Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
Statistical Approaches to Combining Models andObservations
Mark Berliner
Department of Statistics
The Ohio State University
SIAM UQ12, April 5, 2012
Outline
I. Bayesian Hierarchical Modeling
II. Examples: Glacial dynamics; Paleoclimate
III. Using Large-scale Models
IV. Example: Ocean Forecasting
1
I. Bayesian Hierarchical Modeling (BHM)
• Bayesian Analysis:(1) UQ via prob. modeling;
(2) update via Bayes’ Theorem;
(3) infer, predict & make decisions via prob. (“risk”) analysis
• Hier. Model: Sequence of conditional probability distributions(corresponding to a joint distribution)
• Framework for modeling:Observations Y Processes (State Variables) X Parameters θ
1. Data Model [Y | X, θ]
2. Prior Process Model [X | θ]
3. Prior Parameter Model [ θ ]
• Bayes’ Theorem: [X, θ | Y]
2
Strategies
1. Incorporating physical models
Physical-statistical modeling (Berliner 2003 JGR)
From “F=ma” to process model [ X |θ ]
2. Qualitative use of theory
(e.g., Pacific SST model (Berliner et al. 2000 J. Climate)
******************************************************
3. Incorporating large-scale computer model output
4. Combinations
5. Alternate Approaches
3
Ex) Glacial Dynamics (Berliner et al. 2008 J. Glaciol)
• Steady Flow of Glaciers and Ice Sheets
– Flow: gravity moderated by drag (base & sides) & stuff
– Simple models: flow from geometry
• Data
– Program for Arctic Climate Regional Assessments
– Radarsat Antarctic Mapping Project
∗ surface topography (laser altimetry)∗ basal topography (radar altimetry)∗ velocity data (interferometry)
4
North East Ice Stream, Greenland
5
Physical Modeling: Surface: s, Thickness: H, Velocity: u
• Basal Stress: τ = −ρ g H ds/dx (+ ”stuff”)
• Velocities: u = ub + b0 H τ n where ub = k τ p + (ρgH)−q
Our Model
• Random Basal Stress: τ = −ρ g H ds/dx + ηwhere η is a “corrector process” (“model error”)
• Random H: wavelet smoothing prior
• Random s: parameterized model from literature
• Random Velocities: u = ub + b H τ n + ewhere ub = kτ
p + (ρgH)−q or a constant (change-point)
b is unknown parameter, e is a noise process
• Process Model :[u | s, H, η] [η] [s, H]
6
7
8
9
Ex) Paleoclimate: Temperature Reconstruction(Brynjarsdóttir & Berliner 2010 Ann. Applied Stat)
• Use of proxies:
– Inverse problem:
proxy ≈ f(climate) → climate ≈ g(proxy)– Inverse probability problem:
[proxy data | climate] → [climate | proxy data]
• Boreholes: earth stores info about surface temp’s
– Inverse: borehole data ≈ f(surface temp’s)– Model:
∗ Heat equation∗ Infer boundary condition
10
De
pth
[m
]
Relative temperature [ºC]
60
05
00
40
03
00
20
01
00
0
●
●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
SRD−1
14.51●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
SRD−2
15.73●
●
●
●
●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
SRD−3
16.00●
●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
SRD−4
16.17●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
SRD−7
13.86●
●
●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
SRS−3
11.30●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
SRS−4
12.54●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
SRS−5
12.08●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
WSR−1
13.54
San Rafael Desert San Rafael Swell
1ºC
11
Page 1 of 1
5/19/2010http://farm4.static.flickr.com/3580/3330922136_4b2c40edec_b.jpg
Page 1 of 1
5/19/2010http://hellforleathermagazine.com/images/San_Rafael_Swell.jpg12
Data Model : ~Y | ~Tr, q ∼ N(~Tr + T0 ~1 + q ~R(k), σ2y I)
• ~Tr: reduced temperatures (literature: ~Tr = ~T−T0 ~1− q ~R(k))
• T0: reference surface temperature
• q: surface heat flow
• ~R: thermal resistances (∝ k−1, thermal conductivities, adjustedfor rock types, etc.)
Process Model :
• Heat equation applied to ~Tr• B.C.: Surface temp history ~Th (assumed piecewise constant)
• Easy to solve the heat equation~Tr | ~Th, q ∼ N(A ~Th, σ2 I)
~Th | q ∼ N(~0, σ2h I)Parameter Model : next
13
Spatial Hierarchy
• Nine boreholes: 5 in desert region, 4 in swell region
• Extend the hierarchy: for j = 1, . . . , 9
Data Model : ~Yj | ~Trj, qj ∼ N(~Trj + T0j ~1 + qj ~Rj(kj), σ2yj I)Process Model :
~Trj|~Thj, qj ∼ N(Aj ~Thj, σ2j I)
~Thj|qj ∼ N(~0, σ2hj I)Parameter Model
• ~Th1, . . . , ~Th5 conditionally independent N(~µd, γ2d I)
• ~Th6, . . . , ~Th9 conditionally independent N(~µs, γ2s I)
• ~µd & ~µs ∼ N(~µ0, γ20 I)
• q1, . . . , q5 conditionally independent N(νd, η2d)q6, . . . , q9 conditionally independent N(νs, η
2s )
• etc.
14
● ● ●●
● ●●
●
●
●
●
1600 1700 1800 1900 2000
−2−1
01
2
Year
Anom
aly [º
C]SRD−1, SR Desert
● ● ● ●●
●●
●
● ●
●
1600 1700 1800 1900 2000
−2−1
01
2
Year
Anom
aly [º
C]
SRD−2, SR Desert
● ● ●●
●●
●●
●●
●
1600 1700 1800 1900 2000
−2−1
01
2
Year
Anom
aly [º
C]
SRD−3, SR Desert
● ● ●●
●●
● ●
●●
●
1600 1700 1800 1900 2000
−2−1
01
2
Year
Anom
aly [º
C]
SRD−4, SR Desert
● ●●
●
● ●
●
●
●
● ●
1600 1700 1800 1900 2000
−2−1
01
2
Year
Anom
aly [º
C]
SRD−7, SR Desert
● ●● ●
●
●●
●
● ●
●
1600 1700 1800 1900 2000
−2−1
01
2
Year
Anom
aly [º
C]
SRS−3, SR Swell
●●
●
●●
●●
●
●
●
●
1600 1700 1800 1900 2000
−2−1
01
2
Year
Anom
aly [º
C]
SRS−4, SR Swell
●
●
●
●
●
●
●
●●
●
●
1600 1700 1800 1900 2000
−2−1
01
2
Year
Anom
aly [º
C]
SRS−5, SR Swell
●
●
●
●
●
●● ● ●
●
●
1600 1700 1800 1900 2000
−2−1
01
2
Year
Anom
aly [º
C]
WSR−1, SR Swell
15
● ● ●●
●●
●
●
●
●
●
1600 1700 1800 1900 2000
−1.5
−0.5
0.5
1.5
Year
Anom
aly
[ºC]
µD
● ●●
●●
● ●
●
●
●
●
1600 1700 1800 1900 2000
−1.5
−0.5
0.5
1.5
Year
Anom
aly
[ºC]
µS
16
1600 1700 1800 1900 2000
−2
−1
01
2
Year
An
om
aly
[ºC
]
SRD−1, SR Desert
1600 1700 1800 1900 2000
−2
−1
01
2
Year
An
om
aly
[ºC
]
SRD−2, SR Desert
1600 1700 1800 1900 2000
−2
−1
01
2
Year
An
om
aly
[ºC
]
SRD−3, SR Desert
1600 1700 1800 1900 2000
−2
−1
01
2
Year
An
om
aly
[ºC
]
SRD−4, SR Desert
1600 1700 1800 1900 2000
−2
−1
01
2
Year
An
om
aly
[ºC
]
SRD−7, SR Desert
1600 1700 1800 1900 2000
−2
−1
01
2
Year
An
om
aly
[ºC
]
SRS−3, SR Swell
1600 1700 1800 1900 2000
−2
−1
01
2
Year
An
om
aly
[ºC
]
SRS−4, SR Swell
1600 1700 1800 1900 2000
−2
−1
01
2
Year
An
om
aly
[ºC
]
SRS−5, SR Swell
1600 1700 1800 1900 2000
−2
−1
01
2
Year
An
om
aly
[ºC
]
WSR−1, SR Swell
Multi−site Single−site
17
1600 1700 1800 1900 2000
−2
−1
01
2
Year
An
om
aly
[ºC
]
SRD−2, SR Desert
Multi−site Single−site
1600 1700 1800 1900 2000
−2
−1
01
2
Year
An
om
aly
[ºC
]
SRS−5, SR Swell
Multi−site Single−site
18
III. Incorporating Large-Scale Computer Models
Ensembles ~O = O1 = M(T1), . . . , On = M(Tn)
(T’s include “controls, model names, etc.)
• Data Model Treat ~O as ”observations”
– [Y, ~O | X, θ] (include ”bias, offset, model error ..”)– Convenient for design of collection of Y, ~O
• Process Model Use ~O to develop [X | θ]
– Kernel density estimate
Σi αik(x | Oi)
– Gaussian process models; emulators; UQ
Model Output: ~O = (O1 = M(T1), . . . , On = M(Tn))
[O | θ, ~O]→ [X | θ]
• Parameter Model from model output(eg: Berliner, Levine, & Shea 2003 J. Climate)
• Combinations
19
A Bayesian Multi-model Climate Projection(Berliner & Kim 2008 J. Climate)
• Themes
– “climate” = parameters of prob. dist. of “weather”
– build or “parameterize” scales into dynamic model for X
• Future climate depends on future, but unknown, inputs.
• IPCC: construct plausible future inputs, ”SRES Scenarios”(CO2 etc.)
• Assume a scenario and find corresponding projection
20
Hemispheric Monthly Surface Temperatures (X)
• Observations (Y) for 1882-2001.
• Two models (~O): PCM (n=4), CCSM (n=1) for 2002-2197
• 3 SRES scenarios (B1,A1B,A2).
21
Hierarchical Data Model for Model Output
• Scalar climate variable X; m = 1, . . . , M models (time fixed)
• ~Om: ensemble of size nm of estimates of X from model m.
1. Given means µm and variances σ2Ym
, m = 1, . . . , M;~Om are independent and
~Om | µm ∼ Gau(µm~1nm, σ2Ym Inm)
2. Given β, model biases bm and variances σ2µm;
µm are independent and
µm | β, bm ∼ Gau(β + bm, σ2µm)
3. Given X,
β | X ∼ Gau(X, σ2β) and bm | X ∼ Gau(b0m, σ2bm
)
22
Remarks
• Implied marginal dist.: “~O given X”:Integrating out β induces dependence within & across ensembles
• Modify intuition about value of increasing ensemble size:“Infinite” ensembles do not give perfect forecasts:
If all biases = 0, infinite ensembles give the value of β, not X
• Extensions to different model classes (more β’s)and richer models are feasible.
23
Model Overview
1. [Y | X, θ]: measurement error modelGaussian with mean = true temp.
& unknown variance (with a change-point)
2. [X | θ]: Time series models with time varying parameters
Xt = µi(t) +
ηnj(t) 00 ηsj(t)
(Xt−1 − µi(t−1)) + et3. [θ] Climate = parameters of distribution of weather
• Climate-weather: multiscale phenomena• Time evolution: µi(t) slow; ηj(t) moderate;
et fast, but variances of et are slow
• µi = A + B CO2i + noise• Obs period: ηj = C + D |SOI|j + noise
Fore. period: AR model (i.e., SOI not observed)
• Variances of et: AR-like prior
24
1900 1950 2000 2050 210014
14.5
15
15.5
16
16.5
17
17.5
18
year
mea
n
1900 1950 2000 2050 210014
14.5
15
15.5
16
16.5
17
17.5
18
18.5
year
annu
al te
mpe
ratu
re
25
1900 1950 2000 2050 210012.5
13
13.5
14
14.5
15
15.5
year
mea
n
1900 1950 2000 2050 210012.5
13
13.5
14
14.5
15
15.5
16
year
annu
al te
mpe
ratu
re
26
1900 1950 2000 2050 21000
0.02
0.04
0.06
0.08
0.1
0.12
0.14
year
syst
em
variance
27
Model Adapted to Decadal Forecasting
(Kim & Berliner 2012)
1994 1996 1998 2000 2002 200413
13.5
14
14.5
15
15.5
16
Year
Tem
pera
ture
(Cel
cius
)
1994 1996 1998 2000 2002 200413
13.5
14
14.5
15
15.5
16
Year
Tem
pera
ture
(Cel
cius
)
28
IV. Combining Approaches:Mediterranean Ocean Forecasting
1. Winds as a boundary condition for the ocean surface
(Milliff et al. & Bonazzi et al. 2011 Quart. J. Roy. Met. Soc.)
2. Bayesian Multi-model Ensembling for Ocean Forecasting
(Berliner et al. 2012)
Observations
Initial – Boundary conditions
BHMWinds
Model 1
Model 2
BHM Oceans
BHM Ocean Post. Dist.
29
Bayesian Hierarchical Models to Augment The Mediterranean Forecast System (MFS)
Ralph Milliff CoRA Chris Wikle Univ. Missouri Mark Berliner Ohio State Univ.
Nadia Pinardi INGV (I'Istituto Nazionale di Geofisica e Vulcanologia) Univ. Bologna (MFS Director) Alessandro Bonazzi, Srdjan Dobricic INGV, Univ. Bologna
30
Overview:
• MFS: deterministic operational forecasting
system
• Boundary condition/forcing: Surface Winds
• Bayesian Hierarchical Model (BHM) to
quantify Surface Vector Wind (SVW)
distributions
• Ocean Ensemble Forecasting (OEF) using 10
member BHM-SVW ensembles
Key: Exploit abundant, “good” satellite wind
data combined with physical modeling
31
Building
wind dist. (BHM-SVW)
1. Data Stage:
Satellite
(QSCAT)
and
Numerical
Weather
Pred.
Analyses
(ECMWF)
QSCAT
ECMWF
32
2. Process Model: Rayleigh Friction Model
(Linear Planetary Boundary Layer Equations)
Theory (neglect second
order time
derivative)
discretize:
Our
model
33
BHM Ensemble Winds
10 m/s
10 members selected from the Posterior Distribution (blue)
Ensemble mean wind (green); ECMWF Analysis wind (red)
34
BHM-SVW-OEF initial condition spread:
Uncertainty is
concentrated at
mesoscales
Sea Surface Temperature
Sea Surface Height
Initial condition spread
Initial condition spread
35
Sea Surface Height
BHM-SVW-OEF 10 day forecast spread
Initial condition ensemble
spread has
amplified at the 10 day fcst
in mesoscales
Initial condition spread
10-th fcst day spread
36
Th
e f
ore
cast
sp
read
at
10 d
ays
EEPS forced ensemble BHM-SVW ensemble
ECMWF EPS forcing
Ineffective at
producing flow
field changes
at mesoscales
37
Discussion:
• BHM methods produce realistic distributions of surface winds (SVW)
(Milliff et al. 2011 J Roy Met Soc)
• BHM-SVW results used to in a new ocean ensemble forecasting method:
BHM-SVW-OEF
(Bonazzi et al 2011 J Roy Met Soc)
• The BHM-SVW-OEF produces 10-day-forecast spreads at mesoscales and in the upper thermocline
38
2. Bayesian Multi-model Ensembling forOcean Forecasting (Berliner et al. 2012)
• profiles of wintertime (60 days) of temperature T(z,t) (& salinity)(Rhodes Gyre region in Eastern Mediterranean Sea)
• 16 vertical levels z = 0 m to z = 300 m
• BHM Winds produce ensembles from two models:1) Ocean PArallelise (OPA)
2) Nucleus for European Modeling of the Ocean (NEMO)
• Model as in climate example: model-specific biases
• Prior: Analyzed fields
39
40
41
42