Introduction to Integrated Data Analysis R. Fischer - PowerPoint PPT Presentation

7 th Workshop on Fusion Data Processing, Validation and Analysis Introduction to Integrated Data Analysis R. Fischer Max-Planck-Institut für Plasmaphysik, Garching EURATOM Association Frascati, Mar 26-28, 2012

Outline ➢ Why do we need Integrated Data Analysis (IDA)? ➢ Implementation of IDA ➢ Applications at ➢ W7-AS ➢ JET ➢ TJ-II stellarator ➢ ASDEX Upgrade tokamak ➢ Integrated Diagnostics Design (IDD) is closely related to IDA → talk A. Dinklage

IDA for Nuclear Fusion Different measurement techniques (diagnostics: LIB, DCN, ECE, TS, REF, ...) for the same quantities (n e , T e , …) and parametric entanglement in data analysis • Redundant data: ➢ reduction of estimation uncertainties (combined evaluation, “super fit”) ➢ detect and resolve data inconsistencies (reliable/consistent diagnostics) • Complementary data: ➢ resolve parametric entanglement ➢ resolve complex error propagation (non-Gaussian) ➢ synergistic effects (parametric correlations, multi-tasking tools (TS/IF, CXRS/BES) ) ➢ automatic in-situ and in-vivo calibration (transient effects, degradation, ...) • Goal: Coherent combination of measurements from different diagnostics ➢ replace combination of results from individual diagnostics ➢ with combination of measured data → one-step analysis of pooled data ➢ in a probabilistic framework (unified error analysis!)

Conventional vs. Integrated Data Analysis conventional IDA (Bayesian probability theory) n e (ρ), T e (ρ), ... Thomson ECE ... Scattering mapping ρ(x) data data → n e (x), T e (x) analysis analysis D TS (n e (x)),T e (x)) D ECE (n e (x)),T e (x)) addl. n e (x),T e (x) T e (x) information, Thomson constraints, Scattering ECE model params, mapping mapping data d TS data d ECE ... ρ(x) ρ(x) result: p(n e (ρ),T e (ρ) | d TS ,d ECE ) linked result estimates: n e (ρ) ± Δn e (ρ), T e (ρ) ± ΔT e (ρ) Parametric entanglements

Conventional vs. Integrated Data Analysis (2) Drawbacks of conventional data analysis: iterative ● (self-)consistent results? (cumbersome; do they exist?) (Single estimates as input for analysis of other diagnostics?) ● information propagation? (How to deal with inconsistencies?) ● data and result validation? ● non-Gaussian error propagation? (frequently neglected: underestimation of the true error?) (huge amount of data from steady state devices: W7X, ITER, ...) ● difficult to be automated ● often backward inversion techniques (noise fitting? numerical stability? loss of information? ) ● result: estimates and error bars (sufficient? non-linear dependencies?) Probabilistic combination of different diagnostics (IDA) ✔ uses only forward modeling (complete set of parameters → modeling of measured data) ✔ additional physical information easily to be integrated ✔ systematic effects → nuisance parameters ✔ unified error interpretation → Bayesian Probability Theory ✔ result: probability distribution of parameters of interest IDA offers a unified way of combining data (information) from various experiments (sources) to obtain improved results

Probabilistic (Bayesian) Recipe Reasoning about parameter θ : p  (uncertain) prior information prior distribution d = D  + (uncertain) measured data } p  d ∣ likelihood distribution D = f  + physical model + Bayes theorem p ∣ d = p  d ∣× p  posterior distribution p  d  + additional (nuisance) parameter β p ∣ d = ∫ d  p  , ∣ d  marginalization (integration) = ∫ d  p  d ∣ , × p × p  p  d  generalization of Gaussian error propagation laws + parameter averaging (model comparison) p  d ∣ M = ∫ d  p  , d ∣ M = ∫ d  p  d ∣ , M  p  prior predictive value

Bayesian Recipe for IDA: LIB + DCN + ECE + TS Reasoning about parameter n e , T e : p  n e ,T e  (uncertain) prior information prior distribution d LiB = D LiB  n e ,T e  ; p  d LiB ∣ n e ,T e  + experiment 1: d DCN = D DCN  n e  p  d DCN ∣ n e  ; + experiment 2: likelihood d ECE = D ECE  T e  p  d ECE ∣ T e  ; + experiment 3: distributions + experiment 4: d TS = D TS  n e ,T e  ; p  d TS ∣ n e ,T e  + Bayes theorem p  n e ,T e ∣ d TS ,d ECE , d LiB ,d DCN  ∝ p  d TS ∣ n e ,T e  × posterior distribution p  d ECE ∣ T e  × p  d LiB ∣ n e ,T e  × p  d DCN ∣ n e  × p  n e ,T e 

Application: W7-AS W7-AS: n e , T e : Thomson scattering, interferometry, soft X-ray R. Fischer, A. Dinklage, and E. Pasch, Bayesian modelling of fusion diagnostics, Plasma Phys. Control. Fusion, 45, 1095-1111 (2003) Using synergism: Combination of results from a set of diagnostics ∫ = ⊗ dT e Thomson Soft-X-ray Scattering Electron density 30% reduced error → synergism by exploiting full probabilistic correlation structure

Application: JET JET: n e , T e : Interferometry, core LIDAR and edge LIDAR diagnostics O Ford, et al., Bayesian Combined Analysis of JET LIDAR, Edge LIDAR and Interferometry Diagnostics, P-2.150, EPS 2009, Sofia n e : Lithium beam forward modelling D. Dodt, et al., Electron Density Profiles from the Probabilistic Analysis of the Lithium Beam at JET, P-2.148, EPS 2009, Sofia

Application: TJ-II TJ-II: n e : Interferometry, reflectometry, Thomson scattering, and Helium beam B. Ph. van Milligen, et al., Integrated data analysis at TJ-II: The density profile, Rev. Sci. Instrum. 82, 073503 (2011) Full forward model for Interferometry Reflectometry (group delay) Partial forward model for Thomson scattering Helium beam

Application: ASDEX Upgrade (1) n e , T e : ➢ Lithium beam impact excitation spectroscopy (LiB) ➢ Interferometry measurements (DCN) ➢ Electron cyclotron emission (ECE) ➢ Thomson scattering (TS) ➢ Reflectometry (REF) ➢ Equilibrium reconstructions for diagnostics mapping R. Fischer et al., Integrated data analysis of profile diagnostics at ASDEX Upgrade, Fusion Sci. Technol., 58, 675-684 (2010) (2) Z eff : ➢ Bremsstrahlung background from various CXRS spectroscopies ➢ Impurity concentrations from CXRS S. Rathgeber et al., Estimation of profiles of the effective ion charge at ASDEX Upgrade with Integrated Data Analysis, PPCF, 52, 095008 (2010)

LIB + DCN: Temporal resolution #22561, 2.045-2.048 s, H-mode, type I ELM IDA: Lithium beam + DCN Interferometry LIN: Lithium beam only density profiles with temporal resolution of • 1 ms (routinely written) • 50 μs (on demand)

IDA: LIB + DCN + ECE ➔ simultaneous: ✔ full density profiles ✔ (partly) temperature profiles → pressure profile ➔ ➔ n e > 0.95*n e,cut-off → masking of ECE channels ➔ opt. depth ~ n e T e → masking of ECE channels

Summary: IDA ➢ Probabilistic modeling of individual diagnostics ✔ forward modeling only (synthetic diagnostic) ✔ probability distributions: describes all kind of uncertainties ✔ multiply probability distributions, marginalization of nuisance parameters ✔ parameter estimates and uncertainties ➢ Probabilistic combination of different diagnostics ✔ systematic and unified error analysis is a must for comparison of diagnostics ✔ error propagation beyond single diagnostics ✔ more reliable results by larger (meta-) data set (interdependencies, synergism) ✔ redundant information → resolve data inconsistencies ✔ advanced data analysis technique → software/hardware upgrades ➢ Applications at W7-AS, JET, TJ-II, and ASDEX Upgrade