On the origin of the twilight color index maximum and its application to cloud-height retrieval

Abstract. A number of previous studies have demonstrated the capability of detecting high-altitude clouds during twilight using the color index (CI), defined as the ratio of zenith intensities at two different wavelengths, typically selected in the visible range or near infrared (NIR). When high clouds are present, a maximum or minimum (depending on the wavelengths selection) is observed in the CI signal (Sarkissian et al., 1991; Toledo et al., 2016). These studies also showed that the solar zenith angle (SZA) at which the CI maximum or minimum occurs (SZA_max) strongly depends on cloud altitude, enabling cloud-height retrieval through comparison with radiative transfer (RT) simulations. Twilight conditions require RT simulations in spherical geometry, which are computationally expensive. In this work, we introduce a single-scattering formulation of the CI that provides a physically transparent framework for identifying the mechanisms that determine the SZA of the CI maximum and, consequently, the inferred cloud altitude. The simplified formulation is explicitly compared with Monte Carlo RT simulations in spherical geometry and is shown to accurately reproduce the behavior of SZA_max over a wide range of conditions relevant for high-altitude clouds. In particular, the model provides reliable cloud-height estimates for cloud optical depths up to τ_C ≲ 0.3. Within this single-scattering formulation, we demonstrate that SZA_max occurs at the SZA for which the relative SZA-variations of the zenith intensity at the two selected wavelengths become equal, thereby explaining the emergence of the extremum in differential terms. However, achieving a well-defined CI extremum requires selecting two wavelengths with sufficient spectral separation, typically spanning distinct regions of the visible--NIR spectrum. To overcome this spectral dependence, we introduce a Rayleigh-referenced color index (CI^R), defined as the ratio between the measured zenith intensity and the corresponding intensity expected for a purely Rayleigh-scattering atmosphere at the same wavelength. This index reproduces the characteristic extrema associated with high-altitude clouds while requiring simulations and observations at only a single wavelength. The proposed formulation facilitates extensive sensitivity studies and provides greater flexibility in spectral selection, particularly in regions affected by gas absorption.