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

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)/ ∑ _{r}A(p,r)) cm^{3}s^{-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)/ ∑ _{r}A(p,r) | ratio of transition probabilities: excited ion can radiatively decay by more than one route |

Φ(T) | Amount of plasma at temperature T: |

- Plasma is opticially thin in observed lines
- Elemental abundances are constant over full depth of plasma
- Plasma is in steady-state of ionisation balance
- Plasma has Maxwellian electron distribution
- All atomic processes have been included in
*G(T)* - Atomic data used are of adequate accuracy
- Observations include good intensity calibration
- User has reliable method of inverting the intensity integral
- (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.)

(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)

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

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

- Integral discretisation – trapezoidal method
- Smoothing – regularisation (entropy method)
- Degree of smoothing – chi squared constraint

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

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

- Integral discretisation – trapezoidal method?
- Smoothing – regularisation
- Degree of smoothing – weighted least squares fit

-- ElizabethAuden - 22 Feb 2005

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