Avalanching is a prominent source of accumulation on glaciers that have high and steep valley-walls surrounding their accumulation zones. These glaciers are typically characterised by an extensive supraglacial debris cover and a low accumulation area ratio. Despite an abundance of such glaciers in the rugged landscapes of the High Himalaya, attempts to quantify the net avalanche contribution to mass balance and its long-term variation are almost missing. We first discuss diagnostic criteria to identify strongly avalanche-fed glaciers. Second, we develop an approximate method to quantify the magnitude of the avalanche accumulation exploiting its expected control on the dynamics of these glaciers. The procedure is based on a simplified flowline model description of the glacier concerned and utilises the known glaciological mass-balance, velocity and surface-elevation profiles of the glacier. We apply the method to three Himalayan glaciers and show that the data on the recent dynamics of these glaciers are consistent with a dominant contribution of avalanches to the total accumulation. As a control experiment, we also simulate another Himalayan glacier where no significant avalanche contribution is expected, and reproduce the recent changes in that glacier without any additional avalanche contribution.