Rician Distribution as a Physically Interpretable Model for Wind-Speed Statistics

C. P. Connaughton; J. E. Sardonia; M. M. Bandi; S. E. Lakhal; S. Mitra

REVIEW 3 major objections 4 minor 53 references

Rician Distribution as a Physically Interpretable Model for Wind-Speed Statistics

T0 review · 3 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper derives a Rician distribution for wind speed from two orthogonal Gaussian velocity components and shows it outperforms the Gaussian and matches the Weibull, with parameters that map to mean flow and turbulence.

desk verdict Rician fits wind data like Weibull, but the physical-interpretability claim doesn't survive the step from instantaneous speeds to 10-minute averages. read the letter →

arxiv 2607.25840 v1 pith:K7KMMANY submitted 2026-07-28 physics.data-an physics.ao-phphysics.flu-dyn

S. Mitra , S. E. Lakhal , C. P. Connaughton , J. E. Sardonia , M. M. Bandi This is my paper · ORCID

classification physics.data-anphysics.ao-phphysics.flu-dyn PACS 92.60.Gn

keywords RiciandistributionwindspeedstatisticsWeibullGaussianmaximumlikelihoodestimationenergyprobabilitydensityturbulenceintensity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes the Rician distribution as a two-parameter model for wind speed, derived from a physical picture: wind velocity consists of a Gaussian component along a preferred direction with mean μ and a perpendicular Gaussian component with zero mean and the same variance σ. Fitted to multi-year 10-minute data from four wind farms, the Rician distribution consistently yields better fits than the Gaussian and is competitive with the Weibull distribution, including in the high-wind tail. The parameters μ and σ map onto the coherent mean flow and fluctuation intensity, and their ratio μ/σ interpolates between Gaussian-like and Weibull-like (Rayleigh) regimes. This gives wind-resource modeling a compact, physically interpretable alternative to the purely empirical Weibull form.

What carries the argument

The central object is the Rician (Rice) distribution, which arises as the magnitude of a vector whose two components are independent Gaussian variables with equal variance. The derivation is carried by the change of variables to polar coordinates and the integral representation of the modified Bessel function I₀. The physically meaningful parameter ratio μ/σ is the key control: for μ/σ ≫ 1 the distribution approaches a Gaussian; for μ/σ ≪ 1 it reduces to the Rayleigh distribution, a special Weibull case with shape β=2 and scale η=√2 σ. Thus the Rician distribution functions as a two-parameter bridge between the canonical Gaussian and Weibull wind-speed models.

What would settle it

Collect a dataset of wind velocity vectors (not just speeds) from an anemometer or lidar, and check empirically whether the two horizontal components are Gaussian, independent, and share the same variance. If the component distributions deviate substantially—or if the fitted Rician rarely matches the empirical speed distribution on a large held-out set while a more flexible distribution such as the generalized gamma does—the physical derivation is falsified and the Rician reduces to a descriptive curve. A second, simpler check: compare the independently measured mean wind speed and turbulence

Watch

Extended reading notes

Core claim

Starting from the assumption that the instantaneous wind velocity can be decomposed into two independent Gaussian components—one along a time-varying preferred direction with mean μ, the other perpendicular to it with zero mean and the same variance σ—the paper derives the Rician probability density f(v) = (v/σ²) exp[-(v²+μ²)/(2σ²)] I₀(vμ/σ²) for the wind speed v. Maximum-likelihood fits of this density to wind-speed records from four utility-scale wind farms show that it outperforms the Gaussian model on bulk and tail metrics (KL divergence, KS distance) and remains broadly comparable to the Weibull model, while providing parameters with direct physical meaning. The ratio μ/σ acts as a cont

Load-bearing premise

The derivation relies on the assumption that the wind velocity components along and perpendicular to the preferred direction are independent, Gaussian, and have equal variance σ; the paper tests only the resulting speed distribution, not whether the actual velocity components satisfy these properties.

Editorial extensions

If this is right

Wind-resource assessment can track seasonal and geographic variability through the physically interpretable pair (μ, σ) rather than through the phenomenological Weibull parameters.
Because the Rician distribution is a two-parameter model that handles the heavy upper tail, it offers a single distribution for both the bulk and extreme events in wind-speed records.
The interpolation property means a single fitted model can describe wind regimes ranging from strongly forced, low-fluctuation flows (Gaussian-like) to fluctuation-dominated, weak-mean flows (Weibull-like), with the ratio μ/σ as a regime indicator.
The derivation suggests a direct route from velocity-component statistics to speed statistics, so site-specific mean flow and turbulence intensity estimates can be translated into the speed distribution.
The model's parameters could be used in wind-power modeling: since power scales as the cube of speed, the Rician speed distribution can be transformed to a power distribution, potentially improving power-fluctuation forecasts.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The equal-variance and independence assumptions on the velocity components are testable with vector anemometry (e.g., sonic or Doppler lidar). If components are anisotropic or intermittent, the derived Rician form is an approximation, and the physical interpretation of μ and σ weakens; a generalized Rice or non-Gaussian component model would be needed.
The paper's monthly-window analysis suggests that the Rician parameters μ and σ themselves could be modeled as slow processes (e.g., a Markov or regression model), enabling stochastic wind-speed simulation that preserves both the seasonal signal and the short-term distribution.
Because the ratio μ/σ completely characterizes the regime, the paper implies a dimensionless index for wind conditions that could be compared across sites without rescaling; this might help in clustering wind farms by flow regime.
The derivation depends only on the Gaussian assumption, so the same logic could be adapted to 3D wind velocity (adding a third component) to yield a generalized Rician (or Hoyt-like) distribution for magnitude; the paper doesn't consider this, but it's a natural extension.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this paper is a decent empirical study showing the Rician distribution fits 10-minute averaged wind-speed data about as well as Weibull and better than Gaussian. The derivation of the Rician from two Gaussian velocity components is textbook, and the paper does a clean job with the asymptotics. What's new is the application to four wind farms, the monthly seasonal decomposition of the Rician parameters, and the claim that μ and σ map onto mean flow and fluctuation intensity.

The paper deserves credit for an honest comparison: Table II shows Weibull still wins on two of four farms in KL and KS, and the paper says "competitive" rather than "superior." The monthly analysis across turbines is a nice way to check stability.

The weak spots, in order of severity. First, the derivation is for instantaneous speed, but the data are 10-minute averaged speeds. The paper never addresses that gap. A 10-minute block average of instantaneous speeds is a different random variable; even if the instantaneous velocity components were perfectly Gaussian, the averaged speeds would not follow Eq. 6. That means the fitted μ and σ cannot be interpreted directly as the coherent mean flow and fluctuation intensity — they are effective parameters for a phenomenological fit. This is the paper's central physical-interpretability claim, so it needs to be fixed with an argument or a test (e.g., checking whether the fitted parameters recover the hourly mean and variance).

Second, the Gaussian-component assumption is never validated. Real turbulence is intermittent and anisotropic, so the Rician is an approximation, not a derived law. The paper should at least look at the velocity components or acknowledge this.

Third, no error bars on the goodness-of-fit metrics, no significance tests, and the full-record fits are done on one representative turbine per farm. The monthly analysis uses all turbines, which helps, but a reader can't tell whether the differences between Weibull and Rician are meaningful.

Fourth, data and code are proprietary, so the headline numbers are not independently checkable.

Overall, the paper is clearly written and the math is right. But the averaging gap weakens the physical interpretation, and the lack of uncertainty bounds weakens the empirical comparison. It's a useful contribution for wind-energy analysts who want a two-parameter alternative to Weibull, but it needs revision before I'd trust the "physically interpretable" framing.

I'd send it to peer review — a good referee can push the authors on the averaging issue and the validation of the Gaussian components. It's not a desk reject.

Referee Report

3 major / 4 minor

Summary. The paper derives a Rician distribution for wind speed from the assumption that the instantaneous two-dimensional wind velocity has independent Gaussian components, with a nonzero mean μ in a preferred direction and zero-mean fluctuations with variance σ² in both directions. It fits the resulting Rician model, alongside Gaussian and Weibull models, by maximum likelihood to 10-minute averaged wind-speed records from one representative turbine at each of four wind farms, using KL divergence, KS distance, and a tail-restricted log-L2 metric. The paper reports that Rician consistently outperforms Gaussian and is competitive with Weibull, that the fitted μ and σ vary seasonally and geographically in an interpretable way, and that the Rician distribution interpolates between Gaussian-like (μ/σ large) and Weibull/Rayleigh-like (μ/σ small) regimes. Appendix B supplies the asymptotic transitions.

Significance. If the central claim holds, the work offers a compact two-parameter model for wind-speed statistics whose parameters have a direct fluid-mechanical interpretation, bridging the Gaussian and Weibull families. The mathematical derivation of the Rician form from the stated Gaussian-component assumptions is correct, and the asymptotic expansions in Appendix B are carefully executed. The empirical analysis spans four geographically distinct farms and includes both full-record and monthly window comparisons, which is a genuine strength. However, the physical interpretability claim rests on an unexamined match between an instantaneous-velocity model and the 10-minute averaged scalar speed data, and the component-level Gaussianity/independence assumptions are never tested. The empirical claims are also somewhat stronger than the reported aggregate metrics support. These issues are load-bearing for the paper's main contribution, but they are addressable.

major comments (3)

[Sections I, III, IV; Eq. (A1); Table I] The derivation applies to the instantaneous speed v = sqrt(v∥² + v⊥²) of instantaneous Gaussian vector components, but the data are explicitly 10-minute averaged wind speeds (Sec. I, Table I). A block average of instantaneous speeds is a different random variable whose distribution is not generally Rician, even when the instantaneous components are exactly Gaussian; averaging narrows the distribution and changes the tail. The paper provides no argument or test that Eq. (6) transfers to the averaged data. Consequently, the fitted μ and σ are not established to be the coherent mean flow and fluctuation intensity of the underlying velocity field, which is the paper's central physical-interpretability claim. The authors should either fit the derivation to appropriately matched data, derive the distribution of the averaged speed, or substantially weaken the physical interpretation.
[Table II; Section IV] The claim that Rician is 'competitive with Weibull' is only partially supported by the reported numbers. The KS distance for Weibull is smaller than for Rician on all four farms (0.026 vs 0.027; 0.016 vs 0.019; 0.007 vs 0.014; 0.014 vs 0.023), and the tail-L2 is smaller for Weibull on farms 1, 2, and 4 (0.731 vs 1.076; 0.004 vs 0.006; 0.026 vs 0.031). KL values are comparable but not uniformly favorable. No confidence intervals, standard errors, p-values, or bin-width sensitivity analyses are given, so the aggregate ranking is not statistically quantified. The 'consistently outperforms Gaussian' claim is supported, but the Weibull comparison needs qualification and uncertainty quantification.
[Section III, Appendix A, Section V] The physical derivation assumes v∥ ∼ N(μ, σ²) and v⊥ ∼ N(0, σ²), independent and with equal variance (Eq. A1). The paper fits only the scalar speed magnitude and never checks these component-level assumptions against vector wind data. Real atmospheric turbulence is often intermittent and anisotropic, so the Rician form may be an empirical approximation rather than a physically derived law. If component data are unavailable, the authors should state this limitation explicitly and discuss how violations of the component assumptions affect the interpretation of μ and σ.

minor comments (4)

[Eq. (8)] The discrete KL divergence uses histogram probabilities f_k and g_k; the bin width and the treatment of empty bins are not specified. Since the KS distance is histogram-free, the KL comparisons would be more informative with a sensitivity analysis over bin choices.
[Fig. 7] There is a typo in the panel label: 'Wind Fram 1' should be 'Wind Farm 1'.
[Section VI] The text says the Rician distribution approaches a 'Weibull-type form' with β ≈ 2; it would be clearer to state that the limit is exactly the Rayleigh distribution, which is a special case of the Weibull family, rather than implying convergence to the general two-parameter Weibull family.
[Section IV] The full-record fits use 'one representative turbine' per farm (Table II). The statement that the ranking 'remains qualitatively the same across turbines' is not accompanied by the corresponding results or a quantitative summary, so the reader cannot verify that claim from the paper as written.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Rician derivation is explicit, the parameters are MLE-fitted on the same data rather than presented as independent predictions, and the model comparisons are empirical.

full rationale

The paper derives the Rician speed distribution in Sec. III and Appendix A from an explicit two-component Gaussian velocity model (Eq. A1: v∥ ~ N(μ,σ²), v⊥ ~ N(0,σ²), independent), and the derivation to Eq. A7 is a standard integration of the resulting joint density. This is a mathematical derivation from stated assumptions, not a fit dressed as a derivation. The parameters μ and σ are estimated by maximum likelihood (Eq. 7) on the same wind-speed records that are later compared with Gaussian and Weibull fits; the paper does not claim to predict held-out data or to compute μ and σ from independent measurements. The physical interpretation of μ and σ as coherent mean flow and fluctuation intensity is a restatement of the generative model's definitions, not an independent empirical result, so it is definitional but not circular in the sense of a fitted value being renamed as a prediction. The self-citations (e.g., Refs. [6,7,16,17,23]) provide background and prior context and are not load-bearing for the central derivation, and no uniqueness theorem or ansatz is imported from the authors' prior work. The most substantive concern is that the empirical data are 10-minute averaged speeds while the derivation applies to instantaneous speed; that is an unexamined modeling-assumption gap, not a circular reduction, because block averaging is not constructed from the Rician formula or from the fitted μ,σ. Overall, the derivation, fitting, and benchmarks are self-contained and do not reduce to the paper's own inputs by construction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The model rests on a two-parameter Gaussian-component assumption whose validity is not independently tested. The derivation from that assumption is mathematically standard and non-circular; the main epistemic burden is whether the physical decomposition is real. No new entities are introduced.

free parameters (3)

μ (Rician coherent mean-flow parameter) = e.g., 5.44 m/s (Farm 1, full record)
MLE-fitted per turbine/month; central to the claimed physical interpretation.
σ (Rician fluctuation intensity) = e.g., 2.90 m/s (Farm 1, full record)
MLE-fitted per turbine/month; assumed equal in both velocity components, unvalidated.
Tail threshold v_min = 10 m/s
Hand-chosen cutoff for the tail-restricted L2 metric; affects the reported tail comparisons.

assumptions (5)

domain assumption Wind velocity components are independent Gaussians with equal variance: v∥ ∼ N(μ, σ²), v⊥ ∼ N(0, σ²).
Eq. A1, Sec. III. This is the generative physical model; never validated on component data.
domain assumption The preferred direction is well-defined even though not fixed in Earth referential.
Sec. III, second paragraph. If wind direction rotates within a fitting window, the parallel/perp decomposition is ambiguous.
domain assumption Monthly windows are quasi-stationary.
Sec. II. Used to define fitting subsets and the seasonal parameter evolution.
standard math Standard Gaussian/polar-coordinate derivation and the integral representation I0(z) = (1/π) ∫₀^π e^{z cosθ} dθ.
Appendix A, Eq. A6; standard analysis.
standard math Asymptotic expansion I0(x) ~ e^x / √(2πx) for large x.
Appendix B, Eq. B7; used to derive the Gaussian and Weibull limits.

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Cite this review

Pith. "Pith review of Rician Distribution as a Physically Interpretable Model for Wind-Speed Statistics." pith.science (2026). https://pith.science/paper/K7KMMANY

@misc{pith2026260725840,
  author       = {Pith},
  title        = {Pith review of: Rician Distribution as a Physically Interpretable Model for Wind-Speed Statistics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K7KMMANY}},
  note         = {Machine review of arXiv:2607.25840}
}

abstract

The statistics of atmospheric wind variations are commonly modeled using Gaussian or Weibull forms, which often trade physical interpretability against statistical accuracy, especially in the distribution tails. Here we derive a Rician distribution for wind speed from a simple physical model based on two orthogonal Gaussian velocity components with a non-zero mean in the preferred direction. Using wind-speed records from four geographically distinct wind farms, we show that the Rician model consistently outperforms the Gaussian model and remains competitive with the Weibull model. The same behavior persists when the data are partitioned into monthly windows, where the Rician parameters also provide a transparent description of seasonal and geographic variability, compared to Weibull parameters. In addition, the model naturally connects Gaussian-like and Weibull-like regimes through the Rician parameter ratio $\mu/\sigma$, making the Rician distribution a compact and physically interpretable two-parameter model for wind-speed statistics.

Figures

Figures reproduced from arXiv: 2607.25840 by the authors.

**Figure 2.** FIG. 2: Histograms of the one-year and monthly [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** FIG. 4: Monthly averaged KL-divergence values for [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Figures from the paper (2 more)

**Figure 3.** Figure 3: FIG. 3: Empirical wind-speed (a) PDF, (b) CDF, and [PITH_FULL_IMAGE:figures/full_fig_p005_3.png]

**Figure 6.** Figure 6: FIG. 6: Turbine ensemble averaged Rician parameters [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

53 extracted references · 1 linked inside Pith

[1]

W eibull (Rayleigh) limit:λ≪1 For smallλ, expand each factor in Eq. B1 [47]: exp ( −λ2 2 ) = 1− λ2 2 +O(λ 4) I0(λz) = 1 + λ2z2 4 +O(λ 4) Multiplying, fR(z;λ) =ze −z2/2 [ 1 +λ2 4 (z2−2) +O(λ 4) ] (B2) The leading term is exactly the Rayleigh PDF inz, fRay(z) =ze−z2/2 (B3) which isa Weibull distributionwith shape parameterβ= 2and scale parameterη= √ 2σ. Hen...
[2]

Quasi-Gaussian and Gaussian limits:λ≫1 Rewrite Eq. B1 as fR(z;λ) =ze −(z−λ)2/2F(λz)(B5) where F(x) =e −xI0(x)(B6) Using the asymptotic expansion [47], asx→∞ F(x) = 1√ 2πx [ 1 + 1 8x +O ( 1 x2 )] (B7) Substitutingx=λzgives fR(z;λ) = √z 2πλe−(z−λ)2/2 [ 1 + 1 8λz +O ( 1 (λz)2 )] (B8) Dropping higher-order terms yields the quasi-Gaussian approximation, f QG R...
[3]

Pobočíková, Z

I. Pobočíková, Z. Sedliačková, and M. Michalková, Pro- cedia Engineering192, 713 (2017), 12th international scientific conference of young scientists on sustainable, modern and safe transport

2017
[4]

M. A. Syaefudin and A. M. Shiddiqi, in2025 15th Inter- national Conference on Information & Communication Technology and System (ICTS)(2025), pp. 1–6

2025
[5]

K. M. Ahmed, M. A. Khan, I. Siddiqui, S. Khan, M. Shoaib, and I. Zia, in2022 Global Conference on Wireless and Optical Technologies (GCWOT)(2022), pp. 1–8

2022
[6]

Yamaguchi and T

A. Yamaguchi and T. Ishihara, Atmosphere12, 1 (2021)

2021
[7]

Apt, Journal of power sources169, 369 (2007)

J. Apt, Journal of power sources169, 369 (2007)

2007
[8]

G. Bel, C. Connaughton, M. Toots, and M. Bandi, New Journal of Physics18, 023015 (2016)

2016

Show all 53 references

[9]

M. M. Bandi, Physical review letters118, 028301 (2017)

2017
[10]

M. M. Bandi and J. Apt, Applied Sciences6, 262 (2016)

2016
[11]

Mei, C.-W

S.-J. Mei, C.-W. Liu, D. Liu, F.-Y. Zhao, H.-Q. Wang, and X.-H. Li, Science of The Total Environment565, 1102 (2016), ISSN 0048-9697

2016
[12]

Liu, Fire Safety Journal141, 103974 (2023)

N. Liu, Fire Safety Journal141, 103974 (2023)

2023
[13]

H. Chum, A. Faaij, J. Moreira, G. Berndes, P. Dhamija, H. Dong, B. Gabrielle, A. G. Eng, W. Lucht, M. Mapako, et al., Cambridge University Press, Cambridge. United Kingdom and New York, NY, USA (2011)

2011
[14]

Apt and P

J. Apt and P. Jaramillo,Variable renewable energy and the electricity grid(Routledge, 2014)

2014
[15]

Schmietendorf, J

K. Schmietendorf, J. Peinke, and O. Kamps, The Euro- pean Physical Journal B90, 1 (2017)

2017
[16]

Smith, O

O. Smith, O. Cattell, E. Farcot, R. D. O’Dea, and K. I. Hopcraft, Science advances8, eabj6734 (2022)

2022
[17]

D. J. MacKay,Sustainable Energy-without the hot air (Bloomsbury Publishing, 2016)

2016
[18]

Bel and M

G. Bel and M. M. Bandi, Physical Review Applied12, 024032 (2019)

2019
[19]

Bel and M

G. Bel and M. Bandi, Physical Review Applied21, 034019 (2024)

2024
[20]

Z. Chen, L. Wu, and M. Shahidehpour, IEEE Transac- tions on Sustainable Energy6, 188 (2015)

2015
[21]

O. M. Ajami, M. S. Alkhusaibi, R. H. Tan, F. A. Ja- 9 maludin, and M. Nadarajah, Global Energy Interconnec- tion (2026)

2026
[22]

Bevrani,Robust power system frequency control (Springer, 2009)

H. Bevrani,Robust power system frequency control (Springer, 2009)

2009
[23]

X. Chen, M. Zhang, Z. Wu, L. Wu, and X. Guan, IEEE Transactions on Industrial Informatics20, 6825 (2024)

2024
[24]

D. G. Padhan and S. Majhi, ISA Transactions52, 242 (2013)

2013
[25]

S. E. Lakhal, J. E. Sardonia, and M. M. Bandi, PRX En- ergy(2026), URLhttps://link.aps.org/doi/10.1103/ vms3-ng8z

2026
[26]

Faranda, G

D. Faranda, G. Messori, T. Alberti, C. Alvarez-Castro, T. Caby, L. Cavicchia, E. Coppola, R. V. Donner, B.Dubrulle, V.M.Galfi, etal., Phys.Rev.E110, 041001 (2024)

2024
[27]

Carbone, D

F. Carbone, D. Telloni, A. G. Bruno, I. Hedgecock, F. De Simone, F. Sprovieri, L. Sorriso-Valvo, and N. Pir- rone, Atmosphere10, 611 (2019)

2019
[28]

von Brandis, G

A. von Brandis, G. Centurelli, J. Schulte, L. Vollmer, B. Djath, and M. Dörenkämper, Wind Energy Science8, 589 (2023)

2023
[29]

Jiang, J

Q. Jiang, J. D. Doyle, T. Haack, M. J. Dvorak, C. L. Archer, and M. Z. Jacobson, Geophysical Research Let- ters35(2008)

2008
[30]

M. J. Dvorak, C. L. Archer, and M. Z. Jacobson, Renew- able Energy35, 1244 (2010)

2010
[31]

Akdağ, H

S. Akdağ, H. Bagiorgas, and G. Mihalakakou, Applied Energy87, 2566 (2010)

2010
[32]

Carta, P

J. Carta, P. Ramírez, and S. Velázquez, Renewable and Sustainable Energy Reviews13, 933 (2009)

2009
[33]

Carta and P

J. Carta and P. Ramírez, Renewable Energy32, 518 (2007)

2007
[34]

S. A. Akdağ and A. Dinler, Energy conversion and man- agement50, 1761 (2009)

2009
[35]

Lencastre, A

P. Lencastre, A. Yazidi, and P. G. Lind, Energies17, 2621 (2024)

2024
[36]

H. Shi, Z. Dong, N. Xiao, and Q. Huang, Frontiers in Energy Research9, 769920 (2021)

2021
[37]

Wadi, inPower Electronics Converters and their Control for Renewable Energy Applications, edited by A

M. Wadi, inPower Electronics Converters and their Control for Renewable Energy Applications, edited by A. Fekik, M. Ghanes, and H. Denoun (Academic Press, 2023), pp. 237–263

2023
[38]

Celik, Renewable Energy31, 105 (2006)

A. Celik, Renewable Energy31, 105 (2006)

2006
[39]

A. N. Celik, Journal of Wind Engineering and Industrial Aerodynamics91, 693 (2003)

2003
[40]

T. P. Chang, Applied Energy88, 272 (2011)

2011
[41]

P. K. Chaurasiya, S. Ahmed, and V. Warudkar, Alexan- dria Engineering Journal57, 2299 (2018)

2018
[42]

Chellali, A

F. Chellali, A. Khellaf, A. Belouchrani, and R. Khan- niche, Renewable and Sustainable Energy Reviews16, 379 (2012)

2012
[43]

Famoye, Technometrics37, 466 (1995)

F. Famoye, Technometrics37, 466 (1995)

1995
[44]

IEC,Tech.Rep.IEC61400-12-1:2017, InternationalElec- trotechnical Commission, Geneva, Switzerland (2017)

2017
[45]

Newman, Contemporary Physics46, 323–351 (2005)

M. Newman, Contemporary Physics46, 323–351 (2005)

2005
[46]

Alstott, E

J. Alstott, E. Bullmore, and D. Plenz, PLoS ONE9, e85777 (2014)

2014
[47]

Cohen, American Statistician - AMER STATIST56, 331 (2002)

A. Cohen, American Statistician - AMER STATIST56, 331 (2002)

2002
[48]

Majumdar and G

S. Majumdar and G. Schehr,Statistics of Extremes and Records in Random Sequences(2024), ISBN 9780198797333

2024
[49]

Balakrishnan,Mathematical Physics: Applications and Problems(2020), ISBN 978-3-030-39679-4

V. Balakrishnan,Mathematical Physics: Applications and Problems(2020), ISBN 978-3-030-39679-4

2020
[50]

S. L. Brunton and J. N. Kutz,Dynamics and Control (Cambridge University Press, 2022)

2022
[51]

Sabhapandit,Extremes and records(2019), 1907.00944, URLhttps://arxiv.org/abs/1907.00944

S. Sabhapandit,Extremes and records(2019), 1907.00944, URLhttps://arxiv.org/abs/1907.00944

2019 arXiv
[52]

Yang and B

X. Yang and B. Fei,Mri denoising(2013)

2013
[53]

Karakuş, E

O. Karakuş, E. E. Kuruoğlu, and A. Achim, IEEE Trans- actions on Geoscience and Remote Sensing60, 1 (2022)

2022

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

W eibull (Rayleigh) limit:λ≪1 For smallλ, expand each factor in Eq. B1 [47]: exp ( −λ2 2 ) = 1− λ2 2 +O(λ 4) I0(λz) = 1 + λ2z2 4 +O(λ 4) Multiplying, fR(z;λ) =ze −z2/2 [ 1 +λ2 4 (z2−2) +O(λ 4) ] (B2) The leading term is exactly the Rayleigh PDF inz, fRay(z) =ze−z2/2 (B3) which isa Weibull distributionwith shape parameterβ= 2and scale parameterη= √ 2σ. Hen...

[2] [2]

Quasi-Gaussian and Gaussian limits:λ≫1 Rewrite Eq. B1 as fR(z;λ) =ze −(z−λ)2/2F(λz)(B5) where F(x) =e −xI0(x)(B6) Using the asymptotic expansion [47], asx→∞ F(x) = 1√ 2πx [ 1 + 1 8x +O ( 1 x2 )] (B7) Substitutingx=λzgives fR(z;λ) = √z 2πλe−(z−λ)2/2 [ 1 + 1 8λz +O ( 1 (λz)2 )] (B8) Dropping higher-order terms yields the quasi-Gaussian approximation, f QG R...

[3] [3]

Pobočíková, Z

I. Pobočíková, Z. Sedliačková, and M. Michalková, Pro- cedia Engineering192, 713 (2017), 12th international scientific conference of young scientists on sustainable, modern and safe transport

2017

[4] [4]

M. A. Syaefudin and A. M. Shiddiqi, in2025 15th Inter- national Conference on Information & Communication Technology and System (ICTS)(2025), pp. 1–6

2025

[5] [5]

K. M. Ahmed, M. A. Khan, I. Siddiqui, S. Khan, M. Shoaib, and I. Zia, in2022 Global Conference on Wireless and Optical Technologies (GCWOT)(2022), pp. 1–8

2022

[6] [6]

Yamaguchi and T

A. Yamaguchi and T. Ishihara, Atmosphere12, 1 (2021)

2021

[7] [7]

Apt, Journal of power sources169, 369 (2007)

J. Apt, Journal of power sources169, 369 (2007)

2007

[8] [8]

G. Bel, C. Connaughton, M. Toots, and M. Bandi, New Journal of Physics18, 023015 (2016)

2016