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

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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

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**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