Predicting daily flow duration curves (FDCs) in ungauged catchments remains a major challenge in hydrology and is critical for effective water resources management. The FDCs typical were predicted by relating FDC parameters or percentiles to catchment properties using statistical or machine learning-based models. Such models often suffer from limited interpretability and transferability across hydroclimatic conditions. In this study, we propose a process-informed, interpretable machine learning framework for predicting daily FDCs by integrating multivariate adaptive regression splines (MARS) with the Budyko theory, which provides a physically based representation of long-term water–energy constraints on catchment behaviour. Assuming the FDC follows a log-normal distribution determined by three parameters, MARS is used to explore relationships between FDC parameters and 19 catchment characteristics using data from 347 catchments across Australia. Results indicate that the proposed framework can satisfactorily predict the mean and deviation of daily streamflow, while prediction skill for ratio of non-zero flow days remains comparatively weaker. Incorporating hydrological constraints through Budyko theory improves the physical interpretability and robustness of model predictions, particularly in revealing the dominant hydroclimatic controls on streamflow regimes. Prediction performance is found generally higher in wetter catchments than in drier ones, mainly due to limitations of the models in predicting non-zero flow ratio and the lognormal assumption. To further improve FDC prediction in ungauged catchments, catchment characteristics more closely related to groundwater processes may be required, in addition to the adoption of more advanced modelling approaches.
- Open Access
- Article
Predicting Streamflow Regimes in Ungauged Catchments with Process-Informed Machine Learning
- Hongxing Zheng 1,*,
- Ruirui Zhu 2,
- Lu Zhang 3,
- Francis Chiew 1
Author Information
Received: 13 Nov 2025 | Revised: 01 Jan 2026 | Accepted: 09 Jan 2026 | Published: 23 Jan 2026
Abstract
Keywords
flow duration curve | process-informed machine learning (PIML) | MARS model | prediction in ungauged basins (PUBs)
References
- 1.
Abril, M.; Muñoz, I.; Casas-Ruiz, J.P.; et al. Effects of water flow regulation on ecosystem functioning in a Mediterranean river network assessed by wood decomposition. Sci. Total Environ. 2015, 517, 57–65.
- 2.
Booker, D.J.; Snelder, T.H. Comparing methods for estimating flow duration curves at ungauged sites. J. Hydrol. 2012, 434–435, 78–94.
- 3.
Brown, A.E.; McMahon, T.A.; Podger, G.M.; et al. A Methodology to Predict the Impact of Changes in Forest Cover on Flow Duration Curves; CSIRO Land and Water Science Report 8/06; CSIRO: Canberra, NSW, Australia, 2006.
- 4.
Castellarin, A.; Galeati, G.; Brandimarte, L. Regional flow duration curves: Reliability for ungauged basins. Adv. Water Resour. 2004, 27, 953–965.
- 5.
Pyron, M.; Neumann, K. Hydrologic alterations in the Wabash River watershed, USA. River Res. Appl. 2008, 24, 1175–1184.
- 6.
Smakhtin, V.U. Low flow hydrology: A review. J. Hydrol. 2001, 240, 147–186. 10.1016/S0022-1694(00)00340-1.
- 7.
Vogel, R.M.; Fennessey, N.M. Flow duration curves I: New interpretation and confidence intervals. J. Water Resour. Plan. Manag. 1994, 120, 485–504.
- 8.
Hrachowitz, M.; Savenije, H.H.G.; Blöschl, G.; et al. A decade of predictions in ungauged basins (PUB)—A review. Hydrological sciences journal 2013, 58, 1198–1255.
- 9.
Mohamoud, Y.M. Prediction of daily flow duration curves and streamflow for ungauged catchments using regional flow duration curves. Hydrol. Sci. J. 2008, 53, 706–724.
- 10.
Razavi, T.; Coulibaly, P. Streamflow prediction in ungauged basins: Review of regionalization methods. J. Hydrol. Eng. 2013, 18, 958–975.
- 11.
Shu, C.; Ouarda, T. Improved methods for daily streamflow estimates at ungauged sites. Water Resour. Res. 2012, 48, WO2523.
- 12.
Sivaplan, M.; Takeuchi, K.; Franks, S.W.; et al. IAHS decade on predictions in ungauged basins (PUB), 2003–2012: Shaping an exciting future for the hydrological sciences. Hydrol. Sci. J. 2003, 48, 857–880.
- 13.
Mosley, M.P. Delimitation of New Zealand hydrologic regions. J. Hydrol. 1981, 49, 173–192.
- 14.
Vandewiele, G.L.; Elias, A. Monthly water balance of ungauged catchments obtained by geographic regionalization. J. Hydrol. 1995, 170, 277–291.
- 15.
Castellarin, A. Regional prediction of flow-duration curves using a three dimensional kriging. J. Hydrol. 2014, 513, 179–191.
- 16.
Robson, A.; Reed, D. Flood Estimation Handbook—3: Statistical Procedures for Flood Frequency Estimation; Institute of Hydrology: Wallingford, UK, 1999.
- 17.
Li, M.; Shao, Q.X.; Zhang, L.; et al. A new regionalization approach and its application to predict flow duration curve in ungauged basins. J. Hydrol. 2010, 389, 137–145.
- 18.
Atieh, M.; Taylor, G.; Sattar, A.M.; et al. Prediction of flow duration curves for ungauged basins. J. Hydrol. 2017, 545, 383–394.
- 19.
Fouad, G.; Loáiciga, H.A. Independent variable selection for regression modeling of the flow duration curve for ungauged basins in the United States. J. Hydrol. 2020, 587, 124975. https://doi.org/10.1016/j.jhydrol.2020.124975.
- 20.
Leong, C.; Yokoo, Y. A step toward global-scale applicability and transferability of flow duration curve studies: A flow duration curve review (2000–2020). J. Hydrol. 2021, 603, 16. https://doi.org/10.1016/j.jhydrol.2021.126984.
- 21.
Burn, D.H.; Boorman, D.B. Estimation of hydrological parameters at ungauged catchments. J. Hydrol. 1993, 143, 429–454.
- 22.
Gupta, H.V.; Bastidas, L.A.; Sorooshian, S.; et al. Parameter estimation of a land surface scheme using multicriteria methods. J. Geophys. Res. 1999, 104, 19491–19503. https://doi.org/10.1029/1999JD900154.
- 23.
Nijssen; B; Schnur, R.; Lettenmaier, D.P. Global retrospective estimation of soil moisture using the variable infiltration capacity land surface model, 1980–1993. J. Clim. 2001, 14, 1790–1808. https://doi.org/10.1175/1520-0442(2001)014,1790:GREOSM.2.0.CO;2.
- 24.
Bhasme, P.; Vagadiya, J.; Bhatia, U. 2022. Enhancing predictive skills in physically-consistent way: Physics informed machine learning for hydrological processes. J. Hydrol. 2022, 615, 128618.
- 25.
Mohammadi, B.; Gao, H.; Pilesjö, P.; et al. Integrating machine learning with process-based glacio-hydrological model for improving the performance of runoff simulation in cold regions. J. Hydrol. 2025, 656, 132963. https://doi.org/10.1016/j.jhydrol.2025.132963.
- 26.
Holmes, M.G.R.; Young, A.R.; Gustard, A.; et al. A region of influence approach to predicting flow duration curves within ungauged catchments. Hydrol. Earth Syst. Sci. 2002, 6, 721–731. https://doi.org/10.5194/hess-6-721-2002.
- 27.
Mendicino, G.; Senatore, A. Evaluation of parametric and statistical approaches for the regionalization of flow duration curves in intermittent regimes. J. Hydrol. 2013, 480, 19–32. https://doi.org/10.1016/j.jhydrol.2012.12.017.
- 28.
Müller, M. F. and Thompson, S. E.: Comparing statistical and process-based flow duration curve models in ungauged basins and changing rain regimes, Hydrol. Earth Syst. Sci., 20, 669–683, https://doi.org/10.5194/hess-20-669-2016, 2016.
- 29.
Requena, A.I.; Chebana, F.; Ouarda, T.B. A functional framework for flow-duration-curve and daily streamflow estimation at ungauged sites. Adv. Water Resour. 2018, 113, 328–340. https://doi.org/10.1016/j.advwatres.2018.01.019.
- 30.
Costa, V.; Fernandes, W. Regional modeling of long-term and annual flow duration curves: Reliability for information transfer with evolutionary polynomial regression. J. Hydrol. Eng. 2021, 26, 12. https://doi.org/10.1061/(ASCE)HE.1943-5584.0002051.
- 31.
Wolff, W.; Duarte, S.N. Toward geostatistical unbiased predictions of flow duration curves at ungauged basins. Adv. Water Resour. 2021, 152, 13. https://doi.org/10.1016/j.advwatres.2021.103915.
- 32.
Lan, T.; Zhang, J.; Li, H.; et al. Flow duration curve prediction: A framework integrating regionalization and copula model. J. Hydrol. 2025, 647, 132364. https://doi.org/10.1016/j.jhydrol.2024.132364.
- 33.
Friedman J.H. Multivariate adaptive regression splines. Ann. Stat. 1991, 19, 1–141.
- 34.
Budyko, M.I. The Heat Balance of the Earth’s Surface; U.S. Department of Commerce: Washington, DC, USA, 1958.
- 35.
Friedman, J.H.; Roosen, C.B. An introduction to multivariate adaptive regression splines. Stat. Methods Med. Res. 1995, 4, 197–217. https://doi.org/10.1177/096228029500400303.
- 36.
Shao, Q.; Traylen, A.; Zhang, L. Nonparametric method for estimating the effects of climatic and catchment characteristics on mean annual evapotranspiration. Water Resour. Res. 2012, 48, W03517. https://doi.org/10.1029/2010WR009610.
- 37.
Sharda, V.N.; Prasher, S.O.; Patel, R.M.; et al. Performance of Multivariate Adaptive Regression Splines (MARS) in predicting runoff in mid-Himalayan micro-watersheds with limited data. Hydrol. Sci. J. 2008, 53, 1165–1175. https://doi.org/10.1623/hysj.53.6.1165.
- 38.
Beard, L.R. Statistical analysis in hydrology. Trans. Am. Soc. Civ. Eng. 1943, 108, 1110–1160.
- 39.
Post, D.A. A new method for estimating flow duration curves: An application to the Burdekin River Catchment, North Queensland, Australia. In Proceedings of the iEMSs 2004 International Congress ‘‘Complexity and Integrated Resources Management”, Osnabrueck, Germany, 14–17 June 2004.
- 40.
Shao, Q.; Wong, H.; Xia, J.; et al. Models for extremes using the extended three-parameter Burr XII system with application to flood frequency. Hydrol. Sci. J. 2004, 49, 702. https://doi.org/10.1623/hysj.49.4.685.54425.
- 41.
Johnson, N.L.; Kotz, S.; Balakrishnan, N. Lognormal Distributions, Continuous univariate distributions. In Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, 2nd ed.; John Wiley & Sons: New York, NY, USA, 1994; Volume 1; ISBN 978-0-471-58495-7.
- 42.
Milborrow, S. Package “Earth”. 2018. Available online: https://cran.r-project.org/web/packages/earth/earth.pdf (accessed on 1 July 2025).
- 43.
Fu, B.P. On the calculation of evaporation from land surface in mountainous areas. Sci. Meteorol. Sin. 1996, 16, 328–335. (In Chinese)
- 44.
Zhang, L.; Dawes, W.R.; Walker, G.R. The response of mean annual evapotranspiration to vegetation changes at catchment scale. Water Resour. Res. 2001, 37, 701–708.
- 45.
Zhang, L.; Hickel, K.; Dawes, W.R.; et al. A rational function approach for estimating mean annual evapotranspiration. Water Resour. Res. 2004, 40, W02502. https://doi.org/10.1029/2003WR002710.
- 46.
Nash, J.E.; Sutcliffe, J.V. River flow forecasting through conceptual models part I: A discussion of principles. J. Hydrol. 1970, 10, 282–290.
- 47.
Zhang, Y.Q.; Viney, N.; Frost, A.; et al. Collation of Australian Modeller’s Streamflow Dataset for 780 Unregulated Australian Catchments; CSIRO: Canberra, NSW, Australia, 2013; 115p.
- 48.
Ukkola, A.M.; Prentice, I.C.; Keenan, T.F.; et al. Reduced streamflow in water-stressed climates consistent with CO2 effects on vegetation, Nat. Clim. Chang. 2015, 6, 75–78.
- 49.
Jeffrey, S.J.; Carter, J.O.; Moodie, K.M.; et al. Using spatial interpolation to construct a comprehensive archive of Australian climate data. Environ. Model. Softw. 2001, 16, 309–330.
- 50.
Zhu, Z.; Bi, J.; Pan, Y.; et al. Global Data Sets of Vegetation Leaf Area Index (LAI)3g and Fraction of Photosynthetically Active Radiation (FPAR)3g Derived from Global Inventory Modeling and Mapping Studies (GIMMS) Normalized Difference Vegetation Index (NDVI3g) for the Period 1981 to 2011. Remote Sens. 2013, 5, 927–948. https://doi.org/10.3390/rs5020927.
- 51.
MaKenzie, N.J.; Jacquier, D.W.; Ashton, L.J.; et al. Estimation of Soil Properties Using the Atlas of Australian Soils; CSIRO Land and Water, Report 11/00; CSIRO: Canberra, NSW, Australia, 2000.
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