From a theoretical perspective you can use the NIST values but you would need to use an appropriate interpolation function to determine the shape of the absorption coefficient between tabular points. Your equation above should formally be an integral equation so as you have written it, there must be an implicit dE on your discretization. Actually, now that I look at it, I am not sure what you mean by F1(E1), F2(E2), ... Perhaps you could define the nomenclature more clearly.

In any case, from an experimental perspective this is a very difficult problem if the form of the x-ray spectrum is unknown, and particularly when the spectrum is not smooth (e.g., significant line radiation). In this latter case it is also a poorly posed problem mathematically. As well, the energy dependence of the detector used to record x-ray signals can be a significant part of the problem and it will appear in any equation used to determine the spectrum.

Are you interested in the mathematical formulation of the problem, an method to employ experimentally, or both?

Dear Mcnaney, Thank you so much for this answer. Actually I used the Laplace Inverse Transform method (Archer, Benjamin R., and Louis K. Wagner. "A Laplace transform pair model for spectral reconstruction." Medical physics 9.6 (1982): 844-847.) to get an x-ray spectrum from the Transmission Data. To begin with I chose a case with no characteristic lines on the spectrum. I used the NIST values for the mass attenuation coefficients. After getting the result I used it to calculate the Transmission Data backward just to check how accurate the spectrum I got. My result for the calculated Transmission Data does not match with the measured Transmission Data. So I came up with this question.

You are absolutely right! The set of equations in my problem should be formally expressed as an integral equation. 

The F1(E1) mentioned in my question represents the number (relative) of photons in the spectrum on the energy interval between E1 and E1+dE. While using the spectrum to calculate the Transmission Data I chose an equal energy bin width of dE.

Thank you for pointing the energy dependence of the detector used to record x-ray signals. I will further study about this.

P.S. I am interested in both the mathematical formulation of the problem and a method to employ experimentally.

While I am not familiar with the paper you are referencing, there are a  few comments worth making.

Thank you again. I am not quite clear about the spectral sensitivity (or energy dependence) of the detector. Is there any technique to quantify such sensitivities through the experiments like Half Value Layer measurement or Transmission Data measurement or different kinds of experiment?

Unless you built your own detector, there may already be information available. For example, most solid state detector materials, common scintillators and many x-ray films have been looked at in the past. 

Another issue: I am not familiar with this particular table, but some of the x-ray data tabulated at NIST comes from calculations of not too high quality (relativistic Hartree-Fock or something like this), which may easily include errors of a few percent. So you should definitely do an error analysis to be sure that your solution is reasonably robust.

Well - to me this approach seems to be at least ambitious. Without any knowledge of either the spectral density function of the primary x ray source, the detector response or the composition of the irradiated specimen the solution would be ambigeous. How you would for instance disentangle intensity coming from a higher concentration or layer thickness of element A from a higher intensity coming simply from a higher primary x ray intensity or from a higher detector response in this energy regime ? You would need a detailed knowlede of your x ray machine and finally also need to calculate non linear effects, i.e. secondary excitation etc.

Estimating the x-ray spectrum from the response of various detectors behind different filters was discussed quite a bit in the literature, for relatively higher-energy photons between say 100 keV and 1 MeV.  The paper I know best is "Differential absorption spectrometer for pulsed bremsstrahlung: RSI v.64, p.1835 (1993).", which you can download from www.ecopulse.com. It uses a brute-force iteration method to get to one (of possibly many) satisfactory responses. Subsequent papers by others deal with the mathematics of the problem (which is indeed an ill-posed matrix inversion problem) go through the math more extensively. One by Dave Fehl comes to mind, but there are others.

To get the attenuation you need it may well be necessary to do a Monte Carlo x-ray transmission computation, with codes such as MCNP, ITS, NRCEGS, Penelope, and others if your x-rays are in the MeV range: for softer x-rays below a few 100 keV this may not be needed, but, it's always good to check whether multiple scattering could be important.

Finally, you have to think about whether you really want to to know the spectrum. To compute the response in some specific detector it may not matter much which spectrum you deduct from your filtered measurements; any uncertainty could well disappear when you do the integration needed for the response.

You might also want to have a look at a database called XRAYLIB. It contains a lot of data and it has bindings to most common programming languages.

You can try using FFAST tabulated data (http://www.nist.gov/pml/data/ffast/) of x-ray mass attenuation coefficients, which is the most accurate tabulation (I found) compared with experiment. Experimental geometry is an issue, where x-rays are attenuated by background matters (in between the monitor and the detector), detector gas and detector windows, that can be modelled to remove their effects; the efficiency of a detector can be estimated using attenuation data measured using that particular detector. Of course, theoretical data of x-ray mass attenuation coefficients do not provide information about the oscillatory part (XAFS) around an absorption edge; it just provides the background comparable for regions away from the structural parts!