A mathematically rigorous understanding of the geometric structure of wall-turbulence, along with universality laws that can be established for certain statistical measures of the turbulent flow field can guide estimation and control design using reduced-order models at high Reynolds numbers. In this vein, recent model-based studies have investigated the efficacy of the linearized Navier-Stokes equations in predicting the convective velocity and structural features associated with energetically dominant flow scales. On par with recent efforts in the development of physics-aware data-driven models, we build on work by Zare, Jovanovic, and Georgiou (J. Fluid Mech., vol. 812, 2017) to overcome the shortcomings of linearized models by introducing data-enhanced dynamical refinements to the linearized equations. In addition to matching the one-dimensional energy spectra, such models have shown to provide good estimates of two-point correlations of the turbulent velocity field. The frequency response of the resulting data-informed model can be used to estimate the convection velocity for various spatial length scales and as a function of the wall-normal distance. We demonstrate the utility of this approach in studying the geometric self-similarity and wall-attachment of coherent structures that have dominant influence on the transportation of turbulent channel flows with different Reynolds numbers.