Geomorphic and geophysical evidence supports a debris-covered glacier origin for a suite of landforms at the mid-latitudes of Mars, including lobate debris aprons (LDA), lineated valley fill (LVF), and concentric crater fill (CCF). These large reservoirs of ice and their near-surface structure provide a rich record for understanding the planet's climate and history of global volatile exchange over the past billion years. LDA, LVF, and CCF are also potential sites for future robotic and human missions but the accessibility of glacial ice for direct sampling and in situ resource utilization depends largely on the geotechnical properties of the surface debris ("supraglacial debris"), including its thickness, grain sizes, and density structure. The physical properties of this supraglacial debris layer have been poorly constrained. We use images of morphology, digital elevation models, thermal inertia data, and radar sounding data to probe the near surface of LDA, LVF, and CCF in Deuteronilus Mensae in order to place constraints on the sources, grain sizes, thickness, and stratigraphy of supraglacial debris. We find evidence for at least a two-layer stratigraphy. Layered mantle consisting of atmospherically emplaced dust and ice superposes boulder-rich sediment sourced by rockfalls glacially transported downslope. High thermal inertia, boulder-rich termini and debris bands reminiscent of medial moraines are found throughout the study region, supporting a rockfall origin for at least a fraction of the debris exposed at the surface. This supraglacial debris layer would have thickened with time from sublimation of glacial ice and liberation of englacial sediment and dust. At present, the entire supraglacial debris package is a minimum of a few meters in thickness and is likely tens of meters in thickness in many locations, possibly thinning regionally at lower latitudes and locally thinning toward the headwalls. The lack of terracing or interior structures in craters formed within LDA, LVF, and CCF and the absence of near-surface reflectors in SHARAD radar data further suggest that no strong contrasts in permittivity or strength occur at the interface of the layers.