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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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**Bayesian Hierarchical Models **

Main Themes Goal: Develop probability distributions for unknowns of interest by combining information sources: Observations, theory, computer model output, past experience, etc. Approaches: Bayesian Hierarchical Models Incorporate various information sources by modeling priors data model or likelihoods Emphasis on using model/theory explicitly or quantitatively, as opposed to qualitatively 2

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**Bayesian Hierarchical Models**

Skeleton: Data Model: [ Y | X , q ] Process Model Prior: [ X | q ] Prior on parameters: [ q ] Bayes’ Theorem: posterior distribution: [ X , q | Y] Compare to “Statistics”: [ Y | q ] [ q ] “Physics”: [ X | q (Y) ]

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**Approaches Stochastic models incorporating science**

Physical-statistical modeling (Berliner 2003 JGR) From ``F=ma'' to [ X | q ] Qualitative use of theory (eg., Pacific SST model; Berliner et al J. Climate) Incorporating large-scale computer models From model output to priors [ q ] Model output as samples from process model prior [ X | q ] almost ! Model output as ``observations'' (Y) Combinations 4

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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) 5

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**Modeling: surface – s, thickness – H, velocity - u**

Physical Model Basal Stress: t = - rgH ds/dx (+ “stuff”) Velocities: u = ub + b0 H tn where ub = k tp + ( rgH )-q Our Model Basal Stress: t = - rgH ds/dx + h where h is a ``corrector process;” H, s unknown Velocities: u = ub + b H t n + e where ub = k t p + ( rgH )-q or a constant; b is unknown, e is a noise process 8

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**Wavelet Smoothing of Base**

Also, a parameterized, smooth surface model

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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) 13

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**Y | Tr, q ~ N( Tr + T0 1 + q R(k), s2 I)**

Modeling Data Model: Y | Tr, q ~ N( Tr + T0 1 + q R(k), s2 I) true temp Adjustments for rock types, etc. Process Model: heat equation applied to Tr with b.cond. surface temp history Th Tr | Th ,q ~ N( BTh , s2 I) Th | q ~ N( 0 , s2 I) Y “reduced” temp. Heat eq. is solvable. r h 15

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**Combining with other sources and proxies**

In progress: Combining boreholes (parameters and b.cond as samples from a distribution) Combining with other sources and proxies 19

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**Bayesian Hierarchical Models to Mediterranean Forecast System (MFS)**

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

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**Bayesian Modeling in Support of Massive Forecast Models**

MFS is an Ocean Model A Boundary Condition/Forcing: Surface Winds 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)) 23

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**“Rayleigh Friction Model” for winds**

(Linear Planetary Boundary Layer Equations) (neglect second order time derivative) discretize: Theory Our model

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**10 members selected from the Posterior Distribution (blue)**

BHM Ensemble Winds A snapshot depiction of MFS-Wind-BHM output is shown here. Ten realizations of the SVW process from the estimate of the posterior distribution are shown for a single time during the analysis period for the Western Med. At each output grid location, a cluster of 10 blue vectors depicts the realizations from the BHM. Their mean (i.e. a somewhat crude estimate of the posterior mean) is shown in green. The red vectors are the corresponding ECMWF analysis vectors included here for comparison. As we have seen, we should not expect (or even desire) that the analysis vectors and the realizations from MFS-Wind-BHM agree. The QuikSCAT data stage has endowed the BHM winds with more realistic kinetic energy content. Note that the spread in each cluster of realizations from the BHM is a function of space (and as we will see next, a function of time as well). This is a representation of a space-time uncertainty function. In regions where wind speed are weak, the error model for the QuikSCAT retrievals has larger amplitude, and the ensemble spread can be large <point S of Sicily>. In regions with relatively strong winds, the clusters are more aligned and uncertainty is therefore reduced. 10 m/s

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**Approaches Stochastic models incorporating science**

Physical-statistical modeling (Berliner 2003 JGR) From ``F=ma'' to [ X | q ] Qualitative use of theory (eg., Pacific SST model; Berliner et al J. Climate) Incorporating large-scale computer models From model output to priors [ q ] Model output as samples from process model prior [ X | q ] almost ! Model output as ``observations'' (Y) Combinations 26

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**Part B) Information from Models**

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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**[ Y | X , q ] is measurement error model:**

Model output as realizations of prior “trends” Process Model Prior X = O + hq where h is “model error”, “bias”, “offset” [ Y | X , q ] is measurement error model: Y = X + eq Substitution yields [ Y | O , hq , q ] Y = O + hq + eq Modeling h is crucial (I have seen h set to 0) Seems odd to have model ouptut as condition in data model, but “reality” is there through eta

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**[ O | X , q ] to include “bias, offset, ..” [ X | O , q ]**

Model output as “observations” Data Model: [ Y, O | X , q ] ( = [ Y | X, q ] [ O | X , q]) [ O | X , q ] to include “bias, offset, ..” Previous approach: start by constructing [ X | O , q ] This approach: construct [ O | X , q] 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 Data Model: Gaussian with mean = true temp. & unknown variance (with a change-point) Two models (O): PCM (n=4), CCSM (n=1) for , and 3 SRES scenarios (B1,A1B,A2). Data Model: assumes O’s are Gaussian with mean = bt + 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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**Ojk = b + bj + ejk Notes (Freeze time) b is common to both Models**

Data model for kth ensemble member from Model j: Ojk = b + bj + ejk b is common to both Models bj is Model j bias E( ejk ) = 0 and variances of e’s depend on j Computer model model: b = X + e where E(e) = 0 Priors for biases, variances, and X Extensions to different model classes (more b’s) and richer models are feasible.

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**Temp: mean and ENSO variations plus weather**

Temp: mean and ENSO variations plus weather. This is based on biases which may change every 25 years

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IPCC (global) Us (NH) Note that we messed up a color code: our green is IPCC red, Our red is IPCC green (our blue is their purple). We didn’t do the constant thing (IPCC’s orange).

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**Both n-hemi. Left: basis period = 25, right = no change.**

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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 | q ] may arise Model output as “observations” Combining models: Just like different measuring devices; Nice for analysis & mixed (obs’ & comp.) design Need a prior [ X | q ] Model output as realizations of prior “trends” Most common among Bayesian statisticians Combining models: like combining experts 36

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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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**Part C) Combining approaches**

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! 38

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