ON THE PERFORMANCES OF GASEOUS
ABSORPTION MODELS (LBL, CKD, AND ALD) IN
A RADIATIVE FORWARD MODEL FOR
ATMOSPHERIC REMOTE SENSING

Antoine Rimboud, N. Mourtaday, François Thieuleux, Céline Cornet, J. Riedi, L. C.-Labonnote, Frédéric André

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Antoine Rimboud, N. Mourtaday, François Thieuleux, Céline Cornet, J. Riedi, et al.. ON THE PERFORMANCES OF GASEOUS ABSORPTION MODELS (LBL, CKD, AND ALD) IN A RADIATIVE FORWARD MODEL FOR ATMOSPHERIC REMOTE SENSING. 11th International Symposium on Radiative Transfer RAD-25 June 15-20, 2025, Kuşadasi, Turkiye, Jun 2025, Kuşadasi, Turkey. pp.199-206, 10.1615/RAD-25.260 . hal-05468759

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
ON THE PERFORMANCES OF GASEOUS ABSORPTION MODELS (LBL,
CKD, AND ALD) IN A RADIATIVE FORWARD MODEL FOR
ATMOSPHERIC REMOTE SENSING

A. Rimboud, N. Mourtaday, F. Thieuleux, C. Cornet, J. Riedi, L. C.-Labonnote, F. André*

Univ. Lille, CNRS, UMR 8518 – LOA, Lille, France

ABSTRACT. Models of gaseous absorption are either formulated in terms of absorption coefficients (high resolution Line-By-Line spectra, *k*-distributions, ...) or transmissivities (statistical narrow band models, *l*-distributions, ...). The objective of the present work is to investigate the impact of the gas model form on the performances (CPU cost, accuracy, ...) of radiative transfer calculations in realistic atmospheric remote sensing scenarios. Using the high-fidelity Monte Carlo 3DMCPOL model, four configurations are studied: a purely gaseous atmosphere, two cases with one single cloud layer (high cloud with a small optical thickness, low cloud with a high optical thickness) and, eventually, one case where both high and low cloud layers are considered. The transmissivity-based augmented *l*-distribution model is more accurate and requires a number of photons smaller than CKD (a factor up to 40% smaller in some of the cases studied) to reach a level of precision compatible with atmospheric remote sensing applications.

# 1. INTRODUCTION

Fast though accurate Radiative Transfer Models (RTMs) are essential in space remote sensing studies. Indeed, RTMs are required to prepare space missions (in order to analyze in advance the capability of some imaging device to provide pertinent information about the state of the atmosphere) but also, in an operational context, to serve as forward models to infer scalar fields (temperature, gaseous species concentrations, cloud locations and water content, *etc*) from data measured by satellite instruments. RTMs specifically devoted to remote sensing applications in the atmosphere need to account for all the phenomena that occur in the observed scenes: this may include (as it obviously depends on the spectral interval considered) absorption and emission by the gaseous species, scattering, absorption and emission by cloud and aerosols, influence of the ground surface, *etc*.

We consider in this work the 3DMCPOL atmospheric RTM model developed at the Laboratory of Atmospheric Optics (University of Lille and CNRS) since 2010. The main characteristics of the model are described in Section 2. 3DMCPOL was primarily developed for radiative transfer studies in the visible range (for the analysis of 3-dimensional cloud covers) but the model was recently modified to allow statistical radiative transfer calculations in the thermal infrared (3DMCPOL was used previously in the thermal range in Ref. [1] but both emission locations and spectral integration were treated deterministically). The strategy adopted to extend the 3DMCPOL model from the visible to the thermal range is described briefly in Section 3. Based on these developments, an analysis of the gas model form (absorption coefficient- or transmissivity-based) on the performance (in terms

<footnote>*Corresponding author: frederic.andre@univ-lille.fr</footnote>
of CPU cost and accuracy) of 3DMCPOL is then proposed in Section 4, in a set of configurations representative of actual space missions.

## 2. DESCRIPTION OF THE 3DMCPOL MODEL

The 3DMCPOL model [2] was initially developed to simulate radiative transfer in 3-dimensional scattering atmospheres, accounting for the polarization of light. Its main application is the simulation of realistic cloudy scenes, as measured by remote sensing instruments such as MODIS [3], 3MI [4] or C3IEL [5]. In its "original" version, mostly restricted to the visible range of the electromagnetic spectrum, the only source of radiation is the Sun. Photons originating from the Sun propagate in the atmosphere before being detected by optical sensors located on satellites. 3DMCPOL uses the Monte Carlo Method (MCM) to simulate statistically photon paths and account for the extinction of radiative energy due to the interaction with the participating atmosphere. The MCM used in 3DMCPOL is an analog forward model of radiation transport formulated in terms of Stokes vectors to handle polarization. The local estimate method [6] that consists of evaluating contributions to the simulated signal at each scattering event, is used to accelerate the calculations. Moreover, 3DMCPOL is parallelized for additional computational efficiency. The model was compared with other atmospheric models (MYSTIC, *etc*) in the visible range in Refs. [7, 8]. Its application together with LBL, a CKD (Correlated K-Distribution, [9]) model and the *l*-distribution method [10] was described in Ref. [11] in the case of the O₂ A-band. However, the results reported in this reference were based on deterministic versions, for the spectral integration, of the gas radiation models. More recently, 3DMCPOL was also used to study the propagation of lightning sources in cloudy scenes [12].

In order to extend the 3DMCPOL model from the visible to the infrared, we have adopted the following strategy. In the infrared, contrary to the visible where the Sun is the only source of radiation, thermal emission occurs everywhere in the computational domain. However, as soon as the location of a source term is selected, the transport of radiation is mostly the same in the visible and in the infrared, as only the radiative properties of the propagation medium are different due to their spectral dependency. Consequently, it is sufficient to define a strategy to sample source locations (and their emission wavenumbers) to extend the method used by 3DMCPOL from the visible to the thermal infrared. The next section focuses on this specific issue.

## 3. THEORY: ABSORPTION COEFFICIENT AND TRANSMISSIVITY FORMULATIONS IN THE THERMAL RANGE

This section is devoted to the mathematical description of the Monte Carlo sampling method applied in the present work to select thermal source locations and wavenumbers for absorption coefficient- and transmissivity-based models. Due to limited space, we restrict the derivation to a line-of-sight calculation. This rather simple scenario is sufficient to introduce the main tools required to treat realistic 3-dimensional configurations. The extension of the sampling method from line-of-sight to more realistic 3-dimensional cases is discussed briefly at the end of the section. No element provided here can be considered as actually new. However, the derivation enlightens the modeling choices made in the present work. These choices may have some impact on the results described and discussed later in the paper.

Let us consider a non-uniform (both in terms of temperature, species concentrations and total pressure) ray located between two locations 0 and S inside a gaseous medium. The spectral absorption coefficient of the gas at wavenumber *v* and at a location *s* in [0, S] is $\kappa_v(s)$. The thermal radiation emitted by the gaseous medium is measured by an optical sensor located in S. This sensor is characterized by a spectral filter response function $\phi$. The optical filter is narrow enough to treat the blackbody intensity
at any location along the non-uniform path as constant. The spectral absorption coefficient of the gas all along the non-uniform path is strictly positive over the whole band defined by the optical filter (in other words, no transparency region of the gas spectrum is located over the band). The radiative intensity measured by the sensor at location S, assuming no thermal emission in s = 0, is defined as:

$$I(S) = \int_{0}^{+\infty} \phi(v) \left( \int_{0}^{S} \kappa_{v}(s) I_{b,v}(s) \exp \left[ -\int_{s}^{S} \kappa_{v}(s') ds' \right] ds \right) dv \quad (1)$$

where $I_{b,v}(s)$ is the blackbody intensity at location s along the non-uniform path.

The radiative intensity $I(S)$ measured by the optical sensor is below the maximum $I_M(S)$ (this maximum value corresponds to the optical thin limit in Eq. (1), for which all exponentials at the RHS are equal to 1) defined as:

$$I(S) \leq I_M(S) = \int_0^S \bar{k}_\phi(s) I_{b,\phi}(s) ds \quad (2)$$

where $I_{b,\phi}(s) = \int_0^{+\infty} \phi(v) I_{b,v}(s) dv$ is the filter averaged value of blackbody intensity at location s and $\bar{k}_\phi(s) = \frac{\int_0^{+\infty} \phi(v) \kappa_v(s) I_{b,v}(s) dv}{I_{b,\phi}(s)}$ is the mean absorption coefficient of the gas averaged over the filter response function at the same location s. Eq. (1) can be reorganized to yield:

$$\frac{I(S)}{I_M(S)} = \int_0^S \frac{\bar{k}_\phi(s) I_{b,\phi}(s)}{I_M(S)} \left( \int_0^{+\infty} \frac{\phi(v) \kappa_v(s) I_{b,v}(s)}{\bar{k}_\phi(s) I_{b,\phi}(s)} \exp \left[ -\int_s^S \kappa_v(s') ds' \right] dv \right) ds \quad (3)$$

Eq. (3) can be rewritten as:

$$\frac{I(S)}{I_M(S)} = \int_0^S \frac{\partial F(s)}{\partial s} \int_0^{+\infty} \frac{\partial G(v, s)}{\partial v} \exp \left[ -\int_s^S \kappa_v(s') ds' \right] dv ds \quad (4)$$

in which two functions, $F$ (of the single variable $s$) and $G$ (of the two variables $s$ and $v$), are introduced. These functions are defined by the following two relationships:

$$F(s) = \int_{0}^{s} \frac{\bar{k}_{\phi}(s') I_{b,\phi}(s')}{I_{M}(S)} ds', \quad G(v,s) = \int_{0}^{v} \frac{\phi(v') \kappa_{v'}(s) I_{b,v'}(s)}{\bar{k}_{\phi}(s) I_{b,\phi}(s)} dv' \quad (5)$$

From these definitions, both $F$ and $G$: 1/ take a null value when either $s$, for $F$, or $v$ at any given $s$, for $G$, are set to 0, 2/ are equal to 1 when either $s$ is equal to $S$, for $F$, or when $v \to +\infty$ at any location $s$, for $G$, 3/ are strictly increasing functions of $s$ in the case of $F$ or of the wavenumber $v$ at any location $s$ in the case of $G$. Consequently, $F$ and $G$ are distribution functions [13]. Eq. (4) becomes:

$$\frac{I(S)}{I_M(S)} = \int_0^1 \int_0^1 \exp \left[ -\int_{s(\xi_1)}^S \kappa_v(\xi_1, \xi_2)(s') ds' \right] d\xi_1 d\xi_2 \quad (6)$$
where $s(\xi_1)$ and $v(\xi_1, \xi_2)$ are respectively defined as solutions of $F[s(\xi_1)] = \xi_1, \xi_1 \in [0, 1]$ and $G[v(\xi_1, \xi_2), s(\xi_1)] = \xi_2, \xi_2 \in [0, 1]$.

The double integral appearing in Eq. (6) can be readily estimated using a Monte Carlo algorithm.

The present formulation is appropriate for models of gaseous radiative properties formulated in terms of absorption coefficients (the derivation is given for the case of high-resolution spectra but its adaptation to k-distribution methods is straightforward as it only requires replacing wavenumbers by pseudo-wavenumbers $g$ in Eq. (6)).

The method needs to be modified slightly to treat gas radiation models formulated in terms of transmissivities. For this purpose, we simply rewrite Eq. (1) as (where $\tau_v(s, S)$ is the spectral transmissivity of the non-uniform path $[s, S]$):

$$I(S) = \int_{0}^{+\infty} \phi(v) \left( \int_{0}^{S} -\frac{\partial \tau_{v}(s, S)}{\partial s} I_{b,v}(s) ds \right) dv \quad (7)$$

With the same assumptions and following the same steps as for the previous case of absorption coefficient-based models, we then obtain:

$$\frac{I(S)}{I_M(S)} = \int_0^S \frac{\partial F(s)}{\partial s} \left[ -\frac{1}{\bar{k}_\phi(s)} \frac{\partial \bar{\tau}_\phi(s, S)}{\partial s} \right] ds \quad (8)$$

where the filter averaged transmissivity along the non-uniform path $[s, S]$ is:

$$\bar{\tau}_{\phi}(s, S) = \frac{\int_{0}^{+\infty} \phi(v) \exp \left[ -\int_{s}^{S} \kappa_{v}(s') ds' \right] I_{b,v}(s) dv}{\int_{0}^{+\infty} \phi(v) I_{b,v}(s) dv} \approx \frac{\int_{0}^{+\infty} \phi(v) \exp \left[ -\int_{s}^{S} \kappa_{v}(s') ds' \right] dv}{\int_{0}^{+\infty} \phi(v) dv} \quad (9)$$

As for Eq.(6), Eq. (8) can be rewritten as an integral over $[0, 1]$:

$$\frac{I(S)}{I_M(S)} = - \int_0^1 \frac{1}{\bar{k}_\phi[s(\xi_1)]} \frac{\partial \bar{\tau}_\phi[s = s(\xi_1), S]}{\partial s} d\xi_1 \quad (10)$$

A direct comparison of Eqs.(6) and (10) immediately shows that using a model of gas radiative properties formulated in terms of absorption coefficient requires evaluating an integral in dimension 2 whereas, as could be expected, using a gas radiation model formulated in terms of transmissivities only requires a single integral (over the locations of radiative emission). This difference between the two kinds of models is likely to have some influence on the CPU cost in realistic 3-dimensional calculations. Notice that, for 3-D cases, volumes need to be sampled instead of locations along lines-of-sight as detailed above. This sampling of emission volumes mostly follows the same methodology as described in this section, but the $F$ function needs to be adapted for this purpose (formally, this mostly requires minor modifications to the $F$ function set by Eq. (5) to sample elementary volumes $V$ instead of locations along the path $s$ - this needs also minor changes in the definition of $I_M$). A
set of test cases is considered in the next section to evaluate the impact of the gas model form on the CPU cost of radiative transfer calculations. A few words about the accuracy of the methods are also provided.

# 4. APPLICATION

## 4.1 Description of the test configurations

In this section, we investigate the observation of water vapor H₂O using the METimage instrument. More specifically, we study the METimage / EPS-SG channels VII-35 centered at the wavelength 8.538 µm. The simulations are performed under standardized atmospheric conditions (U.S. Standard Atmosphere, 1976), with 49 sub-layers. The clouds are modeled in the simulations as plane-parallel layers, with droplets following a generalized gamma size distribution characterized by an effective droplet radius of 10 µm and an effective variance of 0.25 [14]. The Mie theory is used to calculate the optical properties of water droplets, considered as spherical particles. Polarization effects are not considered in the present analysis.

The performance analysis of the gas models in 3DMCPOL is conducted across a variety of cloudy scenes with realistic altitudes and optical thicknesses. Four distinct configurations are used following the same strategy as in Ref. [11]. The first case involves a pure gaseous atmosphere without clouds. The second and third scenarios consider either a high, thin cloud with a Cloud Optical Thickness (COT) of 2 located between 10 and 11 km, or a low, thick cloud with a COT of 10 situated at an altitude of 1 to 2 km. The fourth configuration combines both cloud layers. To account for surface interactions, a Lambertian surface with a low reflectivity value of R = 0.01 is assumed across all simulations.

Finally, the atmosphere is observed at a viewing azimuth angle $\phi$ of 0° and zenith angles $\theta$ ranging from 0° to 60°. In 3DMCPOL, a 1D plane-parallel atmospheric model is used, which does not account for Earth's curvature. Beyond $\theta = 60°$, this assumption becomes invalid [15], limiting the simulations to this angular range.

## 4.2 Description of the gas radiation models considered

There exist many methods to approximate the radiative properties of gaseous species, both in uniform and non-uniform configurations [16]. In the present work, we have considered two models formulated in terms of absorption coefficients: a high resolution LBL model (12,961 spectral values at a resolution of 0.01 cm⁻¹ between 1,106.474 cm⁻¹ and 1,236.084 cm⁻¹ are used) and a CKD model based on a Gauss-Legendre quadrature with 16 nodes (CKD-16). The transmissivity-based model is the Augmented $\ell$-distribution (written ALD in the following) described in Ref. [17]. ALD is applied here in tabulated form. Look-up tables contain 20,000 values per mapping function (see for instance Ref. [10] for additional details about the definition and use of these mapping functions).

## 4.3 Results and discussion

*Accuracy Analysis.* Table 1 presents the relative errors of the CKD and ALD methods compared to the LBL reference model, computed with 20G = 2.10¹⁰ photons to minimize statistical uncertainty, ensuring that only model limitations are assessed. Overall, the ALD model is more accurate than CKD, with errors up to an order of magnitude smaller, except for the high thin cloud at $\theta_2$ for which both models achieve comparable accuracy.
<table><caption>Table 1: Errors (%) of CKD and ALD models relative to the reference values obtained using the LBL model. Simulations are performed using 20G photons.</caption><thead><tr><th>Observation angle</th><th>Atmosphere</th><th>CKD relative error</th><th>ALD relative error</th></tr></thead><tbody><tr><td rowspan="4">θ₁ = 0°</td><td>Pure gaseous</td><td>0.058</td><td>0.007</td></tr><tr><td>High thin cloud</td><td>0.030</td><td>0.019</td></tr><tr><td>Low thick cloud</td><td>0.039</td><td>0.000</td></tr><tr><td>Both clouds</td><td>0.033</td><td>0.015</td></tr><tr><td rowspan="4">θ₂ = 60°</td><td>Pure gaseous</td><td>0.071</td><td>0.011</td></tr><tr><td>High thin cloud</td><td>0.003</td><td>0.005</td></tr><tr><td>Low thick cloud</td><td>0.045</td><td>0.005</td></tr><tr><td>Both clouds</td><td>0.034</td><td>0.010</td></tr></tbody></table>

The tendencies observed in Table 1 are consistently reflected in Figure 1, that provides a complementary analysis while extending the results to a range of angles between $\theta_1$ and $\theta_2$, specifically focusing on the most complex case involving two cloud layers. This figure confirms that ALD matches or surpasses CKD in accuracy, with CKD's systematic biases—compared to ALD—suggesting its inherent model limitations rather than statistical noise. Notice though that the levels of accuracy reported here (systematically below 0.1%) are much higher than what is usually needed in remote sensing applications (of the order of 0.5% in most scenarios).

<figure><img src="image_2.png" /><figcaption>Figure 1: Errors (%) of CKD and ALD models relative to the reference values obtained using the LBL model, for multiple observation angles in the test configuration with both clouds. Simulations are performed using 20G photons.</figcaption></figure>

*Efficiency Analysis.* Table 2 provides a comparative analysis of the LBL, CKD and ALD models in terms of computational efficiency. The key metric is the number *N* of photons required to achieve a given statistical uncertainty of 0.5% for our test configurations, at two observation angles ($\theta_1$ = 0° and
$\theta_2 = 60^{\circ}$). To determine this photon count $N$, a simulation is performed with a photon count $N_{sim}$, resulting in an uncertainty $\epsilon_{sim}$. Since the uncertainty scales as $\epsilon \sim 1/\sqrt{N}$ for Monte Carlo methods, the required photon count is calculated using the well-known formula $N = N_{sim} \cdot (\epsilon_{sim}/0.5\%)^2$.

<table><caption>Table 2: Comparison of radiative transfer models in terms of photon number $N$ required to achieve 0.5% numerical uncertainty and corresponding efficiency ratios for CKD and ALD relative to LBL (taken as the reference). Results are obtained using a single Monte Carlo for each configuration.</caption><thead><tr><th>Angle</th><th>Atmosphere</th><th>N<sub>LBL</sub></th><th>N<sub>CKD</sub></th><th>N<sub>ALD</sub></th><th>N<sub>CKD</sub>/N<sub>LBL</sub></th><th>N<sub>ALD</sub>/N<sub>LBL</sub></th></tr></thead><tbody><tr><td rowspan="4">θ<sub>1</sub> = 0°</td><td>Pure gaseous</td><td>1,710,204</td><td>1,727,745</td><td>1,341,278</td><td>1.01</td><td>0.78</td></tr><tr><td>High thin cloud</td><td>9,669,180</td><td>9,853,488</td><td>6,591,004</td><td>1.02</td><td>0.68</td></tr><tr><td>Low thick cloud</td><td>9,244,374</td><td>9,273,147</td><td>8,326,820</td><td>1.00</td><td>0.90</td></tr><tr><td>Both clouds</td><td>16,332,928</td><td>16,370,642</td><td>14,113,004</td><td>1.00</td><td>0.86</td></tr><tr><td rowspan="4">θ<sub>2</sub> = 60°</td><td>Pure gaseous</td><td>1,609,210</td><td>1,624,555</td><td>1,008,762</td><td>1.01</td><td>0.63</td></tr><tr><td>High thin cloud</td><td>29,047,618</td><td>29,169,912</td><td>22,721,851</td><td>1.00</td><td>0.78</td></tr><tr><td>Low thick cloud</td><td>19,021,629</td><td>19,146,470</td><td>16,315,564</td><td>1.01</td><td>0.86</td></tr><tr><td>Both clouds</td><td>46,227,046</td><td>46,268,570</td><td>41,767,371</td><td>1.00</td><td>0.90</td></tr></tbody></table>

The results in Table 2 show that LBL and CKD models yield comparable statistical performance, with CKD/LBL photon count ratios near unity across all scenarios. This is expected, as Monte Carlo convergence is assessed with respect to photon number $N$, making computational time primarily dependent on per-photon CPU cost, which varies with factors like code optimization and compiler efficiency—not considered here. From a Monte Carlo perspective, the lack of performance gain in terms of $N$ from LBL to CKD aligns with the expectation that statistical weight variability is largely unaffected by spectral discretization. In contrast, ALD consistently achieves higher photon efficiency. In a purely gaseous atmosphere, it reduces photon count by 22% at $\theta_1$ and 37% at $\theta_2$ compared to LBL. For the high thin cloud scenario, reductions are 32% at $\theta_1$ and 22% at $\theta_2$. These gains underscore ALD's efficiency in low-scattering regimes. Even in complex, multiple-scattering scenarios (low thick cloud and combined clouds), ALD outperforms LBL: photon reductions reach 10–14% depending on the angle. Overall, these findings not only validate the ALD model—relatively new in the radiative transfer literature compared to the more established LBL and CKD models—in scattering configurations, but also highlight its gain in performance across various configurations representative of atmospheric remote sensing. This can be attributable to the reduced dimensionality of transmissivity-based radiative transfer as emphasized in Section 3, which improves Monte Carlo convergence by lowering weight variability.

## 5. CONCLUSION

This study demonstrates the efficiency and accuracy of the ALD model for radiative transfer calculations compared to the CKD and LBL models. The key findings are that ALD reduces photon counts by up to ~40% (Table 2) while achieving superior accuracy compared to CKD in most cases by up to an order of magnitude (Table 1 and Figure 1), including complex configurations with multiple scattering. The greatest benefits are observed in purely absorbing cases, whereas ALD performance tends to degrade with increasing optical depth in the presence of multiple scattering. Several research directions can be pursued to further build on these findings. First, the reduced dimensionality of the ALD method, which is key to its efficiency, makes it particularly well-suited for future extensions to quasi-Monte Carlo methods, offering the potential for further reduction in computational effort. Second, regarding the 3D and polarization aspects, similar analyses need to be conducted in three-dimensional geometries, along with the inclusion of polarization effects, to evaluate the impact on
both efficiency and accuracy. This work provides a foundation for further advancements in efficient, high-precision radiative transfer simulations for atmospheric remote sensing.

ACKNOWLEDGEMENT

This work was supported by the French National Research Agency under the grant ANR-20-CE04-0005 ASGARD.

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