Modeling and analyzing scalar fields -- such as terrains or volumetric data -- are central problems in scientific visualization and geospatial computing. These data describe continuous physical phenomena but are typically represented in discrete forms such as regular grids or Triangulated Irregular Networks (TINs), which limit differentiability and complicate multi-scale topological analysis. Conversely, continuous analytical models -- such as spline or PDE-based formulations -- achieve smoothness and compactness but are often computationally expensive, sensitive to sampling irregularities, and unable to guarantee consistent topological structures.

This dissertation proposal aims to advance modeling and topological analysis of terrain and volumetric data by bridging the gap between discrete representations and implicit neural representations (INRs). It first develops a scale-space framework for TINs, grounded in piecewise-linear Morse theory and scale-space theory, to characterize the evolution of critical features across multiple scale levels. This scale-space framework provides a discrete foundation for understanding terrain morphology and topological persistence defined over scale evolution. Building on these insights, the dissertation develops ImplicitTerrain, a continuous neural representation based on classical Morse theory that reconstructs continuous scalar fields from discrete samples with high differentiability and storage compactness. ImplicitTerrain offers an alternative formulation that reintroduces topological analysis into the continuous domain, enabling morphological reasoning through the derivatives of a learned scalar field.

Together, these studies form a cohesive progression -- from discrete topological understanding to continuous neural modeling -- establishing a conceptual and computational basis for topology-aware representation of geospatial and scientific data. In this dissertation proposal, several research directions are planned, including 3D volumetric extensions of the terrain INR framework (ImplicitTerrain), a comparative survey of INR architectures for scalar-field analysis, and topology-regularized INR learning.