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DEM Methods: the 1991 RAL Report on Intensity Integral Inversion

Differential emission measure – measure of the amount of plasma at a given temperature T. The following information is from the report RAL-91-092, “Intensity Integral Inversion Techniques: a Study in Preparation for the SOHO Mission,” edited by Richard Harrison and Alan Thompson in December 1991 and provided by Jim Lang. The report details a comparison exercise between 6 methods of modelling differential emission measure. As stated in the report’s conclusions, all methods have three commonalities: 1) the intensity integral is discretised, 2) smoothing is applied, and 3) the degree of smoothing must be chosen carefully. It’s also worth noting that while many of the codes use polynomials in the smoothing process, the polynomial order was varied for best fit by a human judge – this could lead to difficulties with automation.

DEM Equations

DEM can be calculated by inverting the intensity integral for an optically thin spectral line in a low density plasma:

I = 1/4pi Ab(Z) ∫ G(T) Φ(T) dT photons cm-2s-1st-1

Ab(Z)Abundance of element (assumed constant over full depth of plasma
G(T)Atomic physics parameters relevant to transition (kernel function):
G(T) = (n(H)/n(e)) (n(z)/n(Z)) {∑ g(n(g)/n(z)) X(g,p)} (A(p,q)/ ∑ rA(p,r)) cm3s-1
n(H)/n(e)ratio of density of hydrogen atoms and ions to density
n(z)/n(Z)fraction of an element’s ions in relevant stage of ionisation wrt total elemental density
g(n(g)/n(z)) X(g,p)collisional excitation processes: excitation from ground level and metastable levels
A(p,q)/ ∑ rA(p,r)ratio of transition probabilities: excited ion can radiatively decay by more than one route
Φ(T)Amount of plasma at temperature T:

DEM Assumptions

  1. Plasma is opticially thin in observed lines
  2. Elemental abundances are constant over full depth of plasma
  3. Plasma is in steady-state of ionisation balance
  4. Plasma has Maxwellian electron distribution
  5. All atomic processes have been included in G(T)
  6. Atomic data used are of adequate accuracy
  7. Observations include good intensity calibration
  8. User has reliable method of inverting the intensity integral
  9. (For diagnostic line ratios) Plasma emitting lines used for temperature or density diagnostics has uniform temperature and density. (Note: not necessary to make this assumption with DEM method.)

DEM Models Examined

(Note: these method lists for integral discretisation, choice of smoothing, and degree of smoothing were made after an initial pass-through of the RAL-91-092 report. Please correct any errors. – ECA)

Glasgow Code

  1. Integral discretisation – product integration.
  2. Smoothing – regularisation
  3. Degree of smoothing – data based method of Golub, Heath, and Wahba (1979)

MPI Lindau Code

  1. Integral discretisation - replace the G(T) function with a delta function (cannot resolve structures small than the width of the G(T) function.)
  2. Smoothing – polynomial
  3. Degree of smoothing – user choice of polynomial order

Arcetri Code

  1. Integral discretisation – trapezoidal method
  2. Smoothing – regularisation (entropy method)
  3. Degree of smoothing – chi squared constraint

Wroclaw Code

  1. Integral discretisation – 3rd order spline interpolation
  2. Smoothing – iterative processing from flat to best fit solution (includes positivity)
  3. Degree of smoothing – chi squared constraint

MSSL Code

  1. Integral discretisation – Chebychev polynomial
  2. Smoothing – polynomial, regularisation (log method) (includes positivity)
  3. Degree of smoothing – user choice of polynomial order

Cambridge Code

  1. Integral discretisation – trapezoidal method?
  2. Smoothing – regularisation
  3. Degree of smoothing – weighted least squares fit

-- ElizabethAuden - 22 Feb 2005

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Topic revision: r1 - 2005-02-22 - ElizabethAuden
 
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