Hale Collage. Spectropolarimetric Diagnostic Techniques!!!!!!!! Rebecca Centeno

Hale Collage Spectropolarimetric Diagnostic Techniques Rebecca Centeno March 1-8, 2016

Today Simple solutions to the RTE Milne-Eddington approximation LTE solution The general inversion problem Spectral line inversion codes Levenberg-Marquardt techniques Principal Component Analysis

Solutions to the RTE: Milne-Eddington Approximation Radiative Transfer Equation: d I dτ c = K( I S) Let s assume: Polarization due to Zeeman effect. The elements of { K are constant: K = K 0 S = (S 0 + S 1 τ) (1, 0, 0, 0) T Then: I(0) = 0 e K 0τ c K 0 ( S 0 + S 1 τ c )dτ c = S 0 + K 0 1 S1 is the Unno-Rachkovsky [ solution to the RTE and it is analytical in nature ] [ ] I(0) = S 0 + 1 η I (ηi 2 + ρ2 Q + ρ2 U + ρ2 V )S 1 Q(0) = 1[ [ ηi 2 η Q + η I (η V ρ U η U ρ V )+ρ Q (η Q ρ Q + η U ρ U + η V ρ V ) ] S 1 ] U(0) = 1[ ηi 2 η U + η I (η Q ρ V η V ρ Q )+ρ U (η Q ρ Q + η U ρ U + η V ρ V ) ] S 1 U(0) = 1[ ηi 2 η V + η I (η U ρ Q η Q ρ U )+ρ V (η Q ρ Q + η U ρ U + η V ρ V ) ] S 1 with: = η 2 I (η 2 I η 2 Q η 2 U η 2 V + ρ 2 Q + ρ 2 U + ρ 2 V ) (η Q ρ Q + η U ρ U + η V ρ V ) 2

Solutions to the RTE: Milne-Eddington Approximation Model Parameters: Line-to-continuum absorption: η 0 Doppler width: λ D Damping parameter: a Magnetic field: B, θ, φ Source function: S 0, S 1 LOS velocity: v LOS Magnetic filling factor: α Macroturbulent velocity: v MAC No thermodynamical information No velocity gradients so no asymmetries NCP = 0 No magnetic field gradients If you want to synthesize spectral lines in a Milne-Eddington atmosphere: http://www.iac.es/proyecto/inversion/online/milne_code/milne.php

Solutions to the RTE: Local Thermodynamical Equilibrium LTE hypothesis: the plasma is in thermodynamic equilibrium at local values of temperature and density. Hence: Maxwellian distribution of velocities Saha and Boltzmann give the populations of different atomic species Kirchhoff s Law applies: j = B ν (T) (η I, η Q, η U, η V ) T [ Absorption profiles have the same shape as emission profiles complete redistribution of frequencies [ Also assumes hydrostatic equilibrium and Zeeman induced polarization. And the RTE still looks like this: d dz I η I η Q η U η V I B ν (T ) Q U = η Q η I ρ V ρ U Q η U ρ V η I ρ Q U V η V ρ U ρ Q η I V

Solutions to the RTE: Local Thermodynamical Equilibrium Model atmosphere: temperature: T(z) pressure: P(z) LOS velocity: v LOS (z) Magnetic field: B(z), θ(z), φ(z) Macroturbulent velocity: v MAC Macroturbulent velocity: v MIC 9 physical quantities stratified atmosphere: 9 x Nz free model parameters Too many free parameters Need to constrain stratification to a limited number of nodes (from Fontenla et al 1999)

Spectral Line Inversions Forward modeling: Set of physical parameters (T, P, ρ, v LOS, B..) Reasonable assumptions (LTE, Milne-Eddington, Zeeman...) Stokes profiles (I, Q, U, V) Solve Radiative Transfer Eq. Inverse problem: given a set of Stokes profiles (data), what are the physical conditions in the atmosphere?

Spectral Line Inversions fitting metric + educated guess (from Vitticchié et al, 2011) open circles: observations tokes MISMA inversion of HINODE SOT/SP Stokes profiles. For details of the layout of the fi solid line: synthetic Stokes profiles

Spectral Line Inversions: General Problem RADIATIVE TRANSFER feedback solve ATMOSPHERIC MODEL + EXTERNAL FIELDS atomic populations solve radiation output compare OBSERVATIONS STATISTICAL EQUILIBRIUM feedback

Spectral Line Inversions: Local Thermodynamic Equilibrium RADIATIVE TRANSFER feedback ATMOSPHERIC MODEL EXTERNAL FIELDS atomic excitation solve radiation output compare OBSERVATIONS feedback

Spectral Line Inversions: Milne-Eddington Approximation RADIATIVE TRANSFER feedback ATMOSPHERIC MODEL EXTERNAL FIELDS solve radiation output compare OBSERVATIONS feedback

Spectral Line Inversion Methods Let s assume we know how to solve the RTE. Inversion methods Levenberg-Marquardt methods Blind trial and error?? (least squares fitting) Principal Component Analysis techniques (pattern recognition)

Spectral Line Inversions: the merit function Let s assume our model atmosphere is characterized by a series of Np parameters, a. The solution to the RTE in the model a gives us a set of synthetic Stokes profiles, which we can compare to the observed ones. We can measure the difference using a merit function: χ 2 = 1 N f s λ [ I obs s (λ) Is syn (λ) ] 2 ω 2 s Where the number of degrees of freedom: Nf = Ns x Nλ - Np I syn and I obs are the synthetic and observed Stokes profiles ωs are some weighting factors (related to measurement error). The sums are over wavelength and Stokes parameters.

Spectral Line Inversions: the merit function χ 2 = 1 N f s λ [ I obs s (λ) Is syn (λ) ] 2 ω 2 s χ 2 is a hyper-surface of N p dimensions. It quantifies the goodness of the fit (the distance between the observed and synthetic Stokes vector) with one number The whole inversion problem boils down to minimizing χ 2

Spectral Line Inversions: Levenberg-Marquardt Techniques The problem boils down to the minimization of χ 2. The first derivative is given by: [ ] χ 2 a i = 2 N f s λ [ I syn s (λ) Is obs (λ) ] ωs 2 I syn (λ) a i (i = 0,, N p ) And the second derivative of χ 2 is given by: 2 χ 2 a i a j = 2 N f ( I syn s (λ) Is syn (λ) ωs 2 a s ( j a i λ [ + [ I syn s (λ) Is obs (λ) ] 2 I syn ) (λ) a i a j When close to the minimum of χ 2, we can ( expect [I syn - I obs ) ] 0 2 χ 2 a i a j 2 N f s λ ω 2 s ( I syn s (λ) a j Is syn ) (λ) a i

Spectral Line Inversions: Levenberg-Marquardt Techniques Let s assume the model a is close to the minimum of χ 2, so there is a perturbation δa that ( takes us directly to [ the minimum. We can use a quadratic approximation, such that: a min ( ) a current χ 2 (a + δa) χ 2 (a)+δa T ( χ 2 + H δa) where H ij = 1 2 2 χ 2 a i a j is half the Hessian matrix (dimensions N p x N p )

( Spectral Line Inversions: Levenberg-Marquardt Techniques When one is really close to the minimum, the second order approximation is adequate, and we can equate to zero the term in parenthesis: ( χ 2 + H δa) =0 δa = H 1 χ 2 This involves inverting the Hessian matrix thus we obtain a better approximation to the minimum of χ 2 by shifting in the parameter space an amount δa. When we re far from the minimum, we can get closer to it following the gradient (first order approximation): δa = k χ 2 with k small enough

Spectral Line Inversions: Levenberg-Marquardt Techniques Marquardt had two insights: The diagonal elements of the Hessian matrix give us a sense of what good values for k could be (it s a dimensional argument). fudge factor λ = k δa i = 1 H ii χ 2 = k δa i = 1 λh ii χ 2 The two methods can be combined into one equation, that allows to vary smoothly between the gradient and the Hessian approaches: χ 2 + Hδa = 0 where H ij { (1 + λ)h ij if i = j H ij if i j λ gradient (first order) method λ hessian (second order) method

Spectral Line Inversions: Levenberg-Marquardt Techniques Evaluate χ 2 (a ini ) for the initial guess model Take modest value of λ (λ=10-3 ) Solve equation for δa Evaluate χ 2 (a+δa) If χ 2 (a+δa) χ 2 (a) Do not update a Increase λ: (λ new = λ*10) If χ 2 (a+δa) χ 2 (a) Decrease λ: (λ new = λ/10) Update a: a new = a+δa Stop when χ 2 barely decreases once or twice in a row. This algorithm is explained in detail in Numerical Recipes by Press et al.

Spectral Line Inversions: Levenberg-Marquardt Techniques white = observations red = synthetic fit

Spectral Line Inversions: Levenberg-Marquardt Techniques white = observations red = synthetic fit

Levenberg-Marquardt Techniques: Issues H can be quasi-singular due to different sensitivity of χ 2 to the various model parameters. But it has to be inverted Singular Value Decomposition methods: H is real and symmetric Y such that: H = Y T W Y and YY T = Y T Y = 1 So H -1 = Y T W -1 Y If W k we set 1/W k = 0, so a k does not contribute to the model perturbation. Global vs. local minima of χ 2 Levenberg-Marquardt techniques can lead to local (rather than global) minima depending on the location of the initial guess χ 2 surface

Principal Component Analysis (PCA) Techniques Let s assume we have a set of observations S ij = S j (λ i ), i =1,...,N ; j =1,...,M S Stokes vector (I,Q,U,V) N wavelengths M pixels They are independent realizations of the Stokes profile S(λ), so the average: S(λ i )= 1 M M S ij, j=1 i =1,...,N We can define a covariance matrix: C ij = M [S il S(λ i )][S jl S(λ j )], i,j =1,...,N real and symmetric l=1 Which can be diagonalized by an orthogonal transformation: Cf (k) = e (k) f (k), k =1,...,N, where f (k) are the eigenvectors that form a basis for the residual S j (λ) - S(λ) So that: S j (λ) S(λ) N k=1 c (k) j f (k)

Principal Component Analysis (PCA) Techniques average 1 st PC 2 nd PC 3 rd PC 4 th PC 5 th PC f (0) f (1) f (2) f (3) f (4) f (5) Casini et al 2012 0.95 0.90 0.85 0.80 0.75 0.70-5 -0-5 -0.20 0.2 - Stokes I -0.2 0.2 - -0.2 0.2 - -0.2 - -0.2-0.3 wavelength 008 006 004 002 000 0.2 - Stokes Q 0 5 0-5 -0-5 -0.20 - - -0.2-0.3 - -0.2 wavelength Stokes U Stokes V 1 10-5 5 10-6 -5 10-6 0-1 10-5 0.2-0 5 0-5 -0-5 -0.20 - - -0.2 0.2 - -0.2 wavelength 1 10-4 5 10-5 -5 10-5 0-1 10-4 0.2 - -0.2 0.2 - -0.2 0.2 - -0.2 0.2-0.2 - -0.2 wavelength

be retained for spectro-polarimetric inversions relies on two fundamental assumptions. The first assumption is that the error bars of the profiles are dominated by photon noise, and the second one that the photon counts is large enough that the associated poissonian noise can be treated as a random variate, so that its variance can simply be added to that of the model. On the other Principal Component Analysis (PCA) Techniques hand, the systematic errors due to deviations of the observations from the line formation model, especially in the case of complicated atmospheric structures, is very likely to dominate the inversion errors. in practical cases, we should not expect that retaining such a high number orders BuildThus, a complete database Determine a good setofof necessarily improves the goodness of the profile fits from the inversion. We will come back to this of Stokes profiles Principal Components argument in the next section. 3. Indexing of PCA inversion databases Calculate the Decompose observations We created of an the inversion database of 0.75 million models spanning theeigen-basis same parameter space eigen-profiles database on the as the eigenbasis of Figure 1. The profile information in this database is encoded in the expansion coefficients given by Equation (5). The inversion database is constructed by a strategy of filtered Monte Carlo sampling, where each new randomly selected point in the parameter space is tested for proximity to previously included models in the database. The testing parameter is the PCA observations compare For each S (λ ) in the database i k distance between two models, i and j, which is defined as follows, for each of the four Stokes (k) (k) eigen-coefficients c j calculate eigen-coefficients c i parameters: m %1/2 # $ " (k) (k) 2 dij = ci cj, (7) k=1 where m is the maximum number of orders retained for the reconstruction of the Stokes profiles. For each observation j The filtering criterion is to reject models for which the cumulative PCA distance for the four dij PCA distance between model i and observation j index k sums over truncated set of eigen-coefficients choose model i that minimizes dij

Principal Component Analysis: Pros and Cons Pros fast (searches best fit in a pre-built database of models) stable (always finds best fit: no problems of local minima) model independent (universal search/minimization algorithm) Cons no solution refinement (can be fixed by increasing the density of the database) database can become unmanageably large (dimensionality of parameter space, parameter ranges; partial mitigation from optimally sampling the parameter space) All PCA stuff is from Roberto Casini

Spectral line inversions from HMI SDO/HMI

Spectral line inversions from HMI Courtesy: R. Bogart and K. Hayashi