Physics-constrained random forest for spatial prediction of shallow equivalent thickness from three-component microtremor data

The thickness of sediments above bedrock controls seismic site response and groundmotion amplification, but is difficult to map at high resolution in urban areas with sparse boreholes. This control is captured by the shallow equivalent thickness—the effective sediment-layer thickness that reproduces the observed near-surface resonance, rather than the purely geometric depth to bedrock. Conventional horizontal-to-vertical (H/V) inversions assume a single homogeneous layer, while purely data-driven models lack physical constraints and may produce physically inconsistent predictions under limited samples. This study proposes a physics-constrained random forest (RF) machine learning model for spatial prediction of the shallow equivalent thickness from threecomponent microtremor data, guided by quarter-wavelength theory. At 101 sites and 11 co-located boreholes in the Youjiang river-valley basin, the H/V fundamental frequency and amplification factor, three-component energy ratios, a waveform factor and site coordinates form a multi-source feature vector. A site-specific fundamental-frequency–thickness power law is calibrated against boreholes ( R 2 = 0.96; mean error 5.32%) and embedded as the dominant feature of the random forest. Under 5- fold cross-validation the RF achieves R 2 = 0.857, RMSE = 1.81 m and MAE = 1.02 m, outperforming support vector regression, radial basis function and k-nearest neighbour methods ( R 2 = 0.726–0.778) and traditional interpolation ( R 2 < 0.16). Predicted thickness lies between 10 and 26 m and varies inversely with the fundamental frequency (2.29–6.71 Hz), consistent with quarter-wavelength theory; removing it reduces R 2 by 0.283 and inflates RMSE and MAE by 72.4% and 122.5%. The framework validates that integrating physical constraints with data-driven learning enables high-precision, interpretable urban-scale subsurface characterization under limited sample conditions.