Exploring Uncertainty Propagation in Coupled Hydrologic and Hydrodynamic Systems via Distribution-Agnostic State Space Analysis

Abstract

Accurate overland runoff and infiltration predictions are critical for effective water resources management, in particular for urban flood management. However, the inherent uncertainty in rainfall patterns, soil properties, and initial conditions makes reliable flood forecasting a challenging task. This paper presents a framework for quantifying the impact of these uncertainties on hydrologic and hydrodynamic simulations via a state space approach based on a differential algebraic equation (DAE) formulation that couples surface and subsurface constraints with the governing dynamics. Under this formulation, the complex interactions between overland flow and infiltration dynamics are captured in realtime. To account for uncertainty in inputs and parameters, the proposed framework quantifies and propagates these uncertainties through the DAE model formulation under partial measurements. The effectiveness of the approach is demonstrated through a series of numerical experiments on synthetic and real world catchments, highlighting its ability to provide probabilistic estimates of watershed state conditions while accounting for uncertainty. An important aspect of the proposed methods is that they are distribution-agnostic, i.e., they only require covariances of uncertainty and not specific types of distributions. The proposed framework is further validated against Monte Carlo (MC) ensemble simulations while providing probabilistic state estimates for measured and unmeasured watershed states under partial gauging.

Figures (12)

• Figure 1: Conceptual framework representing the proposed realtime covariance propagation framework for 2D overland flow and infiltration watershed modeling under uncertainty; (a) illustrates the watershed sensing configuration with flumes, stream gauges, and rain gauges, together with the five uncertainty sources and their respective probabilistic distribution models; (b) shows the two propagation pathways: the proposed realtime covariance propagation framework under partial measurements and an MC ensemble pathway used as reference; and (c) compares resulting uncertainty intervals at gauged and ungauged locations, illustrating that the proposed framework provides tighter measurement informed confidence intervals at monitored locations while still providing uncertainty estimates at unmeasured locations.
• Figure 2: Numerical case study catchment (V-Tilted): synthetic topography with smooth hillslopes and a rougher central channel. The hillslopes and channel are assigned Manning roughness values of $n=0.015$ and $n=0.15$. The cross-slopes are $S_x=5\%$ and $S_y=2\%$. The outlet boundary condition is prescribed as normal depth with slope $0.02$. The grid resolution is $20~\mathrm{m}$.
• Figure 3: Walnut Gulch Experimental Watershed, Tombstone, Arizona, USA. The main panel shows the watershed boundary overlaid on the digital elevation model (DEM), with elevation ranging approximately from $1220\;\mathrm{m}$ to $1930\; \mathrm{m}$. Red triangular markers denote the locations of stream gauges within the watershed. The left panels illustrate the land use/land cover (LULC) classification and corresponding area distribution across the 24 vegetation cover types. The dominant classes are shrubs with grass ($63.0\;\mathrm{km}^2$, $42.7\%$), shrubs and sparse grass ($33.7\;\mathrm{km}^2$, $22.8\%$), and grass with scattered shrubs ($20.4\;\mathrm{km}^2$, $13.8\%$), together accounting for approximately $79\%$ of the watershed. Minor classes include built up and disturbed areas ($11.9\;\mathrm{km}^2$), grass ($10.8\;\mathrm{km}^2$), woody riparian cover along channels ($6.1\;\mathrm{km}^2$), and upland oak and juniper woodland ($1.8\;\mathrm{km}^2$). The inset map (bottom right) indicates the geographical location of WGEW within Arizona, USA.
• Figure 4: Hydrograph comparison for the V-Tilted and Walnut Gulch watersheds under the corresponding rainfall patterns $\boldsymbol R(k)$ [$\mathrm{mm\,h^{-1}}$]. (a) V-Tilted rainfall pattern $\boldsymbol R(k)$ and (b) the respective outlet discharge trajectories from the proposed DAE formulation, explicit CA routing, and local inertia solvers. (c) Walnut Gulch rainfall pattern $\boldsymbol R(k)$ and (d) the respective outlet discharge trajectories from the proposed DAE formulation, explicit CA routing, and local inertia solvers.
• Figure 5: Time domain uncertainty envelopes for the V-Tilted case study at the mid-channel and near-outlet cells. The figure shows the nominal DAE trajectory $\boldsymbol x_k^0$, the $\pm1\sigma$ band, and the $95\%$ confidence interval for water depth $h$ [mm], cumulative infiltration $F$ [mm], and directional discharge $Q_d$ [m$^3$ s$^{-1}$] at each location, with the corresponding event rainfall $R(k)$ [mm h$^{-1}$] shown in each column. Confidence interval width grows during the active runoff period and narrows during recession, consistent with the covariance mapping in \ref{['eq:Kxx']}, with wider envelopes at the near-outlet cell reflecting upstream uncertainty accumulation through the coupled routing dynamics in \ref{['eq:diff_states']} and \ref{['eq:algebraic_constraint']}.

Theorems & Definitions (6)

• Definition 1: Differentiation index Volker2005
• Definition 2: Regularity Volker2005
• Proposition 1: Semi-explicit index-1 structure
• Proposition 2
• proof : Proof of Proposition \ref{['thm:index1_local']}
• proof : Proof of Proposition \ref{['thm:pse_solution']}