Operator-Adapted Modal Spaces for Reduced-Order Forward Modeling and Parameter Inversion in Layered Consolidation

Identifying the consolidation state of building foundations forms part of structural health monitoring and condition assessment of infrastructure in service. Recovering the coefficient of consolidation of each layer from pore water pressure monitoring requires repeated solution of the layered consolidation equation during a parameter search. The forward space is therefore required to remain fixed over the entire inversion and to be adapted to the flux condition at material interfaces; classical layered eigenbases and the global sine space each satisfy only one. This study constructs two operator-adapted modal spaces that satisfy both. The permeability operator basis is generated by an unweighted eigenvalue problem for the permeability operator, whereas the frozen storage basis solves a generalized eigenvalue problem with a storage weight proportional to the vertical permeability. Both depend only on the known layer positions and the relative permeability profile, not on its absolute magnitude. Under a unified projection, propagation, and full-order finite element reference, the three spaces are compared in a high-contrast four-layer configuration. Among the tested dimensions, the operator-adapted spaces first reach a full-period relative error of 10−2 at 20 modes and the global sine space at 80. After the initial layer t < 10−3, the errors of the operator-adapted spaces decrease approximately as K−2.1 to K−4.1, compared with about K−1 for the sine space, and neither operator-adapted basis is uniformly more accurate than the other. In a fixed four-layer geometry, the reduced-order gain increases with the permeability contrast. Within the same operator-adapted space, direct matrix propagation combined with bounded least squares yields lower parameter and field errors than a neural parameterization of the time coefficients. These results provide a quantitative basis for offline space construction and online solver selection in parameter identification for foundation consolidation monitoring.