# BLP3-SP: A Bayesian Log-Pearson Type III Model with Spatial Priors for Reducing Uncertainty in Flood Frequency Analyses

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Log-Pearson Type III Distribution

_{1}, Q

_{2}, …, Q

_{N}} are distributed as a Log-Pearson Type III distribution, X = log(Q) distributes as a Pearson Type III distribution, with a probability density function (pdf):

_{p}(γ) is the pth quantile of the LP3 distribution with mean 0, standard deviation 1, and skewness γ, named as the frequency factor. It can also be approximated by the Wilson–Hilferty transformation for |γ| < 2 [25]:

_{p}is the pth quantile of the standard normal distribution.

#### 2.2. Bayesian Theorem for LP3 Distribution

_{1}, x

_{2}, x

_{3}, …, x

_{s}} (posterior) is proportional to the product of the probability of 𝜃 (prior) and the probability of X given 𝜃 (likelihood). Assuming the independence between the observations, the posterior can be calculated as below:

_{X}( ) is the pdf for X. In this study, 𝜃 comprises mean μ, standard deviation 𝜎, and skewness 𝛾 in Equation (2).

#### 2.3. Prior Distribution

#### 2.4. Parameter Estimation

#### 2.5. Proposal Distribution

## 3. Case Study Area and Data

#### 3.1. Study Area and Gauge Station Data

^{2}. Flood frequency can be estimated using the annual maximum series (AMS) or partial duration series (PDS). The AMS consists of records of the annual peak discharge, while the PDS is based on all floods exceeding a predefined base line [1]. If minor floods are considered (AEP > 0.10), PDS is more appropriate than AMS. However, for floods with an annual exceedance probability (AEP) less than 0.10, there is no significant difference between the AEP estimation using AMS or PDS [36]. Meanwhile, due to its wide availability and longer data length, AMS has also been used in many studies [16,20,37]. Therefore, AMS was used in this study.

#### 3.2. Spatial Data for Prior Estimation

## 4. Results

#### 4.1. Estimated Prior Information from Spatial Regression

^{2}and standard deviation for each potential model are summarized in Table 4. For the models to estimate μ and σ, the one with the greatest R

^{2}and the smallest standard deviation was selected. Therefore, the model for estimating the prior μ distribution is the 4-NN SLM, the model for estimating the prior σ distribution is the 2nd Queen SEM, and the model for estimating the prior γ distribution is the 2nd Queen SLM.

#### 4.2. Posterior Distribution and Flood Quantiles

## 5. Discussion

#### 5.1. Compared with Other Spatial Prior Methods

#### 5.2. Effects of Length of Observations

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Data availability of the entire set of streamflow gauges used in this study (the gray row is the site for testing and validation and the red dots are the 10-year time series used in the model).

**Figure 4.**The means and 95% confidence limits for the three scenarios: non-informative prior and 54-year data, non-informative prior and the last 10-year data, and spatial regression prior and the last 10-year data.

**Figure 5.**The 95% confidence interval for four scenarios: non-informative prior, mean prior, areal interpolation priors, and spatial regression prior.

Site ID | Site No. | Latitude | Longitude | Watershed Area (km ^{2}) | Series Length (year) |
---|---|---|---|---|---|

1 | 08075780 | 29.95 N | 95.52 W | 18.76 | 55 |

2 | 08074150 | 29.85 N | 95.49 W | 15.90 | 53 |

3 | 08068000 | 30.24 N | 95.46 W | 2158.28 | 85 |

4 | 08075400 | 29.62 N | 95.45 W | 48.36 | 55 |

5 | 08068500 | 30.11 N | 95.44 W | 1052.12 | 82 |

6 | 08069000 | 30.04 N | 95.43 W | 737.34 | 77 |

7 | 08075900 | 29.96 N | 95.42 W | 86.42 | 54 |

8 | 08074500 | 29.78 N | 95.40 W | 227.43 | 85 |

9 | 08076500 | 29.86 N | 95.33 W | 69.14 | 67 |

10 | 08076000 | 29.92 N | 95.31 W | 166.20 | 67 |

11 | 08070500 | 30.26 N | 95.30 W | 271.47 | 76 |

12 | 08075500 | 29.67 N | 95.29 W | 150.76 | 67 |

13 | 08075770 | 29.79 N | 95.27 W | 47.79 | 56 |

14 | 08071000 | 30.23 N | 95.17 W | 307.53 | 56 |

15 | 08070000 | 30.34 N | 95.10 W | 859.51 | 81 |

ID | Area (km ^{2}) | Elevation (m) | Slope (%) | Tree Canopy (%) | Imperviousness (%) |
---|---|---|---|---|---|

1 | 18.76 | 37.64 | 24.22 | 8.79 | 39.94 |

2 | 15.90 | 28.61 | 24.37 | 3.36 | 52.30 |

3 | 2158.28 | 87.00 | 59.96 | 52.58 | 2.18 |

4 | 48.36 | 19.90 | 27.95 | 5.00 | 30.34 |

5 | 1052.12 | 69.91 | 40.04 | 47.49 | 5.88 |

6 | 737.34 | 54.79 | 20.71 | 11.92 | 9.41 |

7 | 86.42 | 33.75 | 23.95 | 13.54 | 34.05 |

8 | 227.43 | 29.25 | 27.29 | 6.29 | 44.21 |

9 | 69.14 | 24.25 | 18.84 | 9.90 | 35.42 |

10 | 166.20 | 29.42 | 25.31 | 12.73 | 33.59 |

11 | 271.47 | 88.01 | 56.58 | 52.83 | 1.94 |

12 | 150.76 | 16.02 | 29.34 | 6.83 | 28.67 |

13 | 47.79 | 14.90 | 19.38 | 4.66 | 51.31 |

14 | 307.53 | 69.16 | 46.66 | 69.56 | 1.04 |

15 | 859.51 | 88.35 | 61.48 | 71.02 | 0.60 |

**Table 3.**Summary of the p-values for each spatial regression model (the significant ones are bolded).

Weight Type | μ | σ | γ | |||
---|---|---|---|---|---|---|

SEM | SLM | SEM | SLM | SEM | SLM | |

First-order Queen weight | 0.024 | 0.646 | 0.110 | 0.563 | 0.974 | 0.555 |

Second-order Queen weight (including the lower order) | 0.618 | 0.497 | 0.008 | 0.646 | 0.187 | 0.039 |

4-NN | 0.126 | 0.023 | 0.237 | 0.244 | 0.057 | 0.280 |

Distance band (Max-Min distance) | 0.187 | 0.117 | 0.058 | 0.073 | 0.706 | 0.645 |

Distance band (15,240 m) | 0.359 | 0.669 | 0.775 | 0.908 | 0.994 | 0.085 |

Distance band (60,960 m) | 0.389 | 0.273 | 0.799 | 0.144 | 0.526 | 0.589 |

Distance band (45,720 m) | 0.137 | 0.068 | 0.536 | 0.142 | 0.196 | 0.223 |

Triangular kernel with 3-NN adaptive bandwidth | 0.001 | 0.428 | 0.071 | 0.001 | 0.385 | 0.239 |

**Table 4.**Summary of the R

^{2}and standard deviation of models with significant spatial coefficient (p-value < 0.05).

Model | R^{2} | STD | |
---|---|---|---|

μ | 1st Queen SEM | 56.18% | 0.20 |

4-NN SLM | 60.57% | 0.18 | |

Triangular kernel SEM | 33.20% | 0.30 | |

σ | 2nd Queen SEM | 78.03% | 0.01 |

Triangular kernel SLM | 47.69% | 0.03 | |

γ | 2nd Queen SLM | 59.02% | 0.06 |

μ | σ | γ | |
---|---|---|---|

mean | 7.8957 | 0.6993 | −0.3969 |

variance | 0.1778 | 0.0143 | 0.0635 |

μ | σ | γ | ||||
---|---|---|---|---|---|---|

Mean | Variance | Mean | Variance | Mean | Variance | |

Spatial regression prior with 10-year data | 7.5338 | 0.0278 | 0.5264 | 0.0183 | −0.3133 | 0.2516 |

Non-informative prior with 54-year data | 7.0798 | 0.0093 | 0.6624 | 0.0098 | −0.2349 | 0.3011 |

**Table 7.**Estimation of the discharge (m

^{3}/s) for certain design floods with 95% confidence interval.

Return Period | 10 Year | 25 Year | 50 Year | 100 Year | 200 Year |
---|---|---|---|---|---|

Non-info prior and 10-year data | 82.9 (63.3–175.6) | 96.1 (71.8–307.0) | 106.1 (75.9–475.4) | 115.5 (78.1–744.3) | 125.2 (79.2–1027.6) |

Spatial regression and 10-year data | 97.5 (70.0–176.2) | 118.9 (79.3–248.8) | 134.5 (84.1–312.0) | 149.6 (87.8–390.1) | 164.7 (90.6–482.6) |

Reduction in confidence interval | 5.35% | 27.93% | 42.95% | 54.62% | 65.26% |

μ | σ | γ | |||||
---|---|---|---|---|---|---|---|

Mean | Variance | Mean | Variance | Mean | Variance | ||

Mean Prior | Prior | 8.2205 | 0.4291 | 0.9052 | 0.0690 | −0.2822 | 0.1627 |

Posterior | 7.5374 | 0.0309 | 0.4848 | 0.0374 | −0.2190 | 0.3650 | |

Areal Interpolation Prior | Prior | 9.1225 | 0.2963 | 0.8840 | 0.0136 | −0.8983 | 0.1506 |

Posterior | 7.6258 | 0.0670 | 0.7340 | 0.0220 | −0.6832 | 0.8131 | |

Spatial Regression Prior | Prior | 7.8957 | 0.1778 | 0.6993 | 0.0143 | −0.3969 | 0.0635 |

Posterior | 7.5338 | 0.0278 | 0.5264 | 0.0183 | −0.3133 | 0.2516 |

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**MDPI and ACS Style**

Tian, D.; Wang, L.
BLP3-SP: A Bayesian Log-Pearson Type III Model with Spatial Priors for Reducing Uncertainty in Flood Frequency Analyses. *Water* **2022**, *14*, 909.
https://doi.org/10.3390/w14060909

**AMA Style**

Tian D, Wang L.
BLP3-SP: A Bayesian Log-Pearson Type III Model with Spatial Priors for Reducing Uncertainty in Flood Frequency Analyses. *Water*. 2022; 14(6):909.
https://doi.org/10.3390/w14060909

**Chicago/Turabian Style**

Tian, Dan, and Lei Wang.
2022. "BLP3-SP: A Bayesian Log-Pearson Type III Model with Spatial Priors for Reducing Uncertainty in Flood Frequency Analyses" *Water* 14, no. 6: 909.
https://doi.org/10.3390/w14060909