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Using a simplified two-stage ice sheet model, we explore the potential of statistical data assimilation methods to improve predictions of glacier melt, which has significant implications for reducing uncertainty in projections of sea level rise. Through twin experiments utilizing artificial data, we find that the ensemble Kalman filter improves simulations of glacier evolution initialized with incorrect initial conditions and parameters, providing us with better predictions of future glacier melt. We explore the number of observations necessary to produce an accurate model run. We also explore optimal observation assimilation schemes, and determine that deviations from the true glacier response that stem from having few data points in the pre-satellite era can be corrected with modern observation data. Our results show that statistical data assimilation methods have great potential to improve complex glacier models using real-world observations.