Background function polynomial

Background removal routines typically employ polynomial splines of some order (typically second or third order). These are defined over a series of intervals with the constraint that the function and a stipulated number of derivatives be continuous at the intersection between intervals. In addition, the observed EXAFS oscillations need to be normalized to a single-atom value and this is generally done by normalizing the data to the edge jump. 

The presence of the central spot (the primary beam) and diffuse rings Idiff from the film support brings significant errors into estimated intensities. The shape of the primary beam feam can be approximated by one of several peak-shape functions such as pseudo-Voigt, Gaussian or Lorentzian [16], The diffuse background can be described by a polynomial function of order 12. Then equation (1) becomes... 

Therefore, if the total phase shift is known as a function of the energy, the positions of the resonances can be determined by fitting it with a linear combination of arctangent functions plus a smooth polynomial background. See, for example, Ref. [74] and references therein. 

Alternative background correction schemes can be incorporated for more complicated situations. For example, if the background signal is curved and multiple valleys are available in the spectrum, it may be possible to fit a polynomial function... 

When peak shape functions and their parameters, including Bragg reflection positions, are known precisely and the background is modeled by a polynomial function with j coefficients, the solution of Eq. 6.6 is trivial because all equations are linear with respect to the unknowns (Bj, see Eq. 4.1, and / ). It facilitates the use of a linear least squares algorithm described in section 5.13.1. In practice, it is nearly always necessary to refine both peak shape and lattice parameters in addition to Bj and h to achieve a better precision of the resultant integrated intensities. Thus, a non-linear least squares minimization technique (see next section) is usually employed during full pattern decomposition using Eq. 6.6. 

The result illustrated in Figure 6.7 shows a satisfactory fit and the next groups of parameters included in the refinement were a more complex background (third order polynomial instead of a linear function) and peak asymmetry. The fit further improves and when the calculated and observed... 

The parameters U, V and W are specific to the diffractometer in question. This function was established almost 50 years ago on a neutron diffractometer. It actually has no theoretical background and its expression is essentially empirical, so for some diffractometers, the angular resolution function can stray significantly from this second degree polynomial. However, the expression established by Caglioti and his colleagues is still commonly used. 

With this background, consider how we might achieve the best fit to a set of data points. The usual criterion for a best fit curve is the least squares fit in which the rms deviation between the data points and the fitted curve is minimized. A small enough section of the data must be considered for each segment so that it can be fitted by a polynomial. This will be a tedious process for a complex function like the FID from a multiline spectrum. 

Here, x is the channel location at which the function is evaluated. Hj, is the amplitude computed for the reference line of the element j. The symbols p. and CT represent the centroid and width, calculated at the X-ray energy of the line being fitted. The background points are chosen in selected channel regions by determining the local minimum for each region. The background fit at channel x is calculated (by second or third-order polynomial) as ... 

Excel has built-in capability to generate customized functions using Visual Basic for Applications (VBA). This is a powerful tool that can save time without becoming an expert in programming as it opens the possibilities to run loops and conditionals on the background. This capability also allows the user to build relatively large equations that are used in several areas of the worksheet (e.g., polynomials for the estimation of specific heat of components) and allows the user to read the calculations... 

---