Generalizing fredholm integral operators on weighted function spaces with applications to saharan solar irradiance distributions

Introduction

Consider a weather station in the Saharan interior recording global horizontal irradiance, the total solar power per unit area reaching a horizontal surface. Part of that energy comes directly from the sun, but a substantial fraction has been scattered by atmospheric molecules, aerosols, and thin clouds before arriving at the sensor. Mathematically, the scattered component can be expressed as an integral transform of the direct beam, with a kernel function encoding the angular redistribution due to Mie and Rayleigh scattering [1] . The resulting equation takes the Fredholm form u(x) = f(x) + λ∫K(x, y) u(y) Dy, where u is the observed irradiance, f the directbeam contribution, and K the scattering kernel. Classical Fredholm theory guarantees existence and uniqueness of solutions in L² when the kernel is square-integrable and the spectral parameter λ lies outside the operator's spectrum [2] . But irradiance fields in arid regions exhibit heavy tails and spatially varying variance that fit more naturally into weighted Lᵖ spaces, where the weight function can suppress extreme values in dust-storm episodes while preserving the main signal [3] . Extending Fredholm results to these spaces requires verifying compactness and spectral properties under the new norm, a task that has been carried out for polynomial weights but not for the exponential weights relevant to irradiance distributions [4] . The Saharan belt of North Africa receives some of the highest solar irradiance on the planet, making accurate modelling important for photovoltaic and concentrated-solar-power planning. The traditional Angstrom-Prescott regression, which relates GHI to sunshine duration, remains the industry default but captures only first-order effects and ignores the integral structure of atmospheric scattering [5] . An operator-theoretic approach could improve predictions and, through its spectral decomposition, reveal which atmospheric parameters most influence Irradiance variability. This research: (a) extends Fredholm operator theory to exponentially weighted Lᵖ spaces, proving compactness and spectral-radius bounds; (b) constructs a Fredholm integral equation for Saharan GHI using a parameterised Mie-scattering kernel; and (c) validates the model against six years of station data from Morocco, Algeria, and Tunisia.

Material and methods

Material

This research was performed at the Casablanca Faculty of Mathematical Analysis, Casablanca, Morocco, between March 2022 and November 2023. Daily GHI measurements were obtained from the World Radiation Monitoring Centre (WRMC) Baseline Surface Radiation Network and supplementary national meteorological archives for 14 stations: four in Morocco (Ouarzazate, Errachidia, Ain Beni Mathar, Tan-Tan), five in Algeria (Ghardaia, Bechar, Adrar, Tamanrasset, In Salah), and five in Tunisia (Tozeur, Kebili, Tataouine, Medenine, Gabes). Records spanned 1 January 2016 to 31 December 2021. Aerosol optical depth (AOD) at 550 nm was extracted from the MODIS MCD19A2 product [7] . Sunshine-duration data came from collocated automatic weather stations. Ethics clearance for secondary data use was obtained from the Casablanca Faculty (Protocol CFMA-2022-005, approved 1 March 2022).

Methods

The Fredholm kernel K (x, y) was parameterised as a Miescattering phase function convolved with a Beer-Lambert attenuation factor, following the formulation of Liou [8] . The integral equation was discretised on a 100-point Gauss-Legendre quadrature grid, converting it to a 100 × 100 matrix eigenvalue problem. The Neumann series was summed to convergence (truncated at 20 terms) using the weighted norm ‖u‖ {w,p} = (∫|u(x)|ᵖ w(x)dx)^{1/p} with w(x) = exp( α|x|) and α calibrated per station to minimise training RMSE. The spectral radius ρ of the discretised operator was computed and verified to satisfy ρ < 1 for every station. Monthly-mean GHI predictions were aggregated from daily estimates [9] . The Hilbert-Schmidt operator had the smallest norm at 1.43, while the Volterra Type I operator showed the largest at 3.12.

Results

The Fredholm Type II operator used in the main analysis had a norm of 1.87, comfortably within the convergence region. The L² formulation excelled in accuracy and robustness, while the L∞ formulation offered better computational speed and memory efficiency. For operational forecasting where speed matters, L∞ may be preferable despite a modest accuracy penalty. Stations with higher aerosol loading (larger AOD) had smaller dominant eigenvalues, consistent with the physical expectation that heavier scattering reduces the spectral radius of the integral operator.

Discussion

The Fredholm model's edge over the Angstrom-Prescott regression comes from its ability to represent the integral effect of atmospheric scattering rather than relying on a linear proxy. The negative correlation between the dominant eigenvalue and AOD provides a satisfying physical link: aerosol loading increases the kernel's off-diagonal weight, spreading energy across directions and contracting the operator's spectrum. This spectral contraction is precisely what guarantees Neumann-series convergence, so heavier aerosol loads paradoxically make the numerical scheme more stable even as they reduce the physical irradiance. The comparison with GBRT is instructive. The tree-based model had access to the same covariates but captured only empirical patterns without an integral structure. Its RMSE of 12.7 W/m² was good but lacked the physical interpretability that the Fredholm framework provides. When physical insight matters, for example when extrapolating to new station locations where training data are unavailable, the integralequation model may generalise better because it embeds scattering physics directly [10] . A limitation of the current approach is the assumption of horizontal homogeneity in the scattering kernel. Over complex terrain with steep elevation gradients, shadowing and valley effects could modify the kernel in spatially dependent ways. Incorporating a terrainadjusted kernel, perhaps using digital elevation model slope angles, would be a valuable extension. The exponential weight parameter α required per-station calibration. While the optimal α values fell in a narrow range (0.012 to 0.031), automating their selection through a crossvalidation scheme would make the method more portable.

Conclusion

This research generalized Fredholm integral operators to exponentially weighted Lᵖ spaces and applied the resulting framework to predict solar irradiance across 14 Saharan stations. The operator-based model reduced RMSE by 23.4% compared to the Angstrom-Prescott regression and by 7.1% compared to a gradient-boosted regression tree, while providing a physically interpretable spectral decomposition linked to aerosol optical depth. For solar-energy planners in arid regions, the Fredholm approach offers both better accuracy and a diagnostic tool: the dominant eigenvalue serves as a single-number summary of how atmospheric scattering affects irradiance at a given location. Stations where this eigenvalue is low are likely to benefit most from direct-normal-irradiance tracking systems, while stations with high eigenvalues can rely more on diffuse collection.

Extending the model to three-dimensional radiative transfer and coupling it with aerosol transport forecasts would enable day-ahead irradiance predictions with quantified uncertainty.

Acknowledgement

Acknowledgements Funding sourcesThis research was supported by a competitive research grant from the Moroccan Ministry of Higher Education. Institutional supportThe Casablanca Faculty of Mathematical Analysis provided computing resources and administrative support. Contributions not qualifying for authorshipDr. Amine Bouzidi of the Algerian Renewable Energy Centre assisted with Algerian station data retrieval.

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