Breaking the Depth–Radius Degeneracy in Joint GPR–Quantum-Gravimetry Subsurface Void Detection via Multi-Frequency Mie Dispersion Sensing

Subsurface void detection (tunnels, sinkholes, mine workings, buried cavities) is commonly approached by combining Ground Penetrating Radar (GPR), which locates dielectric boundaries with high spatial resolution but is agnostic to mass, with quantum cold-atom gravimetry, which measures the true mass deficit of a void but is comparatively slow and spatially coarse. We derive the joint Fisher Information Matrix (FIM) for estimating void depth (d), radius (r), and density contrast (Δρ) from combined GPR travel-time and gravimetric residual-acceleration data, and compute the Cramér-Rao Bound (CRB) — the best possible estimator precision — across six representative underground scenarios. We show that the simplest joint model contains a near-exact depthradius degeneracy: GPR travel time constrains only the combination u = d−r, leaving d and r jointly poorly determined (tens of centimeters to tens of meters of uncertainty, depending on scenario) even with gravimetric data included. We then show that a naive x adding single-frequency GPR reection amplitude as a second data channel appears to resolve the degeneracy by a factor of 2040×, but this improvement is illusory: once the antenna/ground calibration constant (A0) is honestly treated as an unknown nuisance parameter, the amplitude channel's contribution collapses exactly back to the degenerate baseline. We show this null result is a structural property, not a modeling failure: the log-amplitude channel's residual sensitivity to radius is a constant oset across survey positions, structurally identical to the calibration nuisance direction a size/calibration degeneracy directly analogous to the ux/distance degeneracy in photometric astronomy. Finally, we show that a genuine multi-frequency GPR sweep (100900 MHz), analyzed through full Mie scattering theory for the void's frequency-dependent backscattering eciency Qback(xs), provides real, non-degenerate shape information that no single unknown calibration constant can mimic. This restores 24 orders of magnitude improvement in depth/radius precision (sub-centimeter CRB across all six scenarios) even after honestly marginalizing the unknown calibration, and indirectly improves density-contrast precision by an order of magnitude as a consequence of the now well-constrained radius. All instrument noise oors are drawn from published eld-measured or commercial-instrument sources rather than assumed round numbers, and we state explicitly where no such source exists. We present this as a theoretical and simulation-based feasibility case for combining multi-frequency GPR with quantum gravimetry as a joint sensing modality, with an explicit statement of the modeling assumptions under which the result hold