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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 Themes 1)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.priors 2.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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Approaches A.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 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 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 = u b + 0 H n where u b = k p + ( gH ) -q Our Model Basal Stress: = - gH ds/dx + where is a ``corrector process; H, s unknown Velocities: u = u b + H n + e where u b = 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 temps Model: Heat equation Borehole data f(surface temps) Infer boundary condition (initial cond. is nuisance)

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Modeling Data Model: Y | T r, ~ N( T r + T q R(k), 2 I ) true temp Adjustments for rock types, etc. Process Model: heat equation applied to T r with b.cond. surface temp history T h T r | T h, ~ N( BT h, 2 I ) T h | ~ 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 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 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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Approaches A.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 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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Part B) Information from Models 1)Develop prior from model output Think of model output runs O 1, …, O n as samples from some distribution Do data analysis on Os to estimate distribution Use result (perhaps with modifications) as a prior for X Example: Os are spatial fields: estimate spatial covariance function of X based on Os. 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 (CO 2 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 Os are Gaussian with mean = t + model bias t (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 k th ensemble member from Model j: O jk = + b j + e jk – is common to both Models – b j is Model j bias – E( e jk ) = 0 and variances of es depend on j Computer model model: = X + e where 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)

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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 Xs and Os. 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, Contd 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 its nice to be flexible in incorporating their information Thank You!

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