance calculation accuracy. Profile 81 has the lowest variable values among sample profiles, and here the threshold's effect is strongest. Compared with the transmittance deviation calculated without constraints, the fast transmittance calculated using a threshold constraint closely matches that from LBLRTM. In Fig. 5b, channel 1239 (central wavenumber 1423.75 cm⁻¹) represents water vapor absorption in the mid-level atmosphere. For profiles with above-average water vapor content (water-rich profiles), forward model accuracy matches that in Fig. 5a. For water-poor profiles below the average, the threshold effect becomes stronger. As water vapor content decreases along the profile, the altitude where threshold-induced transmittance deviation increases gradually rises. For profile 81, the highest altitude where the threshold method improves transmittance deviation does not exceed the channel detection height. In Fig. 5c, channel 1372 (central wavenumber 1506.875 cm⁻¹) lies within the 6.7 µm absorption line used to detect upper-level water vapor. The effect of the threshold on forward modeling accuracy is similar to Fig. 5a and b, but its influence is more pronounced.

Figure 6 presents a comparative analysis of transmittance versus atmospheric pressure for the channel at the right end of the 6.7 µm absorption line. Within this spectral range, only the weak absorption lines of NO₂ and CH₄ lie in the near-wing region of the 6.7 µm absorption line. The transmittances calculated using the threshold method and LBLRTM show nearly identical distributions across atmospheric pressure, consistent with Fig. 5. This suggests that trace gases in either the far-wing or near-wing regions do not affect the accuracy improvement achieved by the threshold method.

For 83 atmospheric profiles, level-to-space atmospheric transmittance was calculated at six satellite zenith angles, producing a total of 498 atmospheric samples. At each isobaric surface, this resulted in a system of 498 linear equations. Analysis indicates that when the cumulative transmittance from a level to the top of the atmosphere is less than 10⁻⁴, the cumulative transmittance from the level below to the top of the atmosphere may become negative. In consideration of both instrument characteristics and transmittance variation patterns, this study adopts a semi-empirical threshold of 10⁻⁴ as the screening criterion to filter out physically meaningless transmittance. Please see Appendix A (Figs. A1 to A3) for further comparison errors between the LSM series models (LSM1, LSM1.1, LSM1.2) with varying thresholds and the LBLRTM (true value) under typical layer, full atmospheric column, and strong absorption channel conditions, respectively. The 10⁻⁵ threshold suffers severe numerical instability caused by division by small values and yields poor fitting accuracy, whereas switching to 10⁻⁴ substantially reduces calculation bias. Further analysis reveals that an excessively large threshold prematurely discards low-level atmospheric samples and introduces notable errors in transmittance calculation for the middle and lower layers of dry atmospheric profiles; by contrast, the 10⁻⁴ threshold exerts negligible adverse effects on weakly absorbing channels with slowly decaying transmittance. Based on the above physical mechanism analysis and quantitative error evaluation, the present threshold was finalized to balance numerical stability and the integrity of atmospheric profile samples.

Figure 7 shows that for each profile, there is an identified position indicating the pressure level above which the profile participates in the training of the corresponding channel, and below which the profile does not participate in the training of the corresponding channel: blue for channel 1169, brown for channel 1239, green for channel 1372, yellow for channel 1596, and red for channel 1729. On the vertical axis, 0 indicates the top of the atmosphere, and 100 indicates the surface. As pressure increases, the number of samples used decreases gradually. Applying the threshold method to filter samples reveals a strong correlation between the number of profiles used in calculations and the absorption characteristics of the channels in these levels. The stronger the absorption, the fewer profiles are included in the training for the middle and lower tropospheric layers. All upper-atmosphere profiles can be retained; however, in the lower atmosphere, retainable samples may drop to fewer than ten. The number of retained samples is higher for channels in absorption line wing regions than for those in peak regions. Moreover, retained sample altitudes are lower for wing-region channels (up to 800 hPa) than peak-region channels (up to 600 hPa). This occurs because, as a channel approaches an absorption line peak region, more samples below its detection altitude have near-zero transmittance and are thus filtered out by thresholding. This can be explained as follows: as the channel moves toward the absorption line peak region, the number of samples with near-zero transmittance below the channel detection altitude gradually increases, which results in more samples being affected by threshold filtering. Additionally, the number of retained samples varies between the left and right ends of the peak region.

In Fig. 8, the surface emissivity for all channels is uniformly set to 0.98 to eliminate its influence on forward modeling results across different schemes. The deviations between the 6.7 µm absorption line spectrum transmittances at the top of the atmosphere calculated by LSM0 and LSM1 for all channels and the LBLRTM results are shown in Fig. 8a and b. In the figure, gray indicates the transmittance deviation of all profiles calculated by LSM0 and LSM1 relative to LBLRTM. The LSM0 forward simulation exhibits positive deviations in the absorption line peak and near-wing regions, while negative deviations occur in the distant wing regions. The deviation distribution of LSM1 is more uniform, though negative deviations increase in the peak region. The standard deviation of LSM1 is significantly reduced from approximately 0.1 K for LSM0 to less than 0.05 K, with a uniform distribution across spectral channels. Unlike LSM0, the standard deviation in the absorption line wings does not exceed that in the peak regions. The deviation distribution shows similar characteristics to the standard deviation. Figure 8c and d show the deviation distributions of brightness temperature in the mid-wave channels for LSM0 and LSM1 forward modeling, respectively. As shown in Fig. 8, for both LSM0 and LSM1 calculations of brightness temperature, the maximum deviation from LBLRTM simulated brightness temperature exceeds 0.5 K across all 83 profiles. Regarding average deviation, LSM1 shows an average value around -0.1 K, which is significantly smaller than the -0.2 K deviation of LSM0. At the left end of the 1350 cm⁻¹ range, the deviations between the two methods are comparable. In terms of standard deviation and root mean square error distributions, LSM1 also significantly outperforms LSM0. These results collectively demonstrate that the threshold method effectively improves forward model accuracy.

In the LSM2 experiments, a weight constraint was additionally applied to the threshold-constrained training profiles, as shown by the right-hand side term in Eq. (6). In Fig. 9, the difference between LSM1 and LSM2 is negligible in the far-wing region to the left of the 6.7 µm absorption line, and their correction effects on transmittance are identical to those shown in Fig. 5a. For the extremely dry atmospheric conditions represented by profile 81, LSM2 shows greater discrepancies from LBL transmittance at lower atmospheric levels and near the channel detection altitude. At 1506.875 cm⁻¹, LSM2 exhibits negative deviations relative to LBL results across all altitudes from 300 to 600 hPa, clearly contrasting with thepositive deviations produced by LSM1. Moreover, below 700 hPa, LSM1 maintains a constant transmittance of 0.38, differing significantly from LBL transmittance, which monotonically approaches zero. In this region, LSM2 results better match the actual atmospheric transmittance distribution, indicating that LSM2 significantly improves the forward modeling accuracy of transmittance in the lower atmosphere for the 6.7 µm absorption line peak region channel under extremely dry conditions.

Figure 10 shows the transmittance distribution for the right-end channel of the 6.7 µm absorption line. For extremely dry atmospheres, the deviation between LSM2 and LBL remains smaller than that of LSM1 below 250 hPa. Between 300 and 600 hPa, the deviation between LSM2 transmittance and LBL gradually decreases. Below 600 hPa, it essentially overlaps with the LBL transmittance profile, with the largest discrepancy occurring in the low-vapor-content atmosphere of profile 81. The maximum deviation between LSM0/LSM1 and LBL transmittance calculations occurs below 900 hPa, increasing at lower altitudes. However, the difference between LSM2 and LBL remains largely constant with height and is smaller than that of LSM1.

The configuration in Fig. 11 is essentially identical to that of Fig. 8, showing the distribution of channel transmittance and simulated brightness temperature deviations for LSM0, LSM2, and LBL across channels at the top of the atmosphere. The transmittance deviation between LSM0 and LBL exceeds 0.2 in most channels, while the brightness temperature bias and standard deviation remain within 1 K. After applying thresholding combined with weighting coefficient correction, the maximum channel transmittance deviation falls below 0.1, as shown in Fig. 11b. The mean and standard deviation of transmittance in mid-wave spectral channels are less than 0.02. As shown in Fig. 11d, the mean and standard deviation of brightness temperature differences are both below 0.2 K. In Fig. 11b, the transmittance deviation distribution of LSM2 at the left end of 1380 cm⁻¹ is identical to that of LSM1, indicating that weight correction does not significantly improve forward modeling accuracy for overlapping absorption spectra of trace gases and water vapor. In other mid-wave spectral channels, LSM2 shows a marked improvement in forward modeling accuracy compared to LSM0. Furthermore, compared to the forward modeling accuracy of LSM1 in Fig. 8b and d, LSM2 exhibits a more uniform deviation distribution across the mid-wave spectrum, eliminating the large deviations observed in some spectral channels of LSM1.

Figure 12 shows the effect of LSM3 on the transmittance profile of the left-end channel of the 6.7 µm absorption peak compared with LSM2, LSM1, and LSM0, under the same settings as in Fig. 5. As shown in Fig. 12, the improvement in transmittance by LSM3 remains concentrated in the low-to-mid troposphere with low water vapor content, and the altitude of influence closely correlates with the channel detection altitude. The improvements of LSM3 compared to LSM0 and LSM1 are significant. The transmittance profile calculated by LSM3 almost coincides with that of LSM2. Figure 13 compares transmittance for the right-end channel of the 6.7 µm absorption peak, using the same settings as in Fig. 6.

Figure 14 shows the statistical deviation of total channel transmittance and brightness temperature for the LSM0 and LSM3 schemes, using the same settings as in Fig. 8. As shown, compared to the transmittance deviation in LSM2 from Fig. 11, the deviation in LSM3 is further reduced. In the brightness temperature deviation statistics in Fig. 8d, the mean RMSE for the mid-wave spectrum (1300–1900 cm⁻¹) of LSM1, LSM2, and LSM3 progressively decreases from 0.1 to 0.05 and 0.045 K, respectively, while the mean maximum value decreases from 0.45 to 0.22 and 0.15 K. This indicates that combining the threshold method with weighting coefficients for least-squares solutions can further reduce forward modeling deviations in HIRAS-II mid-wave channels, with the LSM3 scheme producing the smallest error distribution.