On the Non-Uniqueness of the Settlement-Based Inverse Problem in Recovering Soil Modulus Profiles from Plate Bearing Test Data: The Case for a Simplified, Poisson Ratio-Calibrated Inversion Method

Non-destructive in situ tests, such as the plate bearing (plate load) test, are widely used to estimate the equivalent deformation modulus (e.g., Ev2) of existing road embankments. However, the depth of influence sampled by such a test is governed by the loading plate diameter, so a single test yields only an average, diameter-dependent modulus rather than the actual variation in stiffness with depth. This study investigates whether systematically varying the plate diameter and inverting the resulting settlement–diameter (dispersion) curves can recover the full depth-dependent stiffness profile, E(z). Synthetic settlement–diameter curves were generated using a Boussinesq-based forward model for four families of reference stiffness profiles, representing normal (stiffness increasing with depth) and reverse (stiffness decreasing with depth) linear and exponential trends, combined with six Poisson’s ratios and five profile slopes/exponents (30 cases per profile family, 120 cases in total). Two inversion strategies were applied to back-calculate E(z) from each dispersion curve: a classical Occam-type, smoothness-constrained (Tikhonov-regularized) nonlinear inversion, and a direct, closed-form simplified inversion method (SIM) based on differencing the apparent-modulus-versus-diameter curve. The results were benchmarked against the known reference profiles. Once calibrated so that its governing parameters depend only on Poisson’s ratio and the shape of the measured dispersion curve, SIM could be applied blindly—without knowledge of the reference profile or a starting model, requiring only an assumed Poisson’s ratio and the established calibration—and recovered E(z) with markedly lower error than Occam’s inversion (WAD = 2.2–4.7% and RMSPE = 2.6–5.8%, versus 7.4–19.1% and 9.4–28.3%, respectively, across the four profile families). For the Poisson’s ratio most typical of earth materials, ν=0.3, the calibration further collapses to a single parameter set common to all four families investigated (I=0.66; c=1.3 for stiffness increasing with depth, c=2.5 for stiffness decreasing with depth), which attains WAD ≤ 4.2% across all four families with no calibration equation at all. Notably, Occam’s inversion reproduced the settlement–diameter curve itself with good accuracy in most cases, yet this close data fit did not guarantee an accurate stiffness profile—a direct manifestation of the intrinsic non-uniqueness of the settlement-based inverse problem. These findings are bounded by their evidence base: noise-free data from the same forward operator used in the inversion, smooth profiles, a calibration evaluated on the cases that produced it, and an Occam comparison specific to L-curve-selected regularization. Within these limits, SIM is a promising alternative to regularized inversion; measurement noise, layered profiles and field validation are the next steps.