Measurement Techniques, VoL 38, No. 12, 1995

PHYSICOCI-IEMICAL MEASUREMENTS

BACKGROUND-INVARIANT DETECTION AND MEASUREMENT OF ANALYTICAL SIGNALS O F X-RAY S P E C T R O M E T E R S

G. M. Bashin, A. N. Dmitrienko, and A. T. Savichev

UDC 543.42

Statistical tests for the detection and determination of background-invariant trace elements have been synthesized for Poisson x-ray fluxes in spectrum-analyzer channels. Cases of total and partial a priori indeterminacy of the intensities of x-ray fluxes were considered. The tests were used on an energy-dispersion spectrometer to determine lanthanum aqueous solutions.

The growing complexity of objects analyzed and the volume of data from x-ray spectrometry have stimulated interest in digital methods of processing analytical signals. These methods make it possible to consider the analysis of a given chemical composition as a combination of operations on an analytical system [1]. The use of x-ray spectrometric instruments to determine small contents (traces) of chemical elements is usually limited by the loss of the visual standard of analytical lines in the spectrum of characteristics, while the complex structure of the spectrum, because of matrix and instrumental effects, substantially limits the possibilities of isolating analytical signals and compensating for the random background of the observations [2, 3]. Since the informative parameters (i.e., the intensities) of weak analytical signals and the background are unknown in an experiment in a system consisting of the object under study and the x-ray spectrometer, the problems of detection (qualitative analysis) and measurement (quantitative analysis) of traces of chemical elements must be solved in the course of statistical interpretation of the x-ray spectra. The latter are usually represented by positive integral values of the states of x-ray counters with a Poisson distribution [3], the intensity of which form the unknown vector parameter of the problem. Qualitative and quantitative x-ray analyses are reduced to the detection of changes in analytical signals (spectral components) and measurement of the extent of those changes. Using those general methods of the theory of verification of statistical hypothesis under the conditions of the a priori indeterminacy of the parameters of Poission distributions [4, 5], we obtained statistical tests for the detection of traces of chemical elements that cannot be measured directly but remain constant under unknown and variable measuring conditions. Those tests presume the existence of a training sample (x-ray spectra that do not contain the chemical elements being determined), use no a priori information about the models of the characteristic x-ray spectra, and ensure that the checked probability of false detection of traces of chemical elements is independent of the intensity of the background radiation. The matrices of the object being analyzed and of the training object are similar [6], as a result o f which the intensity of the background radiation is the same for both objects at a point of the x-ray spectrum that corresponds to the line of the element being analyzed. This holds with sufficient practical accuracy for x-ray spectrometry, e.g., for aqueous solutions of chemical elements. Our purpose here is to synthesize background-invariant tests for the detection and measurement of traces of chemical elements under the conditions of total and partial a priori indeterminacy of the intensities of Poisson x-ray fluxes, when the lower (nonzero) limit and finite upper limit of the average intensity of those fluxes are known.

Translated from Izmeritel'naya Tekhnika, No. 12, pp. 47-49, December, 1995. 1388

0543-1972/95/3812-1388512.50 ~

Plenum Publishing Corporation

c?.,, E~J), ppm /t7080

..... / J

60 z~7 2O 0

-

0

~

.....

I

2O

4~0

60

mtf~]

Fig. 1. Calibration curve C(mt[t]) of the concentration C of an aqueous lanthanum solution as a function of the mean value ml[t] of the statistic t. The amplitudes of the spectral components being analyzed are assumed to be given by the states of the counters of the x-ray photons detected in the channels of the x-ray spectrometer, and the intensities of those Poisson x-ray fluxes are determined by the unknown content of the chemical element. Let us consider two successive time intervals T and To, during which Poisson x-ray fluxes of the object under study and the "dummy" object (containing none of the chemical element to be determined) are observed with the distribution densities in one spectrometer channel, corresponding to the characteristic line of the chemical element being determined; those fluxes are

m(k/X)= (xr)* ~ exp(--XT"), ~o(kolXo)=

(xor.) k~

/lot

exp(--~oTo),

(1) (2)

where k and/Co are the states of the x-ray counters for the object under study and the "dummy" object, and X and ho are the average intensities of the fluxes. The problem of determining traces of chemical elements reduces to checking the hypothesis H0: h = X0 against the alternative Hi: h _ X0 for Xo' -< X0 < X" and AX' < A X _< AX", where AX = X - - Xo. As the optimization test we choose the Neumarm-Pearson test, which ensures maximum probability B that traces of chemical elements are detected correctly, given an upper bound % for the probability e~ of false detection for Xo' < Xo < Xo". If the parameter X0 is a pr/or/determinate, the optimal test for detection of an analytical signal has the form t~>%,

(3)

and the threshold level co (a positive integer) is determined by

exp(--~oT) t=co ~ ~(X~

<:%'

(4)

from the condition of minimization of its left and right parts. Here and below the parameter c (in this case with different indices) denotes the threshold level and/or its components. Upon satisfaction of conditions (3), the decision is made that an analytical signal has been detected with a probability of

t~-c, it 1389

,r

8'<'c)

too

5O

0

)

O

20

,

i

i

I

40

60

80

.? 100 C,p p m

Fig. 2. Dependence of the relative errors of 8(C) and 8'(C) of lanthanum determination on the concentration C of the solution: 1) 8; 2) 6'; e = 0.2. Since the threshold level in (3) is a positive integer, randomization must be employed to accurately fix the probability of false determination of an analytical signal, i.e,, drawing of a random quantity • which is equal to 0 or 1 and has a probability distribution density pOt) = (1 -o0)8(x - I) + o~(x), where ~o = (c~-oq)/(o~2 - o q ) , (~l and oz2 are the values of right side (4) for i = co and co -- 1, respectively, eq __. o~ ___ o~2, and 8(x) is the delta function. For x = I and x = 0 the threshold level is set equal to c o and co - - 1, respectively. In the case of a priori indeterminacy of the average intensities of the x-ray fluxes of the analyzed and "dummy" objects, synthesize tests for checking hypothesis Ho against the alternative Ht can be synthesized by using an adaptive Bayes approach (with the threshold level set on the basis of the most likely estimate of the parameter Xo), the method of the ratio of maximum likelihood [7] suP[W(l~/MwdkdXo) l!~uP[W(k/X,)wdkofXo) l>c ~,,),o

;%

and a method ( < X > -- < N) > ) < *( < X0 > ) is based on the most likely estimates < ), > and < Xo > of the parameters X and Xo being compared, where ~ < Xo > ) is the threshold function which ensures a fLxed probability of false detection of an analytical signal, and c is the threshold level (not necessarily integral). For all the approaches mentioned, above and an arbitrary parametric a priori indeterminacy (including partial indeterminacy) the optimal tests for the detection of an analytical signal, constructed on minimal sufficient statistics k and ko, has the form [8]: k >_ fl,ko), where the nonlinear functionf(x) is determined by the optimization test and the degree of a priori indeterminacy, and under the conditions of total a priori indeterminacy the statistics k and/q are proportional to the most likely estimates of X and X0. The adaptive Bayes approach leads to the following test for the detection of an analytical signal

(5)

k>:c'o +c.,

where the integral threshold level includes a constant component co' and the component c0, which is determined in the interval T from (4) upon substitution of ~ by its most likely estimate < Xo > = ko/To. Equation (4) becomes

exp(--k~T/To) ~ (k~ i~co

(6)

ii

<%"

For AN << Xo the value of co can be found by using the total time interval Tl = To + T, when the < X0 > previously introduced is replaced by < < X0 > > = (k + ko)/(T + To), and Eq. (6) becomes I390

exp

--

(k+k~176] ~ [(k+ko)'I'/Txl z "~t ] t=c~ i! 4a~

The constant component co' of the threshold level is introduced because of the errors in the estimation of ho, which affect the control of the probability of false detection of an analytical signal. The test for the maximum likelihood ratio leads to the following test for detection of an analytical signal kin(<~,>T) +/tolnko-- (k +/to)lnf/t +/to) +k-- <~,> F>co, provided that X > Xo and when that condition is not met, it leads to the test kl nk+k01 n/to - (/t +/%) In(k.+.ko) >c, which for XoTo > > t, ~ T > >

1, XT > > 1 and AX/AXo < < 1 becomes k--koT /To>c'o[k+l%(T /To) ] 112 .

(7)

Here

[lilt for biT><7%>; = t < ~ > for k/T~< , and the threshold constant c"0 can be an integer. Comparison of the most likely estimates leads to a test for the detection of an analytical signal

~-ko'r/ro>c~ +co,

(8)

where, as in (9), the threshold level is in the form of adaptive and constant components, co and Co'. For XoTo > > 1, hoT > > 1, and kT > > I, the distributions (1) and (2) are normalized and the test (8) detection of a signal under the conditions of a priori indeterminacy becomes

~-kor:ro>4+% C ~

,

(9)

where Co" is the constant of the adaptive component of the threshold level and for total a priori indeterminacy the threshold constant is co' = O. This test is invariant under the displacement parameter and for co = 0 it is also invariant under the scale parameter. For AX/X << 1 in the test (9) it is necessary to replace/co under the radical by ko + (To/1)k while tests (7) and (.9) coincide for hoTo > > 1, hoT > > 1, and XT > > 1. Tests (5) and (7)-(9) are equally effective for XoTo > > 1, hoT > > I, and XT > > 1. For the limitations previously set for ~ and AX, the statistics ko and k are replaced by k'0To for k0~"0To

'%=

and k'= h;T/To+

I AT~'T for lt--koT/To<.AVT h--koT~Te for AL'T:A~"T,

which are proportional to the estimates of X0 and zXk under the conditions of partial a priori indeterminacy. The latter can be displaced, primarily at the limits of determination of X~ and AX. 1391

It can be easily shown that for ~oTo > > 1, koT > > 1, and XT > > 1 the analytical threshold signal is

,~=A~,T/(7,oT)V~=tl) -I (~t)+q) -I (~)

(io)

under the conditions of total a priori determinacy and v=(i+rlTo)v 2 [o -~ (~)+o -~ (13)I

(tl)

under the conditions of total a priori indeterminacy. Here @-l(x) is the function inverse to the Laplace integral |

-tn ~ Jr

exp(--~/2)dt.

It follows from (10) and (II) that for XoTo > > 1, ~0T > > 1, and XT > > i that the relative losses in the analytical threshold signal under partial apriori indeterminacy relative to total apriori indeterminacy range from 0 to 10 log(l + T/T0), where the lower and upper limits correspond to the cases of total a priori determinacy and indeterminacy. These losses are negligible when there is a substantial training sample (the spectrum of a "dummy" object, obtained with a considerable exposure time). In the case ~To--, oo, koT ~ oo, and kT --- oo with fixed ~o' and ho", which is of practical interest, inclusion of a priori data does not result is more efficient detection of an analytical signal in relation to the case of total a priori determinacy, since the ratio of the mean square deviation of the statistics k/T--ko/To to the size ~o' -- ko" of the region indeterminacy tends to zero. The detection of an analytical signal can be made more efficient only when that ratio _> 1. We note that in the case of the alternative hypothesis Ht: X ;~ ko, which corresponds to the detection and measurement of the degree of difference of content of chemical elements in the objects studies, the tests obtained remain valid for comparison of the determinant statistics with two threshold levels, the lower of which has the same structure as the upper level, as given above. The determinant statistics obtained were used to determine lanthanum in aqueous solutions. The lanthanum determination was carried out on a TEFA-III energy-dispersion spectrometer (manufactured by EGG ORTEC) with a lowtemperature semiconductor Si(Li) x-ray detector having a resolution of 165 eV for the Mn (5.9 keV) line. The 1024-channel pulse-height analyzer incorporated into the spectrometer has a limiting resolution of 10 eV/channel. Solutions of the concentrations analyzed were in open cylindrical polyethylene vessels with a 32-ram diameter and a bottom of 4-/xm-thick polystyrene film. The volume of material studied was 2-3 cm3. The samples were excited with an 241Am source (activity 3.7-10 l~ see - I , quantum energy 59 keV) with the analyzed and "dummy" samples irradiated for T = To = 50 see with a specmma analyzer resolution of 80 eV/chararet. The maximum of the lanthanum K~ line corresponded to channel No. 419. Figure 1 shows linear smoothing (by the method of least squares) of the calibration curve for T/To = t (i.e., for the same exposure time for the analyzed and "dummy" samples) for aqueous lanthanum solution at a concentration C plotted against the mean value m~ It] of the statistic t = k -- ko. Similar graphs for the statistic t' = k' - - k'o agree with the one given. In the experiment we assumed that A~,'--.~Akrn~nq-_--(Akmax--Ak,.nla), A~,"=Ahr~ax--~(A~,rnax--A~,rnln), where 0 <__e <__0.5 is a dimensionless limiting parameter and , ~ and Akm~ are the minimum and maximum values of 500 independent values of the parameter A h = (k - - ko)T for each of the aqueous lanthanum solutions analyzed. The agreement of the calibration curves is explained by the symmetry of the chosen boundaries of the intervals of the partial a priori indeterminacy of the parameters of the x-ray fluxes relative to the mean values of their Poisson distributions, normalized for ~ , X >_ 10. The graphs of the relative errors #(C) = a(C) and a'(C) = a'(C) of the lanthanum determination from the statistics t and t' with variances oa and (a') z with a limitation parameter ~ = 0.2 are given in Fig. 2. Depending on the concentration of the solution, therefore, the accuracy of lanthanum determination when the statistic t' is used can be increased, as was to be expected. This is attributed to inclusion of the a priori information about possible values of the parameters of Poisson x-ray

1392

fluxes. The accuracy of determination is increased substantially for (AX"-AX')/(2XoT) 1/2 < < 1, when the difference of the limiting intensities of those fluxes is comparable to the mean-square deviation of the statistic t. The graphs of the relative errors of lanthanum determination (Fig. 2) can be used to choose the degree of limitation on the observations in order to increase the accuracy of the methods considered for measuring analytical signals. The accuracy of determination is also enhanced by optimal unification of the lines of the x-ray spectrum. In conclusion, we point out that the simple choice of degree of limitation of observations, as considered above, is effective only for sufficiently long exposure, when the Poisson distribution of the intensity of the spectral lines has become well normalized about the mean. When the exposure is for a shorter time, the growing asymmetry of that distribution requires an optimal choice of the boundaries of the limitation, which is a problem in itself. The experimental investigations were carried out with the financial support of the Russian Fund for Fundamental Research, Grant No. 94--03--08756. The research was also made possible in part by Grant No. N5V000 from the International Scientific Foundation.

REFERENCES 1.

2. 3. 4. 5. 6o 7. 8.

M. A. Sharaf, D. L. Illm6n, and B. R. Koval'skii, Chemometrics [in Russian], Khimiya, Leningrad (1989). R. Muller, Spectrochemical Analysis by X-Ray Fluorescence, Plenum Press, New York (1972). R. Waldset, Applied X-ray Spectrometry [Russian translation], AtomizdaL, Moscow (1977). G. M. Bashin, A. T. Savichev, and M. S. Khots, Zh. Anal. Khim., 48, No. 5, 827 (1993). G. M. Bashin and A. T. Savichev, Zh. Anal. Khim., 48, No. 11, 26 (1993). V. P. Afonin et al., Zh. Anal. Khim., 37, No. 7, 1239 (1982). M. G. Kendall and A. Stuart, Statistical Inference and Relations [Russian translation], Nauka, Moscow (1973). A. N. Dmitrienko, Radiotekh. t~lektron., 31, No. 8, 1541 (1986).

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