Process-based models for inversion and forward estimation have supported our understanding of earth, environment, and hazards processes. These methods are often applied on spatial data without accounting for their spatial nature and uncertainty. Using an example of reconstructing past volcanic eruption characteristics and associated tephra fallout from different sets of field observation, we demonstrate the importance of making the best use of data-related uncertainty and spatial information in inversion and forward estimation. We present strategies for: (1) the selection of appropriate cost functions accounting for their behaviour and implied distribution of residuals, (2) the treatment of differential uncertainty when combining multiple data, and (3) the leveraging of both model and data when estimating the spatial distribution of output. Results show that a data-informed choice of cost function and accounting for uncertainty and spatial characteristics of data leads to consistent improvements in model predictive performance for both inversion and forward models.