Monday, June 27, 2011 CEDAR-GEM joint workshop: DA tutorial 1 UNDERSTANDING DATA ASSIMILATION APPLICATIONS TO HIGH-LATITUDE IONOSPHERIC ELECTRODYNAMICS Tomoko Matsuo University of Colorado, Boulder Space Weather Prediction Center, NOAA
Monday, June 27, 2011 CEDAR-GEM joint workshop: DA tutorial 2 What is data assimilation? Combining Information prior knowledge of the state of system empirical or physical models (e.g. physical laws) complete in space and time x observations directly measured or retrieved quantities incomplete in space and time y
Monday, June 27, 2011 CEDAR-GEM joint workshop: DA tutorial 3 Bayes Theorem - Bayesian statistics provides a coherent probabilistic framework for most of DA approaches [e.g., Lorenc, 1986] prior p(x ) N (x b, P b ) x = x b + b observation likelihood p(y x ) N (Hx, R) x = Hx + b probability distribution of y when x have a given value posterior p(x y) p(y x )p(x )
Monday, June 27, 2011 CEDAR-GEM joint workshop: DA tutorial 4 Bayes Theorem - Bayesian statistics provides a coherent probabilistic framework for most of DA approaches [e.g., Lorenc, 1986] prior p(x ) N (x b, P b ) x = x b + b observation likelihood p(y x ) N (Hx, R) y = Hx + y probability distribution of y when x have a given value posterior p(x y) p(y x )p(x ) p(x y) N (x a, P a ) where x a = x b + K(y Hx b ) P a =(I KH)P b K = P b H T (HP b H T + R) 1
Monday, June 27, 2011 CEDAR-GEM joint workshop: DA tutorial 5 What is covariance? two variables case Bayes theorem p(x y) p(y x )p(x ) prior p(x ) N (x b, P b ) x b =(2.3 2.5) P b = 0.225 0.05 0.05 0.15 observation-likelihood p(y x ) N (Hx, R) H = (1 0) x 1 : observed x 2 : unobserved
Monday, June 27, 2011 CEDAR-GEM joint workshop: DA tutorial 6 What is covariance? in spatial sense lag distance lag distance P 1 (µ =0.5, γ =0.05) P 2 (µ =1.5, γ =0.1)
Monday, June 27, 2011 CEDAR-GEM joint workshop: DA tutorial 7 What is covariance? in spatial sense x 1 N (0, P 1 ) x 2 N (0, P 2 ) 3 3 3 1 1 112 3 3 3 3 3 1 1 112 3 3 3 3 3 1 1 112 3 3
Monday, June 27, 2011 CEDAR-GEM joint workshop: DA tutorial 8 What is covariance? in spatial sense x 1 N (0, P 1 ) x 2 N (0, P 2 ) 3 1 1 2 3 3 1 1 2 3 2-D Correlation Function HP 3 1 1 2 3 0.2 0.2 0.1 0.1 0.5 0.5 0.3 0.3 0.5 0.5 0.3 0.3 0.2 0.2 0.1 0.1 0.5 0.5 0.3 0.3 0.2 0.2 0.1 0.1
Monday, June 27, 2011 CEDAR-GEM joint workshop: DA tutorial 9 Assimilative Mapping of Ionospheric Electrodynamics Inverse procedure to infer maps of E I J E,, I, J, B From observations of B IS or HF radar, Satellites IS radar Satellite or ground-based magnetometers [Richmond and Kamide, 1988]
Monday, June 27, 2011 CEDAR-GEM joint workshop: DA tutorial 10 Assimilative Mapping of Ionospheric Electrodynamics [Richmond and Kamide, 1988] prior observations
Monday, June 27, 2011 CEDAR-GEM joint workshop: DA tutorial 11 [Lu et al., 1998]
Monday, June 27, 2011 CEDAR-GEM joint workshop: DA tutorial 12
Monday, June 27, 2011 CEDAR-GEM joint workshop: DA tutorial 13 AMIE relationship among electromagnetic variables Inverse procedure to infer maps of E,, I, J, B From observations of E I J B IS or HF radar, Satellites IS radar Satellite or ground-based magnetometers linear relationship (for a given ) F( E) =, I, J, B E I J I = Φ = Σ E = I, J ΔB Biot-Savart s law
Monday, June 27, 2011 CEDAR-GEM joint workshop: DA tutorial 14 AMIE basis functions as forward operator functional analysis E Φ = Ψ x Ψ : x : spherical harmonics coefficients = Ψ x forward operator y = Hx = F ( Ψ) x linear relationship (for a given ) F( E) =, I, J, B E I J I = Φ = Σ E = I, J ΔB Biot-Savart s law
Monday, June 27, 2011 CEDAR-GEM joint workshop: DA tutorial 15 Ideas to improve AMIE: Adaptive covariance x a = x b + K(y Hx b ) K = P b (α)h T (HP b (α)h T + R) 1 Maximum likelihood Method [Dee 1995, Matsuo et al., 2005] d = y Hx b d N (0, S) where Find alpha that maximizes the following pdf p(d α) = S(α) =R + HP b (α)h T 1 2π J dets(α) exp dt S 1 (α)d 2
Monday, June 27, 2011 CEDAR-GEM joint workshop: DA tutorial 16 Ideas to improve AMIE: Adaptive covariance For given observations x a = x b + K(y Hx b ) K = P b (α)h T (HP b (α)h T + R) 1
Monday, June 27, 2011 CEDAR-GEM joint workshop: DA tutorial 17 Ideas to improve AMIE: Multi-resolution basis functions Spherical Harmonics Multi-resolution bas Σ = WH 2 W T [Nychka et al., 20 A 1-D wavelet basis. Father wavelet (scaling) functions Mother wavelet functions Wavelets 0e+00 4e 04 8e 04 1e 03 0e+00 1e 03 Father wavelet (scaling) functions Mother wavelet functions Mother wavelet functions Mother wavelet functions 0e+00 4e 04 8e 04 1e 03 0e+00 1e 03 0.004 0.000 0.004 0.015 0.000 0.015.6 0.8 1.0
Monday, June 27, 2011 CEDAR-GEM joint workshop: DA tutorial 18 Ideas to improve AMIE: what to minimize Inverse procedure to infer maps of E,, I, J, B B or E How to take advantage of new space-based observations? From observations of E I J B IS or HF radar, Satellites IS radar Satellite or ground-based magnetometers Science Objective: Understand the global electro- T
Monday, June 27, 2011 CEDAR-GEM joint workshop: DA tutorial 19 Summary Bayesian statistics as an overarching framework for many of DA methods Assumptions: Gaussian distribution, Linear H Role of Covariance in spatial interpolation Applications to high-latitude electrodynamics (AMIE) Functional analysis of E, Φ, I, J, B Kalman update of spherical hamonics coefficients Current issues Resolution: global function ->> compactly supported functions New space-based magnetometer observations ->> Improve conductance models Adaptive Covariance E or B