The research, led by Gaurav Kumar Gugliani of the Jaypee Institute of Information Technology together with Arnab Sarkar of the Indian Institute of Technology (B.H.U.) Varanasi, Najmeh Nakhaei Rad of the University of Pretoria, and Christophe Ley of the University of Luxembourg, focuses on the Weibull distribution, the workhorse of wind speed modeling. The Weibull distribution describes wind climates with just two numbers: a shape parameter, which captures how variable or steady the wind is, and a scale parameter, which reflects the characteristic wind speed of a site. Nearly every wind resource assessment, turbine selection study, and wind loading calculation in the world passes through these two parameters. If they are biased, everything downstream inherits that bias.

The problem begins with the Indian Meteorological Department’s operational protocol. Wind speeds are recorded in decimal knots, converted to kilometers per hour, and then disseminated as integers, with fractional parts of 0.55 or greater rounded up and everything below rounded down. The authors point out that this is not a uniquely Indian quirk. Germany’s national weather service, the Deutscher Wetter Dienst, supplies wind data on a 0.1 meter-per-second grid even though a single anemometer pulse corresponds to 0.12 meters per second, meaning that recorded pulses of 0.12, 0.24, and 0.36 meters per second are logged as 0.1, 0.2, and 0.4, and the value 0.3 meters per second never appears in the record at all. Such discretization breaks the statistical continuity of the underlying physical process, and no amount of careful fitting downstream can fully undo it.

To quantify the damage, the team turned to Monte Carlo simulation, the statistical equivalent of a wind tunnel for data. They generated synthetic wind speed samples of one million values drawn from a known Weibull distribution, deliberately chose parameters inspired by real Indian wind regimes, with a shape parameter of 1.5 and a scale parameter of 10.8 kilometers per hour, and then pushed those samples through a numerical replica of the entire IMD pipeline: discretization to 0.1 knots, conversion to kilometers per hour, and integer rounding. When the Weibull probability density function was fitted to the original high-resolution data, it matched beautifully. When it was fitted to the disseminated integer data, it visibly underfit the histogram. The rounding had left fingerprints all over the distribution.

Having established that the bias is real, the researchers set out to reverse it. They repeated the simulation across an enormous grid of parameter combinations, varying the true shape parameter from 0.50 to 4.00 in steps of 0.05 and the true scale parameter from 2.0 to 25.0 kilometers per hour in steps of 0.10. For each pair, they estimated Weibull parameters from both the pristine and the corrupted samples using maximum likelihood estimation, generating a vast catalogue of matched biased and unbiased estimates. The goal was audacious in its simplicity: learn the mathematical mapping that takes the biased numbers an analyst actually receives and returns the unbiased numbers that would have been obtained from the original measurements.

The mapping they settled on is a second-order bivariate polynomial response surface, one for each Weibull parameter. The choice was deliberate. Linear surfaces proved too crude to capture the nonlinear interplay between the biased shape and scale estimates, while higher-order polynomials offered only marginal gains at the cost of complexity and overfitting risk. The quadratic form struck the right balance between accuracy, stability, and, crucially, practical usability: a practitioner can simply plug the biased estimates into two explicit equations and read off corrected values, no iterative numerical procedures required. The coefficients were determined by ordinary least squares fitting across the full simulation grid.

The validation is where the results become genuinely striking. To test whether the correction surfaces were merely memorizing their training data, the team generated an entirely independent validation grid, offset by half a step from the training nodes so that no validation point coincided with any training point, and drew fresh million-value samples with new random seeds. The reconstructed shape parameters matched the true values with a coefficient of determination of 0.9998, a mean absolute percentage error of just 0.85 percent, and a root mean square error of 0.0147. The scale parameter, which is more sensitive to the absolute magnitude of the wind speeds and therefore suffers compounded distortion through the sequential knot-to-kilometer-per-hour-to-integer conversions, was reconstructed with a coefficient of determination of 0.9723 and a mean absolute percentage error of 4.17 percent. These are, by any statistical standard, excellent reconstruction figures.

Applied to real data from 37 Indian meteorological stations, the correction revealed systematic patterns. The biased shape parameter generally overestimated the corrected value, and the biased scale parameter consistently exceeded its corrected counterpart. The magnitude of the distortion varied with the wind regime: stations with low or moderate shape parameters, such as Amritsar, Jaipur/Sanganer, Cochin, and Calcutta/Dum Dum, showed noticeable corrections in both parameters, while stations with higher shape parameters, such as Tuticorin H.P., Indore, and New Kandla, saw smaller shape corrections but still substantial scale adjustments. This matters because surface-level wind measurements, the kind taken at ten meters near airports, are precisely the ones with low shape parameters, where the flow is heavily modified by buildings, vegetation, and terrain.

The most dramatic consequences emerge when the biased parameters are used to estimate wind power density, the standard measure of the energy available in the wind, which depends on the cube of the wind speed and therefore amplifies any parameter error. Across the full simulation domain, the wind power density computed from disseminated data always underestimated the true value, and in the low-shape-parameter region the ratio dropped to nearly 0.40, meaning wind power potential could be underestimated by almost 60 percent. After applying the correction equations, the ratio of corrected to biased wind power density moved close to unity over most of the parameter domain, generally falling between about 0.90 and 1.18, with residual variation only at the extreme low-shape boundary where the quantity is inherently hypersensitive.

The authors are careful to delineate what their method does and does not fix. The correction removes only the statistical bias introduced at the dissemination stage; it does not compensate for terrain-induced effects from topography, surface roughness, or nearby obstacles, which must be handled by separate, well-established terrain-normalization procedures. The framework was also calibrated specifically for the IMD protocol and the two-parameter Weibull distribution, so applying it to other reporting systems or other distributions would require recalibration. Still, the broader message resonates far beyond Indian meteorology: in an era when climate archives feed directly into renewable energy investment, building codes, and structural safety assessments, the humble act of rounding a number can propagate into decisions worth fortunes. This study provides the first explicit inverse correction framework for recovering unbiased Weibull parameters from rounded wind data, and it suggests that every national weather agency disseminating discretized wind records should be asking the same uncomfortable question about its own archives.

Subject of Research: Correction of quantization-induced sampling bias in Weibull parameter estimation for wind speed modelling

Article Title: Assessing and correcting the effects of biased sampling on Weibull parameter estimation for wind speed data modelling

Article References: Gugliani, G. K., Sarkar, A., Nakhaei Rad, N., & Ley, C. (2026). Assessing and correcting the effects of biased sampling on Weibull parameter estimation for wind speed data modelling. Theoretical and Applied Climatology, 157(10), Article 668. https://doi.org/10.1007/s00704-026-06576-2

Image Credits: AI Generated

DOI: 10.1007/s00704-026-06576-2

Keywords: Weibull distribution, wind speed data, sampling bias, Monte Carlo simulation, wind power density, Indian Meteorological Department, quantization error, maximum likelihood estimation, response surface, wind resource assessment, surface wind observations, statistical correction