A Spatial Bayesian Hierarchical Model for a Precipitation Return - - PowerPoint PPT Presentation

A Spatial Bayesian Hierarchical Model for a Precipitation Return Levels Map

Daniel Cooley1,2

Douglas Nychka2, Philippe Naveau2,3

1Department of Applied Mathematics, University of Colorado at Boulder 2Geophysical Statistics Project, National Center for Atmospheric Research 3Laboratoire 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– no uncertainty estimates – outdated extremes techniques – 30 more years of data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

−109 −108 −107 −106 −105 −104 −103 −102 37 38 39 40 41 longitude latitude 1000 1500 2000 2500 3000 3500 4000

• Denver
• Ft. Collins

Colo Spgs Pueblo Limon Breckenridge

• Grand Junction

WYOMING NEBRASKA NEW MEXICO KANSAS UTAH

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.
• Characterizing spatial de-

pendence in the data.

• Short-range dependence.
• Max Daily Prcp 2000

Boulder

• Ft. Collins

Greeley Denver 3.6 cm 2.3 cm 4.8 cm 4.6 cm

• Modeling return levels.
• Characterizing dependence

in the distributions.

• Longer-range dependence.
• 25 Year Return Level

Boulder

• Ft. Collins

Greeley Denver 8.6 cm 9.4 cm 6.1 cm 8.4 cm

Modeling Climatological Extremes

• We use a POT approach, and assume exceedances over a

threshold u are described by a GPD(σu, ξ).

• We assume the climatological extreme precipitation is char-

acterized by a latent process – σu and ξ are functions of location

• Return levels:

zr(x) = u(x) + σu(x) ξ(x)

• (rnyζu(x))ξ(x) − 1
• .

– ny is the number of observations in a year. – ζu(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-

• ld exceedances and exceedance rates.

Exceedances Model - Data Level

Let Zj(xi) be the precipitation amount recorded at the station located at xi on day j. We assume that precipitation events Zj(xi) which exceed a threshold u = .45 inches are GPD, whose parameters depend on the station’s location. P{Zj(xi) − u > z|Zj(xi) > u} =

• 1 +

ξ(xi)z exp φ(xi)

−1/ξ(xi)

Exceedances Model - Process Level

φ(x): Modeled with standard geophysical methods → Gaus- sian process µφ(x) = f(αφ, covariates(x)) = αφ,0 + αφ,1(elevation) (for example) kφ(x, x′) = βφ,0 ∗ exp(−βφ,1 ∗ ||x − x′||2)

Exceedances Model - Process Level

φ(x): Modeled with standard geophysical methods → Gaus- sian process µφ(x) = f(αφ, covariates(x)) = αφ,0 + αφ,1(elevation) (for example) kφ(x, x′) = βφ,0 ∗ exp(−βφ,1 ∗ ||x − x′||2) ξ(x): Modeled in three ways

• 1. as a single value ξ for the whole region
• 2. as separate values ξmtn, ξplains
• 3. as a Gaussian process as above

Exceedances Model - Priors

Priors for αφ,·: Non-informative αφ,· ∼ Unif(−∞, ∞)

Exceedances Model - Priors

Priors for αφ,·: Non-informative αφ,· ∼ Unif(−∞, ∞) Priors for βφ,·: Based on empirical information – difficult to ellicit prior information βφ,0 ∼ Unif(0.005, 0.03) βφ,1 ∼ Unif(1, 6)

• 0.0

0.5 1.0 1.5 2.0 2.5 3.0 0.000 0.005 0.010 0.015 0.020 0.025 0.030 climate space distance variogram estimate

Model Schematic

Zj(x) φ(x) ξ(x)

✲ ✛

Model Schematic

Zj(x) φ(x) ξ(x)

✲ ✛

αφ βφ

❅ ❅ ❅ ❘

• ✒

αξ βξ

• ✠

❅ ❅ ❅ ■

Model Schematic

Zj(x) φ(x) ξ(x)

✲ ✛

αφ βφ

❅ ❅ ❅ ❘

• ✒

αξ βξ

• ✠

❅ ❅ ❅ ■

Prior Prior

✲ ✲

Prior Prior

✛ ✛

Model Schematic

Zj(x) φ(x) ξ(x)

✲ ✛

αφ βφ

❅ ❅ ❅ ❘

• ✒

αξ βξ

• ✠

❅ ❅ ❅ ■

Prior Prior

✲ ✲

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 βφ,1.

−106.0 −105.0 37 38 39 40 longitude latitude 1500 2000 2500 3000 3500

• Denver

Boulder

• Ft. Collins

Colo Spgs Pueblo Greeley

• lon/lat space

trans elev 1 2 3 4 2 4 6 8 trans msp

• ●
• climate space

Threshold Selection: Quantile or Level?

Boulder Station

Precip Amount frequency 20 30 40 50 60 50 100 150 200 250

• ●
• ●
• ●
• ●
• 20

30 40 50 60 10 20 30 40 50 60 Threshold Modified Scale

• ●
• ● ●
• ●
• 20

30 40 50 60 −0.1 0.0 0.1 0.2 0.3 0.4 Threshold Shape

Simulation experiment: Parameter bias least if threshold is chosen in the middle of the precision interval. Threshold chosen at .45 inches for all stations ⇒ ζu(x) modeled spatially ⇒ Exceedance Rate Model

Exceedance Rate Model

Data Layer: Assume each station’s number of exceedances Ni is a binomial random variable with mi trials each with a probability of ζ(xi) Process Layer: Assume logit(ζ(x)) is a Gaussian process, with mean and covariance µζ(x) = fζ(αζ, covariates(x)) Cov(ζ(x), ζ(x′) = kζ(x, x′) = βζ,0 ∗ exp(−βζ,1 ∗ ||x − x′||2) Priors:

• αζ,· ∼ Unif(−∞, ∞),
• βζ,0 ∼ Unif(0.005, .2)
• βζ,1 ∼ Unif(1, 6)

Model Schematic for Return Levels

Exceedances Model zr(x) φ(x) ξ(x)

αφ βφ

❅ ❅ ❅ ❘

• ✒

✲ ✛

αξ βξ

• ✠

❅ ❅ ❅ ■

Prior Prior

✲ ✲

Prior Prior

✛ ✛

ζ(x)

αζ βζ

Prior Prior

✻

• ✒

❅ ❅ ■ ✻ ✻

Exceedance Rate Model

Interpolation Method

Draw values from [φ(x)|φ(x1), . . . , φ(x56), αφ, βφ].

−106.0 −105.0 37 38 39 40 longitude latitude 3.2 3.3 3.4 3.5 3.6 3.7 3.8

• Denver

Boulder

• Ft. Collins

Colo Spgs Pueblo Greeley −106.0 −105.0 37 38 39 40 longitude latitude 3.0 3.2 3.4 3.6 3.8

• Denver

Boulder

• Ft. Collins

Colo Spgs Pueblo Greeley

Exceedance Models Tested

Baseline Model ¯ D pD DIC Model 0: φ = φ 112264.2 2.0 112266.2 ξ = ξ Models in Latitude/Longitude Space ¯ D pD DIC Model 1: φ = φ + ǫφ 98533.2 33.8 98567.0 ξ = ξ Model 2: φ = α0 + α1(msp) + ǫφ 98532.3 33.8 98566.1 ξ = ξ Model 3: φ = α0 + α1(elev) + ǫφ 98528.8 30.4 98559.2 ξ = ξ Model 4: φ = α0 + α1(elev) + α2(msp) + ǫφ 98529.7 29.6 98559.6 ξ = ξ Models in Climate Space ¯ D pD DIC Model 5: φ = φ + ǫφ 98524.3 27.3 98551.6 ξ = ξ Model 6: φ = α0 + α1(elev) + ǫφ 98526.0 25.8 98551.8 ξ = ξ Model 7: φ = α0 + α1(elev) + ǫφ 98524.0 26.0 98550.0 ξ = ξmtn, ξplains Model 8: φ = α0 + α1(elev) + ǫφ 98518.5 79.9 98598.4 ξ = ξ + ǫξ ǫ· ∼ MV N(0, Σ) where [σ]i,j = β·,0 exp(−β·,1||xi − xj||)

Posterior Distributions

3.4 3.5 3.6 3.7 3.8 3.9 2 4 6 8

phi

density 0.05 0.10 0.15 5 15 25

xi

density 0.000 0.010 0.020 0.030 20 40 60

beta_0 (Sill)

density 1 2 3 4 5 6 0.0 0.4 0.8

beta_1 (Range)

density

Spatial Coherence of φ

−106.0 −105.0 37 38 39 40 longitude latitude 3.0 3.2 3.4 3.6 −106.0 −105.0 37 38 39 40 longitude

Traditional vs Climate Space

25-year Return Level Point Estimate

−106.0 −105.0 37 38 39 40 longitude latitude 3 4 5 6 7 8 9 10

• Denver

Boulder

• Ft. Collins

Colo Spgs Pueblo Greeley −106.0 −105.0 37 38 39 40 longitude latitude 3 4 5 6 7 8 9 10

• Denver

Boulder

• Ft. Collins

Colo Spgs Pueblo Greeley

Results for Model 3: Return Level Uncertainty

−106.0 −104.5 37 38 39 40 4 6 8 10

• Denver

Boulder

• Ft. Collins

Colo Spgs Pueblo Greeley

−106.0 −104.5 37 38 39 40 4 6 8 10

• Denver

Boulder

• Ft. Collins

Colo Spgs Pueblo Greeley

−106.0 −104.5 37 38 39 40 1.5 2.0 2.5 3.0

Madogram on Colorado Data

• 20

40 60 80 100 0.00 0.05 0.10 0.15 0.20 distance madogram

• Renormalized annual max data.
• Shows very short range dependence in annual max obser-

vations.

• Like to apply the madogram to GPD data.

Conclusions and Future Work

• Used a Bayesian hierarchical model to produce maps which

characterize climatological extreme precipitation.

• Method accounts for uncertainty due to both parameter

estimation and interpolation.

• Dealt with issue of low precision by modeling exceedance

rate spatially.

• Performed spatial analysis in a non-traditional climate space.
• Extend idea to other data (ozone levels).
• Model for all duration periods.