Smoothing Earth’s surface: the complexity of soil texture class transitions

Soil depth functions are essential for analysing, modeling, understanding and visualising soil profiles. While robust methods existed for continuous properties, soil texture is typically reported as discrete classes, and no established approach exists to interpolate soil categorical information with depth. Here, we introduced phySplines, a physics-informed, analytically solvable spline for interpolating soil categorical information. Soil texture classes were mapped to a latent numerical space and continuously interpolated by minimising the depth-weighted Jacobian energy via the exact analytic integral of Euler’s cumulative mass process, which encoded depth-dependent resistance and enforces non-parametric physically consistent smoothness. PhySplines achieved kappa values of 0.96, 0.94 and 0.96 at global, provincial and local scales, respectively. By embedding pedological theory within a fully continuous and interpolation framework, the function avoided over representation of dominant classes, captured previously unmodelled transitional states, mitigated the drift effect and generalised across missing layers. PhySplines maintained mass and energy continuity (infinitely differentiable C^∞) without over-constraining the solution, allowing for greater flexibility for future work into numerical classification and the quantification of soil thermodynamical multifunctionality. Ultimately, minimising potential energy and maintaining the mass contuninity, phySplines transformed complex soil profiles into dynamic, interpretable narratives, allowing users to “see” between horizons.