Semi-arid regions across Sub-Saharan Africa face a paradoxical hydroclimatic challenge: intense convective storms generate devastating flash floods, followed by prolonged agricultural droughts that threaten food security [1]. In Kitui County, Kenya, where this study is situated, annual rainfall averages 500 - 800 mm but exhibits extreme inter-annual variability, with coefficients of variation exceeding 30% [2]. The region’s ephemeral river channels, dry for most of the year, can transform into torrents within minutes during storm events, overwhelming communities with limited early warning capacity.

The March 2023 flood event in Kyangwithya village exemplifies this challenge: a 45-minute storm with 65 mm of rainfall generated floodwaters that claimed livestock, destroyed crops, and nearly took a child’s life. The existing early warning system, based on rainfall thresholds alone, provided only 15 minutes of notice with insufficient spatial specificity. This event motivated the development of the Solar-Water-Ecosystem Digital Twin (SWEDT) framework presented in this paper.

Semi-arid flood dynamics concentrate multiple mathematical challenges into a single system that tests the limits of standard numerical methods:

Nonlinear wave propagation: Flash floods propagate as shock waves through initially dry channels, requiring Riemann solvers that capture discontinuities without introducing spurious oscillations [3][4].Moving boundaries: The wetting-drying front represents a free-boundary problem where the computational domain evolves dynamically. Standard Riemann solver assumptions break down at the interface between wet and dry cells [5].Complex source terms: Steep topography, infiltration losses, and vegetation drag introduce source terms that must be balanced exactly with flux gradients to preserve steady states, the so-called C-property [6][7].Multi-scale dynamics: Convective storms operate on hourly timescales while catchment response extends over days, requiring numerical schemes that efficiently capture both scales.

The shallow water equations capture these dynamics, but their numerical solution demands careful treatment to avoid diffusion of shocks, violation of steady states, and crashes at wetting-drying fronts [4].

This paper advances the state of the art in three specific ways that distinguish it from standard shallow water equation implementations:

1) Formal asymptotic analysis: Beyond simple scaling arguments, we perform a rigorous perturbation expansion that quantifies the precise order of neglected terms and identifies the conditions under which the hydrostatic approximation remains valid for semi-arid ephemeral channels.

2) Implicit-Explicit friction treatment with asymptotic-preserving property: We develop a novel time integration scheme for the Manning friction term that remains stable and accurate as water depth approaches zero, with a proof that the scheme preserves the correct asymptotic limit.

3) Truncation error analysis linkingnumericsto physics: We derive the modified equation for our MUSCL-Roe scheme, showing how the leading-order truncation error depends on bathymetric gradients—explaining why mesh resolution is critical in regions of steep topography.

Section 2 presents the complete mathematical derivation from the Navier-Stokes equations to the final governing system, with each assumption justified through scaling and asymptotic analysis. Section 3 describes the numerical implementation, including the novel IMEX scheme and truncation error analysis. Section 4 details the Earth Observation data used for model validation. Section 5 presents validation results and scenario analyses. Section 6 discusses the implications of the mathematical analysis and formulates testable hypotheses for future field campaigns. Section 7 concludes the paper.

We begin with the incompressible Navier-Stokes equations for a Newtonian fluid [8]. These represent the most general description of fluid motion relevant to our problem:

where u , v , w are velocity components in the x , y , z directions respectively, p is pressure, ρ is fluid density, ν is kinematic viscosity, g is gravitational acceleration, and ∇ 2 is the Laplacian operator.

These equations are too complex for direct numerical simulation of catchment-scale floods. We must systematically simplify using physical assumptions justified by scaling analysis.

To understand the relative importance of each term, we introduce characteristic scales derived from Kitui catchment morphology and historical flow records (Kitui gauge, 2002-2024) (Table 1).

Table 1.Characteristic scales for semi-arid floods in Kitui County.

Define dimensionless variables (denoted by superscript asterisk):

The vertical velocity scale follows from the continuity equation: w ~ ( H 0 / L 0 ) U 0 , which is much smaller than horizontal velocities, our first indication of shallowness.

Substituting these scales into the x -momentum Equation (2) and dividing by U 0 2 / L 0 yields the dimensionless equation:

where the key dimensionless numbers are:

From Kitui gauge records and catchment analysis:

F r ≈ 0.3   -   0.8 (subcritical to transcritical flow). R e ≈ 10 5   -   10 7 (fully turbulent). ε ≈ 10 − 3 (very shallow).

These magnitudes provide crucial physical insight:

F r ~ 1 : Gravity and inertia are equally important. The full nonlinear momentum equations must be retained. R e ≫ 1 : Viscous terms are negligible compared to advection. Turbulence must be modeled, not resolved. ε ≪ 1 : Vertical variations are much smaller than horizontal. This justifies the hydrostatic approximation.

We now go beyond simple scaling to perform a formal asymptotic expansion. Let ε = H 0 / L 0 ≪ 1 be the small parameter. We expand all variables in powers of ε :

The expansion for vertical velocity begins at O ( ε ) due to continuity. Substituting these expansions into the dimensionless Navier-Stokes equations and collecting terms of equal order in ε yields a hierarchy of problems.

2.3.1. Leading Order ( ε 0 )

At leading order, the vertical momentum Equation (4) reduces to the hydrostatic balance:

where η * is the dimensionless free surface elevation. The horizontal momentum equations at leading order are:

2.3.2. First Order ( ε 1 )

At first order, we obtain corrections to the hydrostatic pressure:

Proposition 2.1 (Validity of the Hydrostatic Approximation)The hydrostatic approximation is valid to O ( ε ) when:

For Kitui parameters ( ε ≈ 10 − 3 , F r 2 ≈ 0.25 , R e ≈ 10 6 ), these conditions yield:

Both are orders of magnitude less than unity, rigorously justifying the hydrostatic assumption for this application.

With R e ≫ 1 , the flow is fully turbulent. Direct Numerical Simulation resolving all turbulent eddies is computationally impossible for a 150 km² catchment. Following standard practice [9], we decompose velocity and pressure into mean and fluctuating components:

Assumption 2.2 (Reynolds Decomposition)The instantaneous velocity and pressure can be decomposed as:

where the time average of fluctuations is zero: u ′ ¯ = v ′ ¯ = w ′ ¯ = p ′ ¯ = 0 .

Time-averaging the equations yields the Reynolds-Averaged Navier-Stokes (RANS) equations [9]:

The terms − ρ u ′ 2 ¯ , − ρ u ′ v ′ ¯ , etc., are Reynolds stresses; they represent momentum transfer due to turbulent fluctuations. These introduce new unknowns, creating the closure problem.

To close the RANS equations, we model Reynolds stresses as proportional to the mean strain rate, by analogy with molecular viscosity [9][10].

Assumption 2.3 (BoussinesqHypothesis)The Reynolds stresses are related to the mean velocity gradients via an eddy viscosity ν T :

where k = 1 2 u ′ i u ′ i ¯ is the turbulent kinetic energy and δ i j is the Kronecker delta.

For shallow flows, we adopt a simple algebraic eddy viscosity model [10]:

where u * = τ b / ρ is the shear velocity and α is an empirical coefficient. Based on previous studies in semi-arid channels, we use α = 0.7 in this work.

For catchment-scale applications, resolving the vertical structure of velocity is unnecessary and computationally prohibitive. We integrate the equations over depth [3][11].

Assumption 2.4 (Depth-Averaging)Flow properties are nearly uniform over depth, so we replace them with depth-averagedvalues.

Define total water depth H and depth-averaged velocities:

where z b ( x , y ) is the bed elevation.

2.6.1. Depth-Averaged Continuity

Integrating the continuity Equation (19) from the bed z b to the free surface η and applying the Leibniz integral rule yields [11]:

This represents exact conservation of mass in the depth-averaged framework.

2.6.2. Depth-Averaged Momentum

Integrating the horizontal momentum equations similarly and using the hydrostatic pressure relation p = ρ g ( η − z ) gives:

where τ b x and τ b y are the bed shear stress components.

The bed shear stress must be related to the depth-averaged flow variables. For rough turbulent flow in natural channels, Manning’s equation is empirically robust and widely validated [12][13].

Assumption 2.5 (Manning Friction)Bed shear stress follows Manning’s formula:

where n is Manning’s roughness coefficient with units [s/m^1/3], determined from land cover classification.

To model the complete hydrological cycle, we add source terms representing rainfall, infiltration, and evapotranspiration [14].

Assumption 2.6 (Mass Sources)Rainfall adds mass to the system, while infiltration and evapotranspiration remove mass:

where R is rainfall rate [m/s], I is infiltration rate [m/s], and E is evapotranspiration rate [m/s].

For infiltration, we adopt the Green-Ampt model for unsaturated soils [14][15]:

where K s is saturated hydraulic conductivity [m/s], ϕ is porosity, θ i is initial soil moisture content, ψ f is wetting front suction head [m], and F ( t ) is cumulative infiltration depth [m].

Combining all components, the complete system solved by SWEDT is:

These are nonlinear hyperbolic conservation laws with source terms; a class of problems with rich mathematical structure and well-established numerical methods.

Theorem 2.7 (Hyperbolicity)The homogeneous system(ignoring source terms and diffusion) is strictly hyperbolic. For any direction ( n x , n y ) , the flux Jacobian has real eigenvalues[3]:

This ensures well-posedness of the Riemann problem and enables Godunov-type shock-capturing methods.

Theorem 2.8 (Conservation Form)The system can be written in conservative form:

with conserved variable vector U = [ H , H u ¯ , H v ¯ ] T , flux vectors F and G , and source term vector S . This form ensures that mass and momentum are conserved exactly in the numerical scheme [4].

We partition the domain Ω into non-overlapping control volumes Ω i (cells). Integrating the conservative form (35) over Ω i and applying the divergence theorem [4]:

where n is the outward unit normal vector and d S is the surface element.

Defining the cell average U i = 1 | Ω i | ∫ Ω i U d Ω , we obtain the semi-discrete form:

where the sum is over all faces f bounding cell i , L f is the face length, and F f is the numerical flux through the face.

The intercell flux F f ⋅ n f is the key to capturing shocks without oscillations. At each face, we solve a Riemann problem between left and right states U L and U R . Roe’s approximate Riemann solver [3][16] balances accuracy and efficiency:

where α ˜ k are the wave strengths, λ ˜ k are the eigenvalues, and e ˜ k are the right eigenvectors of the Roe-averaged flux Jacobian.

3.2.1. Roe-Averaged Variables

For the shallow water equations, the Roe averages are [3]:

3.2.2. Wave Speeds and Eigenvectors

The eigenvalues (wave speeds) in direction n = ( n x , n y ) are:

The corresponding right eigenvectors are [3]:

3.2.3. Wave Strengths

The wave strengths α ˜ k are obtained by projecting the jump Δ U = U R − U L onto the eigenvector basis [3]:

Lemma 3.1 (Consistency)The Roe flux(38)is consistent: F f ( U , U ) = F ( U ) ⋅ n f [3].

Lemma 3.2 (Conservation)The numerical flux is conservative: F f ( U L , U R ) = − F f ( U R , U L ) [4].

First-order schemes are too diffusive for sharp flood fronts. To achieve second-order accuracy in space, we reconstruct values at cell faces using a linear interpolation with slope limiting [3][17].

The gradient ∇ U i is computed using the Green-Gauss method:

Slope Limiting (TVD Condition)

Godunov’s theorem states that linear first-order schemes are either oscillatory or overly diffusive. To prevent spurious oscillations near discontinuities while maintaining accuracy in smooth regions, we apply a slope limiter ϕ ( r ) [17][18].

where r i measures the ratio of successive gradients:

We adopt the van Leer limiter [17]:

Theorem3.3 (TVD Property)The MUSCL scheme with van Leer limiter satisfies the Total Variation Diminishing(TVD)condition [19]:

where T V ( U ) = ∑ i | U i + 1 − U i | is the total variation. This ensures that no new extrema are created, preventing spurious oscillations.

We use a second-order Strong Stability Preserving (SSP) Runge-Kutta scheme [20][21].

where L ( U ) represents the right-hand side of (37).

The Manning friction term becomes stiff for shallow depths, potentially causing numerical instability. Standard explicit treatments require prohibitively small time steps as H → 0 . We develop a novel Implicit-Explicit (IMEX) scheme that treats friction implicitly while keeping the flux explicit.

3.5.1. Stiffness Analysis

Consider the simplified one-dimensional momentum equation with only friction:

Linearizing around a reference state, the characteristic relaxation time is:

For typical values in a drying channel ( H = 0.05 m, u = 1 m/s, n = 0.03 ), τ fric ≈ 0.4 s. This can be smaller than the Courant-Friedrichs-Lewy (CFL)-limited time step ( Δ t CFL ≈ 2.5 s for Δ x = 10 m). Explicit treatment would require Δ t ≤ τ fric , destroying computational efficiency.

3.5.2. IMEX Scheme Derivation

We split the right-hand side into explicit (flux and bed slope) and implicit (friction) components:

For the friction term, this yields:

These can be solved explicitly for the velocities:

Theorem 3.4 (Asymptotic-Preserving Property)The IMEX scheme(65) - (66) is asymptotic-preserving: in the limit H → 0 , it yields the correct“dry bed”behavior u n + 1 → 0 and v n + 1 → 0 without numerical instability, and the scheme remains well-defined for all H ≥ 0 .

Proof. As H n + 1 → 0 , the denominator in (65) remains positive due to the friction term:

provided H n > 0 . The numerator remains bounded, ensuring u n + 1 → 0 . For cells that are completely dry ( H n = 0 and H n + 1 = 0 ), the momentum equation reduces to 0 = 0 , which is satisfied with u n + 1 = 0 .

To understand why mesh resolution critically affects accuracy, we derive the modified equation for our MUSCL-Roe scheme. This reveals how the leading-order truncation error depends on flow variables and topography.

Consider the one-dimensional scalar conservation law for illustration:

The MUSCL-Roe scheme with van Leer limiter can be written as:

where the numerical flux F i + 1 / 2 = F ( U i + 1 / 2 L , U i + 1 / 2 R ) uses reconstructed values.

Proposition 3.5 (Modified Equation for MUSCL-Roe Scheme)The MUSCL-Roe scheme with van Leer limiter approximates the conservation law(68)with leading truncation error:

where the numerical diffusion coefficient is:

with λ max ( U ) = | F ′ ( U ) | and ϕ ( r ) the van Leer limiter function.

For the shallow water system, this generalizes to:

Proposition 3.6 (Modified Equation for SWE System)The MUSCL-Roe schemeapproximates the shallow water equations with leading truncation error:

where the numerical diffusion matrix D ( U ) has eigenvalues:

This analysis yields several insights:

The leading error is dispersive ( ∂ x x x ), explaining why numerical oscillations appear at shocks. The error scales with | λ max | , meaning regions of high velocity or deep water are more sensitive to mesh resolution. The limiter ϕ ( r ) modulates the error; in smooth regions ϕ ( r ) ≈ 1 , while near discontinuities ϕ ( r ) < 1 , reducing error but also locally reducing accuracy to first order.

The explicit treatment of fluxes imposes a CFL condition for stability [22].

Theorem 3.7 (CFL Condition for SWEDT)For the IMEX scheme with explicit fluxes and implicit friction, the time step must satisfy:

with CFL ≤ 0.5 for the explicit RK2 scheme [21]. The implicit friction treatment removes the stiffness constraint Δ t ≤ τ fric , allowing time steps determined solely by the explicit fluxes.

Proof. For a linearized system, von Neumann stability analysis yields the amplification factor G ( θ ) . The condition | G ( θ ) | ≤ 1 for all Fourier modes θ leads to (74). For nonlinear SWE, the condition ensures that the numerical domain of dependence contains the true physical domain of dependence [4].

The bed slope source term − g H ∂ z b / ∂ x must be discretized carefully to preserve steady states (lake at rest). A naive discretization would cause artificial flow even on flat water. We use the surface gradient method [7][23].

Theorem 3.8 (C-Property Preservation)The discretization (75) exactly balances the pressure gradient for quiescent flow ( u ¯ = v ¯ = 0 , H + z b = c o n s t a n t ) [6][7].

Dry channels become wet, then dry again; a moving boundary problem that violates standard Riemann solver assumptions. We implement the modified Riemann solver approach [5].

At wet-dry interfaces, the Riemann solver is modified to prevent negative depths:

To verify correct implementation, we use the Method of Manufactured Solutions (MMS) [24][25]. We choose analytic functions that satisfy the equations with added source terms:

Substituting into the governing equations yields analytic source terms. The numerical solution should converge to the manufactured solution at the expected rate.

Table 2. Method of Manufactured Solutions convergence study for one-dimensional test case.

The observed convergence rate ≈ 1.9 confirms second-order accuracy (Table 2), consistent with the theoretical analysis [24].

The final governing equations presented in Section 2.9 (Equations (31) - (33)) include all physical terms derived from first principles. However, for the production validation runs reported in Section 5, a simplified subset of these terms was active. This subsection explicitly states which terms were included and which were omitted, aligning the mathematical derivation with the actual implementation.

3.12.1. Active Terms (Included in Production Runs)

The following terms were active in all production simulations:

1) Continuity equation (Equation(31)):

where R is rainfall rate [m/s] and I is infiltration rate [m/s].

2) Momentum equations (Equations(32)-(33)), with the following active terms:

Local acceleration: ∂ ( H u ¯ ) ∂ t .Advection: ∂ ∂ x ( H u ¯ 2 ) + ∂ ∂ y ( H u ¯ v ¯ ) .Pressure gradient: − g H ∂ η ∂ x (via surface gradient method).Manning friction: − g n 2 u ¯ ‖ u ¯ ‖ H 1 / 3 .Bed slope source term: − g H ∂ z b ∂ x .

3.12.2. Inactive Terms (Omitted from Production Runs)

The following terms, while present in the full derivation, were not active in the production validation runs:

1) Eddy viscosity diffusion term: ∇ ⋅ ( H ν T ∇ u ¯ ) was set to zero.

Rationale: The characteristic Reynolds number for the Kitui catchment is R e ≈ 10 5   -   10 7 , indicating that advective transport dominates over diffusive transport. For shallow, high-velocity flood flows in semi-arid channels, the diffusion term is negligible compared to advection and friction [3][4]. Including the diffusion term would require estimation of ν T at each time step with minimal impact on results (estimated contribution < 2% of momentum balance based on scaling analysis). Verification: A sensitivity test comparing runs with and without diffusion for the November 2022 event showed a difference of 0.02 m in peak depth (1.5% of 1.35 m) and 0.01 in F1 score, confirming negligible impact.

2) Evapotranspiration termE: Set to zero in the continuity equation.

Rationale: The flood event windows are 7 days (Section 4.1), during which cumulative evapotranspiration (E ≈ 4.5 mm/day × 7 days = 31.5 mm) is an order of magnitude smaller than cumulative rainfall (peak events 65 - 120 mm) and infiltration. Evapotranspiration accounts for <5% of the mass balance during the flood window. The reduced system R − I captures > 95% of the net mass input. Justification: For the semi-arid ephemeral channels studied, flood hydrodynamics are dominated by rainfall-runoff and infiltration processes; evapotranspiration becomes important only during the recession limb (days 5 - 7) and for post-flood water balance, not for peak flood prediction.

3.12.3. Reduced System Actually Solved by the Code

The system actually solved by the SWEDT code for production runs is:

This reduced system excludes the eddy viscosity diffusion term and the evapotranspiration term. All production results reported in Section 5 are based on this reduced system.

3.12.4. Summary Table: Active vs. Inactive Terms (Table 3)

Table 3. Active and inactive terms in production validation runs.

3.12.5. Code Implementation Verification

The OpenFOAM solver “swedtFoam” implements the reduced system. A verification test comparing the reduced system against the full system (including diffusion and evapotranspiration) for the November 2022 event confirmed:

Peak depth difference: <0.02 m.Peak timing difference: <0.5 hours.F1 score difference: <0.01.

Thus, the simplifications do not compromise the validation conclusions.

Introduction to Validation Setup

To ensure reproducibility of the validation results, this subsection documents the complete model setup for the three flood events analyzed in Section 5.

4.1.1. Model Domain and Topography

The computational domain covers the 150 km² Kitui catchment, delineated from 30 m resolution Shuttle Radar Topography Mission (SRTM) GL1 data [29]. The domain boundaries follow the natural topographic divides identified through hydrological analysis in QGIS (GRASS GIS r.watershed). The main channel network was extracted using a 10 m resolution Digital Elevation Model (DEM) derived from SRTM data via cubic interpolation, with a flow accumulation threshold of 0.5 km² for channel initiation.

4.1.2. Inlet and Outlet Locations

Boundary Location Definitions

The computational domain boundaries were defined based on topographic divides. The specific locations and boundary conditions for inflow, outflow, and lateral boundaries are as follows:

Inflow boundary: Located at the upstream end of the main river channel (coordinates: approximately 1˚24'S, 38˚10'E), where the contributing area upstream is 42 km². At this boundary, a time-varying discharge hydrograph derived from the Kitui gauge records was imposed. Outlet boundary: Located at the catchment pour point (coordinates: approximately 1˚32'S, 38˚18'E). A free-flow (zero-gradient) boundary condition was applied for water depth, with a critical flow condition for velocity. Lateral boundaries: All remaining boundaries coincide with topographic divides. A reflective (no-flow, no-transmission) boundary condition was applied, consistent with the closed catchment assumption.

4.1.3. Event Windows and Initial Conditions

Simulation Event Windows

Three flood events with available Sentinel-1 SAR coverage were simulated. Table 5 lists the start and end dates for each event, along with the initial water depths representing dry-season baseflow conditions (0.03 - 0.05 m).

Table 5. Simulation event windows and initial conditions.

For each event, the simulation started 24 hours before the first observed rainfall to allow for model spin-up. Initial water levels were set to a uniform 0.03 - 0.05 m depth representing baseflow conditions in the ephemeral channel network, informed by dry-season field observations. Initial velocities were set to zero throughout the domain. The initial soil moisture for the Green-Ampt infiltration model was set to θ i = 0.15 (volumetric), based on in-situ measurements at five locations in the catchment during the dry season preceding each event.

4.1.4. Boundary Condition Specifications

Boundary Condition Mathematical Formulation

The following boundary conditions were applied at each boundary type to ensure physical consistency and numerical stability:

Upstream(inflow) boundary: Discharge hydrograph Q ( t ) derived from Kitui gauge data, converted to unit-width discharge q ( t ) = Q ( t ) / W , where W = 15 m is the estimated channel width at the inflow location. The hydrograph was linearly interpolated from hourly gauge records to the model time step (variable, Δ t ≈ 0.5   -   2   s ).Downstream (outlet) boundary: Critical flow condition:

with zero-gradient for water depth: ∂ H / ∂ x = 0 .

Lateral (no-flow) boundaries: Reflective condition:

where n is the direction normal to the boundary.

Wetting-drying threshold: H min = 0.01 m, as specified in Equation 76.

4.1.5. Rainfall Forcing Product Selection

Rainfall Forcing Product Selection

Since no single satellite rainfall product is perfect for all event types, we selected the most appropriate product for each event based on comparative assessment against the Kitui gauge. Table 6 shows that GPM IMERG v07 was used for convective extreme events (November 2022, May 2024), while CHIRPS v2.0 was used for the prolonged moderate rainfall event (April 2023).

Table 6.Rainfall forcing product selection by event.

For all events, the rainfall time series was applied at 30-minute intervals (GPM) or daily intervals (CHIRPS, linearly interpolated to model time steps). No spatial variability in rainfall was imposed across the 150 km² domain, consistent with the scale of convective storm cells in the region (typical storm diameter 10 - 20 km).

Introduction to Model Parameters

This subsection documents the provenance of all model parameters, distinguishing between fixed values derived from literature or field data and calibrated values. Table 7 summarizes the complete parameter set.

Table 7. Parameter provenance, sources, and calibration status.

4.2.1. Manning’s n: Land Cover-Based Assignment

Manning’s roughness coefficients were assigned exclusively from land cover classification rather than calibrated against observed data. This approach was deliberately chosen to preserve the predictive capability of the model for ungauged scenarios, which is the ultimate goal for application in data-scarce regions. The Copernicus CGLS-LC100 product [30] at 100 m resolution was used to classify the Kitui catchment into three primary land cover classes relevant to flood propagation:

Mainchannel (permanent water bodies): n = 0.030 —assigned to all cells classified as “River/Water body”. This value is consistent with Manning’s n for natural channels with moderately clean, winding streams [12]. Floodplain (grassland): n = 0.045 —assigned to “Grassland” class, representing dense grass cover along channel margins. Floodplain (woodland): n = 0.060 —assigned to “Woody vegetation” and “Shrubland” classes.

No calibration of Manning’s n was performed. The consistent positive bias in simulated flood volumes (+4.7% across events) is hypothesized to result from a systematic underestimation of main channel roughness, suggesting that the true Manning’s n for the ephemeral Kitui channels may be 0.033 - 0.035 rather than 0.030. This hypothesis is offered for future field validation (see Section 6.4, Hypothesis 1).

4.2.2. Green-Ampt Infiltration Parameters

The Green-Ampt model [13][14] parameters were derived from:

Soil texture: Sandy loam, based on the Kenya Soil Survey classification for the Kitui region. K s = 5.0 × 10 − 6 m/s: Saturated hydraulic conductivity for sandy loam from the Kitui County soil survey database (KMD, 2024). ϕ = 0.45 : Porosity for sandy loam, standard value from [14]. θ i = 0.15 : Initial volumetric soil moisture measured at five dry-season field locations in November 2022 (depth: 0 - 30 cm). ψ f = 0.11 m: Wetting front suction head for sandy loam, from Rawls et al. (1983) as compiled in [14].

All Green-Ampt parameters were fixed; no calibration was applied.

4.2.3. Evapotranspiration

Evapotranspiration was modeled using the simple reference evapotranspiration approach:

where:

E T 0 = 4.5 mm/day—average annual reference evapotranspiration for Kitui region, derived from MODIS MOD16 product (2002-2024 average). K c = 0.65 —crop coefficient for dry-period sparse vegetation, following FAO-56 guidelines [34]. This value is appropriate for the semi-arid conditions during the dry season preceding flood events.

Evapotranspiration was applied uniformly over the domain during the simulation windows. The contribution of E to the mass balance is small compared to rainfall ( R ) and infiltration ( I ) during flood events (typically E < 0.1 mm/hour vs. R up to 50 mm/hour). As noted in Section 3.12, E was inactive in production runs.

4.2.4. Numerical Parameters: Eddy Viscosity and Wetting-Drying

Numerical Parameters: Eddy Viscosity and Wetting-Drying

The numerical stability and accuracy of the model depend on several key parameters, which were fixed based on literature values:

Eddyviscosity coefficient α = 0.7 : This value controls the magnitude of turbulent diffusion in the algebraic eddy viscosity model ν T = α u * H . As noted in Section 3.12, the diffusion term was inactive in production runs. Minimum water depth H min = 0.01   m : Standard threshold used in shallow water models to distinguish wet from dry cells [5]. Below this depth, cells are considered dry and excluded from flux calculations, preventing negative depths and ensuring numerical stability.

Both parameters were fixed; sensitivity to α was not analyzed in this study, though the range α = 0.5   -   1.0 is noted as a direction for future work.

4.2.5. Calibration Statement

No calibration was performed on any model parameter. All parameters were either:

1) Derived from open-access land cover or soil survey data (Manning’s n, Green-Ampt soil parameters),

2) Taken from published literature (eddy viscosity coefficient, wetting-drying threshold, evapotranspiration coefficients), or

3) Prescribed as scenario parameters (vegetative buffer n values).

This “no-calibration” approach demonstrates the model’s applicability to ungauged basins, which is essential for deployment in data-scarce regions like Kitui County. The systematic positive bias of +4.7% is therefore an unbiased estimate of model performance under fully predictive conditions. Hypotheses for improving accuracy through calibration (e.g., adjusting Manning’s n) are provided in Section 6.4.

Introduction to SAR Validation Workflow

This subsection describes the complete workflow for validating SWEDT-simulated flood extents against Sentinel-1 Synthetic Aperture Radar (SAR) imagery. The workflow addresses acquisition timing, flood extraction rules, grid matching, and the exact definitions of validation metrics.

4.3.1. Acquisition Timing Relative to Flood Peaks

Three flood events with available Sentinel-1 SAR imagery were selected for validation. Table 8 reports the acquisition timing relative to the simulated flood peak.

Table 8. Sentinel-1 SAR acquisition timing and flood peak alignment.

Sentinel-1 revisits the Kitui region every 6 - 12 days (combined ascending and descending orbits). Consequently, exact alignment with flood peaks is not possible. The reported F1 scores represent flood extent comparison at the specific SAR acquisition times, not at peak flood. This temporal offset explains part of the variability in F1 scores (0.79 - 0.84) across events.

4.3.2. Flood Extent Extraction from SAR Imagery

Sentinel-1 Ground Range Detected (GRD) products in Interferometric Wide (IW) swath mode were processed using the Sentinel Application Platform (SNAP) version 9.0. The extraction workflow consisted of:

1) Preprocessing: Thermal noise removal, radiometric calibration (sigma0), and speckle filtering (Refined Lee filter, window size 7 × 7).

2) Geometric correction: Range-Doppler terrain correction using SRTM 30 m DEM, output resolution 10 m.

3) Water body detection: Bimodal thresholding of VH polarization backscatter coefficients. The threshold was determined automatically using Otsu’s method [35] applied to the histogram of VH sigma0 values.

4) Water classification: Pixels with VH backscatter below the threshold were classified as “water”; all others as “land”.

5) Post-processing: Morphological closing (disk radius = 1 pixel) to remove isolated speckle noise, followed by removal of water bodies smaller than 3 × 3 pixels (900 m²).

The water fraction f water for each model grid cell was computed as the proportion of 10 m SAR pixels classified as water within the cell.

4.3.3. Grid Matching between SAR and Model

The model uses a 10 m uniform Cartesian grid, identical to the Sentinel-1 pixel resolution after terrain correction. Grid matching was direct (one-to-one pixel-to-cell mapping), requiring no interpolation. For each model cell i :

SAR-observed water extent: Binary indicator W i SAR = 1 if f water ≥ 0.5 (majority vote), else 0. Model-simulated water extent: Binary indicator W i model = 1 if simulated water depth H i ≥ H min ( H min = 0.01 m), else 0.

The domain for validation excluded cells classified as “permanent water bodies” in the Copernicus CGLS-LC100 product to avoid bias from reservoirs and perennial river segments.

4.3.4. Validation Metrics: Exact Definitions

Three metrics were used to quantify model performance against SAR-derived flood extents:

F1 Score (F1)

The harmonic mean of precision and recall, defined as:

where:

T P (True Positives) = cells correctly identified as flooded ( W i model = 1 and W i SAR = 1 ). F P (False Positives) = cells incorrectly identified as flooded ( W i model = 1 and W i SAR = 0 ). F N (False Negatives) = cells incorrectly identified as dry ( W i model = 0 and W i SAR = 1 ).

F1 ranges from 0 (worst) to 1 (best). A value of 0.82 indicates 82% agreement weighted toward balanced precision and recall.

RMSE Depth (Root Mean Square Error)

For cells where SAR indicates flooding ( W i SAR = 1 ), RMSE depth quantifies the discrepancy between simulated and “observed” water depth:

where H i SAR is not directly measured by SAR. Instead, the observed water depth was estimated from the Kitui gauge stage-discharge relationship converted to water depth using the local rating curve: