rhessi imaging spectroscopy a regularization ... - unipa.itdaa_erice07/contributors/massone.pdf ·...

Electron flux maps of solar flares: a regularization approach to

RHESSI imaging spectroscopy

Anna Maria Massone

CNR-INFM LAMIA, Genova, Italy

[email protected]

In collaboration with :

• Michele Piana, Dipartimento di Informatica, Università di Verona, Italy

• Marco Prato, Dipartimento di Matematica Pura ed Applicata, di Modena e Reggio Emilia, Italy

• Gordon Emslie, Dep. of Physics, Oklahoma State University, US

• Gordon Hurford, Space Sciences Laboratory, University ofCalifornia at Berkeley, US

• Richard Schwartz, NASA Goddard Space Flight Center, Greenbelt, US

• Eduard Kontar, Dep. of Physics & Astronomy, The University, Glasgow, UK

People involvedPeople involved

The RHESSI mission has been launched by NASA on February 5, 2002in order to understand the high-energy processes at the core of the solar flare phenomena :

A solar flare is the most energetic explosion in the solar system.

The energy released during a flare is typically on the order of 1027 erg per

second (~1019 KW). Large flares can emit up to 1032 erg of energy.

Significant electromagnetic emission, particularly in the X-ray range.

RHESSI MissionRHESSI Mission((Reuven Ramaty High Energy Solar Spectroscopic Imager))

Scientific objective: to study the processes of electron and ion acceleration in solar flares through the hard X-ray and gamma-ray radiation that the

accelerated particles produce

RHESSI observationsRHESSI observations

• High spatial resolution X-ray images

� arc-sec-quality images

� image restoration: CLEAN, MEM, forward-fitting

• High energy resolution X- and γ-ray spectra

� 1 keV spectral resolution

� energy range: 1 keV – 17 MeV

High-resolution spectroscopy at each point of the X-ray imageImaging Spectroscopy

e +H

e

X-ray

spectra/images/imaging spectroscopy

RHESSI

Photon space Photon space vsvs Electron spaceElectron spaceX-ray emission: Bremsstrahlung

1. Photon spectra electron spectra (inversion of the Bremsstrahlung eq.)

2. Photon images local photon spectra local electron spectra

Is it possible to build images in the electron space?

• nine bi-grid collimators• nine Ge detectors

� planar array of equally spaced, X-ray-opaque slats separated by transparent slits

� nine different widths for the slit

RHESSI HardwareRHESSI Hardware

• θ is the incident angle between the photons and the collimator axis

• duration of a completeRHESSI rotation: 4s

As the spacecraft rotates, imaging information is encoded as rapid time-variations of the detected flux

RHESSI GeometryRHESSI Geometry

Modulation profilesModulation profiles

point source

different θ

more off-axis

source distribution

bigger source distribution

real profile

smaller intensity

Roll angle and aspect phaseRoll angle and aspect phase

xsun

ysun

Spacecraft axis

α

β

• roll angle (α): defines the grid direction with respect to acoordinate system

For each detector:

{ }sunsun yx ,

• aspect phase (β) : measures the position of a reference point near the source with respect to the spacecraft axis

α

β

tq ↔

( )βαt ,↔

( )βαq ,↔Data stacking

Data stacking IData stacking I

RHESSI data are light curves, i.e. photon-induced counts

recorded while the collimators rotate

For each time point, a roll angle

and a phase are definedco

unts

coun

tsco

unts

roll

angl

e (d

eg)

phase (deg)

For different rotations, differentphases correspond to the same

roll angle

The counts corresponding tothe same roll and phase bin

are stacked in a histogram (32 rollbins, 12 phases)

Data stacking IIData stacking II

Data stacking IIIData stacking III

For each roll bin, the count profile, as a function of the phase bin, is fitted by a

Fourier series

If A and B are the first two Fourier components, the complex number whose real part is A and

imaginary part is B is called visibility

Visibilities IVisibilities I

(u,v) spatial frequency components(x,y) point in the sourceI(x,y;ε) photon spectrum at point (x,y) and energy εV(u,v;q) observed visibility at energy qD(qj,εk) Detector Response Matrix (DRM)

Formal definition:

A RHESSI visibility is a complex observable number that can be derived from RHESSI data and which represents a

measurement of a single Fourier component of the source distribution measured at a specific spatial frequency and

energy- and time-range.

∫∫∑+=

yx kk

vyuxiπkkjj dxdyεeεyxIεqDqvuV ∆),,(),(),,( )(2

where

∫∫∑+=

yx kk

vyuxiπkkjj dxdyεeεyxIεqDqvuV ∆),,(),(),,( )(2

∫∞

=ε

dEEεQEyxFyxNRπ

εyxI ),();,(),(41);,( 2

∫∞

=q

dEEqKEvuWqvuV ),();,();,(

∫∫+=

yx

vyuxiπ dxdyeEyxFyxNEvuW )(2);,(),(:);,(

kk

kkjj EQqDR

EqK εεεπ ∑ ∆= ),(),(41:),( 2

Count Count �� Electron Visibilities IElectron Visibilities I

Electron Visibilities

∫∞

=q

dEEqKEvuWqvuV ),();,();,(

Count Count �� Electron Visibilities IIElectron Visibilities II

Visibility inversion problem: determine the electron visibilities, W(u,v;E), from the

observed count visibilities V(u,v;q)

The relation between the measured count visibilities and the electron visibilities is described by a Volterra integral equation of the first kind

{ }N1=kkkk v,u;σ

2. Fix λ by means of some optimality criterion

The Singular Value Decomposition of the operator A gives the set of triplets:

The regularized solution: ‡”1

2 )(N

kkk

k

kλ uvg

λσσW

=

⋅+

=

VAW =

The Tikhonov Method:

1. Solve the minimum problem min- 22 =+ WλVAW

fidelity smoothness

Solution strategy: Tikhonov regularization method

Visibility inversionVisibility inversion

∫∞

=q

dEEqKEvuWqvuAW ),();,();,)((

The The algorithmalgorithm

2002, February 20 - 11:02:08 – 11:14:20 UT

Time range selected: 11:06:02 – 11:06:34 UT

• For each detector and each (u,v) point:

� construct the count visibility spectrum (count visibility vs count energy)

� apply regularized inversion to obtain an electronvisibility spectrum (electron visibility vs count energy)

• For each detector and each electron energy:

� construct the electron visibilities

Selected flare:

Count energy range selected: 10 – 50 keV

Count energy binning selected: 4 keV

RHESSI data: RHESSI data: countcount visibilitiesvisibilities

Detector 1

(u1(1),v1

(1)) Re(V(u1(1),v1

(1);q)) Im(V(u1(1),v1

(1);q))

(u32(1),v32

(1)) Re(V(u32(1),v32

(1);q)) Im(V(u32(1),v32

(1);q))

Detector 9

(u1(9),v1

(9)) Re(V(u1(9),v1

(9);q)) Im(V(u1(9),v1

(9);q))

(u32(9),v32

(9)) Re(V(u32(9),v32

(9);q)) Im(V(u32(9),v32

(9);q))

Fixed energy channel [q,q+∆q]

VisibilityVisibility--basedbased countcount mapsmaps

10-14 keV 14-18 keV 18-22 keV 22-26 keV 26-30 keV


Imaging from visibilities: Maximum Entropy Method (MEM)

CountCount visibilityvisibility spectraspectra

Spatial frequencies(ui

(j),vi(j))

Re(V(ui(j),vi

(j),q1)) Im(V(ui(j),vi

(j),q1))

Re(V(ui(j),vi

(j),qN)) Im(V(ui(j),vi

(j),qN))

• For each detector and each (u,v) point:

� construct the count visibility spectrum (count visibility vs count energy)

Electron Electron visibilityvisibility spectraspectra

Spatial frequencies(ui

(j),vi(j))

Re(W(ui(j),vi

(j),E1)) Im(W(ui(j),vi

(j),E1))

Re(W(ui(j),vi

(j),EM)) Im(W(ui(j),vi

(j),EM))

∫+∞

=q

ji

ji

ji

ji dEEqK EvuWqvuV ),();,();,( )()()()(

Electron energy range 10 – 90 keV

� apply regularized inversion to obtain an electron visibility spectrum(electron visibility vs count energy)

Electron visibilities reach energies higher than photon visibilities(thanks to bremsstrahlung)

Electron Electron visibilitiesvisibilities

Detector 1 (u1

(1),v1(1)) Re(W(u1

(1),v1(1);E)) Im(W(u1

(1),v1(1);E))

(u32(1),v32

(1)) Re(W(u32(1),v32

(1);E)) Im(W(u32(1),v32

(1);E))

Detector 9 (u1

(9),v1(9)) Re(W(u1

(9),v1(9);E)) Im(W(u1

(9),v1(9);E))

(u32(9),v32

(9)) Re(W(u32(9),v32

(9);E)) Im(W(u32(9),v32

(9);E))

Fixed energy channel [E,E+∆E]

VisibilityVisibility--basedbased electron electron mapsmaps





PhotonPhoton mapsmaps vsvs electron electron mapsmaps

70-74 keV 74-78 keV 78-82 keV 82-86 keV


rhessi imaging spectroscopy a regularization ... - unipa.itdaa_erice07/contributors/massone.pdf ·...

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