Inferring aerosol optical depth at unmeasured wavelengths from ground-based spectral photometer data: uncertainty-consistent regression, sensitivity tests, and application to real data

Aerosol optical depth (τ) is routinely reported at wavelengths that are not directly measured by ground-based sun photometers, in particular at 550 nm for satellite validation and at longer wavelengths for short-wave infrared applications. These values are typically obtained by spectral interpolation or extrapolation, most often using linear or quadratic regressions in logarithmic space. However, the uncertainty structure of such regressions is frequently treated incorrectly, because measurement uncertainties in τ are absolute and approximately wavelength-independent in linear space, and therefore become wavelength-dependent in logarithmic space. As a result, the measurement covariance matrix must be explicitly accounted for in log–log regression, although this is rarely done in practice. This study provides both a formal and a practical framework for estimating τ at non-measured wavelengths together with its associated uncertainty. A rigorous formulation is presented for linear and quadratic regression in logarithmic space, including the propagation of random and systematic (bias-related) errors from the original spectral measurements to the interpolated or extrapolated wavelength.

Sensitivity analyses based on synthetic aerosol optical depth spectra generated with the GRASP forward model are used to compare six different approaches for deriving τ(550) and τ(2000), including linear and quadratic regressions over different spectral ranges as well as the GRASP-AOD method. When the covariance matrix is treated correctly, quadratic log–log regression is found to be the most robust method for estimating τ(550), and its results become essentially independent of the chosen spectral range. In contrast, when the covariance matrix is neglected, the same regression becomes highly sensitive to the selected wavelengths, and artificially improved performance is obtained when restricting the fit to the central AERONET channels. These findings are confirmed using real AERONET observations. When the full covariance treatment is applied, differences between estimates obtained using different spectral ranges remain below 0.002 at all sites analysed. When it is ignored, root-mean-square differences exceeding 0.01 are observed at sites dominated by fine-mode aerosols.

Finally, the uncertainty propagation framework is applied to real data and shows that the uncertainty of interpolated τ follows the expected Beer–Lambert law governing sun-photometer measurements, scaling with optical air mass. This provides an independent validation of the formal error model. Overall, this work establishes a consistent methodology for spectral interpolation and extrapolation of τ, ensuring both accurate values and physically meaningful uncertainties for satellite validation and related applications.