Analytical scaling of error propagation in coastal shallow-water hydrodynamic models under two idealised regimes: implications for point-based validation

Error propagation in coastal shallow-water hydrodynamic models controls the spatial extent over which validation results at discrete observation points can be meaningfully interpreted. In engineering practice, model validation is commonly performed at a limited number of observation stations, whereas the reliability of model predictions in unobserved areas remains difficult to quantify. This study derives an analytical scaling framework for error propagation in coastal shallow-water hydrodynamic models under two idealised flow regimes and examines its implications for point-based validation. Starting from the one-dimensional depth-integrated shallow-water equations, we use linear perturbation analysis to derive governing equations for small model errors superimposed on a background flow field. Two contrasting error-propagation regimes are identified. In weakly dissipative deep-channel environments, bed friction is negligible and errors propagate as non-dissipative long waves. The initial error splits according to D’Alembert’s solution and travels at the shallow-water wave celerity with no physical loss of amplitude, implying a broad spatial footprint of point-based validation under this idealised condition. In friction-dominated shallow-flat environments, bed friction controls the spatial evolution of errors, leading to an exponential attenuation with distance. The characteristic decay length scales as the 7/3 power of the total water depth and is inversely proportional to the square of Manning’s roughness coefficient. Therefore, the spatial footprint of point-based validation is regime-dependent and can become highly localised in shallow, rough, and fast-flowing environments. The analytical results are verified using an idealised one-dimensional numerical flume implemented in MIKE 21 Flow Model Hydrodynamic Module (MIKE 21 HD). The simulations reproduce both the non-dissipative propagation of error waves in deep channels and the exponential decay of errors in steady and uniform shallow flows. The derived scaling provides a preliminary physical reference for spatially differentiated validation strategies in coastal hydrodynamic modelling.