Backcalculation inverts falling-weight-deflectometer (FWD) deflection basins into pavement layer stiffness moduli, which drive mechanistic-empirical predictions of pavement life. Tutka et al. (2021) showed, for one three-layer linear-elastic pavement analysed with a dynamic finite-element model, that small random deflection errors which distort the shape of the deflection basin are strongly amplified: a deflection coefficient of variation of 0.74% produced coefficients of variation of 9.18% (asphalt modulus), 27.37% (base course modulus) and 1.92% (subgrade modulus), and 10.20% in predicted pavement life, whereas shape-preserving proportional errors are not amplified. The problem left open by the source, restated but not resolved by the same group in 2024, is to extend this error-sensitivity analysis to realistic pavement structures and material models (modelling factors of real pavements excluded from the 2021 study, such as stress-dependent unbound layers, viscoelastic asphalt, other layer-thickness configurations, temperature effects, interface slip and shallow stiff layers), to convert known measurement-error characteristics into confidence intervals for the backcalculated moduli and - following the source's own propagation of modulus variability to fatigue life - for the resulting predicted pavement life (for example via deflection-basin envelopes), and to mitigate sensitivity to basin-shape distortion in backcalculation, for instance through the averaging of several measurements per road section suggested by the source. The problem remains open.

Background

Nondestructive structural evaluation of road pavements rests on the falling weight deflectometer (FWD), a device that drops a known mass onto a circular plate, applying an impulse load (typically about 50 kN at 850 kPa contact pressure) to the pavement surface while geophone sensors record the maximum surface deflection at several offsets from the load (commonly 0 to 1.5 m). The set of recorded deflections is called the deflection basin. In backcalculation, the layer stiffness moduli (Young's moduli) of a multilayer pavement model - for a flexible pavement typically an asphalt concrete layer, an unbound aggregate base course, and the natural subgrade - are adjusted until deflections computed with a mechanical model (multilayer elastic theory or a finite-element model) reproduce the measured basin. The recovered moduli then feed mechanistic-empirical (M-E) pavement design and assessment, in which critical responses - the maximum horizontal tensile strain at the bottom of the asphalt layer and the maximum vertical compressive strain at the top of the subgrade - are entered into empirical fatigue and permanent-deformation (rutting) transfer functions to predict the number of load repetitions, i.e. the pavement life, that the structure can sustain.

Backcalculated moduli are therefore only as reliable as the deflection measurements and the model class behind the inversion. Tutka, Nagórski, Złotowska and Rudnicki (2021) quantified this for one specific structure: a three-layer flexible pavement (asphalt layer, unbound aggregate base, subgrade as a half-space, i.e. an infinitely deep bottom layer) with linear-elastic, homogeneous, isotropic layers of known thickness, perfectly bonded interfaces, analysed with an axisymmetric dynamic finite-element model including inertia effects (d'Alembert forces and stress-wave propagation) under an FWD-type impulse load. Injecting random deflection errors that distort the shape of the deflection basin (deflection coefficient of variation, CV, of 0.74%) produced CVs of 9.18% for the asphalt modulus, 27.37% for the base course modulus and 1.92% for the subgrade modulus, and - propagated through the fatigue transfer function - a CV of 10.20% for predicted pavement life. By contrast, proportionally scaled deflection errors that leave the basin shape unchanged produced errors of the same order as the data errors themselves (life CV 1.52%). The amplification was traced to a near-invariance of the basin shape under compensating changes of the asphalt and base moduli; the same group's 2024 follow-up measured a correlation of about -0.97 between the two backcalculated moduli under small random errors, with the sensitivity concentrated in the deflections of the first four geophones. The paper closed by suggesting that, with known measurement accuracy, envelopes of the deflection basin could be used to run several backcalculation sets and report moduli as intervals, and that averaging several measurements per road section limits random errors.

The study was deliberately restricted: linear-elastic material behaviour, one layer-thickness configuration, perfect bonding between layers, no temperature effects, and no shallow stiff layer (often called bedrock). Real flexible pavements violate each restriction: asphalt concrete is viscoelastic, meaning its stiffness depends on temperature and loading rate; unbound layers are stress-dependent, meaning their stiffness varies with the stress state imposed by overlying layers and the applied load; layer interfaces may slip; pavement temperature strongly affects measured deflections; and a shallow stiff layer alters the far field of the basin. Forward-modelling studies with such realistic material laws exist (for example finite-element parametric analyses of viscoelastic and stress-dependent pavement response under dynamic FWD loading), and probabilistic formulations of the inversion itself have been explored since at least 2010 - including Bayesian backcalculation, where Markov chain Monte Carlo (MCMC) sampling is used to explore the posterior probability distribution of the moduli given the measured basin and an assumed error model, and conformal prediction, a machine-learning technique that attaches validity-guaranteed confidence measures to regression predictions. What has not been done is to extend the measurement-error sensitivity analysis of the Tutka type to these realistic models and to turn it into a calibrated uncertainty statement that reaches pavement-life predictions.

Problem Statement

As left open by the source (Tutka, Nagórski, Złotowska and Rudnicki, Materials 14(4):873, 2021, and restated by the same group in Materials 17(17):4412, 2024), the problem is to extend the measurement-error sensitivity analysis of layer-modulus backcalculation beyond the single three-layer, linear-elastic, perfectly bonded pavement with a dynamic finite-element model for which it was performed, and to convert the result into a usable uncertainty statement for engineering decisions. The two papers suggest and list the following related open directions: (1) a generalized sensitivity analysis quantifying how random deflection measurement errors - in particular errors that distort the shape of the deflection basin - propagate into backcalculated layer moduli, critical strains and predicted pavement life for realistic flexible-pavement structures and models. The source's own named open items include generalizing beyond the given layer-thickness configuration to a broader range of cases, the presence of a shallow stiff layer (bedrock), and viscoelastic asphalt behaviour (via determination of viscoelastic parameters from deflection time histories); stress-dependent (nonlinear) base and subgrade stiffness, temperature effects, partial interface bonding (slip) and other structural variations within that flexible-pavement family are realistic-modelling factors excluded from the 2021 study that raise the same sensitivity question. (2) A probabilistic calibration that translates the known accuracy characteristics of the deflection measurement into confidence intervals - for example via deflection-basin envelopes running several backcalculation sets - for the backcalculated moduli, as both source papers recommend, and, following the source's own propagation of modulus variability through the fatigue transfer function, for the resulting predicted pavement life, so that backcalculation outputs are reported as intervals reflecting measurement uncertainty rather than as point values. (3) Backcalculation strategies that reduce the sensitivity of the recovered moduli to basin-shape distortion: the 2021 paper suggests taking several measurements in one road section and calculating the average, the 2024 follow-up suggests reporting the dependencies between the elastic moduli of individual layers and applying artificial neural networks to objective-function minimization, and other inversion techniques that handle the correlated asphalt/base modulus trade-off are natural candidates. The overarching objective is to make backcalculated pavement parameters, and the remaining-life predictions built upon them, trustworthy under realistic measurement error and realistic pavement modelling. These three directions are jointly required components of one objective rather than alternatives, and the class of models for direction (1) is the family of flexible (asphalt-layer) pavements, whose named factors are illustrative examples of the required generalization rather than a conjunctive checklist. Each direction has its own sufficiency standard: (1) a sensitivity analysis covering at least one of the named realistic modelling factors and carried through to the backcalculated moduli and the predicted pavement life; (2) calibrated intervals derived from the known measurement-error characteristics, for both the backcalculated moduli and the resulting predicted pavement life; and (3) at least one mitigation strategy addressing the sensitivity to basin-shape distortion, evaluated for its effect on the recovered moduli and the predicted life.

References

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