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Estimating hydrologic model uncertainty in the presence of complex residual error structures

  • S. Samadi
  • D. L. Tufford
  • G. J. Carbone
Original Paper

Abstract

Hydrologic models provide a comprehensive tool to estimate streamflow response to environmental variables. Yet, an incomplete understanding of physical processes and challenges associated with scaling processes to a river basin, introduces model uncertainty. Here, we apply generalized additive models of location, scale and shape (GAMLSS) to characterize this uncertainty in an Atlantic coastal plain watershed system. Specifically, we describe distributions of residual errors in a two-step procedure that includes model calibration of the soil and water assessment tool (SWAT) using a sequential Bayesian uncertainty algorithm, followed by time-series modeling of residual errors of simulated daily streamflow. SWAT identified dominant hydrological processes, performed best during moderately wet years, and exhibited less skill during times of extreme flow. Application of GAMLSS to model residuals efficiently produced a description of the error distribution parameters (mean, variance, skewness, and kurtosis), differentiating between upstream and downstream outlets of the watershed. Residual error distribution is better described by a non-parametric polynomial loess curve with a smooth transition from a Box–Cox t distribution upstream to a skew t type 3 distribution downstream. Overall, the fitted models show that low flow events more strongly influence the residual probability distribution, and error variance increases with streamflow discharge, indicating correlation and heteroscedasticity of residual errors. These results provide useful insights into the complexity of error behavior and highlight the value of using GAMLSS models to conduct Bayesian inference in the context of a regression model with unknown skewness and/or kurtosis.

Keywords

GAMLSS SWAT Bayesian model Predictive uncertainty Residual error distribution Coastal plain watershed 

Notes

Acknowledgements

This research was partially supported by the National Oceanic and Atmospheric Administration (NOAA) Climate Program Office (Grant # NA11OAR4310148) to the Carolinas Integrated Sciences and Assessments. The data and related code are available upon a request to the first author. The analyses were performed in R (R Development Core Team, 2013) by using the contributed package GAMLSS and other add-on packages. The authors and maintainers of this software are gratefully acknowledged.

References

  1. Abbaspour KC (2015) User manual for SWAT-CUP, SWAT calibration and uncertainty analysis programs. Swiss Federal Institute of Aquatic Science and Technology, Eawag, Duebendorf, p 93Google Scholar
  2. Abbaspour KC, Yang J, Maximov I, Siber R, Bogner K, Mieleitner J, Zobrist J, Srinivasan R (2007) Modelling hydrology and water quality in the pre-Alpine/Alpine Thur watershed using SWAT. J Hydrol 333:413–430CrossRefGoogle Scholar
  3. Ajami NK, Duan Q, Sorooshian S (2007) An integrated hydrologic Bayesian multimodel combination framework: confronting input, parameter, and model structural uncertainty in hydrologic prediction. Water Resour Res 43:W01403.  https://doi.org/10.1029/2005WR004745 CrossRefGoogle Scholar
  4. Akaike H (1974) A new look at the statistical model identification. IEEE Trans Automat Control 19(6):716–723CrossRefGoogle Scholar
  5. Amatya KM, Jha MK (2011) Evaluating the SWAT model for a low-gradient forested watershed in Coastal South Carolina. Trans ASABE 54(6):2151–2163CrossRefGoogle Scholar
  6. Arnold JG, Allen PM, Bernhardt G (1993) A comprehensive surface-groundwater flow model. J Hydrol 142:47–69CrossRefGoogle Scholar
  7. ASCE (1993) Criteria for evaluation of watershed models. J. Irrig Drain Eng 119(3):429–442CrossRefGoogle Scholar
  8. Bales JD, Pope BF (2001) Identification of changes in streamflow characteristics. J Am Water Resour Assoc 37(1):91–104CrossRefGoogle Scholar
  9. Bates BV, Campbell AEP (2001) Markov Chain Monte Carlo scheme for parameter estimation and inference in conceptual rainfall-runoff modeling. Water Resour Res 37(4):937–947CrossRefGoogle Scholar
  10. Beven KJ (2008) On doing better hydrological science. Hydrol Process 22:3549–3553.  https://doi.org/10.1002/hyp.7108 CrossRefGoogle Scholar
  11. Beven KJ, Freer J (2001) Equifinality, data assimilation, and uncertainty estimation in mechanistic modeling of complex environmental systems. J Hydrol 249:11–29CrossRefGoogle Scholar
  12. Beven K, Smith PJ, Freer JE (2008) So just why would a modeler choose to be incoherent. J Hydrol 354:15–32CrossRefGoogle Scholar
  13. Box GEP, Tiao GC (1992) Bayesian inference in statistical analysis. Wiley, New York, p 588CrossRefGoogle Scholar
  14. Butts MB, Payne JT, Kristensen M, Madsen H (2004) An evaluation of the impact of model structure on hydrological modelling uncertainty for streamflow simulation. J Hydrol 298:242–266.  https://doi.org/10.1016/j.jhydrol.2004.03.042 CrossRefGoogle Scholar
  15. Clark MP et al (2015) A unified approach for process-based hydrologic modeling: 1. Modeling concept. Water Resour Res 51:2498–2514.  https://doi.org/10.1002/2015WR017198 CrossRefGoogle Scholar
  16. Cole TJ, Green PJ (1992) Smoothing reference centile curves: the LMS method and penalized likelihood. Stat Med 11:1305–1319CrossRefGoogle Scholar
  17. Del Giudice D, Honti M, Scheidegger A, Albert C, Reichert P, Rieckermann J (2013) Improving uncertainty estimation in urban hydrological modeling by statistically describing bias. Hydrol Earth Syst Sci 17(2013):4209–4225CrossRefGoogle Scholar
  18. Del Giudice D, Albert C, Rieckermann J, Reichert P (2016) Describing the catchment-averaged precipitation as a stochastic process improves parameter and input estimation. Water Resour Res 52:3162–3186.  https://doi.org/10.1002/2015WR017871 CrossRefGoogle Scholar
  19. Dunn PK, Smyth GK (1996) Randomised quantile residuals. J Comput Gr Stat 5:236–244Google Scholar
  20. Eilers PHC, Marx BD (1996) Flexible smoothing with B-splines and penalties (with comments and rejoinder). Stat Sci 11:89–121CrossRefGoogle Scholar
  21. El Adlouni S, Bobeé B, Ouarda TBMJ (2008) On the tails of extreme event distributions in hydrology. J Hydrol 355:16–33CrossRefGoogle Scholar
  22. Etemadi H, Samadi S, Sharifikia M (2014) Uncertainty analysis of statistical downscaling techniques in an Arid region. Clim Dyn 42:2899–2920CrossRefGoogle Scholar
  23. Etemadi H, Samadi S, Sharifikia M, Smoak JM (2015) Assessment of climate change downscaling and non-stationarity on the spatial pattern of a mangrove ecosystem in an arid coastal region of southern Iran. Theor Appl Climatol.  https://doi.org/10.1007/s00704-015-1552-5 Google Scholar
  24. Fernandez C, Steel MFJ (1998) On bayesian modelling of fat tails and skewness. J Am Stat Assoc 93:359–371Google Scholar
  25. Fernandez C, Osiewalski J, Steel MFJ (1995) Modeling and inference with v-spherical distributions. J Am Stat Assoc 90(432):1331–1340Google Scholar
  26. Green PJ, Silverman BW (1994) Nonparametric regression and generalized linear models. Chapman and Hall, LondonCrossRefGoogle Scholar
  27. Gupta HV, Sorooshian S, Yapo PO (1998) Toward improved calibration of hydrologic models: multiple and noncommensurate measures of information. Water Resour Res 34(4):751–763CrossRefGoogle Scholar
  28. Guzman JA, Moriasi DN, Gowda PH, Steiner JL, Starks PJ, Arnold JG, Srinivasan R (2015) A model integration framework for linking SWAT and MODFLOW. Environ Model Softw 73:103–116CrossRefGoogle Scholar
  29. Han JC, Huang GH, Zhang H et al (2014) Bayesian uncertainty analysis in hydrological modeling associated with watershed subdivision level: a case study of SLURP model applied to the Xiangxi River watershed, China. Stoch Environ Res Risk Assess 28:973.  https://doi.org/10.1007/s00477-013-0792-0 CrossRefGoogle Scholar
  30. Hantush M, Kalin L (2008) Stochastic residual-error analysis for estimating hydrologic model predictive uncertainty. J Hydrol Eng.  https://doi.org/10.1061/(ASCE)1084-0699(2008)13:7(585)585-596 Google Scholar
  31. Hargreaves GL, Hargreaves GH, Riley JP (1985) Agricultural benefits for Senegal River Basin. J Irrig Drain E ASCE 111:113–124CrossRefGoogle Scholar
  32. Hastie TJ, Tibshirani RJ (1990) Generalized additive models. Chapman and Hall, LondonGoogle Scholar
  33. Hipel KW, McLeod AI (1994) Time series modelling of water resources and environmental systems. Elsevier, Amsterdam. http://www.stats.uwo.ca/faculty/aim/1994Book/
  34. Honti M, Stamm C, Reichert P (2013) Integrated uncertainty assessment of discharge predictions with a statistical error model. Water Resour Res 49(2013):4866–4884CrossRefGoogle Scholar
  35. Joseph JF, Guillaume JHA (2013) Using a parallelized MCMC algorithm in R to identify appropriate likelihood functions for SWAT. Environ Model Softw 46:292–298.  https://doi.org/10.1016/j.envsoft.2013.03.012 CrossRefGoogle Scholar
  36. Katz RW (2010) Statistics of extremes in climate change. Clim Change 100:71–76CrossRefGoogle Scholar
  37. Kim T-W, Valdés JB (2005) Synthetic generation of hydrologic time series based on nonparametric random generation. J Hydrol Eng 105:395–404CrossRefGoogle Scholar
  38. Kuczera G (1983) Improved parameter inference in catchment models, 1. Evaluating parameter uncertainty. Water Resour Res 19(5):1151–1162.  https://doi.org/10.1029/WR019i005p01151 CrossRefGoogle Scholar
  39. Laloy E, Vrugt JA (2012) High-dimensional posterior exploration of hydrologic models using multiple-try DREAM(ZS) and high-performance computing. Water Resour Res 48:W01526.  https://doi.org/10.1029/2011WR010608 Google Scholar
  40. Legates DR, McCabe GJ (1999) Evaluating the use of “goodness-of-fit” measures in hydrologic and hydroclimatic model validation. Water Resour Res 35(1):233–241CrossRefGoogle Scholar
  41. McCuen R, Knight Z, Cutter A (2006) Evaluation of the Nash-Sutcliffe Efficiency Index. J Hydrol Eng.  https://doi.org/10.1061/(ASCE)1084-0699(2006)11:6(597)597-602 Google Scholar
  42. McCullagh P, Nelder JA (1989) Generalized linear models, volume 37 of monographs on statistics and applied probability, 2nd edn. Chapman and Hall, LondonGoogle Scholar
  43. McKay MD, Beckman RJ, Conover WJ (1979) Comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2):239–245Google Scholar
  44. McMillan H, Krueger T, Freer J (2012) Benchmarking observational uncertainties for hydrology: rainfall, river discharge and water quality. Hydrol Process 26:4078–4111CrossRefGoogle Scholar
  45. Melching CS, Bauwens W (2001) Uncertainty in coupled nonpoint source and stream water-quality models. J Water Resour Plann Manag 1276:403–413CrossRefGoogle Scholar
  46. Monteith JL (1965) Evaporation and environment. In: Proceedings of the 19th symposium of the society for experimental biology. Cambridge University Press, New York, pp 205–233Google Scholar
  47. Moore C, Wöhling T, Doherty J (2010) Efficient regularization and uncertainty analysis using a global optimization methodology. Water Resour Res 46:W08527.  https://doi.org/10.1029/2009WR008627 CrossRefGoogle Scholar
  48. Moriasi DN, Arnold JG, Van Liew MW, Binger RL, Harmel RD, Veith T (2007) Model evaluation guidelines for systematic quantification of accuracy in watershed simulations. Trans ASABE 50(3):885–900CrossRefGoogle Scholar
  49. Mosaedi A, Zare Abyane H, Ghabaei Sough M, Zahra Samadi S (2015) Long-lead drought forecasting using equiprobability transformation function for reconnaissance drought index. Water Resour Manag 29:2451–2469CrossRefGoogle Scholar
  50. Nash JE, Sutcliffe JV (1970) River flow forecasting through conceptual models: part 1. A discussion of principles. J Hydrol 10(3):282–290CrossRefGoogle Scholar
  51. Neitsch SL, Arnold JG, Kiniry JR, Williams JR (2001) Soil and water assessment tool user’s manual, version 2000. Grassland, Soil and Water Research Laboratory, Agricultural Research Service, Blackland Research Center, Texas Agricultural Experiment StationGoogle Scholar
  52. Nimmo JR, Healy RW, Stonestrom DA (2005) Aquifer recharge. In: Anderson MG, Bear J (eds) Encyclopedia of hydrological science: part 13. Groundwater, vol 4. Wiley, Chichester, pp 2229–2246.  https://doi.org/10.1002/0470848944.hsa161a Google Scholar
  53. Pourreza-Bilondi M, Samadi S (2016) Quantifying the uncertainty of semiarid runoff extremes using generalized likelihood uncertainty estimation. Special issues on water resources in arid areas. Arab J Geosci.  https://doi.org/10.1007/s12517-016-2650-0 Google Scholar
  54. Pourreza-Bilondi M, Samadi SZ, Akhoond-Ali AM, Ghahraman B (2016) On the assessment of reliability in semiarid convective flood modeling using bayesian framework. ASCE Hydrol Eng.  https://doi.org/10.1061/(ASCE)HE.1943-5584.0001482 Google Scholar
  55. Priestley CHB, Taylor RJ (1972) On the assessment of surface heat flux and evaporation using large-scale parameters. Mon Weather Rev 100(2):81–92CrossRefGoogle Scholar
  56. Rigby RA, Stasinopoulos DM (1996a) A semi-parametric additive model for variance heterogeneity. Statist Comput 6:57–65CrossRefGoogle Scholar
  57. Rigby RA, Stasinopoulos DM (1996b) Mean and dispersion additive models. In: Hardle W, Schimek MG (eds) Statistical theory and computational aspects of smoothing. Physica, Heidelberg, pp 215–230CrossRefGoogle Scholar
  58. Rigby RA, Stasinopoulos DM (2005a) Generalized additive models for location, scale and shape (with discussion). Appl Stat 54:507–554Google Scholar
  59. Rigby RA, Stasinopoulos DM (2005b) Generalized additive models for location, scale and shape. J R Stat Soc Ser C (Appl Stat) 54:507–554.  https://doi.org/10.1111/j.1467-9876.2005.00510.x CrossRefGoogle Scholar
  60. Riggs SR, Ames DV, Brant DR, Sager ED (2000) The Waccamaw drainage system: geology and dynamics of a coastal wetland, Southeastern North Carolina. East Carolina University, Greenville, p 165Google Scholar
  61. Royston P, Altman DG (1994) Regression using fractional polynomials of continuous covariates: parsimonious parametric modelling (with discussion). Appl Stat 43:429–467CrossRefGoogle Scholar
  62. R Development Core Team (2013) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, ISBN 3-900051-07-0. http://www.R-project.org/
  63. Sadegh M, Vrugt JA (2013) Approximate Bayesian computation in hydrologic modeling: equifinality of formal and informal approaches. Hydrol Earth Syst Sci Dis 10(4):p4739CrossRefGoogle Scholar
  64. Samadi S (2016) Assessing the sensitivity of SWAT physical parameters to potential evapotranspiration estimation methods over a coastal plain watershed in the Southeast United States. Hydrol Res.  https://doi.org/10.2166/nh.2016.034 Google Scholar
  65. Samadi S, Meadows EM (2017) The transferability of terrestrial water balance components under uncertainty and non-stationarity: a case study of the coastal plain watershed in the Southeastern United States. River Res Appl.  https://doi.org/10.1002/rra.3127 Google Scholar
  66. Samadi S, Tufford DL, Carbone GJ (2017) Assessing parameter uncertainty of a semi-distributed hydrology model for a shallow aquifer dominated environmental system. J Am Water Resour Assoc (JAWRA) 1–22.  https://doi.org/10.1111/1752-1688.12596
  67. Schoups G, Vrugt JA (2010) A formal likelihood function for parameter and predictive inference of hydrologic models with correlated, heteroscedastic, and non-Gaussian errors. Water Resour Res 46:W10531.  https://doi.org/10.1029/2009WR008933 Google Scholar
  68. Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6(2):461–464CrossRefGoogle Scholar
  69. Sénégas J, Wackernagel H, Rosenthal W et al (2001) Error covariance modeling in sequential data assimilation. Stoch Env Res Risk Assess 15:65.  https://doi.org/10.1007/PL00009788 CrossRefGoogle Scholar
  70. Serinaldi F (2011) Distributional modeling and short-term forecasting of electricity prices by generalized additive models for location, scale and shape. Energy Econ 33(6):1216–1226CrossRefGoogle Scholar
  71. Serinaldi F, Cuomo G (2011) Characterizing impulsive wave-in-deck loads on coastal bridges by probabilistic models of impact maxima and rise times. Coast Eng 58(9):908–926CrossRefGoogle Scholar
  72. Serinaldi F, Kilsby CG (2015) Stationarity is undead: uncertainty dominates the distribution of extremes. Adv Wat Resour 77:17–36CrossRefGoogle Scholar
  73. Sevat E, Dezetter A (1991) Selection of calibration objective functions in the context of rainfall-runoff modeling in a Sudanese savannah area. Hydrol Sci J 36(4):307–330CrossRefGoogle Scholar
  74. Shrestha B, Cochrane TA, Caruso BS, Arias ME, Piman T (2016) Uncertainty in flow and sediment projections due to future climate scenarios for the 3S Rivers in the Mekong Basin. J Hydrol 540:1088–1104.  https://doi.org/10.1016/j.jhydrol.2016.07.019 CrossRefGoogle Scholar
  75. Sikorska AE, Scheidegger A, Banasik K, Rieckermann J (2012) Bayesian uncertainty assessment of flood predictions in ungauged urban basins for conceptual rainfall-runoff models. Hydrol Earth Syst Sci 16:1221–1236.  https://doi.org/10.5194/hess-16-1221-2012 CrossRefGoogle Scholar
  76. Sivapalan M (2009) The secret to ‘doing better hydrological science’: change the question! Hydrol Process 23:1391–1396.  https://doi.org/10.1002/hyp.7242 CrossRefGoogle Scholar
  77. Slater AG, Clark MP (2006) Snow data assimilation via an ensemble Kalman filter. J Hydrometeorol 7(3):478–493CrossRefGoogle Scholar
  78. Sorooshian S, Dracup JA (1980) Stochastic parameter estimation procedures for hydrologic rainfall-runoff models—correlated and heteroscedastic error cases. Water Resour Res 16(2):430–442.  https://doi.org/10.1029/WR016i002p00430 CrossRefGoogle Scholar
  79. Stasinopoulos DM, Rigby RA (2007) Generalized additive models for location scale and shape (GAMLSS) in R. J Stat Softw 23:1–46CrossRefGoogle Scholar
  80. Stasinopoulos DM, Rigby RA (2016) Package ‘gamlss.dist’. https://cran.r-project.org/web/packages/gamlss.dist/index.html
  81. Tian Y, Booij MJ, Xu YP (2014) Uncertainty in high and low flows due to model structure and parameter errors. Stoch Environ Res Risk Assess 28:319.  https://doi.org/10.1007/s00477-013-0751-9 CrossRefGoogle Scholar
  82. Tongal H, Booij MJ (2017) Quantification of parametric uncertainty of ANN models with GLUE method for different streamflow dynamics. Stoch Environ Res Risk Assess 31:993.  https://doi.org/10.1007/s00477-017-1408-x CrossRefGoogle Scholar
  83. USDA-SCS (United States Department of Agriculture–Soil Conservation Service) (1972) National engineering handbook, Section 4 Hydrology, Chapter 4–10, USDA-SCS, WashingtonGoogle Scholar
  84. Van Buuren S, Fredriks M (2001) Worm plot: a simple diagnostic device for modelling growth reference curves. Stat Med 20:1259–1277CrossRefGoogle Scholar
  85. Villarini G, Smith JA, Serinaldi F, Bales J, Bates PD, Krajewski WF (2009) Flood frequency analysis for nonstationary annual peak records in an urban drainage area. Adv Water Resour 32:1255–1266CrossRefGoogle Scholar
  86. Vrugt JA, ter Braak CJF, Clark MP, Hyman JM, Robinson BA (2008) Treatment of input uncertainty in hydrologic modeling: doing hydrology backward with Markov chain Monte Carlo simulation. Water Resour Res 44:W00B09.  https://doi.org/10.1029/2007WR006720 CrossRefGoogle Scholar
  87. Vrugt JA, ter Braak CJF, Gupta HV, Robinson BA (2009a) Equifinality of formal (DREAM) and informal (GLUE) Bayesian approaches in hydrologic modeling? Stoch Environ Res Risk Assess 23:1011.  https://doi.org/10.1007/s00477-008-0274-y CrossRefGoogle Scholar
  88. Vrugt JA, ter Braak CJF, Diks CGH, Robinson BA, Hyman JM, Higdon D (2009b) Accelerating Markov Chain Monte Carlo simulation by differential evolution with self-adaptive randomized subspace sampling. Int J Nonlinear Sci Numer Simul 10(3):271e288CrossRefGoogle Scholar
  89. Wagener T, Gupta HV (2005) Model identification for hydrological forecasting under uncertainty. Stoch Environ Res Risk Assess 19:378.  https://doi.org/10.1007/s00477-005-0006-5 CrossRefGoogle Scholar
  90. Wagener T, Sivapalan M, Troch P, Woods R (2007) Catchment classification and hydrologic similarity. Geogr Compass 1:901–931CrossRefGoogle Scholar
  91. Wang G, Barber ME, Chen S et al (2014) SWAT modeling with uncertainty and cluster analyses of tillage impacts on hydrological processes. Stoch Environ Res Risk Assess 28:225.  https://doi.org/10.1007/s00477-013-0743-9 CrossRefGoogle Scholar
  92. Westra S, Thyer M, Leonard M, Kavetski D, Lambert M (2014) A strategy for diagnosing and interpreting hydrological model nonstationarity. Water Resour Res 50:5090–5113.  https://doi.org/10.1002/2013WR014719 CrossRefGoogle Scholar
  93. Williams JR (1969) Flood routing with variable travel time or variable storage coefficients. Trans ASAE 12:100–103CrossRefGoogle Scholar
  94. Yang J, Reichert P, Abbaspour KC, Yang H (2007) Hydrological modelling of the Chaohe Basin in China: statistical model formulation and Bayesian inference. J Hydrol 340(2007):167–182CrossRefGoogle Scholar
  95. Yang J, Abbaspour KC, Reichert P, Yang H (2008) Comparing uncertainty analysis techniques for a SWAT application to Chaohe Basin in China. J Hydrol 358:1–23CrossRefGoogle Scholar
  96. Zhang HX, Yu SL (2004) Applying the first-order error analysis in determining the margin of safety for total maximum daily load computations. J Environ Eng 1306:664–673CrossRefGoogle Scholar
  97. Zhang X, Srinivasan R, Bosch D (2009) Calibration and uncertainty analysis of the SWAT model using genetic algorithms and Bayesian model averaging. J Hydrol 374:307–317.  https://doi.org/10.1016/j.jhydrol.2009.06.023 CrossRefGoogle Scholar
  98. Zheng Y, Han F (2016) Markov Chain Monte Carlo (MCMC) uncertainty analysis for watershed water quality modeling and management. Stoch Environ Res Risk Assess 30:293.  https://doi.org/10.1007/s00477-015-1091-8 CrossRefGoogle Scholar
  99. Zhenyao S, Lei C, Tao C (2013) The influence of parameter distribution uncertainty on hydrological and sediment modeling: a case study of SWAT model applied to the Daning watershed of the Three Gorges Reservoir Region, China. Stoch Environ Res Risk Assess 27:235.  https://doi.org/10.1007/s00477-012-0579-8 CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringUniversity of South CarolinaColumbiaUSA
  2. 2.Department of Biological SciencesUniversity of South CarolinaColumbiaUSA
  3. 3.Department of GeographyUniversity of South CarolinaColumbiaUSA

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