Optimization routines are completely deterministic, in that, for the same set of parameters, used in the same functional forms, applied to the same set of data, with the same initial and final conditions will result in the same set of parameters on termination. Vary any of the above conditions and the result from the optimization routine will change in some respect. One method for assessing the uncertainty in the parameters for a peak model is to vary these optimization conditions by repeating an experiment on, what are hoped to be, identical samples. Then for each set of data apply the same optimization routine to the same synthetic model and so determine a distribution for the set of parameters used to quantify a sample. Such a procedure will vary almost every aspect of the measurement process and so result in a distribution for the parameters that truly represent the nature of the experiment.
The basis for such an approach as described above lies in the assumption that there exists a set of parameters (only known to nature) that does not depend on any optimization routine nor any other errors introduced into the measurement process, and these true values will lie inside the region of the Ndimensional parameter space defined by the set of outcomes to this sequence of experiments. Obviously, if the synthetic model does not describe a set of parameters in tune with nature, the results may be in more doubt than the measured distribution might suggest. However, given that all is well then the task is to offer a means of understanding the uncertainties in the peak parameters within the context of these parameter distributions.
Peak identification in XPS spectra represents a challenge since synthetic models more often than not involve overlapping lineshapes (Figure 1), the consequence of which is correlated optimization parameters. That is to say, if a single data envelope results from two overlapping peaks and if one of these underlying peaks is reduced in intensity then in order to describe the same data envelope the other must increase in intensity. Adjustments to peak parameters of this nature are inherent to any optimization procedure and the choice between the possible combinations of these peak intensities is made based upon a chisquare or rootmeansquare metric. The problem is therefore to identify the point at which these goodnessoffit metrics fail to produce results that can be believed, and provide some means of illustrating the degree of uncertainty.
Given a distribution for each of the parameters, the best way to describe the nature of the uncertainties is to offer an error matrix as well as a tabulated form for the distributions. The error matrix provides numerical values from which the degree of correlation can be assessed while scatter plots taken from some subset of these distributions allows visual inspection for the same information. Ideally a single number for each parameter would provide the simplest means of stating the uncertainties, but as the old adage goes “To every complex problem there is a simple solution that is wrong”, and so it is for correlated parameters. The unfortunate fact is that if the peaks weren’t correlated then synthetic models would be unnecessary.
Repeating an experiment many times is not always practical
and for most XPS spectra peak models are developed based upon a single
acquisition sequence. Estimates for uncertainties in the peak parameters must
be made using assumptions about the noise typical of XPS spectra. The essence
of
The result of a
Where the x_{sik} are parameter values calculated for each of the simulation steps and each distribution is centered with respect to the mean rather than the initial parameter value.
The standard error in each parameter is given by
and the correlation between parameters i and j is given by
.
An alternative method for estimating uncertainties in the
peak parameters is to quote the inverse of the Hessian matrix used in the
Marquardt Levenberg optimization routine. So the
question is, why bother with
Figure 1: Simulated PVC C 1s Data Envelope.
The best way to introduce the ideas behind
Consider the data envelope in Figure 1. The spectrum is a synthetic envelope created from two GL (50) lineshapes without any background and where the peaks are separated by an energy gap consistent with C 1s lines in a PVC spectrum. In the absence of noise and experimental error the optimization routine always returns the peak parameters tabulated on the spectrum in Figure 1. The next step in the simulation is to introduce noise onto the data envelope that is consistent with noise found in experimental XPS spectra, i.e. variations about the channel intensity of magnitude related to the square root of the counts. Figure 2 shows the data envelope from Figure 1 after noise has been added and the peak parameters refitted.
Figure 2: Simulated C 1s data where noise has been introduced to the system.
This procedure yields the first set of simulation results.
If repeated many times, the output is six distributions, one for each of the
peak parameters involved and these can be used to assess the influence of
experimental noise on these quantities. Note that this differs in some respects
from adopting a purely experimentally determined parameter distribution. The
initial stating point for the peak parameters will not be identical for an
experimental data set since the experimental data may be subject to sample
charging, and any errors in the measurement procedure that can not simply be
described by the random nature of the counting system will be omitted. A
Table 1: Error Matrix for the C 1s spectrum in Figure 1.

1:Area 
1:Pos. 
1:FWHM 
2:Area 
2:Pos. 
2:FWHM 
1:Area 
39.230 
0.122 
0.196 
31.512 
0.128 
0.186 
1:Pos. 
0.122 
0.001 
0.001 
0.120 
0.000 
0.001 
1:FWHM 
0.196 
0.001 
0.002 
0.185 
0.001 
0.001 
2:Area 
31.512 
0.120 
0.185 
38.330 
0.125 
0.202 
2:Pos. 
0.128 
0.000 
0.001 
0.125 
0.001 
0.001 
2:FWHM 
0.186 
0.001 
0.001 
0.202 
0.001 
0.002 
Table 1 is the error matrix that results from a
The uncertainty for correlated parameters can be estimated from plots such as those shown in Figure 3 and Figure 4. By projecting horizontal or vertical lines onto the parameter axes in such a way that the appropriate number of points lie between the projection lines for a given confidence limit, the uncertainty for a parameter can be assessed in the context of others. Again, the uncertainty for a parameter should be viewed in the context of all other correlated distributions, and yet the ellipsoid in 6dimensional parameter space is difficult to quantify; the procedure based on scatter plots should be seen as merely a step in the right direction rather than arriving at the ultimate destination. In the case of the peak area shown in both Figure 3 and Figure4, the estimate taken from the diagonal element of the error matrix would seem to be reasonable. Both scatter plots show that about 65% of the points lie inside projection lines positioned at about 0.95 and 1.05. This interval represents about ±5% of 114.2 CPSeV (see Figure 1) and is not too different from the uncertainty taken from the error matrix ±6.2 CSPeV.
Figure 3: Scatter plot showing the anticorrelation between the peak area parameter distributions.
Figure 4: Scatter plot between Area and FWHM parameters for Peak C 1s 1 in Figure 1
All the parameter distributions are reported relative to the initial parameters used to define the data envelope. As a result the relative parameter distributions may not be symmetrical about the initial value and asymmetric confidence intervals are possible. Note that the error matrix is calculated from the distributions centered on the mean parameter value for the distribution, not the initial values.
One of the real advantages of using