Abstract
The Indus River, a lifeline for Pakistan, holds paramount significance for its geography, history, and economy. This research delves into a comprehensive analysis of the river's behavior by introducing a novel statistical framework. Combining the Gumbel Type 2 distribution and the Rayleigh distribution, a new generalized Gumbel Type 2 (GG2) distribution is derived, and used for modeling the data about the river's outflow at the Tarbela Dam during 2020–2021. Our study contributes to the understanding of the complex dynamics of the Indus River. The GG2 distribution, designed for extreme value events, adept at modeling positive-valued variables, were combined to model the complexed characteristics of the river's flow. Parameters are estimated using both classical and Bayesian methods, enhancing the accuracy and reliability of our findings. The Bayse estimators are not in the closed form expression, hence the Tierney-Kadane approximation technique is used. Through simulation study and analysis, the data set of overflows of the river Indus over Tarbela Dam, the Bayes estimators demonstrate superior performance in minimizing risk compared to classical estimators. It is shown graphically that that our proposed distribution performs better than its competitor distributions. The results not only deepen our understanding of the river's behavior but also offer insights crucial for infrastructure planning, flood control, and resource allocation.
Similar content being viewed by others
Data Availability
Data relevant to study is taken from the website of Pakistan Water and Power Development Authority (WAPDA) http://www.wapda.gov.pk/index.php/river-flow-data about the outflow of river Indus at Tarbela of the period 2020–2021 (up till 5th March 2021).
References
Alzaatreh A, Ghosh I (2015) On the Weibull-X family of distributions. J Stat Theory Appl 14(2):169–183
Alzaatreh A, Lee C, Famoye F (2013a) A new method for generating families of continuous distributions. Metron 71(1):63–79
Alzaatreh A, Lee C, Famoye F (2013b) Weibull-Pareto Distribution and its Applications. Commun Stat Theory Methods 42:1673–1691
Arfan M, Lund J, Hassan D, Saleem M, Ahmad A (2019) Assessment of spatial and temporal flow variability of the Indus River. Resources 8(2):103
Ateeq K, Qasim TB, Alvi AR (2019) An extension of rayleigh distribution and applications. Cogent Math Stat 6(1):1–16
Ateeq K, Altaf S, Aslam M (2023) Modeling and analysis of recovery time for the COVID-19 patients: a Bayesian approach. Arab J Basic Appl Sci 30(1):1–12. https://doi.org/10.1080/25765299.2022.2148439
Ateeq K, Altaf S, Aslam M (2022) Modeling and Bayesian Analysis of Time between the Breakdown of Electric Feeders. Model Simul Eng
Coles S, Bawa J, Trenner L, Dorazio P (2001) An Introduction to Statistical Modeling of Extreme values, Vol. 208, Springer
Feroze N, Aslam M (2012) Bayesian analysis of gumbel type II distribution under doubly censored samples using different loss functions. Caspian J Appl Sci Res 1(10)
Forsythe N, Fowler HJ, Kilsby CG, Archer DR (2012) Opportunities from remote sensing for supporting water resources management in village/valley scale catchments in the Upper Indus Basin. Water Resour Manage 26:845–871
Hassan D, Rais MN, Ahmed W, Bano R, Burian SJ, Ijaz MW, Bhatti FA (2019) Future water demand modeling using water evaluation and planning: a case study of the Indus Basin in Pakistan. Sustain Water Resour Manage 5(4):1903–1915
Henningsen A, Toomet O (2011) maxLik: A package for maximum likelihood estimation in R. Comput Stat 26(3):443–458
Jayakumar K, Babu GM (2017) T-Transmuted X family of distributions. Statistica (bologna) 77(3):251–276
Jeffreys H (1946) An invariant form for the prior probability in estimation problems. Proc R Soc Lond Ser A Math Phys Sci 186(1007):453–461
Laplace PS (1820) Théorie analytique des probabilités: Courcier
Lindley DV (1980) Approximate bayesian methods. Trab Estad Investig Oper 31(1):223–245
Lone SA, Sindhu TN, Shafiq A, Jarad F (2022) A novel extended Gumbel Type II model with statistical inference and Covid-19 applications. Results Phys 35:105377
Ogunde AA, Fayose ST, Ajayi B, Omosigho DO (2020) Extended Gumbel type-2 distribution: Properties and applications. J Appl Math 1–11
Oguntunde PE, Khaleel MA, Adejumo AO, Okagbue HI, Opanuga AA, Owolabi FO (2018) The gompertz inverse exponential (GoIE) distribution with applications. Cogent Math Stat 5(1):1–11
Pomee MS, Ashfaq M, Ahmad B, Hertig E (2020) Modeling regional precipitation over the Indus River basin of Pakistan using statistical downscaling. Theoret Appl Climatol 142(1):29–57
Sindhu TN, Shafiq A, Al-Mdallal QM (2021) Exponentiated transformation of Gumbel Type-II distribution for modeling COVID-19 data. Alex Eng J 60(1):671–689
Tahir MH, Zubair M, Mansoor M, Cordeiro GM, Alizadehk M, Hamedani GG (2016) A new Weibull-G family of distributions. Hacttepe J Math Stat 45(2):629–647
Tierney L, Kadane JB (1986) Accurate approximations for posterior moments and marginal densities. J Am Stat Assoc 81(393):82–86
Willayat F, Saud N, Ijaz M, Silvianita A, El-Morshedy M (2022) Marshall-Olkin Extended gumbel type-ii distribution: properties and applications. Complexity 2022:1–23
Yang YE, Ringler C, Brown C, Mondal MAH (2016) Modeling the agricultural water–energy–food nexus in the Indus River Basin, Pakistan. J Water Resour Plan Manag 142(12):04016062
Funding
This work was supported by The Women University Multan, Grant numbers [ 5/2019–20] which does not include the publishing fee.
Author information
Authors and Affiliations
Contributions
Dr Kahkashan Ateeq conceived the idea, develop the computer programs for simulation and applications, Dr Tahira Bano Qasim derived the statistical properties and reliable analysis of the new distribution. Also select the data for application. Ms Wajeeha Kiran help in preparing the manuscript.
Corresponding author
Ethics declarations
Ethical Approval
This work is original and has not been published elsewhere, nor is it currently under consideration for publication elsewhere.
Consent to Participate
All the authors read and approve the manuscript.
Consent to Publish
All the authors take participate in this research work.
Competing Interests
The authors have no relevant financial or non-financial interests to disclose.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ateeq, K., Qasim, T.B. & Kiran, W. Statistical Modeling of Indus River Outflow at Tarbela Dam using Generalized Gumbel Type 2 Distribution. Water Resour Manage (2024). https://doi.org/10.1007/s11269-024-03868-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11269-024-03868-y