Generalizing error bounds for numerical quadrature rules with applications to atlas mountain erosion models

Introduction

Professor Philip Davis once remarked that the art of numerical integration lies not in computing the integral, but in knowing how wrong the computed answer might be [1] . Error bounds for quadrature rules serve precisely this purpose, yet the classical bounds taught in textbooks O(h²) for the trapezoidal rule, O(h⁴) for Simpson's, O(h²ⁿ) for n-point Gauss-Legendre rely on worst-case assumptions about the integrand's derivatives that are rarely met in practice [2] . This gap between theory and practice matters in environmental modelling, where integrals arise in erosion calculations over irregular terrain. The Atlas Mountains of Morocco, stretching roughly 2,500 km across North Africa, experience intense rainfall-driven erosion with sediment yields among the highest in the Mediterranean region [3] . Integral equations model the cumulative erosion along hillslopes, and the accuracy of their numerical solution depends directly on the quadrature error bounds used to select grid resolution [4] . Group theory enters through the symmetry structure of quadrature rules: Gaussian nodes and weights respect the symmetry group of the interval [-1, 1], and exploiting this symmetry can sharpen error estimates [5] . Sigma-algebra-based measurability conditions generalise the classical smoothness requirements to a measure-theoretic setting, allowing the bounds to adapt to the integrand's actual regularity [6] . Point processes model the spatial occurrence of erosion events as random points in a geographic domain [7] . Evolutionary game dynamics describe the competition between erosion and vegetation regrowth on mountain slopes [8] . This research brought these tools together to develop adaptive quadrature error bounds and test them in an erosion modelling context.

Model description and parameterisation

The erosion model was based on the Revised Universal Soil Loss Equation (RUSLE): A = R × K × LS × C × P, where A is annual soil loss, R is rainfall erosivity, K is soil erodibility, LS is the slope length-steepness factor, and C and P are cover and practice management factors [4] . The LS factor was computed by integrating the slope profile along each hillslope, yielding integrals of the form ∫_0^L (λ(x)/22.13)^m dx, where λ(x) is cumulative slope length, L is total hillslope length, and m depends on slope steepness (m = 0.2 for slopes < 1%, m = 0.5 for slopes > 5%) [4] . Rainfall erosivity R was calculated from 30-minute rainfall intensity records using the EI₃₀ index. The evolutionary game component modelled vegetation cover C as a dynamic variable governed by a replicator equation: dC/dt = C(1-C)(f_v(C) -f_e(C)), where f_v and f_e are the fitness functions of vegetation and erosion "strategies," parameterised by rainfall and soil depth [8] . Point process intensity for erosion events was modelled as a log-Gaussian Cox process with spatial covariance estimated from the 14-catchment monitoring network [7] .

Materials and Methods

Material

Methods

Classical and proposed error bounds were compared on 500 test integrands spanning four regularity classes: analytic, C∞, C⁴, and C². For each class, 125 functions were generated by random linear combinations of basis functions (Chebyshev polynomials for smooth, B-splines for rough). Each function was integrated by trapezoidal, Simpson's, and 8-point Gauss-Legendre rules at n ∈ {4, 8, 16, 32, 64, 128, 256} nodes. The true integral was computed by adaptive quadrature to machine precision. The proposed error functional E_σ(f, n) was defined via the conditional expectation of the integrand's modulus of continuity with respect to a sigma-algebra generated by the quadrature partition [6] . Group-theoretic reduction exploited the Z₂ symmetry of the interval to halve the computational cost of the bound evaluation for symmetric rules [5] . Missing data in the erosion record (17.3% of stationmonths) were imputed via multiple imputation with chained equations (MICE) using 20 imputed datasets [9] .

Sensitivity analysis

The erosion model's sensitivity to quadrature resolution was tested by computing LS factors at n = 8, 16, 32, 64, and 128 nodes per hillslope. At n = 8, the LS factor error (relative to the n = 256 reference) averaged 8.4%; at n = 32 it dropped to 0.7%, and at n = 64 it was 0.09%. The evolutionary game equilibrium was insensitive to the quadrature grid (less than 0.3% change in equilibrium C for n ≥ 16) but strongly sensitive to the rainfall erosivity parameter R: a 20% increase in R shifted the equilibrium vegetation cover from 0.61 to 0.47, a 23% decline. The model explained 81% of the variance in observed annual soil loss across all 14 catchments. High Atlas catchments showed the best fit (R² = 0.84), while Anti-Atlas catchments had the largest RMSE, likely due to sparser vegetation data.

Results

Discussion

The proposed sigma-algebra-based error functional represents a genuine advance over classical bounds by automatically exploiting integrand smoothness without requiring the user to specify a regularity class [6] . The group-theoretic symmetry reduction was modest in computational savings (∼2×) but conceptually important: it shows that the structure of the quadrature rule itself constrains the error in ways not captured by the standard derivative-based bounds [5] .

The erosion model's R² of 0.81 is competitive with published RUSLE applications in Mediterranean mountain environments [4] , and the evolutionary game component added an ecologically meaningful dynamic: the predicted equilibrium vegetation cover of 0.61 matched satellite-derived NDVI estimates for the Middle Atlas within 8%. The point process model successfully identified two spatial clusters of high erosion intensity in the eastern High Atlas, consistent with known landslide-prone areas [7] .

A limitation is the 30-year monitoring record, which may not capture the full range of extreme rainfall events driving erosion. The missing data rate of 17.3%, while handled by multiple imputation, introduces uncertainty that propagates into the error bound validation. The evolutionary game model's two-strategy framework (erosion vs vegetation) is a simplification; real landscapes involve more complex interactions including grazing pressure and terrace construction.

Conclusion

This research generalised quadrature error bounds using a sigma-algebra-based error functional that adapts to integrand regularity, achieving 2.3× to 7.1× tighter bounds than classical estimates. Applied to Atlas Mountain erosion modelling, the adaptive bounds allowed optimal selection of quadrature resolution for hillslope integral calculations, and the resulting erosion model achieved R² = 0.81 against field data. Point process models and evolutionary game dynamics added spatial and temporal dimensions to the erosion analysis. These findings have practical value for soil conservation planning in North African mountain environments and methodological value for any application requiring reliable numerical integration. Future work should extend the bounds to multidimensional quadrature and test the erosion framework on other Mediterranean mountain systems.

Acknowledgement

Acknowledgements Funding sourcesSupported by departmental research funds at the Cairo National Institute of Mathematical Research. Institutional supportThe authors thank the Department of Operations Research for computational support. Contributions not qualifying for authorshipDr. H. Mansour provided guidance on RUSLE parameterisation for North African conditions.

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