1. An Introduction to X-ray Spectral Fitting I: Getting started with XSPEC
Other material
XSPEC manual:
“X-ray school” lecture notes:
Local pages:

2. Syllabus Part I: What is spectral fitting? Why?
How to compare data and model
Running XSPEC
Fitting a model to data
Fitting non X-ray data (e.g. optical)
Simulating data
Part II:
How to test the fit
Confidence regions (uncertainties)
Comparing different models
Few count/bin spectra
Reference material

3. Why spectral fitting?
The recorded spectrum is not the true spectrum of the source:
instrumental response
errors (uncertainties, e.g. Poisson noise)
To test a physical model (e.g. blackbody) against data and make statistical inferences.

4. The detector response
As an observer, what you get is counts per detector channel, C(i):
T is the observation length (in s)
A(E) is the effective area of the telescope + detector (in cm2).
R(i,E) is the redistribution: the probability of photon with energy E being recorded in channel i.
S(E) is the true spectrum
(in units of photons/s/cm2/keV)

5. Redistribution: line spread
No detector is perfect. In real life the spectrum you get out is not exactly what you put in. This plot shows the energy redistribution of an EPIC camera from XMM-Newton. This is the detector output from a monochromatic (6.4 keV) input.
This limits the energy resolution.

6. The response matrix
The response (or redistribution) matrix R(i, E) gives the probability that a photon with energy E will be detected in channel i.

7. Effective area
The effective collecting area of the telescope and detector as function of energy.
Product of :
detector efficiency,
mirror reflectivity,
mirror geometrical cross-section…

8. “uncertainties”: X-ray data
X-ray detectors record individual X-ray events – “counts” – per detector channel. These are subject to Poisson (counting) statistics, i.e. the number of counts is drawn from the distribution:
This gives the probability of observing exactly C (integer) counts if μ is the expected number (true flux).
As the number of counts increases this distribution converges to the familiar Gaussian distribution with a mean of μi and a one sigma “error” of μi.
For an observed number of counts Ci we assign the familiar standard deviation Ci.

9. Poisson vs. Gauss μ=5 μ=10 μ=20 μ=50
Comparison of Poisson distribution (dotted) with Gaussian distribution (solid curve) having the same mean and variance.
Rule of thumb: Poisson distribution is usually ‘close enough’ to Gaussian when μ >20.
Common practice to bin data such that there are ~20 ct/bin in a spectrum, e.g. using “grppha”
μ=5
μ=10
μ=20
μ=50

10. The forward fitting method
Cannot invert the observed counts-channel spectrum to useful flux-energy spectrum –
no unique solution, very noisy, etc…
So convert the model instead!
Define model
Calculate model
Convolve with
response
Change model
parameters
Compare to data

11. Fit statistics: How to compare data and model
The most used measure of the “goodness” of a fit is Pearson’s chi-square statistic – χ2 – which measures the variance (sum of squares) of the “residuals” (i.e. data minus model)
where σ(i) is the error on the counts in channel i and μ(i) is the folded model, i.e. including detector response.
A smaller χ2 means a better fit (i.e. smaller data-model differences).

12. Pearson’s χ2 statistic

13. Multi-mission spectral fitting packages
XSPEC (from GSFC/NASA)
Part of HEAsoft
Been around since 1983/1986
The de facto standard
Many spectral models
Sherpa (from CXC)
Part of CIAO (for Chandra)
Uses XSPEC model library
Some advanced stats routines
Multi-dimensional fitting
ISIS (from MIT)
Distributed computation
Highly programmable
SPEX (from SRON)
Specialised for high-resolution plasma modelling

14. Using XSPEC Loading data Generating a model Fitting the model to data
Saving what you have done
Plotting your results
Using multiple datasets
Simulating data
Examples of real data

15. Input files The data: The instrument:
Source spectrum gives the counts per channel (*.pha)
Background spectrum (*.pha)
The instrument:
Detector “response matrix” – conversion from photons at energies to counts in channels (*.rmf)
“Ancillary” response file (*.arf) –effective area of detector
NB: Sometimes: rmf+arf = *.rsp

16. First steps with XSPEC lhea-setup-new (initialise) xspec (start XSPEC)
data spec.pha (load spectrum)
back back.pha (background)
resp resp.rmf (redist. matrix)
arf resp.arf (ancillary resp.)
cpd /xs (set plot device)
setplot energy (plot in keV)
ignore ** **
plot data (plot data)
show (data info)
(NB: The initialisation command may change)

17. “Raw” X-ray spectral data

18. Non X-ray data
XSPEC can be used to fit any binned, 1-dimensional dataset
optical spectra, power spectrum, …
Use the tool flx2xsp to convert from ASCII file to XSPEC files
Input ASCII file in 4 columns
E_low(i)
E_high(i)
Flux(i) i.e. flux density * [E_high-E_low]
Error(i)
Ensure bins are contiguous
i.e. E_high(i) = E_low(i+1)
Output is a spectrum (*.pha) and a ‘dummy’ response file (*.rsp) ready to load into XSPEC
Although XSPEC works in energy units, your data do not need to be in energy bins… (rename the axis)

19. X-ray spectral models
X-ray spectral models are built up from smaller components. There are two main types of model component:
Additive models – emission components, e.g. blackbody or emission line.
Multiplicative models – modifies the emission spectrum, e.g. absorption by neutral gas

20. Example models Frequently used additive models:
Power law (pow)
Blackbody (bb)
Bremsstrahlung (bremss)
Collisional plasma (mekal)
Gaussian emission line (gaus)
More detailed models such as Compton scattering, thermal accretion disc, …
Frequently used multiplicative models:
Photoelectric absorption due to neutral gas, e.g. in our Galaxy (wabs, phabs, TBabs)
Photoelectric absorption due to ionized (“warm”) gas (absori)
Individual photoelectric edges (edge) and absorption lines (gabs)
Exponential roll-over (expabs)

21. Building a model model wabs*(pow) editmod wabs*(pow+gaus) plot model
wabs = neutral absorption (NH)
pow = power law (Γ, N)
editmod wabs*(pow+gaus)
gaus = Gaussian (E, σ, N)
plot model

22. Fitting a model fit plot ldata del
Adjust model parameters to minimise χ2
fit
Plot data (in log-space) and Δχ residuals
plot ldata del

23. XSPEC12>model mekal
Input parameter value, delta, min, bot, top, and max values for ...
mekal:kT>9 0.01
e e e e+20
mekal:nH>
mekal:Abundanc>
mekal:redshift>0.25
1 switch
mekal:switch>
e e+24
mekal:norm>
=================================================
Model mekal<1> Source No.: 1 Active/On
Model Model Component Parameter Unit Value
par comp
mekal kT keV /- 0.0
mekal nH cm frozen
mekal Abundanc /- 0.0
mekal redshift frozen
mekal switch frozen
mekal norm /- 0.0
_______________________________________________________
Chi-Squared = e+08 using 750 PHA bins.
Reduced chi-squared = for degrees of freedom
Null hypothesis probability = e+00

24. XSPEC12>fit
Chi-Squared Lvl Par #
=================================================
Variances and Principal Axes
2.95E-09 |
1.44E-02 |
4.44E-04 |
Covariance Matrix
1.443e e e-06
-2.200e e e-06
-6.073e e e-09
Model mekal<1> Source No.: 1 Active/On
Model Model Component Parameter Unit Value
par comp
mekal kT keV /
mekal nH cm frozen
mekal Abundanc / E-02
mekal redshift frozen
mekal switch frozen
mekal norm E-02 +/-9.144E-05
_______________________________________________________
Chi-Squared = using 750 PHA bins.
Reduced chi-squared = for degrees of freedom
Null hypothesis probability = e-02

25. Searching χ2-space
The parameters of the model θi are varied and the combination that gives the lowest χ2 is the “best fit”.
XSPEC uses a ‘local’ search algorithm to find the minimum of the space defined by χ2(θ).
Can get lost in local minima.
WARNING: Start in a sensible place and experiment with different initial positions.

26. Searching χ2-space
The parameters of the model θi are varied and the combination that gives the lowest χ2 is the “best fit”.
XSPEC uses a ‘local’ search algorithm to find the minimum of the space defined by χ2(θ).
Can get lost in local minima.
WARNING: Start in a sensible place and experiment with different initial positions.

27. χ2 minimisation issues χ2(θ) θ
The problem: find min[χ2(θ)] with as few iterations as possible. (Imagine searching 10-dimensional parameter space!)
1) Calculate χ2 at present position.
2) Calculate gradient χ2 based on finite differences i.e.
3) Adjust parameter θi θi+1 in direction of lower χ2

28. χ2 minimisation issues χ2(θ) θ -∆θ +∆θ Parameter value θ
Make sure you set parameter step sizes to sensible values!
newpar 1
par1>
Parameter value θ
Parameter step size ∆θ
Hard & soft minimum
Soft & hard maximum

29. χ2 minimisation issues χ2(θ) θ Watch out for local minima!
Try fitting same model from different initial positions
Perturb the current fit using the “error” command
(This last options forces the parameter “uphill” in χ2 which may push it over and into a better maximum”)

30. Why is XSPEC so slow?
At each iteration XSPEC calculates the gradient of the fit statistic χ2 using:
which requires 2P evaluations of χ2
each of which requires a model calculation S(E) followed by a matrix multiplication:
So with a response matrix R and a 6 parameter model that means >107 calculations per iteration (assuming the model is trivial)!

31. Making plots (using IP)
ip (interactive plotter, uses QDP)
la x <x-axis label>
la y <y-axis label>
la f <filename>
la t <title>
time <on/off>
font <normal/roman/script>
cs <font size, e.g. 1.2>
lw <line width, e.g. 3>
plot
hardcopy /ps, (or /gif etc…)
exit
>

32. Multiple spectra data 1:1 spec1.pha resp 1:1 resp1.rmf
XSPEC can fit several spectra simultaneously with the same model. This might be useful if you have, for example, simultaneous observations with Chandra and XMM and you want to use BOTH spectra to constrain the model.
data 1:1 spec1.pha
resp 1:1 resp1.rmf
data 2:2 spec2.pha
resp 2:2 resp2.rmf

33. Simulating data > fakeit none > fakeit bkg.pha
You may need to simulate (“fake”) some data, to see how a given model would look during an observation. First define the model and, for a no-background simulation, use:
> fakeit none
Then enter the response file name(s), the exposure time (in sec) and the output filename.
If you want to include a background spectrum (e.g. bkg.pha) in your simulation:
> fakeit bkg.pha

34. Miscellaneous commands
save files <filename.xcm>
save model <filename.xcm>
save all <filename.xcm>
delcomp <no.> (delete model comp.)
freeze <no.> (stop a param. varying)
thaw <no.> (un-freeze param.)
newpar <no.> (modify param.)
flux <Elo> <Ehi> (flux in E band)
cosmo <H0 , q0 , Λ0> (set cosmology)
lum <Elo> <Ehi> <reshift>
setplot <en, wave, chan, …>
plot <d,ld,del,chi,res,uf,m,…>
exit (quit XSPEC)

35. γ-ray Burst: GRB 031203 model wabs*pow fit
χ2=160.1 for ν=174 (Prej=29%)

36. Seyfert galaxy: Mrk 79 model wabs*(pow+bb+gaus)
freeze 7 (stop line width varying)
fit
χ2=492.0 for ν=485 (Prej=68%)

37. Do try this at home… A worked example showing exactly how to:
Load data
Set-up a model
Adjust the parameters
Get a good fit
Make plots
Get confidence regions
Examine contour plots