Instead of solving the RT equation under the single-scattering approximation, it is formulated the problem in terms of the probability that an instrument located at the surface and pointing at the zenith detects a photon scattered by a cloud at an altitude hC during twilight. If it is assumed that photons undergo at most a single scattering event, then for given values of SZA, hC, and λ, the probability P that the instrument detects a cloud-scattered photon can be written as: 2P(SZA,z=hC,λ)=P1(SZA,z=hC,λ)×P2c(SZA,z=hC,λ)×P3(SZA,z=hC,λ) Here, (i) P1 represents the probability that a direct solar photon reaches the cloud without being scattered or absorbed along its trajectory from the top of the atmosphere to the cloud (path between points (1) and (2) in Fig. a); (ii) P2c represents the probability that the cloud scatters the photon toward the instrument, corresponding to a scattering angle equal to the SZA; and (iii) P3 is the probability that a photon at altitude hC, traveling downward at a zenith angle of 180°, reaches the instrument without undergoing absorption or additional scattering (path between points (2) and (3) in Fig. a). The probabilities P1 and P3 are determined by the properties of the background atmosphere at wavelength λ, whereas P2c depends solely on the cloud scattering optical depth and phase function. Equation () accounts only for photons scattered by the cloud at altitude hC. Therefore, to reproduce the total zenith intensity measured at the surface, contributions from photons scattered at all altitudes and by molecules or other aerosols present in the atmosphere must be considered. In this case, the total probability of detecting a photon with the instrument (PTotal) is obtained by integrating P over altitude (see the right panel of Fig. b): 3PTotal(λ,SZA)=∫zdarkztopP1(z,λ,SZA)×P2(z,λ,SZA)×P3(z,λ)dz, where P2 represents the probability that a photon is scattered at altitude z into the zenith line of sight of the instrument, corresponding to a scattering angle θ=SZA with respect to the incident solar direction. Note that in Eq. (), P2 evaluated at z=hC accounts for photons scattered by the cloud, molecules, and any other aerosols present at that altitude. If the molecular and aerosol opacities are negligible at hC, then P2(hC)=P2c(hC). Here, ztop denotes the altitude of the top of the atmosphere, while zdark, which depends on the SZA, denotes the lowest altitude that still receives direct solar illumination during twilight. Below zdark, direct solar radiation is completely blocked by the Earth's curvature. From Eq. (), it is defined the single-scattering CI as: 4CI∗(SZA)=PTotal(λ2,SZA)PTotal(λ1,SZA), where λ2>λ1. To compute PTotal, explicit expressions for P1, P2, and P3 are required. For a given altitude z above the instrument and a given SZA, the probability P1 can be written as 5P1(z,λ,SZA)=exp⁡-τ1(z,SZA,λ), where τ1(z,SZA,λ) is the optical depth along the line of sight (see Fig. ). To estimate τ1(z,SZA,λ), it is assumed that aerosol and molecular extinction (S) decrease exponentially with altitude, with scale heights HA and HM, respectively: 6S(z,λ)=S0A(λ)exp⁡-zHA+S0M(λ)exp⁡-zHM=τ0A(λ)HAexp⁡-zHA+τ0M(λ)HMexp⁡-zHM, where S0A and S0M are the aerosol and molecular extinction coefficients at the surface, and τ0A and τ0M are the corresponding total aerosol and molecular optical depths. For a given altitude z above the instrument and a given SZA, the optical depth along the line of sight is given by 7τ1(z,SZA,λ)=S0A(λ)∫0l(z,SZA)exp⁡-l′2+(RT+z)2+2l′μ(RT+z)-RTHAdl′+S0M(λ)∫0l(z,SZA)exp⁡-l′2+(RT+z)2+2l′μ(RT+z)-RTHMdl′, where l(z,SZA) is the distance between the point (0,z) and the intersection with the top of the atmosphere (points Qi in Fig. b). To derive Eq. (), it is used the relation r=l2+(RT+z)2+2lμ(RT+z), where r is the radial distance, RT is the planetary radius, and μ=cos⁡(SZA). Using trigonometry, the path length l(z,SZA) can be expressed as a function of z and the SZA: 8l(z,SZA)=(ztop+RT)2+(z+RT)2sin⁡4(α)-cos⁡4(α)+2(z+RT)sin⁡(α)‾(ztop+RT)2-(z+RT)2cos⁡2(α)‾, where α=SZA-90°. It is important to note that, for the estimation of τ1, the opacity of the cloud is not included. This is because, in our model, the cloud has a limited horizontal extent and is localized above the instrument, rather than being treated as a spherical shell. The treatment of the cloud horizontal extent, and its impact on P1, is introduced later in the context of the comparison with the Monte Carlo simulations.

The integrals in Eq. () do not admit analytical solutions and must therefore be evaluated numerically. Since the terms S0A and S0M are outside the integrals, a precomputed look-up table for the integrals in Eq. () as a function of z, SZA, HA, and HM can be constructed. In this case, P1 can be written as 9P1(z,SZA,λ)=exp⁡-JA(z,SZA,HA)S0A(λ)+JM(z,SZA,HM)S0M(λ), where JA and JM denote the integrals in Eq. (7), which are precomputed and therefore do not need to be recalculated when the wavelength or the aerosol scenario is changed.

As indicated above, P2 represents the probability that direct solar radiation reaching an altitude z above the instrument is scattered at a scattering angle equal to the SZA. Consequently, this probability is proportional to the scattering extinction coefficient at altitude z – including contributions from aerosols, molecules, and clouds – and to the phase function p(θ) evaluated at θ=SZA. If the vertical distribution of the cloud is described by a Gaussian height profile, then P2 can be written as: 10P2(z,SZA,λ)=C[S0A(λ)ωA(λ)exp⁡-zHApA(λ,SZA)+S0M(λ)ωM(λ)exp⁡-zHMpM(λ,SZA)+τC(λ)ωC(λ)ΔhC2πexp⁡-2(z-hC)2ΔhC2pC(λ,SZA)], where ωA, ωM, and ωC are the aerosol, molecular, and cloud single-scattering albedos, respectively; pA, pM, and pC are the corresponding phase functions; τC and ΔhC are the cloud total optical depth and geometrical thickness; and C is a normalization constant that, as shown below, does not need to be explicitly evaluated in our analysis. In Eq. (), the only term depending explicitly on the SZA is the phase function. However, given the limited SZA range considered here, the impact of phase-function variations on SZAmax is expected to be small.

The last term to be computed is P3(z,λ), which represents the probability that a photon travels vertically from altitude z to the surface without undergoing absorption or scattering. This probability is given by 11P3(z,λ)=exp⁡-τ2(z,λ), where τ2(z,λ) is the optical depth due to aerosols, molecules, and clouds between altitude z and the surface (see Fig. ), and is computed as 12τ2(z,λ)=τ0A1-exp⁡-zHA+τ0M1-exp⁡-zHM+τC2erf2z-hCΔhC-erf-2hCΔhC. Here, erf denotes the error function, defined as 13erf(x)=2π∫0xe-t2dt. Once P1, P2, and P3 are defined, the total probability PTotal is obtained by integrating their product from zdark to the top of the atmosphere. Scripts to compute P1, P2, P3, and PTotal are publicly available at .

In the following sections, it is first analyzed the behavior of P1, P2, and P3, and their contribution to PTotal, in order to study the properties of the CI as a function of cloud altitude and wavelengths. It is shown that the single-scattering CI∗ reproduces the main features of the CI obtained from full Monte Carlo RT simulations.

Figures  and show the variation of P1, P2, P3, and P with altitude at 450 and 950 nm for different SZAs, considering a cloud layer located at an altitude of 16 km. In these simulations, the phase functions were omitted in the computation of P2, and both ωM and ωC were set to unity. The P2 curves were normalized by PTotal at SZA=90°. As discussed below, for a given cloud scenario, the choice of normalization in Eq. () does not affect the value of SZAmax derived from the model. As shown in Figs.  and , P1 increases with altitude and decreases with SZA as a consequence of the dependence of τ1 (Eq. ) on these parameters. For a given altitude z above the surface, the path segment between points Q and (0,z) intersects progressively lower atmospheric layers as the SZA increases, where the molecular opacity is higher, leading to larger values of τ1. For fixed values of SZA and z, P1 is also larger at 950 nm than at 450 nm, reflecting the decrease of Rayleigh opacity with increasing wavelength. At 450 nm (Fig. ) and for SZA≳93°, most of the photons reaching the instrument originate from scattering events occurring at altitudes above the cloud layer, as indicated by the behavior of P in Fig. . The altitude at which direct solar radiation is completely blocked by the Earth curvature is given by 14zdark=RT1+tan⁡2(α)-1, Thus, for SZA=93 and 94°, zdark is 8.7 and 15.5 km, respectively, which are both below the altitude of maximum cloud opacity (hC=16 km). This indicates that, at 450 nm, and as a result of molecular opacity, the cloud is already effectively in darkness before the surface blocks direct solar radiation at the cloud altitude (SZA>94°). In contrast, at 950 nm (Fig. ) , and for zdark<hC, the maximum in P occurs at the cloud altitude (for SZA=93 and 94°). Therefore, depending on the wavelength, the cloud can enter the dark region of the atmosphere even when hC>zdark. As discussed below, this behavior is one of the primary factors responsible for the emergence of a maximum in the CI in the presence of high-altitude clouds. With regard to P2 and P3, larger values of P2 are obtained at 450 nm than at 950 nm as a consequence of the stronger molecular opacity at shorter wavelengths. However, since P1 decreases much more rapidly at 450 nm than at 950 nm for altitudes above zdark, and because P3 is larger at 950 nm than at 450 nm, the resulting values of P are greater at 950 nm for altitudes near the cloud layer (i.e., for z>zdark).

These results are further illustrated in Fig. a, where the P profiles at 450 and 950 nm are compared for three different SZA values. The horizontal black dashed lines indicate the altitude zdark corresponding to each SZA. When SZA∼90°, the dominant contribution of photons at both wavelengths originates from altitudes around the cloud layer. However, as the SZA increases, the contribution from these altitudes decreases more rapidly at 450 nm than at 950 nm. As a result, PTotal decreases more slowly with increasing SZA at 950 nm than at 450 nm, and it is precisely this effect that causes CI∗ to increase over this SZA range, as shown in Fig. b. The increase in CI∗ continues up to SZA∼93.8°, beyond which CI∗ decreases rapidly with SZA. By comparing the P profiles with the CI∗ signal in Fig. , it is seen that this transition occurs close to the moment when the cloud begins to darken at 950 nm. These results indicate that the maximum in CI∗ occurs at larger SZA values as λ2 increases, since lower molecular opacity delays the onset of cloud darkening. In the limiting case where the molecular opacity at λ2 is negligible, SZAmax approaches SZAdark, which can be obtained from Eq. () by setting z=hC. For the example shown in Fig. , SZAdark=94.06°, which is close to the corresponding value of SZAmax. From this, it is concluded that for a given SZAmax, the cloud altitude must satisfy 15hC≥RT1+tan⁡2(SZAmax-90∘)-1.

If an increase in λ2 results in an increase in the value of SZAmax, the next point to address is the dependence of SZAmax on λ1 for a fixed λ2. It was performed simulations similar to those shown in Fig.  for different values of λ1 and found almost no variation in SZAmax when λ1 ranges between 400 and 500 nm while λ2 is fixed at 950 nm. Although the shape of the CI∗ signal changes, the value of SZAmax remains nearly unchanged. However, for larger values of λ1, a noticeable shift in SZAmax is observed. In the following section, it is further investigated how variations in λ1 influence CImax.

To study the dependence of SZAmax on λ1 and λ2 for a given cloud scenario, the condition defining the maximum of CI∗ is examined. This condition is obtained by differentiating the ratio defined in Eq. () with respect to the SZA and setting the derivative equal to zero. Under this condition, SZAmax satisfies: 161PTotal(λ1,SZA)∂PTotal(λ1,SZA)∂SZA=1PTotal(λ2,SZA)∂PTotal(λ2,SZA)∂SZA Equation () shows that the CI∗ maximum occurs at the SZA for which both wavelengths exhibit identical relative SZA-variations of PTotal. In other words, the maximum does not arise only from the cloud contribution, but from the differential evolution of the zenith intensity with SZA at the two selected wavelengths. Equation () also indicates that any multiplicative constant applied to PTotal in Eq. (), such as a normalization factor, does not modify the value of SZAmax.

As an example, Fig. a shows the relative rate of variation of PTotal(λ,SZA) with respect to SZA at different wavelengths for the same cloud scenario as in Figs.  and . Figure b shows the corresponding CI∗ signals for the same cloud scenario and wavelength combinations as in Fig. a, with λ1=400 nm fixed and λ2 varying. The results at 600 and 650 nm have been omitted for clarity, as they follow the smooth spectral evolution observed between the neighbouring wavelengths (550 and 700 nm) and therefore do not provide additional qualitative information. As expected, SZAmax coincides with the SZA at which the relative SZA-variation curves of PTotal intersect. In the previous section, it was indicated that simulations similar to those shown in Fig. , but considering λ1 values between 400 and 500 nm while keeping λ2=950 nm fixed, exhibit no significant variations in SZAmax. This behaviour is readily understood from the relative SZA-variation curves of PTotal, where the curves corresponding to 400 and 500 nm are nearly indistinguishable for SZA≳93.5°. As a result, their intersection with the λ2=950 nm curve occurs at essentially the same SZA, leading to nearly identical values of SZAmax. However, if instead of using λ2=950 nm it is considered λ2≲700 nm, noticeable variations in SZAmax may arise when λ1 is varied between 400 and 500 nm.

Figure a also shows that, as λ2 increases relative to 400 nm, the intersection between the corresponding relative SZA-variation curves becomes progressively more transversal, which corresponds to a larger difference between their local slopes at the crossing point. As a consequence, the curvature of CI∗ around SZAmax increases, leading to a more pronounced and better-defined maximum in Fig. b. Conversely, when the relative variation curves intersect with nearly parallel slopes, the resulting CI∗ maximum is broader and less pronounced. This aspect is particularly relevant for real zenith radiance measurements, where instrumental noise and atmospheric variability introduce uncertainties in the determination of SZAmax. A sharper maximum enhances the robustness of cloud detection and improves the precision in the inferred cloud altitude.

Although these results suggest that increasing the spectral separation between λ1 and λ2 generally leads to a sharper CI∗ maximum and therefore to potentially more precise determinations of SZAmax, the situation is more complex in practice. The shape of the relative SZA-variation curves of PTotal also depend on the cloud altitude and on the overall atmospheric optical properties. Consequently, identifying an optimal combination of λ1 and λ2 that minimizes the uncertainty in the retrieved cloud height is not straightforward. In the following section, it is introduced an alternative CI-based index specifically designed to improve the robustness of twilight cloud-height estimation.

In the previous section, it was highlighted the difficulty of identifying a globally optimal combination of λ1 and λ2 even under the simplified assumption of a single cloud layer and molecular scattering only. In realistic atmospheric conditions, however, additional components such as aerosol layers or absorbing gases further increase the complexity of twilight RT. An example is the analysis of CI at wavelengths where the absorption of NO2 or O3 is significant . In such cases, the presence of an absorbing layer modifies the total optical depth and therefore affects PTotal. Consequently, its contribution must be explicitly incorporated into the analysis. The absorption opacity due to gases can be directly included in Eqs. () and () if vertical profiles are available, or alternatively parametrized using a Gaussian profile similar to that adopted for the cloud layer. Assuming a Gaussian-type parametrization, the contribution to P1 requires a term analogous to JA and JM, following the same procedure used in Eq. () but with a Gaussian dependence on z. For the calculation of P3 through τ2, the absorption opacity of gases (τAbs) can be incorporated into Eq. () through 17τAbs(z,λ)=τAbsTotal2erf2z-hAbsΔhAbs-erf-2hAbsΔhAbs, where τAbsTotal, hAbs, and ΔhAbs denote the total optical depth, central altitude, and geometrical thickness of the absorbing layer, respectively.

However, in many cases these gases exhibit a pronounced seasonal variability in both their vertical distribution and total column abundance (see, e.g., ), and an inaccurate representation of such temporal cycles may lead to significant errors in the retrieved cloud altitude. To illustrate this point, we performed a series of sensitivity tests considering a Gaussian absorbing layer centred at 20 km, with a geometrical thickness of 10 km, and a total absorption optical depth ranging from 0.01 to 0.1 at λ1=400 nm, while assuming negligible gaseous absorption at λ2. These calculations were not intended to reproduce a particular atmospheric absorber, but rather to provide a generic test case for investigating the influence of gaseous absorption on CI-based cloud-height retrievals. The simulations show that the displacement of SZAmax⁡ generally increases as the spectral separation between the two wavelengths decreases. For example, for a cloud located at 18 km and an absorption optical depth of 0.05, the maximum displacement reached approximately 0.43° for the 400/500 nm wavelength pair, corresponding to an error of about 2 km in the retrieved cloud height. This value should not be regarded as universal, since the impact depends on the selected wavelengths, the cloud altitude, and the optical depth and vertical distribution of the absorbing species. To mitigate these modelling uncertainties, one possible approach is to select λ1 and λ2 outside the main absorption bands of the gases present, thereby simplifying the RT modelling.

However, this criterion – avoiding absorption bands while simultaneously maximizing the spectral separation between λ1 and λ2 in order to enhance the robustness of the retrieval – is not always feasible in practice. Figure  illustrates this limitation by showing, as representative examples, the absorption cross sections of NO2 (green) and O3 (blue), derived from and . The spectral bands selected in previous CI-based retrieval studies are also indicated for reference. Figure  shows that a suitable choice for λ1 is around 400 nm, since the cloud signature at this wavelength is less pronounced than at longer wavelengths, which leads to a more pronounced CI∗ maximum. Figure  shows that selecting λ1 in the blue spectral region is problematic due to the significant absorption of NO2 in this range. Although one could shift λ1 towards 500–550 nm to reduce this effect, such a displacement would require a corresponding shift in λ2 in order to preserve sufficient spectral separation. This would extend the required spectral range beyond the capabilities of many instruments. For instance, employed measurements from a twin (UV/Vis) Multi-Axis Differential Optical Absorption Spectroscopy (MAX-DOAS) system covering the spectral range from 415 to 542 nm, which limits the available wavelength combinations for CI-based retrievals. It is important to note that the absorbers shown in Fig.  are representative of the atmospheric conditions considered in . Under different atmospheric conditions, other absorbers, such as H₂O, may also need to be taken into account if their vertical distribution extends sufficiently high to influence the CI-based retrievals.

The objective of this section is to define a new CI formulation that enables the detection of clouds during twilight and the estimation of their altitude, while avoiding the need to select λ1 in spectral regions that are problematic for RT simulations. As shown in Figs.  and , when λ1=400 nm (or similar) is employed, the CI∗ maximum occurs at an SZA at which the cloud is already completely in shadow. Thus, at λ1 and for SZA values around or larger than SZAmax, PTotal does not differ significantly from that expected in a purely Rayleigh-scattering atmosphere. Indeed, Fig. a and b show the relative SZA-variation curves 1PTotal(λ,SZA)∂PTotal(λ,SZA)∂SZA for both the cloud + Rayleigh scenario and the pure Rayleigh case at 400, 700, and 900 nm. In these cases, it can be seen that the relative variations in the pure Rayleigh atmosphere closely resemble those of the cloud + Rayleigh case at 400 nm in the vicinity of the intersection of the curves that determines SZAmax. From this reasoning, it is defined a Rayleigh-referenced color index, CIR, as 18CIR(SZA)=PTotal(λ,SZA)PTotalR(λ,SZA), where PTotalR(λ,SZA) denotes the total signal simulated (or measured, if radiances are used) under purely Rayleigh-scattering conditions at wavelength λ. The principle of cloud detection using Eq. () is analogous to that of CI∗, but with the advantage that it requires observations at only a single wavelength. This allows its application to instruments operating at a single wavelength (or within a narrow spectral band that does not permit the selection of two widely separated wavelengths), thereby simplifying the choice of spectral regions where the observations are less affected by gas absorption. Moreover, provided that the molecular scale height and background atmospheric composition remain stable over time, the SZA dependence of the twilight intensity at a given wavelength in a purely Rayleigh atmosphere is expected to be nearly invariant. Therefore, the same Rayleigh reference signal can be used for the calculation of CI. To show this, Fig. c illustrates CI∗ computed for λ1=400 nm and λ2=700 and 900 nm, together with CIR calculated at λ=700 and 900 nm for the same cloud scenario as in Fig. . These results clearly demonstrate that, despite being a considerably simpler index, CIR effectively captures the presence of high-altitude clouds during twilight and enables their height to be estimated through RT simulations at a single wavelength.

The analyses presented in the previous sections have allowed us to identify the key processes controlling the behaviour of the CI and have led to the definition of a new index, the Rayleigh-referenced color index, CIR, which may offer practical advantages over the traditional CI based on measurements at two wavelengths. However, it remains necessary to assess how well the single-scattering formulation reproduces the results obtained from a multiple-scattering model in spherical geometry through a comparison of the simulated zenith intensity at twilight. For this purpose, a Monte Carlo RT model was employed, previously used to simulate twilight clouds on Earth , Mars , and Titan . In order to improve the consistency of the comparison, the vertical and horizontal structure of the cloud is the same in both models. The cloud is assumed to follow a Gaussian distribution not only in the vertical, as described in Eq. (), but also in the horizontal direction. In this formulation, the cloud extinction coefficient decreases with the horizontal distance from the observer zenith according to: 19βC(z,ρ,λ)=τC(λ)ΔhC2πexp⁡-2(z-hC)2ΔhC2exp⁡-ρ22RC2, where RC defines the horizontal extent of the cloud and ρ is the horizontal distance from the local vertical axis of the observer. The horizontal distance is approximated as ρ(l)=r(l)ζ(l), where r(l) is the radial distance along the solar path and ζ(l) is the angular deviation from the local vertical. Here, l denotes the path length along the solar trajectory, as defined in Eq. (). For clarity, the dependence of l on z and SZA has been omitted in the notation. The angular deviation ζ(l) is obtained from the spherical geometry of the solar path as: 20ζ(l)=arccos⁡RT+z+lcos⁡(SZA)r(l)=arccos⁡RT+z+lcos⁡(SZA)l2+(RT+z)2+2lcos⁡(SZA)(RT+z). Note that both P2 and P3 are evaluated along the vertical direction of the observer (i.e., at ρ=0). As a result, the horizontal Gaussian distribution does not affect these terms, since the cloud extinction reaches its maximum along the symmetry axis. In contrast, the horizontal structure of the cloud directly impacts the computation of P1, as the incoming solar radiation propagates along a slanted path and samples regions with ρ≠0. However, since this contribution enters as an additional extinction term, it can be naturally incorporated into the computation of P1 by including it in the optical depth integral defined in Eq. (): 21P1(z,SZA,λ)=exp⁡-JA(z,SZA,HA)S0A(λ)+JM(z,SZA,HM)S0M(λ)+∫0l(z,SZA)βC(z(l′),ρ(l′),λ)dl′, Figures a–e show simulations of the variation of the normalized intensity with SZA obtained using the Monte Carlo RT model and the single-scattering approach described in Sect. , for a cloud layer with τC=0.05 and hC=12,14,16,18 and 20 km. For all simulations, the wavelength was fixed at λ=700 nm, the horizontal scale at RC=20 km, and the cloud thickness at ΔhC=2 km. The Monte Carlo simulations are limited to SZA=95° due to the increased computational cost associated with the adopted cloud geometry. In contrast to spherically symmetric cloud layers, the horizontally localized cloud considered here breaks the symmetry between SZA and latitude, requiring independent simulations for each SZA. This range nevertheless fully covers the region of interest, since the SZAmax for each simulated cloud height occurs at SZA below 95°, as indicated by the dashed black line.

The comparison shown in Fig.  demonstrates that the single-scattering model (hereafter SSM) reproduces the Monte Carlo simulations with very good agreement across the full range of cloud altitudes considered. Small differences between both approaches can be observed at larger SZAs (typically above ∼94°), where the deviations slightly increase. However, the SZAs relevant for the determination of the CI maximum occur at lower values, as indicated by the dashed black line marking SZAmax⁡ in each panel. Therefore, within the range of interest, the approximations adopted in the SSM remain well within acceptable limits. These results not only support the validity of the formulation developed in the previous sections for investigating the physical origin of the CI maximum, but also demonstrate that the SSM provides a reliable and computationally efficient alternative for the analysis of observational data.

Figure  shows a comparison similar to that presented in Fig. , but in this case varying the wavelength from 400 to 800 nm in steps of 100 nm, while keeping the cloud height fixed at hC=16 km. The remaining parameters are kept identical to those used in Fig. . In this scenario, the main wavelength-dependent contribution in the model arises from Rayleigh scattering, whose optical depth varies strongly with wavelength. As a result, the attenuation of the signal along the solar path changes accordingly, while the cloud contribution remains unchanged. This behaviour is clearly reflected in the curves, where the cloud-related feature becomes more pronounced as the wavelength increases. This is consistent with the decreasing contribution of Rayleigh scattering at longer wavelengths, which enhances the relative impact of the cloud on the observed signal. As in the comparison shown in Fig. , the SSM reproduces the Monte Carlo simulations with very good agreement across all wavelengths. In particular, the change in the structure of the cloud-induced feature in the intensity profiles is accurately captured by the SSM. Therefore, it is concluded that the main findings derived from the comparison in Fig.  can be consistently extended to different wavelengths.

Finally, a similar comparison was performed by varying the cloud optical depth from τC=0 (pure Rayleigh atmosphere) up to τC=0.3, as shown in Fig. . In all simulations, the cloud height is fixed at hC=16 km and the wavelength at λ=700 nm, while the remaining parameters are kept identical to those used in Fig. . The lowest non-zero value considered for the cloud optical depth was τC=0.01. As discussed in the context of Fig. , the impact of the cloud is expected to be more pronounced at longer wavelengths, which would allow the detection of optically thinner clouds. However, since the objective of this section is to intercompare both models rather than to establish a detection limit, the wavelength and cloud optical depth range were fixed accordingly. The comparison shows that the agreement between both models progressively degrades as the cloud optical depth increases. While the SSM reproduces the Monte Carlo results very well for optically thin clouds (τC≲0.1), noticeable differences appear for larger optical depths. These discrepancies can be attributed to multiple-scattering effects within the cloud, which are not accounted for in the SSM.

In order to assess the impact of multiple-scattering effects on the retrieval of cloud height, Fig.  shows the relative rate of variation of PTotal(λ,SZA) with respect to SZA, computed for τC=0.3 and cloud altitudes of hC=14, 16, and 18 km, together with the corresponding pure Rayleigh atmosphere. The retrieved value of SZAmax⁡ is determined by the intersection between the cloud and Rayleigh relative-variation curves. In these simulations, the Monte Carlo calculations were limited to SZA<94°. This choice is motivated by the high sensitivity of the relative variations of PTotal(λ,SZA) to the SZA sampling, which requires a fine angular resolution (ΔSZA=0.02°). As a consequence, the computational cost increases significantly as ΔSZA decreases, since a larger number of photons must be simulated to maintain a low level of statistical uncertainty. For this reason, the analysis is restricted to the SZA range of interest around the Rayleigh cutoff, where the determination of SZAmax⁡ is most relevant.

The results show that, despite the differences in the simulated normalized zenith intensities between the Monte Carlo model and the SSM observed in Fig. , the relative variations around the Rayleigh cutoff remain very similar in both cases. In particular, differences in SZAmax⁡ are found to be on the order of 0.12–0.15°, which correspond to uncertainties in the retrieved cloud height below 1 km, as will be discussed in the following sections. Thus, it follows that for cloud optical depths below τC∼0.3, the SSM is able to reproduce the Monte Carlo simulations with a satisfactory level of accuracy for the purpose of cloud-height retrieval. For optical depths approaching τC=0.3, the discrepancies increase, leading to uncertainties in cloud height of up to about 1 km. Based on this analysis, it is established that the SSM provides a reliable framework for the interpretation of twilight zenith intensity measurements and the retrieval of cloud heights for τC≲0.3. This range of optical depths is consistent with typical values reported for cirrus and PSCs (see, e.g., ), highlighting the potential of the SSM for application to real observations while avoiding the computational cost of full Monte Carlo simulations.

First, the dependence of the cloud geometrical thickness on SZAmax⁡ is investigated. Figure a shows the relative rate of variation of PTotal with respect to SZA at 700 nm for different values of the cloud geometrical thickness, ΔhC=1,2,3, and 4 km. In each case, SZAmax⁡ is defined by the intersection between the corresponding curve and the pure Rayleigh reference (black dashed line). The results show that increasing the cloud geometrical thickness significantly affects both the depth and the width of the minimum. However, these changes occur at SZA values larger than SZAmax⁡. At SZAs close to SZAmax⁡, all curves nearly overlap, indicating that the position of the CIR maximum is largely insensitive to variations in ΔhC. This behaviour is confirmed quantitatively by the values of SZAmax⁡. For a cloud height of 16 km, variations in ΔhC lead to changes smaller than 0.05°, which is comparable to the angular resolution used in these simulations. For lower cloud heights (12 and 14 km), slightly larger differences in SZAmax⁡ are observed when reducing the geometrical thickness from 2 to 1 km, reaching up to 0.15 and 0.1°, respectively. However, for ΔhC≥2 km, the variations in SZAmax⁡ remain below 0.05° in all cases. These angular differences translate into uncertainties in the cloud height smaller than 1 km.

Next, the impact of the cloud horizontal scale, RC, is investigated. Figure b shows an analysis analogous to that in Fig. a, but varying RC from the baseline value of 20 km to 10, 50, 100, and 200 km. In all cases, the cloud height is fixed at 16 km and the optical thickness at τC=0.05. As expected, RC has a noticeable impact on the relative SZA-variations of PTotal. As RC increases, the extinction experienced by photons scattered along the line of sight becomes larger, reducing the contribution of P1. This results in a deeper minimum and modifies the shape of the curves. However, in the SZA range between approximately 92.5 and 93°, all curves tend to converge. Importantly, this convergence occurs close to the intersection with the pure Rayleigh curve, which defines SZAmax⁡. As a result, the variations in SZAmax⁡ remain limited despite the significant changes in the curve shape. By comparing the different cases, it is obtained a maximum variation of ΔSZAmax⁡≈0.1°, which translates into an uncertainty in cloud height of approximately 800 m. Similar analyses performed for other cloud heights show that the variations in SZAmax⁡ can reach up to 0.2° for hC=12 km. This increase is consistent with the fact that, at lower altitudes, the CI maximum occurs at smaller SZA, where the solar beam propagates more horizontally and traverses a larger portion of the cloud before scattering. Despite these larger angular differences, the resulting uncertainty in cloud height remains of the same order (∼800 m).

To better understand this behavior, Fig. c shows the variation of SZAmax⁡ as a function of cloud height for τC=0.05, τC=0.1, and τC=0.3. The figure also includes the tangent to the τC=0.05 curve at hC=10 km. It can be seen that the hC–SZAmax⁡ relationship systematically lies below this tangent, indicating a sub-linear behavior. This implies that the sensitivity of SZAmax⁡ to cloud height decreases as altitude increases. This explains why, in the RC sensitivity test, a variation of ΔSZAmax⁡≈0.2° at 12 km leads to a cloud-height variation comparable to that produced by a smaller ΔSZAmax⁡≈0.1° at 16 km. Figure c also shows that the hC–SZAmax⁡ relationship remains essentially unchanged when the only parameter varied is the cloud optical thickness. This indicates that, in practical applications, τC can be kept fixed without introducing significant errors in the cloud-height retrieval. Consequently, any residual uncertainty associated with this parameter is expected to arise primarily from the use of the single-scattering model instead of the Monte Carlo approach. As shown in Sect. , these differences remain small and become only noticeable for optically thicker clouds (τC∼0.3), while still leading to cloud-height uncertainties below 1 km.

Finally, the impact of the cloud phase function on SZAmax⁡ is investigated. In particular, phase functions are computed using droxtals (quasi-spherical ice crystals with rounded edges) following the model of , considering effective radii reff=5,10, and 20 µm. In this context, reff does not correspond to the effective radius of a volume-equivalent sphere, but rather to a geometrical parameter of a lognormal distribution defined in terms of the maximum crystal dimension (Dmax⁡), with r=Dmax⁡/2. For the baseline scenario, the resulting values of SZAmax⁡ are 92.92, 92.96, and 93.00° for the three considered values of reff. According to the hC–SZAmax⁡ relationship shown in Fig. c, these variations correspond to changes in cloud height of less than ∼ 300 m. These results indicate that the impact of the cloud phase function on the retrieval is non-negligible, but remains smaller than that of other parameters explored in this study. Therefore, uncertainties associated with the microphysical properties of the cloud particles are expected to have a secondary influence on the retrieved cloud height.

In all simulations presented so far, it has been considered a pure Rayleigh atmosphere together with the presence of a cloud layer. The next step is to investigate the impact of aerosols on SZAmax⁡, and consequently on the retrieved cloud height. To this end, aerosols are introduced as a vertically distributed layer with an exponential decay characterized by a scale height HA. The aerosol optical depth at the surface, τ0A, together with HA, are varied in order to assess their influence on the results.

Figure  shows the variation of SZAmax⁡ with wavelength for a cloud layer with fixed altitude and optical depth of 16 km and 0.05, respectively. Different values of aerosol optical depth, τ0A(550nm)=0,0.05,0.3, and scale height, HA=0.3, 0.6, and 0.9 km, are considered. In these simulations, the spectral dependence of the aerosol optical depth is described using an Ångström exponent of α=1.3. The results show that both the aerosol optical depth and the scale height have a clear impact on SZAmax⁡, and therefore on the inferred cloud height. This effect becomes more pronounced at longer wavelengths (λ≳750 nm), where the Rayleigh optical depth is significantly reduced and the relative contribution of aerosols to the total attenuation becomes more important. However, as shown in Fig. , noticeable differences are also present at shorter wavelengths (e.g. 600 nm), indicating that aerosol effects are not negligible even in the visible range. To investigate the impact of the Ångström exponent on SZAmax⁡, Fig.  shows simulations similar to those in Fig. , but for α=0.5. In this case, the aerosol optical depth decreases more slowly with wavelength over the spectral range considered, resulting in larger values at longer wavelengths. As in the previous case, larger values of τ0A and HA lead to stronger variations in SZAmax⁡, reaching up to ∼0.6° at λ=900 nm for HA=0.9 km and τ0A(550nm)=0 and 0.3. By contrast, only minor differences are observed when varying the Ångström exponent between α=1.3 and α=0.5 (comparison between Figs.  and ), indicating that τ0A and HA are the dominant parameters controlling the variability of SZAmax⁡ in these simulations.

To quantify the impact on the retrieved cloud height, it is used a relationship similar to that shown in Fig. c. Under this assumption, a variation of ΔSZAmax⁡=0.6° corresponds to an uncertainty of approximately 4 km in cloud altitude. This implies that, in a realistic aerosol scenario with τ0A(550nm)=0.3 and HA=0.9 km, neglecting aerosols in the analysis (i.e., assuming τ0A(550nm)=0) would lead to an error of about 4 km in the retrieved cloud height. However, at 650–700 nm the variations in SZAmax⁡ are significantly smaller, with maximum values of ∼0.08-0.14° for HA=0.9 km and τ0A(550nm)=0 and 0.3. These variations in SZAmax⁡ correspond to uncertainties of approximately 300–700 m in the retrieved cloud height. For smaller values of HA, the differences in SZAmax⁡ tend to zero, and therefore the influence of aerosols on the cloud-height retrieval becomes negligible.

These examples demonstrate that, depending on the observational configuration (i.e., number of channels and selected wavelengths), the presence of aerosols and their vertical distribution can significantly affect the retrieval of cloud height. In this context, the selection of wavelengths around 650–700 nm is preferable, as indicated by the results presented above. However, due to the increase in Rayleigh optical depth at shorter wavelengths, the detection of lower-altitude clouds (below ∼16 km) becomes progressively more challenging as shorter wavelengths are considered. Multi-wavelength observations in the 650–900 nm spectral range provide an optimal compromise. Such observations would allow the spectral variation of SZAmax⁡ to be exploited in order to improve the accuracy of cloud-height retrievals, taking into account not only the presence of aerosols but also other cloud properties analyzed in the previous section.

These results highlight that analyses of this type are essential for defining an optimal cloud-height retrieval strategy tailored to a given observational configuration. In this context, the use of the SSM is particularly advantageous due to its negligible computational cost compared to Monte Carlo RT models. In practice, the computational efficiency of the SSM enables extensive sensitivity studies across a wide range of atmospheric scenarios, including variations in aerosol load, vertical distribution, and wavelength selection. Moreover, given its validity for optically thin clouds (τC≲0.3), the SSM allows full spectral simulations over the visible range to be performed at very low computational cost.

Finally, in the previous section, the differences between the SSM and the Monte Carlo model were shown to become significant mainly for cloud optical depths of τC≳0.3, due to multiple-scattering effects within the cloud that are not included in the SSM and that primarily affect the scattering term P2. By contrast, the aerosol contribution enters mainly through P1, for which the SSM reproduces the pure-Rayleigh behaviour accurately. Therefore, as long as the cloud optical depth remains within the range established in Sect. , the agreement between the SSM and the Monte Carlo model is not expected to change substantially in the presence of aerosols. This further supports the use of the SSM as a reliable and efficient tool for exploring the impact of aerosols and other atmospheric parameters on cloud-height retrievals.