We present a general method for approaching inverse problems for bathymetric determination under shoaling waves. We run the Korteweg-de Vries (KdV) model for various bathymetric representations while collecting data in the form of free-surface imagery and time series. The sensitivity matrix provides information on the range of influence of data on the parameter space. By minimizing the parameter variances, three metrics based on the sensitivity matrix are derived that can be systematically used to make choices of experiment design and model parameterization. This analysis provides insights that are useful, irrespective of the minimization scheme chosen for inversion. We identify the characteristics of the data (time series versus snapshots, early time measurements versus long-duration measurements, nearshore measurements versus offshore measurements), and model (bathymetry parameterizations) for inversion to be possible. We show that Bruun/Dean and Exponential bathymetric parameterizations are preferred over polynomial parameterizations. The former can be used for inversion with both time series and snapshot data, while the latter is preferably used only with snapshot data. Also, guidelines for time separation between snapshots and spatial separation between time series measurements are derived.