Improvements to Asteroseismic Fitting of White Dwarfs in the Gaia Era

White dwarf asteroseismology aims to constrain the interior structures of pulsating white dwarf stars by fitting observed pulsation periods to those computed for stellar models. However, there remain several obstacles for achieving reliable results with the model grid-fitting approach. We simulate data for a parameter recovery experiment in one and two dimensions (mass and effective temperature) to demonstrate improved methodologies that address challenges for characterizing degenerate asteroseismic solutions. We show that interpolating model periods onto a finer grid can adequately resolve asteroseismic solutions that would be missed with sparse model grids. Incorporating absolute magnitude from Gaia astrometry into the statistical fitting is shown to reduce solution degeneracy. We define a seismic solution as a solution to the mode identification problem, and we show that fitting to consistent model pulsation modes can isolate each candidate solution in degenerate solution space. A criterion based on $χ^2$ for what should be considered a reasonable fit is adopted, and we identify all combinations of model periods that meet this criterion, not only those nearest to the measured periods. Finally, we demonstrate how fitting Gaussians to probability distributions allows for the robust characterization of each candidate solution, including uncertainties. Our accurate characterization of the degenerate solution landscape is supported by the comparison of our solutions to a direct marginalization of the likelihood function. These fitting approaches can be generalized to higher dimensions, where there are more than two free parameters.