In the past three decades, an abrupt acceleration in the thaw of ice wedges has spurred rapid surface deformation (i.e., thermokarst) in polygonal tundra landscapes spanning the Arctic. The ensuing conversion of low-centered polygons (LCPs) and flat terrain into high-centered polygons (HCPs) has profound impacts on regional hydrology and carbon fluxes between the soil and atmosphere. However, pathways of ice wedge degradation and the stability of the deformed terrain are uncertain, complicating efforts to project feedbacks on global climate change. In this dissertation, I explore trajectories of surface deformation in ice wedge polygons, using a combination of calibration-constrained numerical experiments, remote sensing, and machine learning. In the first two chapters, numerical simulations of the soil hydrologic and thermal regimes reveal that, relative to terrain unaffected by thermokarst, the permafrost beneath HCPs tends to be well-buffered against climate extremes, promoting landscape stability. Ice wedges at HCP boundaries are less vulnerable to thaw during warm summers, reinforcing prior field-based observations that thermokarst is typically a self-arresting process. Simultaneously, the cooling of thermokarst-affected ice wedges in winter tends to be inhibited by snow accumulation in degraded troughs, reducing the likelihood of renewed ice wedge cracking and restoration of LCP microtopography. Overall, these results indicate that the microtopography of polygons already affected by thermokarst will likely remain stable over the next century. In the second half of this dissertation, a novel machine-learning-based tool is introduced for delineating and measuring the microtopography associated with ice wedge polygons in high-resolution digital elevation models. The tool is used to map polygonal geomorphology across ~1,000 kmΒ² of tundra south of Prudhoe Bay, Alaska, visualizing in unprecedented detail the heterogeneous extent to which thermokarst has affected a modern polygonal landscape. This map of polygonal geomorphology provides useful context for upscaling point- to plot-scale observations of gas exchange in ice wedge polygons to landscape-scale estimates of carbon fluxes. It also provides an extensive baseline dataset for quantifying contemporary rates of land surface deformation, through future surveys at the site.Geological Science