Deriving historical equilibrium-line altitudes from a glacier length record by linear inverse modelling

Glaciers have fluctuated in historic times and the length fluctuations of many glaciers are known. From these glacier length records, a climate reconstruction described in terms of a reconstruction of the equilibrium-line altitude (ELA) or the mass-balance can be extracted. In order to derive a climate signal from numerous glacier length records, a model is needed that takes into account the main characteristics of a glacier, but uses little information about the glacier itself. Therefore, a simple analytical model was developed based on the assumption that the change in glacier length can be described by a linear response equation. Historical length observations, the climate sensitivity and the response time of a glacier were needed to calculate historical equilibrium-line altitudes. Both climate sensitivity and length response time were calculated from a perturbation analysis on the continuity equation. The model was tested on 17 European glacier length records. The results revealed that the ELA of most glaciers increased on average 54 m between AD 1920 and 1950. The results of the analytical model were compared to mass-balance reconstructions calculated with a numerical flowline model and derived from historical temperature and precipitation records. The findings lead us to believe that the analytical model could be very useful to gain information about the historical mass-balance rates and ELAs.