Statistical Approaches to Combining Models and Observations

Transcription

1 Statistical Approaches to Combining Models and Observations Outline Mark Berliner Department of Statistics The Ohio State University SIAM UQ12, April 5, 2012 I. Bayesian Hierarchical Modeling II. Examples: Glacial dynamics; Paleoclimate III. Using Large-scale Models IV. Example: Ocean Forecasting

2 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]

3 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 J. Climate) ****************************************************** 3. Incorporating large-scale computer model output 4. Combinations 5. Alternate Approaches

4 Ex) 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)

5 North East Ice Stream, Greenland

6 Physical Modeling: Surface: s, Thickness: H, Velocity: u Basal Stress: τ = ρ g H ds/dx (+ stuff ) Velocities: u = u b + b 0 H τ n where u b = k τ p + (ρgh) q Our Model Random Basal Stress: τ = ρ g H ds/dx + η where η is a corrector process ( model error ) Random H: wavelet smoothing prior Random s: parameterized model from literature Random Velocities: u = u b + b H τ n + e where u b = 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]

7

8

9

10 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

11 Relative temperature [ºC] San Rafael Desert San Rafael Swell Depth [m] ºC SRD 3 SRD 2 SRD 7 SRD SRD SRS SRS 4 SRS 5 WSR 1

12 Page 1 of 1 Page 1 of 1

13 Data Model : Y T r, q N( T r + T q R(k), σ 2 y I) T r : reduced temperatures (literature: T r = T T 0 1 q R(k)) T 0 : reference surface temperature q: surface heat flow R: thermal resistances ( k 1, thermal conductivities, adjusted for rock types, etc.) Process Model : Heat equation applied to T r B.C.: Surface temp history T h (assumed piecewise constant) Easy to solve the heat equation T r T h, q N(A T h, σ 2 I) Parameter Model : next T h q N( 0, σ 2 h I)

14 Spatial Hierarchy Nine boreholes: 5 in desert region, 4 in swell region Extend the hierarchy: for j = 1,..., 9 Data Model : Yj T rj, q j N( T rj + T 0j 1 + q j Rj (k j ), σ 2 yj I) Process Model : T rj T hj, q j N(A j Thj, σ 2 j I) Parameter Model T hj q j N( 0, σ 2 hj I) T h1,..., T h5 conditionally independent N( µ d, γ 2 d I) T h6,..., T h9 conditionally independent N( µ s, γ 2 s I) µ d & µ s N( µ 0, γ 2 0 I) q 1,..., q 5 conditionally independent N(ν d, η 2 d) q 6,..., q 9 conditionally independent N(ν s, η 2 s) etc.

15 SRD 1, SR Desert SRD 2, SR Desert SRD 3, SR Desert SRD 4, SR Desert SRD 7, SR Desert SRS 3, SR Swell SRS 4, SR Swell SRS 5, SR Swell WSR 1, SR Swell

16 µ S µ D

17 SRD 1, SR Desert SRD 2, SR Desert SRD 3, SR Desert SRD 4, SR Desert SRD 7, SR Desert SRS 3, SR Swell SRS 4, SR Swell SRS 5, SR Swell WSR 1, SR Swell Multi site Single site

18 SRD 2, SR Desert SRS 5, SR Swell Multi site Single site Multi site Single site

19 III. Incorporating Large-Scale Computer Models Ensembles O = O 1 = M(T 1 ),..., O n = M(T n ) (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 Σ i α i k(x O i ) Gaussian process models; emulators; UQ Model Output: O = (O 1 = M(T 1 ),..., O n = M(T n )) [O θ, O] [X θ] Parameter Model from model output (eg: Berliner, Levine, & Shea 2003 J. Climate) Combinations

20 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 (CO 2 etc.) Assume a scenario and find corresponding projection

21 Hemispheric Monthly Surface Temperatures (X) Observations (Y) for Two models ( O): PCM (n=4), CCSM (n=1) for SRES scenarios (B1,A1B,A2).

22 Hierarchical Data Model for Model Output Scalar climate variable X; m = 1,..., M models (time fixed) O m : ensemble of size n m of estimates of X from model m. 1. Given means µ m and variances σ 2 Y m, m = 1,..., M; O m are independent and O m µ m Gau(µ m 1 nm, σ 2 Y m I nm ) 2. Given β, model biases b m and variances σ 2 µ m ; µ m are independent and 3. Given X, µ m β, b m Gau(β + b m, σ 2 µ m ) β X Gau(X, σ 2 β ) and b m X Gau(b 0m, σ 2 b m )

23 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.

24 Model Overview 1. [Y X, θ]: measurement error model Gaussian with mean = true temp. & unknown variance (with a change-point) 2. [X θ]: Time series models with time varying parameters X t = µ i(t) + ηn j(t) 0 0 η s j(t) (X t 1 µ i(t 1) ) + e t 3. [θ] Climate = parameters of distribution of weather Climate-weather: multiscale phenomena Time evolution: µ i(t) slow; η j(t) moderate; e t fast, but variances of e t are slow µ i = A + B CO 2i + noise Obs period: η j = C + D SOI j + noise Fore. period: AR model (i.e., SOI not observed) Variances of e t : AR-like prior

25 mean year annual temperature year

26 mean year annual temperature year

27 system variance year

28 Model Adapted to Decadal Forecasting (Kim & Berliner 2012) 16 Temperature(Celcius) Temperature(Celcius)

29 IV. Combining Approaches: Mediterranean Ocean Forecasting 1. Winds as a boundary condition for the ocean surface (Milliff et al. & Bonazzi et al Quart. J. Roy. Met. Soc.) 2. Bayesian Multi-model Ensembling for Ocean Forecasting (Berliner et al. 2012) Observations BHM Winds BHM Oceans Initial Boundary conditions Model 1 Model 2 BHM Ocean Post. Dist.

30 Bayesian Hierarchical Models to Augment ( MFS ) The Mediterranean Forecast System Nadia Pinardi Alessandro Bonazzi, Srdjan Dobricic INGV (I'Istituto Nazionale di Geofisica e Vulcanologia) ( Director Univ. Bologna (MFS INGV, Univ. Bologna Ralph Milliff CoRA Chris Wikle Univ. Missouri Mark Berliner Ohio State Univ.

31 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

32 Building wind dist. (BHM-SVW) 1. Data Stage: Satellite (QSCAT) and Numerical Weather Pred. Analyses (ECMWF) QSCAT ECMWF

33 2. Process Model: Rayleigh Friction Model (Linear Planetary Boundary Layer Equations) Theory (neglect second order time derivative) discretize: Our model

34 BHM Ensemble Winds 10 members selected from the Posterior Distribution (blue) 10 m/s Ensemble mean wind (green); ECMWF Analysis wind (red)

35 BHM-SVW-OEF initial condition spread: Sea Surface Temperature Initial condition spread Sea Surface Height Initial condition spread Uncertainty is concentrated at mesoscales

36 BHM-SVW-OEF 10 day forecast spread Initial condition spread Sea Surface Height 10-th fcst day spread Initial condition ensemble spread has amplified at the 10 day fcst in mesoscales

37 The forecast spread at 10 days EEPS forced ensemble BHM-SVW ensemble ECMWF EPS forcing Ineffective at producing flow field changes at mesoscales

38 Discussion: BHM methods produce realistic distributions of surface winds (SVW) (Milliff et al 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-dayforecast spreads at mesoscales and in the upper thermocline

39 2. Bayesian Multi-model Ensembling for Ocean 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

40

41

42