Spectral fitting

If we indicate with R(i,j) the generic element of the response matrix, i.e. the probability that an incoming photon with energy comprised in the range of input energy channel j gives a count in the PHA channel i, and F(E) the incident photon flux giving the number of incoming photons for each response matrix input energy range j, the corresponding expected value of the number of counts in the channel i is given by:  

$$C(i)=\sum_{j=1}^{n}{F(j)R(i,j)}$$ (10)

, where n is the number of the response matrix input energy bins and  

$$F(j)=\int_{E_{j-1}}^{E_{j}}{F(E)dE}$$ (11)

, E_j-1 and E_j being the energy bounds of the response matrix input energy channel j.

Conventionally, F(E) is expressed in photons cm^-2 s^-1 keV^-1, R(i,j) in cm² (i.e. in terms of effective area) and C(i) in counts s^-1 channel^-1.
As we have discussed, the response matrix R is not diagonal, and thus it cannot be inverted to directly extrapolate the incident photon spectrum F(E) from the measured spectrum C. In other words, this means that different incident photon spectra F(J) may result in the same value of C(I). The standard technique adopted to analyze spectral data of hard X-ray and gamma-ray experiments is the spectral fitting, which consists in assuming a parametrical photon spectrum model M(E,P1,...,Pn), integrate it in each response matrix input energy bin as in eq. 3.11 to obtain M(j,P1,...,Pn), generate a theoretical count spectrum T(i) by using equation 3.10 replacing F(j) with M(j), compute the value of $\chi^{2}$ as:

$$\chi^{2}=\sum_{i=1}^{m}{\frac{(T(i)-C(i))^2}{\sigma(i)^2}}$$ (12)

, where $\sigma(i)$ is the standard deviation of C(i) and m the number of PHA channels, find the parameter values which minimize the $\chi^{2}$ and test the goodness of the fit by means of the $\chi^{2}$ test.
To perform the $\chi^{2}$ minimization and find the best fit parameters, together with associated errors, for a given model standard software packages are used, like XSPEC ([NASA/HEASARC]), developed at HEASARC, or MINUIT ([James 1994]) developed at CERN. It is important to stress that more than one model may fit well the measured spectrum; the choice between models can be done basing on statistical tools (like the F-Test) and/or on scientific interpretation.
It is important to note that C(i) is not measured directly, because the actually measured spectrum is the sum of the background plus the source spectrum, and thus it is calculated as:

C(i)=M(i)-B(i)

, where M(i) is the measured spectrum and B(i) the background spectrum. The error $\sigma(i)$ is then given by:

$$\sigma(i)=\sqrt{\sigma(i)_{M}^{2}+\sigma(i)_{B}^{2}}=\sqrt{M(i)+B(i)}$$ (14)

, because M(i) and B(i) are Poissonian distributed and thus their standard deviation is equal to their mean value.
Finally, the $\chi^{2}$ statistics can be used to test the model hypothesis only if the number of counts in each channel C(i) is greater than about 20, so that its distribution can be assumed Gaussian. Therefore, in case of weak signals, the PHA channels are grouped in order to have a measured spectrum with at least 20 counts in each channel. test must be adopted, e.g. the C test.