A Spatial Bayesian Hierarchical Model for a Precipitation Return Levels Map Daniel Cooley 1 , 2 Douglas Nychka 2 , Philippe Naveau 2 , 3 1 Department of Applied Mathematics, University of Colorado at Boulder 2 Geophysical Statistics Project, National Center for Atmospheric Research 3 Laboratoire des Sciences du Climat et de l’Environnement, IPSL-CNRS, Gif-sur-Yvette, Fr
Project Background Goal: To produce a map which describes potential extreme precipitation for Colorado’s Front Range. • Part of a larger NCAR project on flooding • 1973 NOAA/NWS Precipitation Atlas is currently used – no uncertainty estimates – outdated extremes techniques – 30 more years of data • Current NWS effort to produce updated maps – maps produced for two US regions (not Colorado) – using RFA methodology of Hoskings and Wallis • Precipitation atlases provide return levels measures
Study Region WYOMING NEBRASKA 4000 41 Ft. Collins ● 3500 40 Denver ● 3000 Breckenridge Limon ● ● Grand Junction latitude UTAH ● 39 2500 Colo Spgs ● KANSAS Pueblo 2000 ● 38 1500 37 1000 NEW MEXICO −109 −108 −107 −106 −105 −104 −103 −102 longitude Data: 56 weather stations, 12-53 years of data/station, Apr 1 - Oct 31, 24 hour precipitation measurements
Weather, Climate and Spatial Extremes • Modeling observations. • Modeling return levels. • Characterizing spatial de- • Characterizing dependence pendence in the data. in the distributions. • Short-range dependence. • Longer-range dependence. Max Daily Prcp 2000 25 Year Return Level Ft. Collins Ft. Collins 2.3 cm 9.4 cm ● ● Greeley Greeley 4.8 cm 6.1 cm ● ● Boulder Boulder 3.6 cm ● 8.6 cm ● Denver Denver 4.6 cm 8.4 cm ● ●
Modeling Climatological Extremes • We use a POT approach, and assume exceedances over a threshold u are described by a GPD ( σ u , ξ ). • We assume the climatological extreme precipitation is char- acterized by a latent process – σ u and ξ are functions of location • Return levels: z r ( x ) = u ( x ) + σ u ( x ) ( rn y ζ u ( x )) ξ ( x ) − 1 � � . ξ ( x ) – n y is the number of observations in a year. – ζ u ( x ) is the probability an observation exceeds u .
Model Goals • Utilize extreme value theory (GPD) • Pool the data from the stations into one model – different from RFA • Model should have spatial coherence • Should utilize available covariates – elevation and mean Apr-Oct precipitation • Should be flexible enough to be able to compare models • Produce measures of uncertainty Spatial Bayesian Hierarchical Models: Independent 3-layer (data, process, prior) models for thresh- old exceedances and exceedance rates.
Exceedances Model - Data Level Let Z j ( x i ) be the precipitation amount recorded at the station located at x i on day j . We assume that precipitation events Z j ( x i ) which exceed a threshold u = .45 inches are GPD, whose parameters depend on the station’s location. � − 1 /ξ ( x i ) � ξ ( x i ) z P { Z j ( x i ) − u > z | Z j ( x i ) > u } = 1 + exp φ ( x i )
Exceedances Model - Process Level φ ( x ) : Modeled with standard geophysical methods → Gaus- sian process µ φ ( x ) = f ( α φ , covariates ( x )) = α φ, 0 + α φ, 1 ( elevation ) (for example) k φ ( x , x ′ ) = β φ, 0 ∗ exp ( − β φ, 1 ∗ || x − x ′ || 2 )
Exceedances Model - Process Level φ ( x ) : Modeled with standard geophysical methods → Gaus- sian process µ φ ( x ) = f ( α φ , covariates ( x )) = α φ, 0 + α φ, 1 ( elevation ) (for example) k φ ( x , x ′ ) = β φ, 0 ∗ exp ( − β φ, 1 ∗ || x − x ′ || 2 ) ξ ( x ) : Modeled in three ways 1. as a single value ξ for the whole region 2. as separate values ξ mtn , ξ plains 3. as a Gaussian process as above
Exceedances Model - Priors Priors for α φ, · : Non-informative α φ, · ∼ Unif ( −∞ , ∞ )
Exceedances Model - Priors Priors for α φ, · : Non-informative α φ, · ∼ Unif ( −∞ , ∞ ) Priors for β φ, · : Based on empirical information – difficult to ellicit prior information β φ, 0 ∼ Unif (0 . 005 , 0 . 03) ∼ Unif (1 , 6) β φ, 1 0.030 ● 0.025 ● ● ● 0.020 ● variogram estimate ● ● ● 0.015 ● ● 0.010 ● ● ● 0.005 0.000 0.0 0.5 1.0 1.5 2.0 2.5 3.0 climate space distance
Model Schematic φ ( x ) Z j ( x ) ξ ( x ) ✲ ✛
Model Schematic α φ α ξ ❅ � ❅ � ❅ ❘ ✠ � φ ( x ) Z j ( x ) ξ ( x ) ✲ ✛ � ✒ ❅ ■ � ❅ � ❅ β φ β ξ
Model Schematic α φ α ξ ✲ ✛ Prior Prior ❅ � ❅ � ❅ ❘ ✠ � φ ( x ) Z j ( x ) ξ ( x ) ✲ ✛ ✒ � ❅ ■ � ❅ � ❅ β φ β ξ ✲ ✛ Prior Prior
Model Schematic α φ α ξ ✲ ✛ Prior Prior ❅ � ❅ � ❘ ❅ ✠ � φ ( x ) Z j ( x ) ξ ( x ) ✲ ✛ ✒ � ❅ ■ � ❅ � ❅ β φ β ξ ✲ ✛ Prior Prior Model assumes that the observations are temporally and spa- tially independent (conditional on the stations’ parameters).
Climate Space Problem: Difficult to obtain convergence for β φ, 1 . lon/lat space climate space Ft. Collins ● ● ● ● Greeley ● ● ● ● ● ● 8 ● ● Boulder 3500 40 ● ● ● ● ● ● ● Denver ● ● ● ● ● ● 3000 ● ● ● 6 trans msp latitude ● ● 39 Colo Spgs ● ● ● ● ● 2500 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Pueblo ● ● ● 4 ● ● 2000 ● 38 ● ● ● ● ● ● 1500 2 37 −106.0 −105.0 1 2 3 4 longitude trans elev
Threshold Selection: Quantile or Level? Boulder Station 250 60 0.4 200 ● ● 50 0.3 ● ● ● ● 150 Modified Scale ● ● ● ● ● ● frequency 40 ● ● ● 0.2 Shape ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 100 ● ● ● ● ● ● 30 ● ● ● ● ● ● ● ● 0.1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 20 50 ● ● ● 0.0 ● ● ● 10 −0.1 0 20 30 40 50 60 20 30 40 50 60 20 30 40 50 60 Precip Amount Threshold Threshold Simulation experiment: Parameter bias least if threshold is chosen in the middle of the precision interval. Threshold chosen at .45 inches for all stations ⇒ ζ u ( x ) modeled spatially ⇒ Exceedance Rate Model
Exceedance Rate Model Data Layer: Assume each station’s number of exceedances N i is a binomial random variable with m i trials each with a probability of ζ ( x i ) Process Layer: Assume logit ( ζ ( x )) is a Gaussian process, with mean and covariance µ ζ ( x ) = f ζ ( α ζ , covariates ( x )) Cov ( ζ ( x ) , ζ ( x ′ ) = k ζ ( x , x ′ ) = β ζ, 0 ∗ exp ( − β ζ, 1 ∗ || x − x ′ || 2 ) Priors: • α ζ, · ∼ Unif ( −∞ , ∞ ), • β ζ, 0 ∼ Unif (0 . 005 , . 2) • β ζ, 1 ∼ Unif (1 , 6)
Model Schematic for Return Levels α φ α ξ ✲ ✛ Exceedances Model Prior Prior ❅ � ❅ � ❘ ❅ � ✠ φ ( x ) z r ( x ) ξ ( x ) ✲ ✛ ✒ � ■ ❅ � ❅ ✻ � ❅ β φ β ξ ✲ ✛ Prior Prior ζ ( x ) ✒ � ■ ❅ � ❅ α ζ β ζ Exceedance ✻ ✻ Rate Model Prior Prior
Interpolation Method Draw values from [ φ ( x ) | φ ( x 1 ) , . . . , φ ( x 56 ) , α φ , β φ ]. Ft. Collins 3.8 Ft. Collins Greeley Greeley 3.8 ● ● ● ● Boulder 3.7 Boulder 40 40 ● ● 3.6 Denver Denver ● ● 3.6 3.4 latitude latitude 39 3.5 39 Colo Spgs Colo Spgs ● ● 3.2 3.4 Pueblo Pueblo ● ● 38 38 3.3 3.0 3.2 37 37 −106.0 −105.0 −106.0 −105.0 longitude longitude
Exceedance Models Tested ¯ Baseline Model D p D DIC Model 0: φ = φ 112264.2 2.0 112266.2 ξ = ξ ¯ Models in Latitude/Longitude Space D p D DIC Model 1: φ = φ + ǫ φ 98533.2 33.8 98567.0 ξ = ξ Model 2: φ = α 0 + α 1 (msp) + ǫ φ 98532.3 33.8 98566.1 ξ = ξ Model 3: φ = α 0 + α 1 (elev) + ǫ φ 98528.8 30.4 98559.2 ξ = ξ Model 4: φ = α 0 + α 1 (elev) + α 2 (msp) + ǫ φ 98529.7 29.6 98559.6 ξ = ξ ¯ Models in Climate Space D p D DIC Model 5: φ = φ + ǫ φ 98524.3 27.3 98551.6 ξ = ξ Model 6: φ = α 0 + α 1 (elev) + ǫ φ 98526.0 25.8 98551.8 ξ = ξ φ = α 0 + α 1 (elev) + ǫ φ Model 7: 98524.0 26.0 98550.0 ξ = ξ mtn , ξ plains Model 8: φ = α 0 + α 1 (elev) + ǫ φ 98518.5 79.9 98598.4 ξ = ξ + ǫ ξ ǫ · ∼ MV N (0 , Σ) where [ σ ] i,j = β · , 0 exp( − β · , 1 || x i − x j || )
Posterior Distributions phi xi 8 25 6 density density 15 4 2 5 0 0 3.4 3.5 3.6 3.7 3.8 3.9 0.05 0.10 0.15 beta_0 (Sill) beta_1 (Range) 0.8 60 density density 40 0.4 20 0.0 0 0.000 0.010 0.020 0.030 1 2 3 4 5 6
Spatial Coherence of φ 40 40 3.6 latitude 3.4 39 39 3.2 38 38 3.0 37 37 −106.0 −105.0 −106.0 −105.0 longitude longitude
Recommend
More recommend
