chemical shifts from ”F NMR spectra of the hexafluoroacetone derivatized olefins. The proton decoupled spectra are obtained using the acetone-& solvent as a lock signal and are referenced to the singlet of internal hexafluorobenzene. Values of AvAB for the A3B3pattern may be obtained directly from the spectral width, eliminating the need for computer analysis and simulation of the spectra. Because of the employment of I9F,the NMR technique is sensitive only to the derivative of the olefin and the derivatives need not be purified as done in this study before being subjected to NMR analysis. Mixtures of geometric isomers and isomers of different bond positions could be determined by this technique coupled with GC/MS. T h e HFA derivatization technique has been shown to be useful for linear olefins and, more recently, we have also applied i t t o long chain (>C16)linear fatty acids. Work is currently in progress to expand the application of this analysis t o cyclic olefins and steroids.
LITERATURE CITED (1) C. J. W. Brooks and J. Watson, J. Chem. Soc., Chem. Commun., 952
(1967). (2) C. J. W. Brooks and I. Maclean. J . Chromatogr. Sci., 9, 18 (1971). (3) W. Blurn and W. J. Richter, Helv. Chim. Acta, 57, 1744 (1974). (4) R. E.Wolff, G. Woiff, and J. A. McCloskey, Tetrahedron,22, 3093 (1966). (5) J. A. McCloskey and M. J. McClelland. J . A m . Chem. Soc., 87, 5090 (1965). (6) B. M. Johnson and J. W. Taylor, Org Mass Spectrom., 7, 259 (1973). (7) B. M. Johnson and J. W. Taylor, Anal. Chem., 44, 1438 (1972). (8) M. V. Buchanan. D. F. Hilienbrand and J. W. Taylor, in preparation for submission to J . Magn. Reson. (9) A . A . Bothner-By and S. M. Castellano. in “Computer Programs for Chemistrv”. Vol. I. D. F. Detar. Ed., W. A. Beniamin, Reading, Mass., 1968, p p 10-53. (10) L. Petrakls and C H. Sederholm, J . Chem. Phys.. 35, 1243 (1961). (11) S. Ng and C. H. Sederholm, J . Chem. Phys., 40, 2090 (1964).
RECEIVED for review July 12,, 1977. Accepted September 15, 1977. This research has been supported by the National Science Foundation under grants MPS-74-23569 a n d MPS-75-21059.
Deconvolution and Background Subtraction by Least-Squares Fitting with Prefiltering of Spectra Peter J. Statham’ Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, California 947:?0
Deconvoiution of overlapped peaks in a spectrum Is complicated by the presence of a high background component. A method of background subtraction, whlch involves suppressing the background in both the specimen data and peak modeis wtth a digltai filter before proceeding to a conventlonal least-squares fit, Is analyzed. The dlmensions of a “top-hat’’ filter are found which give a suitable compromlse with regard to statistical accuracy and sensitivity to both background curvature and possible errors in the peak models. The major advantages over conventional techniques are that the shape of the background need not be known explicitly, there Is no need to find suitable points away from peaks for background scaling, and any background that is approximately linear over the range covered by a single peak will be effectively removed.
This paper addresses itself primarily to the problem of finding peak areas in a digitized x-ray energy spectrum obtained with a solid-state detector, although the problem is essentially one of deconvolving overlapped peaks in the presence of a high background component. In the usual linear least-squares procedure, the sum of a number of functions is fitted to the specimen data to find the contribution from each peak. Each function must exactly model the peak, or series of peaks (for example, the Kcu and K(3 peaks for a given chemical element) it is meant to represent, and the procedure is only applicable when the specimen data can be represented by a linear sum of the model functions plus statistical “noise”. In the special case of x-ray spectra excited by electron bombardment of a flat polished specimen, the background Present address, Link Systems, Halifax Road, High Wycomhe, Bucks, England HP12 3SE.
shape can be predicted quite accurately ( I , 2) and the background can therefore be subtracted away provided that suitable points, away from peaks, can be found for scaling, or if the shape function is calculated at every point, the background can be included as an additional function in the fitting procedure. However, when electron-beam-excited specimens are not perfectly flat (for example, particles), or if we are dealing with x-ray fluorescence or y-ray spectra, the background shape is not so predictable and one has to exploit any features which distinguish background from peaks. In this respect, the most obvious feature is the slow variation of the background with energy compared with the faster variation of structure in the peaks. This suggests methods such as interpolating the background beneath the peaks using smooth analytical functions but this may be inaccurate when carried out over the large range required when several peaks overlap (3). T o get around this problem, coefficients for a polynomial function can be included as undetermined parameters in the least-squares analysis, thus removing the need for an explicit fit to available background points. However, the function used must accurately represent the background over the whole fitting range and while regions of high curvature, such as absorption steps, can be accommodated by including higher terms in t h e polynomial, this may make t h e technique unstable with regard to small errors in the peak model functions. An earlier study (3) discussed further methods of background correction where the background is implicitly taken into account in the peak deconvolution process. In the “iterative stripping” approach (3),peaks are removed in stages using a symmetric, zero-area weighting function a t each stage to estimate the area of each peak above local background. This method is quick and very easy to program but is limited to resolving peaks which are greater than 0.66 fwhm (full width half maximum) apart. T h e iterative process demands a ANALYTICAL CHEMISTRY, VOL. 49, NO. ‘14, DECEMBER 1977
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solution which, apart from “noise”, has a smooth residual and can in fact be shown to be closely equivalent to a special case of a least-squares approach ( 4 ) . Thus, a development to overcome the resolution limit involves preconditioning both the measured data and the peak model functions with a digital filter before proceeding to a conventional least-squares fit. Digital filtering is a linear operation so if the background is completely suppressed, peak structure in the specimen data can be represented by a linear sum of the filtered model functions and the least-squares approach is justified. Brouwer and Jansen (5) suggested fitting the first derivative of a Gaussian function to the first derivative of the specimen data, but this only removes the influence of a constant background component. A background which is linear over the range of each peak can be suppressed by convolution with a “top-hat’’ weighting function since this produces a result similar to the negative of the smoothed second derivative; Schamber (6) has used this form of digital filter and demonstrated its successful application in an analysis system for x-ray and y-ray spectra. Since the basic technique is described clearly in ( 6 ) , the present paper will focus on details of the theory and will examine the effects of using different digital filters prior to a least-squares analysis.
DETAILS OF THE METHOD If yi represents the count accumulated in channel i of the histogram representing the x-ray spectrum from the specimen and h, represents a weighting function then, after filtering,
[ Y ’ - P‘TC]TW’[Y’ - P’TC]
(5)
where “T” denotes the transpose of a matrix, results in the normal equations
[P’W‘P’TIC= P’W’Y’
(6)
which can be solved for C. This procedure results in estimates, c,, which have the minimum variance (7) and is therefore the “correct” solution to the present problem. However, from a practical point-of-view, the determination and subsequent inversion of the matrix S’ is not to be taken lightly, since n may typically exceed 100. As a practical compromise, W{ can be set to l / & (Equation 3) and 2 is minimized. Differentiating Equation 4 with respect to each c, and equating the results to zero to define a minimum results in the normal equations in matrix form as
AC=B
(7 1
where ‘=I2
amn
=
,
,
~m,i~n,rWi’ 1=1,
bm
i= i,
=
CPG,~Y,!W: i= i ,
and
with the solution s indicates the extent of non-zero values of h, so that W =
2s + 1 is the “support” of the filter and it is assumed that the integer subscripts denote contiguous channels separated by equal increments of energy. If P,,~ represents the count in channel i that would correspond to the mth peak model, then the model for the filtered peak is
Each count y , is distributed statistically according to a Poisson distribution and for practical purposes the variance can be approximated by yi. Since y,’is a linear sum of independent random variables, the variance is
(3) However, whereas the statistical contribution is uncorrelated between different channels in the original data set yl, the covariance, ci,?,between y; and y,! is non-zero whenever li - jl < 2s + 1 because the filtering operation introduces a channel-to-channel correlation in the statistical noise. In the least-squares procedure, suitable coefficients c, would be chosen to minimize EZ =
i= i ,
m =NP
i = I,
m= 1
e, ( y ; -
c
(8)
ERROR ANALYSIS T h e error analysis is complicated by the fact that the weighting coefficients, W:,depend on the input data so that each c, is not simply a linear sum of independent random variables. The following treatment overlooks the statistical nature of Wi so the implicit assumption is that small changes in the weighting scheme do not alter the results. Although this assumption seems reasonable when Wi varies fairly smoothly over a limited range of values, it is less likely to hold when yi approaches zero in some regions so that W{becomes very large. I t should therefore be understood that although the variance estimates are useful for characterizing t h e method, some caution should be exercised when using them to determine the experimental error in a practical analysis, particularly when the background is very low or statistical noise is great. T h e statistical fluctuations in yi are uncorrelated from channel to channel and if c, is regarded as a linear sum of yi, then the variance in c, (9)
c,p;,i)2w;
(4) where the variance in channel i has been approximated by
where N P is the number of peak model functions, the fitting range extends from channel il to i2 and W [is a weight which in the simplest case would be set to unity. However, when the statistical deviations are correlated from channel-tochannel, as they are for the filtered data y:, minimization of Equation 4 is a n oversimplification. I n this case, if n = i2 il + 1,a n n x n matrix W‘ can be defined as the inverse of the n x n matrix S’ whose ijth element is ud. If matrices Y’ (n X l),C ( N P X 1)and P’ ( N P X n) have elements y:, c, and p,:L respectively, il 5 i 5 i2, 1 5 rn 5 N P , then minimization of 2150
C = A-’B
ANALYTICAL CHEMISTRY, VOL. 49,
NO. 14, DECEMBER 1977
yi. Now,
From Equation 1 we have
a”:+.
1-J
aYi
and from Equations 7 and 8,
so that
1.0
If we let
I
c
then, rearranging the summations and substituting into Equation 9 gives
The evaluation of this formula can be costly in terms of both additional memory storage and computation time when several peaks are involved but it can be used to examine the suitability of various filters, and a useful approximation for practical use is discussed below.
CHOICE OF FILTER Digital filters have been used successfully by a number of investigators (8-10) for peak detection in the presence of background and noise; the data are weighted according to Equation 1 and if the filtered result, yi, exceeds some multiple of CT:, (Equation 3) then it is assumed that a peak is present in the vicinity of channel i. Given the constraint that the filter must completely suppress background, the best filter will be one that allows detection of the smallest peaks while remaining insensitive to statistical fluctuations in regions of the spectrum where there are no peaks. There does not seem to be any closed-form analytical solution for the optimal filter but Hnatowicz (10) has shown t h a t a symmetric, zero-area weighting function, consisting of the sum of a positive Gaussian peak and a negative constant, represents the best of the linear filters so far suggested for detecting Gaussian peaks on a linear background. When the linear filtering is to be followed by a least-squares fit, it is even less obvious how to decide on an optimal filter. Intuitively, one might expect the filter to emphasize spatial frequencies typical of peaks, at the expense of the slow-varying background and rapidly-varying statistical fluctuations. However, the least-squares fitting of an essentially smooth filtered peak profile to the filtered data might also be expected to reduce the sensitivity to statistical fluctuations. With the added complication of a data-dependent weighting scheme, a closed-form solution for the optimum filter seems unattainable, even in rather specialized cases. Furthermore, there are practical considerations to optimize in addition to statistical accuracy so the approach I have taken is one of numerical simulation. Perhaps the simplest of t h e background-suppressing filters is the “top-hat’’ defined by the relationship
y;=
1
2M
j = i+M+N
j=i+M
+ 1j-i-M
j=i-M-N
which corresponds to the definition, 1 .L
for I i l < M 2M+1 1 h.=-forM< I i I < M + N I 2 # N hi = 0 otherwise h. =
j = i+M+ 1
1-
Figure I. Statistical variance for fitted peak when P I B = 1000. Specimen data consist of peak, fwhm -- 7.5, height = 10000 counts on uniform background of 10 counts/channel and is fitted with a single peak, fwhm = 7.5, height = 10 000 counts, thus giving a mean result is the variance when background is zero and prefiltering is of 1. not used
Since this has already been used with success ( 3 , 6 , 9 ) and its behavior is similar to that of the Gaussian-plus-negativeconstant filter (9), the simulations have been designed to find the optimum parameters for a “top-hat’’ filter. T o facilitate discussion, it should be noted that the “top-hat” filter essentially corresponds to taking the average over U = 2M + 1 channels about the channel of interest and subtracting the mean of averages over N channels on either side of this central region; the total “support” of the filter is thus W = 2M + 1 2N. The filter produces a result similar to the negative of the smoothed second derivative of the data so a single peak is transformed into a central positive lobe with negative lobes on either side and a background which is linear is completely suppressed. The effects of using a particular fdter depend to some extent on the form of the specimen data. Therefore, the simulations have been designed to represent extremes which encompass the range of situations which would normally be encountered. The specimen data directly affect the weighting coefficients, Wi, so if these are all set to unity, one can regard the simulation as either not favoring any particular form of the data or else corresponding to a situation where the background is much larger than the peaks. Statistical Accuracy. The statistical accuracy of each fitted peak height can be computed using Equation 14, and Figure 1 shows how the variance depends on filter dimensions when the specimen data consist of a single peak superimposed on a low background; for a given support, W , it is apparent that U should be as small as possible. When the peak-tobackground ratio is very low, the variance in the fitted result directly determines the detection limit for a small peak; Figure 2 shows that for a given support, a large U for the central average is favored in contrast to the situation in Figure 1where the background is almost negligible. In either situation, it is apparent that increasing W much beyond 17 produces only a marginal reduction in the variance and this figure varies in direct proportion t o the fwhm of the peak, which in all the present examples is 7.5 channels. It is interesting to see how the statistical accuracy compares with that of the ideal case where the background level is known exactly and is subtracted from the data before proceeding to a conventional least-squares fit. Although, when the background is very low (Figure l),the variance with prefiltering can be only marginally greater than the theoretical minimum, ua, the variance for a peak on a high background (Figure 2 ) is still about a factor of two greater than ug even with a broad filter support. However, in practice the background is not known exactly and to u?j we must add a term which is roughly inversely proportional to the number of peak-free points that are available for estimating the background. Thus, for ex-
+
ANALYTICAL CHEMISTRY, VOL. 49, NO. 14, DECEMBER 1977
* 2151
w 0.c
1
0
20
0
30
I
U
Figure 2. Statistical variance for fitted peak when P I B = 0. Specimen data: a uniform background of 10 000 counts/channel, is fitted with a peak, fwhm = 7.5, height = 10000 counts, so the mean result is zero. ug is the theoretical limit in accuracy, obtained by fitting a peak to t h e background-subtracted data, assuming the background is known exactly
ample, if the average of 5 channels is used to determine the background which is to be subtracted, then the variance for a conventional least-squares fit in Figure 2 is 5.8 X which is bettered by the prefiltering technique provided W is greater than 17. Sensitivity t o C u r v a t u r e . Usually, the background can be regarded as linear only over a restricted range and any curvature will not be suppressed by the filter. In the case of x-ray energy spectra produced by electron-excitation of a thick, flat sample, severe departures from linearity occur near absorption steps and for energies below 2 keV where x-ray absorption is great. I t is difficult to make any general predictions about curvature a t low energies because so many experimental factors are involved, but absorption steps can be modeled quite well by a step function,
S, = H
Si= 0
for i < i, f o r i > i,
provided i t is convolved with a unit area Gaussian with the same fwhm as the peaks. H thus represents the height of the step which occurs at channel i,. In an x-ray spectrum, a step produced by differential absorption of the continuum occurs near every major peak. Since these steps are not removed by the filter, they are a significant source of error when we are trying to determine whether a trace amount of another element is present in the vicinity of a major element peak. Since the relative positions of the step and the suspected peak are dependent on the atomic numbers of the chemical elements involved, simulations were designed to find the maximum error t h a t could occur. A step of height H = 10000 counts was convolved with a unit area Gaussian with fwhm = 7 . 5 channels and the result used as the specimen data. A single Gaussian peak of height 10 000 counts, fwhm 7.5 was fitted to the data for different values of its centroid position using the prefiltering technique with uniform weighting (W: = 1 in Equation 7). The step would normally be superimposed on the sum of a smooth background and a series of peaks or may represent the residual component in the fitting of several large peaks so a uniform weighting scheme seemed a less-biased choice than one which favored a particular form of the data. T h e results for a particular filter ( M = 3, N = 4) are shown for illustration in Figure 3 where it is apparent that the largest errors occur when the centroid of the suspected peak is about 0.5 fwhm away from the step. The maximum absolute value 2152
ANALYTICAL CHEMISTRY, VOL. 49, NO. 14, DECEMBER 1977
a3
3s
23
C e q t r a i 3 :hapre
cf
‘fez pezk
Figure 3. Results of fitting a single peak, fwhm = 7.5 channels, to a smoothed step using prefiltering with M = 3, N = 4 and uniform weighting
-
-s
I
,
i
z 0.3,
I
I
23
9
35
J
Figure 4. Sensitivity to curvature. Specimen data consist of a step function convolved with a Gaussian function, fwhm = 7.5channels and is fitted with a single Gaussian peak, fwhm = 7.5 channels. For variou filters, plots show maximum fitted height that c a n occur for any centroid
position when uniform weighting
IS
used
which occurs for various filter choices is plotted in Figure 4. As the support, W , is increased, the maximum error also increases, a behavior one might expect because a filter with a broad support cannot suppress curvature as well as a narrow one. As W is reduced, the “top-hat’’ filter produces a result closer to the negative second derivative of the data and eventually, the maximum error which can occur amounts to a peak height less than 1/3 of the height of the step. Sensitivity t o Errors in the Peak Model. Systematic errors arise in the least-squares technique whenever the functions chosen to model the peaks do not truly represent the peak shapes constituting the data. In an x-ray energy spectrometer, this can occur because of drift or count-ratedependent effects in the pulse-measuring electronics. Furthermore, incomplete charge collection in the solid-state x-ray detector results in low-energy tailing and a consequent departure from a Gaussian peak model. Many of these problems can be circumvented by frequent calibration of the spectrometer and by using experimentally-measured peak profiles instead of analytical approximations such as the Gaussian function. However, there are statistical limitations to the accuracy of any calibration procedure, and chemical wavelength shifts can cause the peak centroid for a given chemical element in the specimen to differ from that of the corresponding peak model (11). Therefore, when prefiltering is introduced into the least,-squares technique, it is desirable for small errors in the peak models to only have a marginal influence on the results. The fitted result for a solitary peak is fairly insensitive t o small errors in the peak model (for
$
reig?’ f o r c e r t r 3 i a i t I05
31-
Q r
l
M 3 x i r r u r n n e g a t ye
“ted
r e gn*
I .K
1
I
I
I
3# :
1
Cent,:
IC
c cronne
23 3f
flt’ea
pe3k
Figure 5. Fitted peak height as a function of centroid position for second peak when a single peak centred at 100.25 plus a uniform background is fitted with two peaks, fwhm = 7.5, and the first is centered at channel 100. Prefiltering is used with M = 3, N = 4 and two weighting schemes are employed
example, see plots in (6))but this is not the case when multiple peak overlaps are to be resolved. A particularly bad case occurs when a very small peak is overlapped by a large one. The range of systematic errors which can occur for the small peak in effect defines a detection limit as a function of both separation from the large peak and the likely errors in the peak model ( 3 ) . In the present case, simulations were designed to examine the influence of the filter function on the errors which could be produced by a slight error in peak position. The specimen data consisted of a single Gaussian peak, fwhm = 7.5 channels centered a t a position corresponding to 100.25 channels and superimposed on a uniform background. Two peaks with fwhm = 7.5 were fitted to the data, one with centroid a t 100 and the other with centroid a t a series of positions throughout the range of the data. The fitted heights for the second peak thus give a measure of the sensitivity to a relative shift of 0.25 channel in the specimen spectrum. When the second peak is centered a t channel 101, the fitted peak heights relative to that of the specimen peak are close to 0.75 and 0.25 as might be expected from the ratio of distances from each peak centroid to the true peak position. This result is insensitive to changes in both the peak-tobackground ratio in the specimen data and changes in the filter parameters. As the second peak centroid is shifted to higher channel numbers, the fitted height decreases, goes negative, and eventually reaches a negative maximum whose absolute value is never again exceeded for any further shift to higher channels. This behavior is illustrated by Figure 5 for two extremes of the peak-to-background ratio. When the background is high or the general level of the specimen data is roughly uniform because several peaks overlap (W: = 1 in Figure 5 ) then there is relatively little difference between the performance with various filters; although the positions of the extrema are altered by changes in U and W , the magnitude of the maximum negative error varies only from 5.6% to 4.2% as W is increased from 3 to 17 and changes in Cr have a virtually negligible effect. However, with a specimen peak of height 10000 superimposed on a background of 10 counts ( P / B = 1000) differences are apparent as a consequence of the nonuniform weighting. Since, in practice, peak separations of several channels are commonly encountered, I chose to look at the fitted height for the second peak when centered a t
U
Flgure 6. Sensitivity to peak shift. Specimen data consist of a peak, fwhm = 7.5 channels, centered at 100.25 channels, superimposed on a uniform background of 10 counts. Two peaks of fwhm 7.5 are fitted
to the data: the first is centered at 100 but t h e centroid of the second is varied and the fitted peak height is observed
channel 105 together with the maximum negative fitted height in order to evaluate each filter. The results are shown in Figure 6. As U is increased, the central positive lobe of the filtered peak becomes broader so one might expect the effects of residual errors to extend further. This behavior is seen in the plots for the fitted height for a centroid a t channel 105 where, in most cases, the values increase for high values of U . The maximum negative peak height decreases with increasing U presumably because the weights, W:,play a greater role as the negative lobes of the filtered peak are forced outwards. Optimal Filter. In the figures, the greatest variations are seen in the statistical variance for a zero peak-to-background ratio and in the errors due to peak shift but other effects are fairly insensitive to the choice of filter; from the results I suggest that W = 15, U = 7, ( M = 3, N = 4) is a suitable compromise with regard to the various effects. Representative points for this filter are shown with a cross on each graph and Figures 3 and 5 give a detailed account of the errors which can arise from a residual absorption step or peak shift. Since the parameters will depend directly on the peak fwhm, suitable values in general would be integers which are close to satisfying U = fwhm, W = 2 X (fwhm). The examples chosen are representative of an x-ray energy spectrum recorded a t 20 eV/channel where peak fwhm = 150 eV and 0.25 channel corresponds to a shift of 5 eV. It is obvious to question whether there is a better filter than the “top-hat”. Zero-area functions based on higher derivatives would suppress more curving backgrounds but with a sacrifice in terms of statistical accuracy. Since the shape of the optimal “top-hat” appears to be closely matched to that of the peak, it is quite possible that a slight improvement in statistical accuracy could be obtained by using s positive Gaussian plus negative constant instead. However, it is not clear why any of the other effects should be markedly reduced and one of the strong arguments in favor of the “top-hat’’ is that the convolution does not involve a series of multiplications, and sums can be obtained very rapidly using integer arithmetic in a computer. ANALYTICAL CHEMISTRY, VOL. 49, NO. 14, DECEMBER 1977
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PRACTICAL APPLICATION If the intensity ratios are not significantly altered by differential absorption (3),the model function, P ~ ,can ~ ,consist of the full series of peaks in correct proportion (e.g. K q , Ka2, KP,, etc.) which correspond to ionization of one shell of a particular chemical element. In this way, good results can be obtained for severe overlap situations in x-ray spectra which would be impossible to resolve accurately using a single peak for each model. As emphasized above, it is very important that the peak models be accurate in any least-squares fitting procedure. Although peaks can be modeled with suitable analytical functions, it can be much simpler to use experimentally-obtained peak profiles directly for each peak model (12j. In electron microprobe analysis, this is accomplished by storing reference spectra from standards which contain sufficiently few elements so that the series of peaks for an element of interest can be isolated from the rest of the spectrum. There is an added benefit because absorption steps, while they are not exactly proportional to peak heights, do depend on them so the influence of steps will be canceled out to a large extent when stored standard spectra are used for models. Despite the many possible causes of error, it is still desirable to have some estimate of the statistical error in each result but the evaluation of Equation 14 can be extravagant in terms of both memory storage and computer time when several peaks are involved. If pnl/is approximated by pnj,W'L,and a: is approximated by y,2hi (Equation 3), then Equation 14 can be reduced (F. H. Schamber, personal communication) to give 2
'ern
N
-1
-amm
'
( 2 M + 1).(2N) ( 2 M + 1 + uv)
which is easy to compute when the inverse matrix A-1 is available. This approximation has been tested for a two-peak deconvolution (fwhm = 7.5) over a range of peak separations from 1 to 24 channels using the filter suggested above ( M = 3, N = 4). The variance estimate was always within a range of +67% to -40% of the correct value and thus appears to be a useful approximation.
DISCUSSION The author's program has been used for electron microprobe analysis a t the Department of Mineralogy and Petrology, Cambridge University, and difficult overlap situations such as t h a t involving the titanium K series and barium L series peaks have been successfully resolved (N. Charnley, personal
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ANALYTICAL CHEMISTRY, VOL. 49, NO. 14, DECEMBER 1977
communication). Furthermore, Schamber (6) summarizes published literature which quotes results of x-ray analyses that have been performed using the prefiltering technique. Since the dimensions of the optimal "top-hat'' fortuitously turn out to be close to those suggested on empirical grounds in (3)and (6), the above evidence can be included as experimental verification of the technique. The prefiltering technique is as effective as subtracting a piece-wise linear approximation of the background from the spectrum and, just as in the iterative-stripping approach, one can expect to see systematic errors of the order of 0.3% concentration by weight in electron microprobe analysis for elements with peaks at low energies where background curvature is severe. Methods based on theoretical prediction are potentially capable of higher accuracy but the form of the continuum at low energies (
