Clouds have a significant influence on radiation propagation in atmospheres. In atmospheric remote sensing, band radiative transfer (BRT) models in cloudy atmospheres are crucial and are widely used in climate change, environmental monitoring, weather forecasting, and other research fields. Although the line-by-line (LBL) model is widely acknowledged as the most accurate BRT calculation method, its widespread usage is constrained by its high cost. Recently, correlated k-distribution (CKD) methods have progressed significantly and have emerged as the most promising alternatives in BRT calculations. They provide a better balance of accuracy and efficiency compared to other methods. However, most CKD methods tend to optimize quadrature parameters based on spectral distributions of absorption coefficients in clear atmospheres and use band-averaged cloud optical properties (COPs), ignoring the influence of COP changing with wavenumber. When COPs vary greatly in the wavenumber space, such treatment will cause significant errors. This paper proposes a CKD method suitable for BRT calculation in cloudy atmospheres that takes into account the effect of spectral parameters other than absorption coefficient on radiation.

Given the high cost of band radiative transfer calculations for cloudy atmospheres in remote sensing applications, a maximum correlated k-distribution optimal algorithm for single-scattering parameters (SSP-MCKD) is proposed. First, the CKD theory is extended to multiple spectral parameters for cloudy atmospheres, such as single albedos, asymmetry factors, etc., after analyzing the correlation of their spectral distributions under different environmental conditions. Further, based on the influence of spectral parameters on the single-scattering source function, the maximum correlated parameter of each spectral line is confirmed. A maximum correlation parameter group is formed by spectral lines with the same maximum correlated parameter. Based on the proportion of spectral lines within groups, the quadrature intervals are divided into each maximum correlation parameter group. Further, spectral parameters within the group are rearranged in the order of the maximum correlated parameter. The average equivalent parameters and quadrature weights are then calculated between and within groups. Finally, experiments were conducted to verify the applicability of the proposed method under different conditions.

Fig. 7 shows the mean relative errors of radiation calculated using the Δlog k method, correlated k-distribution with parameterization of cloud optical properties (PCOP-CKD), and the SSP-MCKD method under different number of quadrature intervals. With increasing quadrature intervals, the results of the three methods gradually converge to those calculated using the LBL model. The Δlog k method mainly considers the influence of the extinction coefficient, and the convergence curves are relatively stable. However, when the extinction coefficient is not the maximum correlated parameter in a given band, poor results are obtained, as shown in Fig. 7(c). The PCOP-CKD method ranks COPs based on the atmospheric absorption coefficients, where the correlation in the spectral distributions between the atmospheric absorption coefficients and COPs is maintained. This method ignores the influence of cloud scattering on BRT calculations. When the number of quadrature intervals is small, considerably desirable results are obtained. However, when the number of quadrature intervals increases, its calculation accuracy improves slowly. Hence, this method cannot meet the demand for practical engineering. The method proposed herein comprehensively considers the influence of spectral parameters on BRT calculations. As the number of quadrature intervals increases, its calculation accuracy has the fastest convergence speed.

Table 6 lists the mean relative errors of radiation calculated using the Δlog k, PCOP-CKD, and SSP-MCKD methods in clouds with different heights. The PCOP-CKD method considers cloud atmosphere as two parts: cloud and atmosphere. This method is more accurate in the atmosphere part. This method in scene 1 is more accurate because of the lower cloud and stronger gas absorption. The average error obtained using the SSP-MCKD method is 1.68%, which is improved by 26.92 percentage points and 5.63 percentage points compared with that of the Δlog k and PCOP-CKD methods, respectively. Thus, the proposed method is suitable for BRT calculations in clouds at different heights.

Table 8 lists the mean relative errors of radiation calculated using the three methods for different cloud types. The average error calculated using the SSP-MCKD method is 5.54%, which is improved by 17.73 percentage points and 3.31 percentage points compared with that of the Δlog k and PCOP-CKD methods, respectively.

The proposed SSP-MCKD method can be effectively employed in BRT calculations for cloudy atmospheres in remote sensing applications. This method has faster convergence in absorption, semi-absorption, and transmission bands compared with the other two methods. When the cloud average extinction coefficient and standard deviation are lower than 45 and 15 km^-1, respectively, the proposed method shows good results. Even when the cloud extinction coefficients or its standard deviation increase significantly, this method still outperforms the other two methods. The idea of optimizing and grouping according to maximum correlation parameters can be used as a reference to solve BRT calculation problems in other mixtures containing both gases and particles.