combining observations and models: a bayesian view mark berliner, osu stat dept bayesian...
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Combining Observations and Models: A Bayesian View
Mark Berliner, OSU Stat Dept
• Bayesian Hierarchical Models
• Selected Approaches
• Geophysical
Examples
• Discussion
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Main Themes1) Goal: Develop probability
distributions for unknowns of interest by combining information sources: Observations, theory, computer model output, past experience, etc.
2) Approaches: Bayesian Hierarchical Models Incorporate various information
sources by modeling 1. priors2. data model or likelihoods
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Bayesian Hierarchical Models• Skeleton:
1. Data Model: [ Y | X , ]
2. Process Model Prior: [ X | ]
3. Prior on parameters: [ ]
• Bayes’ Theorem: posterior distribution: [ X , | Y]
• Compare to
“Statistics”: [ Y | ] [ ]
“Physics”: [ X | (Y) ]
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ApproachesA. Stochastic models incorporating science• Physical-statistical modeling (Berliner 2003
JGR) From ``F=ma'' to [ X | ] • Qualitative use of theory (eg., Pacific SST
model; Berliner et al. 2000 J. Climate)
B. Incorporating large-scale computer models
1) From model output to priors [ ]
2) Model output as samples from process model prior [ X | ] almost !
3) Model output as ``observations'' (Y)
C. Combinations
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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)
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Modeling: surface – s, thickness – H, velocity -
u Physical Model
• Basal Stress: = - gH ds/dx (+ “stuff”)
• Velocities: u = ub + 0 H n
where ub = k p + ( gH )-q Our Model
• Basal Stress: = - gH ds/dx + where is a ``corrector process;” H, s unknown
• Velocities: u = ub + H n + e
where ub = k p + ( gH )-q or a constant;
is unknown, e is a noise process
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Wavelet Smoothing of Base
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Results: Velocity
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Results: Stress and Corrector
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Paleoclimate (Brynjarsdóttir & Berliner 2009)
Climate proxies: Tree rings, ice cores, corals, pollen, underground rock provide indirect information on climate
• Inverse problem: proxy f(climate)
Boreholes: Earth stores info on surface temp’s
• Model: Heat equation
Borehole data f(surface temp’s)
• Infer boundary condition (initial cond. is nuisance)
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Modeling
• Data Model:
Y | Tr, ~ N( Tr + T0 1 + q R(k), 2 I)
true temp
Adjustments for rock types, etc.
• Process Model: heat equation applied to Tr
with b.cond. surface temp history Th
Tr | Th , ~ N( BTh , 2 I)
Th | ~ N( 0 , 2 I)
Y
h
r
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In progress:• Combining boreholes (parameters and
b.cond as samples from a distribution)
• Combining with other sources and proxies
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Bayesian Hierarchical Models to Augment the
Mediterranean Forecast System (MFS)Ralph Milliff CoRAChris Wikle Univ. MissouriMark 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
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Bayesian Modeling in Support of Massive Forecast Models
1. MFS is an Ocean Model
2. A Boundary Condition/Forcing: Surface Winds
3. Approach: produce surface vector winds (SVW), for ensemble data assimilation
• Exploit abundant, “good” satellite wind data (QuikSCAT)
• Samples from our winds-posterior ensemble for MFS
(Before us: coarse wind field (ECMWF))
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“Rayleigh Friction Model” for winds (Linear Planetary Boundary Layer Equations)
Theory
(neglect second order time derivative)discretize:
Our model
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BHM Ensemble Winds
10 m/s
10 members selected from the Posterior Distribution (blue)
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ApproachesA. Stochastic models incorporating science• Physical-statistical modeling (Berliner 2003
JGR) From ``F=ma'' to [ X | ] • Qualitative use of theory (eg., Pacific SST
model; Berliner et al. 2000 J. Climate)
B. Incorporating large-scale computer models
1) From model output to priors [ ]
2) Model output as samples from process model prior [ X | ] almost !
3) Model output as ``observations'' (Y)
C. Combinations
![Page 27: Combining Observations and Models: A Bayesian View Mark Berliner, OSU Stat Dept Bayesian Hierarchical Models Selected Approaches Geophysical Examples Discussion](https://reader034.vdocument.in/reader034/viewer/2022051400/55151cb55503465e608b5169/html5/thumbnails/27.jpg)
Part B) Information from Models
1) Develop prior from model output• Think of model output runs O1, … , On as samples
from some distribution• Do data analysis on O’s to estimate distribution• Use result (perhaps with modifications) as a prior
for X• Example: O’s are spatial fields: estimate spatial
covariance function of X based on O’s. • Example: Berliner et al (2003) J. Climate
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2) Model output as realizations of prior “trends”
• Process Model Prior
X = O +
where is “model error”, “bias”, “offset”
• [ Y | X , ] is measurement error model:
Y = X + e
• Substitution yields [ Y | O , , ]
Y = O + + e
• Modeling is crucial (I have seen set to 0)
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3) Model output as “observations”
• Data Model: [ Y, O | X , ] ( = [ Y | X, ] [ O | X , ])
• [ O | X , ] to include “bias, offset, ..”
• Previous approach: start by constructing
[ X | O , ]
This approach: construct [ O | X , ] • Model for “bias” a challenge in both cases• This is not uncommon, though not always
made clear
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A Bayesian Approach to Multi-model Analysis and Climate Projection
(Berliner and Kim 2008, J Climate)
Climate Projection:
– Future climate depends on future, but unknown, inputs.
– IPCC: construct plausible future inputs, “SRES Scenarios” (CO2 etc.)
– Assume a scenario and get corresponding projection
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Hemispheric Monthly Surface Temperatures
• Observations (Y) for 1882-2001.
Data Model: Gaussian with mean = true temp.
& unknown variance (with a change-point)• Two models (O): PCM (n=4), CCSM (n=1) for
2002-2197, and 3 SRES scenarios (B1,A1B,A2).
Data Model: assumes O’s are Gaussian with mean = t + model biast (different for the two models) and unknown, time-varying variances (different for the two models)
• All are assumed conditionally independent
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Notes (Freeze time)
• Data model for kth ensemble member from Model j:
Ojk = + bj + ejk
– is common to both Models
– bj is Model j bias
– E( ejk ) = 0 and variances of e’s depend on j
• Computer model model:
= X + ewhere E(e) = 0• Priors for biases, variances, and X• Extensions to different model classes (more ’s)
and richer models are feasible.
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IPCC (global) Us (NH)
Figure 10.4
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Discussion: Which approach is best?
• Depends on form and quality of observations and models and practicality
• Develop prior for X from scientific model (part A) offers strong incorporation of theory, but practical limits on richness of [ X | ] may arise
• Model output as “observations”– Combining models: Just like different
measuring devices;– Nice for analysis & mixed (obs’ & comp.) design
– Need a prior [ X | ]
• Model output as realizations of prior “trends”– Most common among Bayesian statisticians– Combining models: like combining experts
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Discussion: Models versus Reality
Need for modeling differences between X’s and O’s.
Model “assessment” (“validation”, “verification”) helps, but is difficult in complicated settings:
– Global climate models. Virtually no observations at the scales of the models.
– Tuning. Modify model based on observations.– Observations are imperfect, and are often
output of other physical models.– Massive data. Comparing space-time fields
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Discussion, Cont’d
• Part C) Combining approaches– Example: Wikle et al 2001, JASA. Combined
observations and large-scale model output as data with a prior based on some physics
• Usually, many physical models. No best one, so it’s nice to be flexible in incorporating their information
Thank You!
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