# Bayesian Hierarchical Model Uncertainty Quantification for Future Hydroclimate Projections in Southern Hills-Gulf Region, USA

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. A Hierarchical Uncertainty Analysis Framework

#### 2.1. Bayesian Model Averaging (BMA) Tree

#### 2.2. Hierarchical Bayesian Model Averaging (HBMA)

**D**and models ${M}_{n+1}$ at level n + 1 is:

**D**and models ${M}_{p}$ at level p. $\mathrm{Var}\left(\Delta |D,{M}_{n+1}\right)$ is the prediction variance using models at level n + 1. It includes the within-model variance ${E}_{{M}_{n+1}}\left[\mathrm{Var}\left(\Delta |D,{M}_{n+1}\right)\right]$ and the between-model variance ${\mathrm{Var}}_{{M}_{n+1}}\left[E\left(\Delta |D,{M}_{n+1}\right)\right]$. ${\mathrm{Var}}_{{M}_{n+1}}[\xb7]$ is the variance operator, which calculates the between-model variance using models at level n + 1. For example, given an RCP at level 1 (see Figure 1), $\mathrm{Var}\left(\Delta |D,{M}_{1}\right)$ is the projection variance for Δ using various GCMs (${M}_{2}$) at level 2 under the same RCP. The within-model variance ${E}_{{M}_{2}}\left[\mathrm{Var}\left(\Delta |D,{M}_{2}\right)\right]$ is the expectation of projection variances of individual GCMs, $\mathrm{Var}\left(\Delta |D,{M}_{2}\right)$, under the same RCP. The between-model variance ${\mathrm{Var}}_{{M}_{1}}\left[E\left(\Delta |D,{M}_{1}\right)\right]$ is due to the variation of the mean predicted Δ by different GCMs at level 1.

#### 2.3. Mean and Variance at the Hierarch Level

## 3. Case Study

#### 3.1. Study Area and Model Data

^{2}. It includes 28 parishes of Louisiana and 20 counties of Mississippi. The area contains 26 U.S. Geological Survey (USGS) 8-digit hydrologic units (HUC8 or subbasin) and 728 12-digit hydrologic units (HUC12 or subwatershed) with average sizes of 588 and 33 km

^{2}, respectively. Precipitation at the outcrops in southwestern Mississippi is the main source of water to the deep aquifers in southeastern Louisiana.

#### 3.2. Climate Projections

#### 3.2.1. Downscaled Climate Projection Data Set

#### 3.2.2. Climate Model Evaluation

_{min}is the minimum BIC value among the climate models. Focusing on monthly precipitation and temperature, the BIC can be written as:

#### 3.3. Hydrologic Modeling

#### 3.3.1. HELP3 Input Data

#### 3.3.2. Parallel Computation for High-Resolution Hydrologic Prediction

## 4. Results and Discussion

#### 4.1. Posterior Model Probabilities

#### 4.2. Temporal Analysis with HBMA

#### 4.3. Future Hydrologic Projection Anomalies

#### 4.4. Spatial Analysis

#### 4.5. Contributions of Sources of Uncertainty

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Katz, R.W.; Craigmile, P.F.; Guttorp, P.; Haran, M.; Sansó, B.; Stein, M.L. Uncertainty analysis in climate change assessments. Nat. Clim. Chang.
**2013**, 3, 769–771. [Google Scholar] [CrossRef] - Solomon, S.; Qin, D.; Manning, M.; Marquis, M.; Averyt, K.B.; Tignor, M.; Miller, H.L.; Chen, Z. Climate Change 2007-the Physical Science Basis. Working Group I Contribution to the Fourth Assessment Report of the IPCC; Cambridge University Press: Cambridge, UK, 2007; Volume 4. [Google Scholar]
- Hawkins, E.; Sutton, R. The potential to narrow uncertainty in regional climate predictions. Bull. Am. Meteorol. Soc.
**2009**, 90, 1095–1107. [Google Scholar] [CrossRef] - Yip, S.; Ferro, C.A.T.; Stephenson, D.B.; Hawkins, E. A simple, coherent framework for partitioning uncertainty in climate predictions. J. Clim.
**2011**, 24, 4634–4643. [Google Scholar] [CrossRef] - Chen, J.; Brissette, F.P.; Leconte, R. Uncertainty of downscaling method in quantifying the impact of climate change on hydrology. J. Hydrol.
**2011**, 401, 190–202. [Google Scholar] [CrossRef] - Wilby, R.L.; Harris, I. A framework for assessing uncertainties in climate change impacts: Low-flow scenarios for the River Thames, UK. Water Resour. Res.
**2006**, 42, W02419. [Google Scholar] [CrossRef] - Déqué, M. Frequency of precipitation and temperature extremes over France in an anthropogenic scenario: Model results and statistical correction according to observed values. Glob. Planet. Chang.
**2007**, 57, 16–26. [Google Scholar] [CrossRef] - Kay, A.L.; Davies, H.N.; Bell, V.A.; Jones, R.G. Comparison of uncertainty sources for climate change impacts: Flood frequency in England. Clim. Chang.
**2009**, 92, 41–63. [Google Scholar] [CrossRef] - Görgen, K.; Beersma, J.; Brahmer, G.; Buiteveld, H.; Carambia, M.; Keizer, O.D.; Krahe, P.; Nilson, E.; Lammersen, R.; Perrin, C.; et al. Assessment of Climate Change Impacts on Discharge in the Rhine River Basin: Results of the RheinBlick2050 Project; CHR Rep I-23; CHR: Lelystad, The Netherlands, 2010; 229p. [Google Scholar]
- Bosshard, T.; Carambia, M.; Goergen, K.; Kotlarski, S.; Krahe, P.; Zappa, M.; Schär, C. Quantifying uncertainty sources in an ensemble of hydrological climate-impact projections. Water Resour. Res.
**2013**, 49, 1523–1536. [Google Scholar] [CrossRef] [Green Version] - Chen, J.; Brissette, F.P.; Poulin, A.; Leconte, R. Overall uncertainty study of the hydrological impacts of climate change for a Canadian watershed. Water Resour. Res.
**2011**, 47, W12509. [Google Scholar] [CrossRef] - Kwon, H.H.; Assis de Souza Filho, F.; Block, P.; Sun, L.; Lall, U.; Reis, D.S. Uncertainty assessment of hydrologic and climate forecast models in Northeastern Brazil. Hydrol. Process.
**2012**, 26, 3875–3885. [Google Scholar] [CrossRef] - Habets, F.; Boé, J.; Déqué, M.; Ducharne, A.; Gascoin, S.; Hachour, A.; Martin, E.; Pagé, C.; Sauquet, E.; Terray, L.; et al. Impact of climate change on the hydrogeology of two basins in Northern France. Clim. Chang.
**2013**, 121, 771–785. [Google Scholar] [CrossRef] - Surfleet, C.G.; Tullos, D. Uncertainty in hydrologic modelling for estimating hydrologic response due to climate change (Santiam River, Oregon). Hydrol. Process.
**2013**, 27, 3560–3576. [Google Scholar] [CrossRef] - Dessu, S.B.; Melesse, A.M. Impact and uncertainties of climate change on the hydrology of the Mara River basin, Kenya/Tanzania. Hydrol. Process.
**2013**, 27, 2973–2986. [Google Scholar] [CrossRef] - Knutti, R.; Furrer, R.; Tebaldi, C.; Cermak, J.; Meehl, G.A. Challenges in combining projections from multiple climate models. J. Clim.
**2010**, 23, 2739–2758. [Google Scholar] [CrossRef] - McAvaney, B.J.; Covey, C.; Joussaume, S.; Kattsov, V.; Kitoh, A.; Ogana, W.; Pitman, A.J.; Weaver, A.J.; Wood, R.A.; Zhao, Z.-C.; et al. Model evaluation. In Climate Change 2001: The Scientific Basis; Houghton, J.T., Ding, Y., Griggs, D.J., Noguer, M., van der Linden, P.J., Dai, X., Maskell, K., Johnson, C.A., Eds.; Cambridge University Press: New York, NY, USA, 2001; Volume 8, pp. 471–524. [Google Scholar]
- Pierce, D.W.; Barnett, T.P.; Santer, B.D.; Gleckler, P.J. Selecting global climate models for regional climate change studies. Proc. Natl. Acad. Sci. USA
**2009**, 106, 8441–8446. [Google Scholar] [CrossRef] [Green Version] - Stocker, T.; Dahe, Q.; Plattner, G.-K.; Tignor, M.; Midgley, P. IPCC Expert Meeting on Assessing and Combining Multi Model Climate Projections. In Proceedings of the Intergovernmental Panel on Climate Change, Boulder, CO, USA, 25–27 January 2010; IPCC: Bern, Switzerland, 2010. [Google Scholar]
- Doblas-Reyes, F.J.; Pavan, V.; Stephenson, D.B. The skill of multi-model seasonal forecasts of the wintertime North Atlantic Oscillation. Clim. Dyn.
**2003**, 21, 501–514. [Google Scholar] [CrossRef] - Yun, W.T.; Stefanova, L.; Krishnamurti, T.N. Improvement of the multimodel superensemble technique for seasonal forecasts. J. Clim.
**2003**, 16, 3834–3840. [Google Scholar] [CrossRef] - Schmittner, A.; Latif, M.; Schneider, B. Model projections of the North Atlantic thermohaline circulation for the 21st century assessed by observations. Geophys. Res. Lett.
**2005**, 32. [Google Scholar] [CrossRef] [Green Version] - Connolley, W.M.; Bracegirdle, T.J. An Antarctic assessment of IPCC AR4 coupled models. Geophys. Res. Lett.
**2007**, 34, L22505. [Google Scholar] [CrossRef] - Murphy, J.M.; Booth, B.B.; Collins, M.; Harris, G.R.; Sexton, D.M.; Webb, M.J. A methodology for probabilistic predictions of regional climate change from perturbed physics ensembles. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci.
**2007**, 365, 1993–2028. [Google Scholar] [CrossRef] [Green Version] - Tebaldi, C.; Knutti, R. The use of multi-model ensemble in probabilistic climate projections. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci.
**2007**, 365, 2053–2075. [Google Scholar] [CrossRef] - Waugh, D.W.; Eyring, V. Quantitative performance metrics for stratospheric-resolving chemistry-climate models. Atmos. Chem. Phys.
**2008**, 8, 5699–5713. [Google Scholar] [CrossRef] [Green Version] - Hoeting, J.A.; Madigan, D.; Raftery, A.E.; Volinsky, C.T. Bayesian model averaging: A tutorial. Stat. Sci.
**1999**, 14, 382–401. [Google Scholar] - Clyde, M.A. Bayesian model averaging and model search strategies. Bayesian Stat.
**1999**, 6, 157–185. [Google Scholar] - Raftery, A.E.; Zheng, Y. Discussion: Performance of Bayesian model averaging. J. Am. Stat. Assoc.
**2003**, 98, 931–938. [Google Scholar] [CrossRef] - Raftery, A.E.; Gneiting, T.; Balabdaoui, F.; Polakowski, M. Using Bayesian model averaging to calibrate forecast ensembles. Mon. Weather Rev.
**2005**, 133, 1155–1174. [Google Scholar] [CrossRef] - Sloughter, J.M.L.; Raftery, A.E.; Gneiting, T.; Fraley, C. Probabilistic quantitative precipitation forecasting using Bayesian model averaging. Mon. Weather Rev.
**2007**, 135, 3209–3220. [Google Scholar] [CrossRef] - Tebaldi, C.; Smith, R.L.; Nychka, D.; Mearns, L.O. Quantifying uncertainty in projections of regional climate change: A Bayesian approach to the analysis of multi-model ensembles. J. Clim.
**2005**, 18, 1524–1540. [Google Scholar] [CrossRef] - Min, S.K.; Simonis, D.; Hense, A. Probabilistic climate change predictions applying Bayesian model averaging. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci.
**2007**, 365, 2103–2116. [Google Scholar] [CrossRef] [Green Version] - Buser, C.M.; Künsch, H.R.; Lüthi, D.; Wild, M.; Schär, C. Bayesian multi-model projection of climate: Bias assumptions and interannual variability. Clim. Dyn.
**2009**, 33, 849–868. [Google Scholar] [CrossRef] - Smith, R.L.; Tebaldi, C.; Nychka, D.; Mearns, L.O. Bayesian modeling of uncertainty in ensembles of climate models. J. Am. Stat. Assoc.
**2009**, 104, 97–116. [Google Scholar] [CrossRef] - Ajami, N.K.; Duan, Q.; Gao, X.; Sorooshian, S. Multimodel combination techniques for analysis of hydrological simulations: Application to distributed model intercomparison project results. J. Hydrometeorol.
**2006**, 7, 755–768. [Google Scholar] [CrossRef] - Duan, Q.; Ajami, N.K.; Gao, X.; Sorooshian, S. Multi-model ensemble hydrologic prediction using Bayesian model averaging. Adv. Water Resour.
**2007**, 30, 1371–1386. [Google Scholar] [CrossRef] - Vrugt, J.A.; Robinson, B.A. Treatment of uncertainty using ensemble methods: Comparison of sequential data assimilation and Bayesian model averaging. Water Resour. Res.
**2007**, 43, W01411. [Google Scholar] [CrossRef] - Zhang, X.; Srinivasan, R.; Bosch, D. Calibration and uncertainty analysis of the SWAT model using genetic algorithms and Bayesian model averaging. J. Hydrol.
**2009**, 374, 307–317. [Google Scholar] [CrossRef] - Najafi, M.R.; Moradkhani, H.; Jung, W.I. Combined effect of global climate projection and hydrologic model uncertainties on the future changes of Streamflow. In Proceedings of the World Environmental and Water Resources Congress 2010, Rhode, Island, 16–20 May 2010; pp. 81–91. [Google Scholar] [CrossRef]
- Dong, L.; Xiong, L.; Yu, K.X. Uncertainty analysis of multiple hydrologic models using the Bayesian model averaging method. J. Appl. Math.
**2013**, 2013, 346045. [Google Scholar] [CrossRef] - Liang, Z.; Wang, D.; Guo, Y.; Zhang, Y.; Dai, R. Application of Bayesian model averaging approach to multimodel ensemble hydrologic forecasting. J. Hydrol. Eng. ASCE
**2013**, 18, 1426–1436. [Google Scholar] [CrossRef] - Wagener, T.; Gupta, H.V. Model identification for hydrological forecasting under uncertainty. Stoch. Environ. Res. Risk Assess.
**2005**, 19, 378–387. [Google Scholar] [CrossRef] - Tsai, F.T.-C.; Elshall, A.S. Hierarchical Bayesian model averaging for hydrostratigraphic modeling: Uncertainty segregation and comparative evaluation. Water Resour. Res.
**2013**, 49, 5520–5536. [Google Scholar] [CrossRef] [Green Version] - Maurer, E.P.; Brekke, L.; Pruitt, T.; Duffy, P.B.; Thrasher, B.; Long, J.; Arnold, J. An enhanced archive facilitating climate impacts and adaptation analysis. Bull. Am. Meteorol. Soc.
**2013**, 95, 1011–1019. [Google Scholar] [CrossRef] - USBR. Downscaled CMIP3 and CMIP5 Climate and Hydrology Projections: Release of Downscaled CMIP5 Climate Projections, Comparison with preceding Information, and Summary of User Needs, prepared by the U.S.; Department of the Interior, Bureau of Reclamation (USBR), Technical Services Center: Denver, CO, USA, 2013; 47p.
- Schroeder, P.R.; Dozier, T.S.; Zappi, P.A.; McEnroe, B.M.; Sjostrom, J.W.; Peyton, R.L. The Hydrologic Evaluation of Landfill Performance (HELP) Model: Engineering Documentation for Version 3; EPA/600/R-94/168b; U.S. Environmental Protection Agency Office of Research and Development: Washington, DC, USA, 1994.
- Murphy, J.M.; Sexton, D.M.; Barnett, D.N.; Jones, G.S.; Webb, M.J.; Collins, M.; Stainforth, D.A. Quantification of modelling uncertainties in a large ensemble of climate change simulations. Nature
**2004**, 430, 768–772. [Google Scholar] [CrossRef] - Buono, A. The Southern Hills Regional Aquifer System of Southeastern Louisiana and Southwestern Mississippi; U.S. Geological Survey, Water-Resources Investigations Report 83-4189; US Geological Survey: Reston, VA, USA, 1983; 38p.
- Maurer, E.P.; Wood, A.W.; Adam, J.C.; Lettenmaier, D.P.; Nijssen, B. A long-term hydrologically based dataset of land surface fluxes and states for the conterminous United States. J. Clim.
**2002**, 15, 3237–3251. [Google Scholar] [CrossRef] - Maurer, E.P. Gridded Meteorological Data: 1949–2010, Santa Clara University. 2013. Available online: http://www.engr.scu.edu/~emaurer/gridded_obs/index_gridded_obs.html (accessed on 2 February 2019).
- Beigi, E.; Tsai, F.T.-C. A GIS-based water budget framework for high-resolution groundwater recharge estimation of large-scale humid regions. J. Hydrol. Eng.-ASCE
**2014**, 19, 8. [Google Scholar] [CrossRef] - Beigi, E.; Tsai, F.T.-C. Comparative study of climate-change scenarios on groundwater recharge, southwestern Mississippi and southeastern Louisiana, USA. Hydrogeol. J.
**2015**, 23, 789–806. [Google Scholar] [CrossRef] - Rogelj, J.; Meinshausen, M.; Knutti, R. Global warming under old and new scenarios using IPCC climate sensitivity range estimates. Nat. Clim. Chang.
**2012**, 2, 248–253. [Google Scholar] [CrossRef] - Meinshausen, M.; Smith, S.J.; Calvin, K.; Daniel, J.S.; Kainuma, M.L.; Lamarque, J.F.; Matsumoto, K.; Montzka, S.A.; Raper, S.C.; Riahi, K.; et al. The RCP greenhouse gas concentrations and their extensions from 1765 to 2300. Clim. Chang.
**2011**, 109, 213–241. [Google Scholar] [CrossRef] [Green Version] - Johnson, F.; Westra, S.; Sharma, A.; Pitman, A.J. An assessment of GCM skill in simulating persistence across multiple time scales. J. Clim.
**2011**, 24, 3609–3623. [Google Scholar] [CrossRef] - Khan, M.Z.K.; Sharma, A.; Mehrotra, R.; Schepen, A.; Wang, Q.J. Does improved SSTA prediction ensure better seasonal rainfall forecasts? Water Resour. Res.
**2015**, 51, 3370–3383. [Google Scholar] [CrossRef] [Green Version] - Tsai, F.T.-C.; Li, X. Inverse groundwater modeling for hydraulic conductivity estimation using Bayesian model averaging and variance window. Water Resour. Res.
**2008**, 44, W09434. [Google Scholar] [CrossRef] - Khire, M.V.; Benson, C.H.; Bosscher, P.J. Water balance modeling of earthen final covers. J. Geotech. Geoenviron. Eng.
**1997**, 123, 744–754. [Google Scholar] [CrossRef] - Jyrkama, M.I.; Sykes, J.F. The impact of climate change on spatially varying groundwater recharge in the grand river watershed (Ontario). J. Hydrol.
**2007**, 338, 237–250. [Google Scholar] [CrossRef] - Scibek, J.; Allen, D.M.; Cannon, A.J.; Whitfield, P.H. Groundwater–surface water interaction under scenarios of climate change using a high-resolution transient groundwater model. J. Hydrol.
**2007**, 333, 165–181. [Google Scholar] [CrossRef] - Toews, M.W.; Allen, D.M. Evaluating different GCMs for predicting spatial recharge in an irrigated arid region. J. Hydrol.
**2009**, 374, 265–281. [Google Scholar] [CrossRef] - Calderhead, A.I.; Martel, R.; Garfias, J.; Rivera, A.; Therrien, R. Pumping dry: An increasing groundwater budget deficit induced by urbanization, industrialization, and climate change in an over-exploited volcanic aquifer. Environ. Earth Sci.
**2012**, 66, 1753–1767. [Google Scholar] [CrossRef] - Peyton, R.L.; Schroeder, P.R. Field verification of HELP model for landfills. J. Environ. Eng. ASCE
**1988**, 114, 247–269. [Google Scholar] [CrossRef] - Schroeder, P.R.; Peyton, R.L. Verification of the Hydrologic Evaluation of Landfill Performance (HELP) Model Using Field Datha; EPA/600/S2-87/050; Hazardous Waste Engineering Research Laboratory, Office of Research and Development, U.S. Environmental Protection Agency: Cincinnati, OH, USA, 1987.
- Robbins, K. Southern Regional Climate Center, Louisiana State University. Available online: http://www.srcc.lsu.edu (accessed on 2 February 2019).
- Richardson, C.W.; Wright, D.A. WGEN: A Model for Generating Daily Weather Variables; ARS-8; U.S. Department of Agriculture, Agricultural Research Service: Washington, DC, USA, 1984; p. 83.
- NRCS. Soil Survey Geographic (SSURGO). Soil Survey Staff, Natural Resources Conservation Service; U.S. Department of Agriculture: Washington, DC, USA, 2014. Available online: http://sdmdataaccess.nrcs.usda.gov/ (accessed on 2 February 2019).
- NLCD. USDA/NRCS National Geospatial Center of Excellence. National Land Cover Dataset. 2011. Available online: http://datagateway.nrcs.usda.gov/ (accessed on 2 February 2019).
- Jin, S.; Yang, L.; Danielson, P.; Homer, C.; Fry, J.; Xian, G. A comprehensive change detection method for updating the National Land Cover Database to circa 2011. Remote Sens. Environ.
**2013**, 132, 159–175. [Google Scholar] [CrossRef] - Mu, Q.; Heinsch, F.A.; Zhao, M.; Running, S.W. Development of a global evapotranspiration algorithm based on MODIS and global meteorology data. Remote Sens. Environ.
**2007**, 111, 519–536. [Google Scholar] [CrossRef] - Mu, Q.; Zhao, M.; Running, S.W. Improvements to a MODIS global terrestrial evapotranspiration algorithm. Remote Sens. Environ.
**2011**, 115, 1781–1800. [Google Scholar] [CrossRef] - Prudhomme, C.; Davies, H. Assessing uncertainties in climate change impact analyses on the river flow regimes in the, U.K. Part 2: Future climate. Clim. Chang.
**2009**, 93, 197–222. [Google Scholar] [CrossRef] - Woldemeskel, F.M.; Sharma, A.; Sivakumar, B.; Mehrotra, R. An error estimation method for precipitation and temperature projections for future climates. J. Geophys. Res. Atmos.
**2012**, 117. [Google Scholar] [CrossRef] [Green Version] - Teng, J.; Vaze, J.; Chiew, F.H.S.; Wang, B.; Perraud, J.M. Estimating the relative uncertainties sourced from GCMs and hydrological models in modeling climate change impact on runoff. J. Hydrometeorol.
**2012**, 13, 122–139. [Google Scholar] [CrossRef] - Hosseinzadehtalaei, P.; Tabari, H.; Willems, P. Uncertainty assessment for climate change impact on intense precipitation: How many model runs do we need? Int. J. Climatol.
**2017**, 37, 1105–1117. [Google Scholar] [CrossRef] - Mandal, S.; Breach, P.A.; Simonovic, S.P. Uncertainty in Precipitation Projection under Changing Climate Conditions: A Regional Case Study. Am. J. Clim. Chang.
**2016**, 5, 116–132. [Google Scholar] [CrossRef] - Taylor, R.G.; Kingston, D.G. Sources of uncertainty in climate change impacts on river discharge and groundwater in a headwater catchment of the Upper Nile Basin, Uganda. Hydrol. Earth Syst. Sci.
**2010**, 14, 1297–1308. [Google Scholar] [CrossRef] [Green Version] - Vetter, T.; Reinhardt, J.; Flörke, M.; van Griensven, A.; Hattermann, F.; Huang, S.; Koch, H.; Pechlivanidis, I.G.; Plötner, S.; Seidou, O.; et al. Evaluation of sources of uncertainty in projected hydrological changes under climate change in 12 large-scale river basins. Clim. Chang.
**2017**, 141, 419–433. [Google Scholar] [CrossRef] - Northrop, P.J.; Chandler, R.E. Quantifying Sources of Uncertainty in Projections of Future Climate. J. Clim.
**2014**, 27, 8793–8808. [Google Scholar] [CrossRef]

**Figure 2.**The study area of the Southern Hills-Gulf region, which includes 26 HUC8 (dark gray lines) and 728 HUC12 (light gray lines).

**Figure 3.**A BMA tree of posterior model probabilities for emission pathways and GCMs at levels 1 and 2.

**Figure 4.**Annual recharge, ET and runoff projections (2010–2099) in Southern Hills-Gulf region using the best and the second-best climate models at level 2.

**Figure 5.**Means and one standard deviation (SD) intervals of annual recharge, ET and runoff projections (2010−2099) in the Southern Hills-Gulf region using all climate models under corresponding emission pathways at level 1.

**Figure 6.**Means and one standard deviation (SD) intervals of (

**a**) annual recharge, and (

**b**) ET and runoff projections (2010–2099) in the Southern Hills-Gulf region at the hierarch level.

**Figure 7.**Means of annual ET, recharge, and runoff projections in 2010–2039, 2040–2069, and 2070–2099 at the hierarch level for the Southern Hills-Gulf region.

**Figure 8.**Standard deviations of annual ET, recharge, and runoff projections in 2010–2039, 2040–2069, and 2070–2099) at the hierarch level for the Southern Hills-Gulf region.

**Figure 9.**Uncertainty contribution (%) in 2010–2039, 2040–2069, and 2070–2099 from emission paths and GCMs to the projected recharge in the Southern Hills-Gulf region.

Modeling Center/Group | GCM |
---|---|

National Center for Atmospheric Research, USA | ccsm4.1 |

NOAA Geophysical Fluid Dynamics Laboratory, USA | gfdl-esm2g.1 gfdl-esm2m.1 |

Institut Pierre-Simon Laplace, France | ipsl-cm5a-lr.1 ipsl-cm5a-mr.1 |

Japan Agency for Marine-Earth Science and Technology, Atmosphere and Ocean Research Institute (The University of Tokyo), and National Institute for Environmental Studies, Japan | miroc-esm.1 miroc-esm-chem.1 miroc5.1 |

Meteorological Research Institute, Japan | mri-cgcm3.1 |

Norwegian Climate Centre, Norway | noresm1-m.1 |

**Table 2.**Posterior model probabilities of 10 climate models based on monthly precipitation and temperature data 1950–2006 in the Southern Hills-Gulf region.

Rank | Climate Model | $\sum}_{\mathit{i}}{\scriptscriptstyle \frac{({\mathit{P}}_{\mathit{p},\mathit{i}}^{\left(\mathit{m}\right)}-{\mathit{P}}_{\mathit{i}}^{\mathit{o}\mathit{b}\mathit{s}}{)}^{2}}{{\mathit{\sigma}}_{\mathit{P},\mathit{i}}^{2}}$ | $\sum}_{\mathit{j}}{\scriptscriptstyle \frac{({\mathit{T}}_{\mathit{p},\mathit{j}}^{\left(\mathit{m}\right)}-{\mathit{T}}_{\mathit{j}}^{\mathit{o}\mathit{b}\mathit{s}}{)}^{2}}{{\mathit{\sigma}}_{\mathit{T},\mathit{j}}^{2}}$ | $\mathbf{\Delta}{\mathbf{BIC}}_{\mathit{p}}^{(\mathit{m})}$ | $\mathbf{Pr}\left({\mathbf{M}}_{\mathit{p}}^{(\mathit{m})}|\mathbf{D}\right)$ |
---|---|---|---|---|---|

1 | gfdl-esm2g.1 | 545.1 | 308.6 | 0.0 | 23.0% |

2 | miroc-esm-chem.1 | 569.7 | 322.5 | 38.5 | 19.4% |

3 | miroc5.1 | 566.7 | 387.0 | 99.9 | 14.7% |

4 | ipsl-cm5a-mr.1 | 626.5 | 351.9 | 124.6 | 13.2% |

5 | gfdl-esm2m.1 | 586.6 | 404.7 | 137.6 | 12.4% |

6 | mri-cgcm3.1 | 423.4 | 645.2 | 214.9 | 8.8% |

7 | miroc-esm.1 | 601.9 | 588.7 | 336.8 | 5.1% |

8 | noresm1-m.1 | 736.9 | 610.8 | 493.9 | 2.5% |

9 | ccsm4.1 | 903.5 | 782.9 | 832.6 | 0.6% |

10 | ipsl-cm5a-lr.1 | 654.1 | 1165.7 | 966.0 | 0.3% |

**Table 3.**Anomalies of mean annuals in 2010–2039, 2040–2069, and 2070–2099 to mean annuals of 1950–2009 in recharge, runoff, and evapotranspiration (ET) in the Southern Hills-Gulf region. Only the best and the second best GCMs under each emission path are listed in level 2.

Level | Model | Recharge (%) Mean Annual = 337.4 mm | Runoff (%) Mean Annual = 352.8 mm |
ET (%) Mean Annual = 832.9 mm | ||||||
---|---|---|---|---|---|---|---|---|---|---|

2010–2039 | 2040–2069 | 2070–2099 | 2010–2039 | 2040–2069 | 2070–2099 | 2010–2039 | 2040–2069 | 2070–2099 | ||

2 | rcp26.gfdl-esm2g.1 | +11.6 | +8.9 | +13.4 | −16.3 | −15.8 | −14.7 | +0.5 | +0.3 | +0.5 |

rcp26. miroc-esm-chem.1 | +6.1 | −7.9 | +5.4 | −17.4 | −23.4 | −16.5 | +2.0 | +1.9 | +2.5 | |

rcp45.gfdl-esm2g.1 | −10.5 | −5.5 | −3.0 | −28.5 | −24.4 | −23.3 | −0.5 | +1.0 | −0.1 | |

rcp45. miroc-esm-chem.1 | −3.0 | +2.5 | +3.1 | −23.3 | −19.2 | −18.3 | +1.1 | +3.2 | +3.5 | |

rcp60.gfdl-esm2g.1 | +18.0 | +4.0 | −8.9 | −12.2 | −18.6 | −21.3 | +2.5 | +1.6 | −0.1 | |

rcp60.miroc-esm-chem.1 | +4.8 | −2.0 | −18.4 | −19.9 | −19.1 | −27.0 | +2.3 | +2.5 | +1.9 | |

rcp85.gfdl-esm2g.1 | −4.0 | −1.1 | −16.8 | −20.9 | −19.9 | −27.8 | +1.3 | +2.4 | +1.0 | |

rcp85. miroc-esm-chem.1 | +10.1 | −17.6 | −33.0 | −16.0 | −27.6 | −29.8 | +4.8 | +3.4 | +2.8 | |

1 | rcp26 | +16.9 | +16.3 | +19.6 | −16.2 | −15.2 | −14.5 | +2.1 | +2.9 | +2.9 |

rcp45 | +10.9 | +4.6 | +9.6 | −20.1 | −20.5 | −17.4 | +2.3 | +2.9 | +3.4 | |

rcp60 | +13.3 | +6.9 | −1.3 | −17.8 | −18.6 | −21.3 | +1.9 | +2.6 | +2.9 | |

rcp85 | +11.1 | +2.2 | −16.3 | −17.6 | −19.7 | −25.4 | +3.7 | +3.9 | +3.3 | |

Hierarch | Hierarch | +13.0 | +7.5 | +2.9 | −17.9 | −18.5 | −19.6 | +2.6 | +3.1 | +3.1 |

**Table 4.**Means and standard deviations (SD) of uncertainty contributions in 2010–2039, 2040–2069, and 2070–2099 from emission paths and GCMs to the projected recharge (including all 728 HUC12) in the Southern Hills-Gulf region.

Uncertainty Source | 2010–2039 | 2040–2069 | 2070–2099 | |||
---|---|---|---|---|---|---|

Mean (%) | SD (%) | Mean (%) | SD (%) | Mean (%) | SD (%) | |

Emission Path | 5.4 | 1.5 | 10.7 | 1.8 | 34.0 | 6.4 |

GCM | 94.6 | 1.5 | 89.3 | 1.8 | 66.0 | 6.4 |

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**MDPI and ACS Style**

Beigi, E.; Tsai, F.T.-C.; Singh, V.P.; Kao, S.-C.
Bayesian Hierarchical Model Uncertainty Quantification for Future Hydroclimate Projections in Southern Hills-Gulf Region, USA. *Water* **2019**, *11*, 268.
https://doi.org/10.3390/w11020268

**AMA Style**

Beigi E, Tsai FT-C, Singh VP, Kao S-C.
Bayesian Hierarchical Model Uncertainty Quantification for Future Hydroclimate Projections in Southern Hills-Gulf Region, USA. *Water*. 2019; 11(2):268.
https://doi.org/10.3390/w11020268

**Chicago/Turabian Style**

Beigi, Ehsan, Frank T.-C. Tsai, Vijay P. Singh, and Shih-Chieh Kao.
2019. "Bayesian Hierarchical Model Uncertainty Quantification for Future Hydroclimate Projections in Southern Hills-Gulf Region, USA" *Water* 11, no. 2: 268.
https://doi.org/10.3390/w11020268