Two inversion methods are developed to reconstruct eruption column dynamics from granulometric data of Plinian deposits. They are based on the steady tephra dispersal model from umbrella clouds which spread as gravity currents. The methods provide (1) expansion rate of an umbrella cloud, (2) grain-size distribution at the top of the eruption column, and (3) total amount of ejecta including very fine particles which are extensively dispersed in the atmosphere. The first quantity is correlated to the magma discharge rate. The second quantity provides information about sorting processes during column rise, and hence, about deceleration in the gas thrust region of the eruption column and the timing of column collapse. The third quantity, together with the estimated magma discharge rate, provides the effective duration of the eruption. The two methods are based on the same dispersal model but use different averaging. Combining the two methods can provide additional information about eruption dynamics, such as minor flow structures of the umbrella cloud and fluctuations of eruption rate. The methods are applicable to large-scale eruptions where the radial expansion velocity of the umbrella cloud exceeds wind velocity. The applicability of the methods is tested from the viewpoint of comparison with real-time observations in the companion paper (paper 2). ¿ 2001 American Geophysical Union