📄 earth science

A Hybrid Sunshine–Temperature–Humidity Model for Estimating the Monthly Mean Daily Global Solar Radiation of Tébessa, Algeria

This study demonstrates that adding relative sunshine duration to a temperature and humidity-based model significantly improves the accuracy and reduces the systematic bias of monthly mean daily global solar radiation estimates for Tébessa, Algeria, despite the predictors' high collinearity at the monthly scale.

Original authors: Tidjani Boutora

Published 2026-09-22

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CC BY 4.0

Original authors: Tidjani Boutora

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

To harness the power of the sun, engineers must first know exactly how much energy reaches the ground at a specific location. This measurement, known as global solar radiation, is the foundation for designing any system that converts sunlight into electricity or heat. In an ideal world, scientists would simply place a specialized instrument called a pyranometer at every potential site to record this data directly. However, these devices are expensive to buy, install, and maintain, leaving vast areas of the world, including much of Algeria, without direct records. To fill these gaps, researchers rely on semi-empirical models: mathematical relationships that estimate solar energy using weather variables that are much easier to collect, such as air temperature, humidity, and the number of hours the sun shines. The challenge lies in finding the right combination of these variables to create an estimate that is both accurate and reliable, especially in regions where the climate is dry and the terrain is high.

In the eastern highlands of Algeria, a researcher set out to improve these estimates for the city of Tébessa. They began with a model they had previously developed for a different city, which used only air temperature and humidity to guess the amount of solar radiation. While this approach worked well in some places, it failed to predict the energy levels accurately in Tébessa and other high-altitude, continental sites. The researcher suspected that temperature and humidity alone were not enough because they respond to the sun rather than measuring the atmosphere's transparency to it. To fix this, they introduced a third ingredient: the relative sunshine duration, which is simply the ratio of actual sunshine hours to the maximum possible hours of daylight. By adding this direct measure of how long the sun actually shines, they created a new, four-part formula designed specifically for the cold, semi-arid climate of Tébessa.

The researcher tested their new formula against ten years of recorded weather data, comparing its predictions to the actual measurements taken by national solar monitoring centers. The results showed that the new model was indeed an improvement. It reduced the systematic error, or the consistent tendency to guess too high or too low, by more than half compared to the old method. The overall accuracy also increased, with the average difference between the predicted and actual values dropping to a very small margin. The model performed particularly well during the warmer, clearer months of the year, where the addition of the sunshine data helped correct the previous overestimations. However, the researcher found that the improvement was less noticeable during the winter, when the sky conditions change very little from month to month, making it difficult for any variable to distinguish one period from another.

Despite the success of the new formula, the study revealed a subtle but important limitation in how the data behaved. The three weather variables used in the model—sunshine, temperature, and humidity—were so closely linked to one another in the monthly averages that the statistical analysis could not clearly separate their individual effects. In the dry, high-altitude climate of Tébessa, these three factors rise and fall together in perfect sync with the seasons: as the days get longer and the sky clears, the air warms up and becomes drier all at once. Because they move in lockstep, the model could not definitively prove which single variable was doing the most work, even though the sunshine duration turned out to be the most sensitive factor when tested. This meant that while the new formula worked better, the researcher could not be entirely certain how much credit to give the sunshine data versus the temperature data when looking at monthly averages.

The study concludes that for practical engineering purposes in Tébessa and similar highland regions, the new four-variable formula is the best tool available, offering a more accurate and less biased estimate than previous methods. However, the researcher cautions that to truly understand the unique contribution of sunshine duration, future studies would need to use daily data rather than monthly averages. On a daily basis, the weather can vary enough to break the perfect link between sunshine and temperature, allowing scientists to see exactly how much each factor matters. Until such detailed daily records are analyzed, the new model stands as a significant step forward, providing a reliable way to estimate solar energy in a region where direct measurements are scarce, while highlighting the complex interplay between the sun, the air, and the ground.

Technical Summary: A Hybrid Sunshine–Temperature–Humidity Model for Estimating Monthly Mean Daily Global Solar Radiation in Tébessa, Algeria

Problem Statement
Reliable estimation of global solar radiation ( $H$ ) is a prerequisite for solar energy system design, yet direct pyranometric measurements are scarce across Algeria. While semi-empirical correlations using routine meteorological data offer a practical alternative, previous work by the author demonstrated that a two-variable model based solely on temperature ( $T$ ) and relative humidity ( $RH$ ) performed poorly when transferred to continental, high-altitude sites like Tébessa (error ~6.82%). The author hypothesized that thermal predictors alone fail to capture atmospheric transmissivity in semi-arid highlands because they respond to, rather than determine, incoming radiation. The study posits that the absence of a direct descriptor of atmospheric transparency—specifically sunshine duration—is the cause of this degradation.

Methodology
The study focuses on Tébessa, Algeria (35.404°N, 8.124°E, 850 m a.s.l.), a site with a cold semi-arid (BSk) climate. The dataset comprises a ten-year (2015–2024) monthly climatological normal of global irradiation from the Renewable Energy Development Centre (CDER), supplemented by sunshine duration, temperature, and humidity records from the Algerian National Meteorological Office (ONM).

The core methodological approach involves:

Model Formulation: Extending the previous two-variable correlation ( $H/H_0 = aT^2 + bRH + c$ ) to a four-parameter hybrid model by introducing relative sunshine duration ( $\sigma = S/S_0$ ) as a third predictor:
$H/H_0 = a\sigma + bT^2 + cRH + d$
Candidate Comparison: Six candidate models were calibrated and compared, ranging from the previous thermal-only model to the proposed hybrid model, a classical Ångström–Prescott model, and variations including interaction terms or quadratic sunshine terms.
Calibration and Validation: Coefficients were estimated using Ordinary Least Squares (OLS) on the 12 monthly values. Due to the data representing a climatological normal (where inter-annual variance is negligible), the effective sample size is 12, not 120. Consequently, performance was assessed via leave-one-month-out cross-validation to ensure out-of-sample estimates.
Statistical Rigor: Ten statistical indicators were used, including Root Mean Square Error (RMSE), Mean Bias Error (MBE), and the Student $t$ -statistic. Crucially, the study performed a multicollinearity diagnosis using Variance Inflation Factors (VIF) to assess the independence of predictors.

Key Results
The proposed hybrid model (M3) yielded the following performance metrics for Tébessa:

Accuracy: RMSE = 0.243 MJ·m⁻²·day⁻¹ (nRMSE = 1.31%) and $R^2$ = 0.9984.
Bias Reduction: The systematic bias (MBE) was reduced by 56% compared to the previous thermal-only model (from +0.0072 to +0.0032 MJ·m⁻²·day⁻¹). The Stone $t$ -statistic improved from 0.094 to 0.043, indicating a non-significant bias.
Comparison: The hybrid model improved upon the classical Ångström–Prescott correlation by 31.5% in RMSE.
Sensitivity: Sensitivity analysis revealed that relative sunshine duration ( $\sigma$ ) is the most influential predictor; a ±10% perturbation in $\sigma$ alters the clearness index by ±1.78%, compared to ±1.31% for $RH$ and ±1.24% for $T$ .

Diagnosis of Multicollinearity
A critical finding of the study is the presence of severe multicollinearity among the predictors when using monthly climatological normals. The predictors ( $\sigma$ , $T^2$ , and $RH$ ) are almost perfectly correlated (e.g., correlation between $\sigma$ and $RH$ is −0.962), resulting in extremely high Variance Inflation Factors (181 for $\sigma$ , 633 for $T^2$ , 142 for $RH$ ).

Consequence: While the sunshine term significantly improves the overall bias and accuracy of the estimate, the regression coefficient for $\sigma$ is not statistically significant at the 5% level ( $p = 0.234$ ) in the four-parameter model. The data cannot uniquely distinguish the individual contribution of sunshine from temperature or humidity because, on a monthly normal, all three variables are driven by the same annual solar cycle.

Significance and Claims
The paper claims three primary contributions:

A Validated Hybrid Model: It provides a calibrated, four-parameter correlation for Tébessa that is more accurate and less biased than previous thermal-only formulations, specifically for semi-arid highland sites.
Quantitative Diagnosis: It moves beyond assumption to quantify why the improvement is modest (4.2% reduction in RMSE) and why the sunshine coefficient lacks individual significance: the multicollinearity inherent in monthly climatological normals prevents the separation of predictors.
Practical Recommendation: The study concludes that while the sunshine term should be retained for engineering applications to reduce systematic bias, a definitive separation of the predictors' effects requires calibration on daily data (or individual-year monthly data) where sunshine and temperature can vary independently. The author recommends utilizing satellite-derived daily sunshine products (e.g., CM SAF SARAH-3) for future refinements, noting that such data is now available for the region.

The author maintains a modest stance, acknowledging that while machine learning approaches may offer higher accuracy with large datasets, the proposed parsimonious, interpretable correlation remains the appropriate tool for sites where only monthly climatic normals are available.

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