A simple analytical model for the reflection function of flat glacier ice surfaces and its application for optical remote sensing of glaciers

• It is shown for the first time that the spectral signature of the glacier ice can be modelled using four parameters; it is suggested that these parameters must be reported in the output of respective glacier ice retrieval algorithms; • The spectral reflectance measurements of the glacier ice at Mitdalsbreen Glacier (Norway) has been performed and analysed using the developed theory; • The technique for the determination of the glacier properties and albedo from space using the developed theory has been applied to hyperspectral spaceborne EnMAP observations of Mitdalsbreen Glacier. We propose simple analytical equations for the modeling of clean and dusty flat glacier ice surfaces, which can be used to inversely derive the parameters of microstructure of flat bare glacier ice and snow using both ground – based and spaceborne observations of the hyperspectral solar reflectance. The retrievals are based on the asymptotic radiative transfer equations valid for the case of weak light absorption in the semi-infinite turbid medium. The light reflection at the air - ice and ice - air interfaces is fully accounted for. To demonstrate the validity of the approach, the derived equations are exemplarily applied to both ground – based and EnMAP satellite measurements over the Hardangerjokulen glacier (Norway). A number of important parameters controlling spectral signatures of the snow and glacier ice surfaces have been derived. The ground-based measurements show that the theoretical formulation presented in this work can be used to represent the solar light spectral reflectivity of glaciers. The application to satellite hyperspectral imagery shows that this novel technique allows for the determination of the glacier ice albedo (spectral, broadband) based on spaceborne glacier ice reflectance measurement. Additionally, the results show that not spectrally neutral soot but rather deposited atmospheric dust which enhances the absorption towards UV (due to the presence of iron oxides) is responsible for the light absorption by snow for the case studied. Spatial distribution maps of ice grain diameter and dust concentration are derived over the glacier. These findings show that the analytical theory presented in this work can support further research on the characterization and monitoring of glaciers based on current and upcoming hyperspectral remote sensing satellite missions.