A pattern recognition approach in X-ray fluorescence analysis

Nuclear Instruments and Methods in Physics Research A277 (1989) 619-626 North-Holland, Amsterdam

A PA'ITERN RECOGNITION

APPROACH IN X-RAY FLUORESCENCE

619

ANALYSIS

L o I Y I N a n d J a c o b I. T R O M B K A Code 682, Laboratory for Astronomy and Solar Physics, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA

S t e p h e n M. S E L T Z E R Center for Radiation Research, National Bureau of Standard& Gaithersburg, MD 20899, USA

Received 10 August 1988 and in revised form 7 December 1988

In many applications of X-ray fluorescence (XR.F) analysis, quantitative information on the chemical components of the sample is not of primary concern. Instead, the XRF spectra are used to monitor changes in the composition among samples, or to select and classify samples with similar compositions. We propose in this paper that the use of pattern recognition technique in such applications may be more convenient than traditional quantitative analysis. The pattern recognition technique discussed here involves only one parameter, i.e., the normaliTexlcorrelation coefficient and can be applied directly to raw data. Its computation is simple and fast, and can be easily carried out on a personal computer. The efficacy of this pattern recognition approach is illustrated with the analysis of experimental XRF spectra obtained from geological and alloy samples.

1. Introduction

2. Principles of the pattern recognition technique

X-ray fluorescence (XRF) spectroscopy is a well established technique for qualitative and quantitative clement analysis of materials, in both research and industrial environments. In general, the goal of X R F analysis can be divided into two major categories. In one type of application, X R F is used to obtain quantitative elemental composition of samples. This type of analysis is usually conducted in the laboratory with high-precision instruments having wavelength-dispersive a n d / o r energy-dispersive capabilities as well as interchangeable X-ray anodes and secondary targets. In the other category, the X R F information is used to monitor the composition of samples or to classify samples, either among themselves or against known references. This type of application is often carried out in the field with much less sophisticated instrumentation. An energy-dispersive spectrometer is used almost exclusively in these situations. In this paper we direct our attention to the latter type of application. Specifically, we will discuss the possibility of using pattern recognition in the analysis of X R F spectra to help monitor or classify samples in terms of their chemical compositions. Our preliminary experimental results indicate that the pattern recognition technique is extremely simple, fast, and efficient, which should make it ideally suited for applications in field environments.

The application of pattern rcr,ognition principles to the analysis of energy-dispersive X R F spectra is based upon the fact that materials with similar chemical compositions produce pulse-height spectra of similar shape when excited by incident X-rays with the same spectral distribution. That is, for a given incident X-ray spectral distribution, the shape of a sample's X R F pulse-height spectrum can be considered, to a large extent, as characteristic of the sample's chemical composition. Therefore, in applications where it is more important to know, or to monitor, the degree of similarity among samples in terms of their chemical compositions, rather than the chemical compositions themselves, one may use the X R F spectral shape to characterize the samples. Because shape information is accessible directly from the raw X R F data, it is much easier to obtain such information than to perform the complete inverse process of quantitative elemental analysis. For example, it is well known that a person experienced with energydispersive X R F can often identify or differentiate sample materials simply by visually inspecting the shapes or pattern of their pulse-height spectra. The pattern recognition technique discussed here takes advantage of this phenomenon, but in addition, places it on a quantitative basis. Pattern recognition of the X R F spectra is imple-

0168-9002/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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Lo I Yin et al. / Pattern recognition approach in X-ray fluorescence analysis

mented in the following manner. Each XRF pulse-height spectrum is considered to be an n-dimensional vector in pattern space, where n is the number of channels (2048 in our experiment). Usually, two vectors or patterns are considered similar if their Euclidean distance in pattern space is small. Since the Euclidean distance is the square root of the sum of squares of the componentby-component differences between the two vectors, the Euclidean distance is small when all the components between two vectors are nearly the same in a component-by-component comparison. However, in XRF the spectral pattern of a given sample is unaffected by changes in absolute intensity of the peaks as long as the spectral distribution of the exciting radiation remains constant. For instance, in comparing two spectra from the same sample, one may have a much higher intensity than the other due to a higher incident X-ray flux or a longer integration time. Here, two patterns can still be the same even though the magnitude of their individual components are quite different, and consequently their Euclidean distance in pattern space is large. Clearly, in such cases, a noumetric similarity discriminant function is required. It will be shown that the normalized correlation coefficient [1] is just such a discriminant function. Consider two XRF spectra with different intensities from the same sample. Although the magnitude of the individual components differ, they are nevertheless proportional to each other by a constant factor. This means that in pattern space the two vectors lie along the same direction from the origin. The parameter which can best characterize this directional similarity is cos 0:

cos 0 = a T b / [ ( a r a ) ( b T b ) ] i/2,

(1)

where 0 is the angle between the two n-dimensional column vectors a and b, and where a T and bx are their transposes. The numerator is simply the dot product of vectors a and b, and the dominator is the product of their magnitudes, (aTa) and (brb). Because the numerator is also the correlation coefficient between a and b, and the denominator is the normalization factor, cos 0 is also referred to as the normalized correlation coefficient (NCC). It can be easily seen that the value of cos 0 remains unchanged if we form new vectors a ' and b' such that all the components of the new vectors are proportional to the old components. Moreover, the value of cos 0 will always be close to 1 for two spectra with similar structures, independent of their intensities.

manufactured by the Kevex X-ray Tube Division *, can be powdered by a 9-12 V battery. The end-window X-ray tube has a tungsten anode, a 0.05 mm thick beryllium window, and a focal spot of approximately 0.25 mm × 0.25 ram. The generator can be operated up to 70 kV with anode currents up to 100 ~A. For the present experiment, a liquid-nitrogen cooled Si(Li) detector having a sensitive area of 28 mm2 and a resolution of 173 eV at 5895 eV was used as the energydispersive detector. (It is anticipated that either a Peltier-cooled Si(Li) or an ambient-temperature HgI 2 detector will be used for the XRF spectrometer on the Mars Rover.) The X-ray generator and the Si(Li) detector were arranged close together, facing the sample, with the normal of the sample bisecting the 40 ° angle formed by the X-ray beam and the detector axis. The X-ray source-to-sample distance was 15 cm and the sample-to-detector distance was 8 cm. A standard nuclear spectroscopy amplifier with a shaping time constant of 12 ~ts was used to feed the pulse-height spectra into a 2048-channel multichannel analyzer. All spectra were taken in air. Consequently, elements lighter than K were inaccessible due to air absorption of their fluorescence X-rays. For the Mars Rover simulation [3] we had analyzed some geological samples supplied to us by the US geological survey to study the feasibility of the pattem recognition technique. In order to test the general applicability of the pattern recognition approach, we have analyzed, in addition, a selection of alloy samples provided by the Materials Branch of the Goddard Space Flight Center. The XRF spectra of these two types of samples will be discussed in turn.

3.1. Geological samples

3. Experimental illustrations

The standard rock and soil samples were in the form of pressed pellets consisting of 50% cellulose and 50% sample. Fig. 1 shows the details of a typical XRF spectrum from the geological sample AGV (Andesite). A composite of five spectra is shown in fig. 2 to compare the patterns among the different samples. The five samples are: AGV (Andesite), simulated Martian soil, BIR (Basalt), G-2 (Granite), and GSP (Granodiorite). All the spectra were obtained with the X-ray generator operated at 45 kV, 15 ~tA anode current, and 10 min accumulation time. Common to all spectra are the pair of strong peaks at 6.40 and 7.06 keV corresponding to the K~ and Ka doublet of Fe, and the structures between 8 and 12 keV which are due to the L-series lines of W produced in the W anode and

Our experimental setup was meant to simulate a portable, battery-operated XRF instrument for possible use on the Mars Rover to perform in-situ chemical analysis of the Martian soil [2,3]. The X-ray generator,

* Commercial identification is used for informational purposes only, and does not imply endorsement by any agency of the US Government.

Lo I Yin et aL / Pattern recognition approach in x-ray fluorescence analysis AGV

621

coherently scattered by the sample into the detector. To the left of the Fe peaks (see fig. 1) are the peaks due to Ar K~ (2.96 keV), K K s (3.31), Ca K s and K~ (3.69, 4.01), and Ti K~ and Kt3 (4.51, 4.93). The K~ peaks of Ar and K are not visible because of their overlap with the K~ peak of the next element. Ar was excited from surrounding air, not from the sample. The weak structures above 13 keV are due to Rb, Sr, Y, and

=0 0

Zr.

Energy,

keV

Fig. 1. XRF spectrum of AGV (Andesite). The markers at the top indicate positions of the K,, and KI3 doublets of prominent chemical elements. The W L lines originate from the W anode of the X-ray tube which are then coherently scattered by the sample into the detector. The region of interest chosen for pattern recognition covers the range from 3.130 to 6.097 keV; this includes K, Ca, Ti, Cr, and Mn, but excludes Fe.

0

N 0

% o

%

% ' ' ' ' l l l l l l l l l l l l l l

5

10

15

I

20

Energy, keY Fig. 2. Composite XRF spectra from five geological samples: AGV (Andesite) MARS (simulated Martian soil), BIR (Basalt, G2A (Granite), and GSP (Granodiorite). The spectra of G2A and GSP are found to be quite similar by the pattern recognition technique, with a normalized correlation coefficient of 97.98.

It is immediately clear from fig. 2 that the region with the most structural changes lies below 8 keV, which is also the region containing the most geochemical information. In this region the peaks are fairly distinct with reasonably good statistics. Therefore we shall apply the pattern recognition technique to this region below 8 keV. Note that in a semi-log display a sample's spectral shape can be considered as characteristic of its chemical composition because the shape remains invariant with changes in intensity. In all the spectra, the Fe K , and Kp doublet at 6.40 and 7.06 keV is 20 to 100 times more intense than those of K, Ca and Ti. In addition, the shape of the Fe doublet does not vary from sample to sample. Consequently when they are included in the evaluation of eq. (1), their contribution completely overwhelms those of the other peaks, making cos 0 close to unity for all pairs of samples. Therefore we have excluded the Fe peaks and selected the spectra region of interest to be 3.130-6.097 keV, coveting K, Ca and Ti, with minor contributions from Cr and Mn. This region in the spectrum is equivalent to a 294-component vector in pattern space. We first applied eq. (1) to 16 XRF spectra from the same sample (AGV) under various conditions. All 16 spectra were taken with the X-ray generator operating at 45 kV. Six of the spectra were obtained at different anode currents and accumulation times so that the absolute intensities vary by more than a factor of 4. The other 10 spectra were acquired under identical conditions to assess the statistical spread of the normalized correlation coefficient (NCC). The only data processing performed on the raw data prior to the application of eq. (1) was a crude background subtraction whereby the lowest count in the region of interest (3.130-6.097 keV) was subtracted from all the channels in the region. The results are tabulated in the form of the triangular matrix shown in table 1. For easy readability the values of the NCC are multiplied by 100 to convert them into percentages. Table 1 shows that the correlation is excellent in all cases, independent of intensity, with the minimum value of 97.09. This corresponds to a maximum deviation of 13.9 ° between any pair of the 16 vectors in pattern space. We next calculated the NCC in the region of interest for over 70 XRF spectra obtained from 16 geological samples over a period of several months, using the same high-voltage setting on the X-ray generator. The value

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

100.00 98.90 99.05 98.97 97.90 98.80 98.83 98.47 98.36 98.50 98.83 98.98 99.08 99.15 99.07 99.19

1

100.00 99.07 99.00 98.72 98.97 98.99 99.07 98.92 98.85 98.56 98.88 98.78 99.08 98.99 99.01

2

100.00 99.32 98.74 99.37 99.53 99.10 99.08 88.23 98.25 98.55 98.70 98.78 98.94 98.97

3

100.00 98.66 99.30 99.38 99.17 99.10 99.11 98.06 98.44 98.49 98,79 98.79 98.88

4

100.00 98.81 98.86 99.18 99.06 98.93 97.09 97.81 97.61 98.33 98.16 97.92

5

100.00 99.34 99.24 99.10 99.14 97.99 98.34 98.37 98.74 98.75 98.72

6

100.00 99.26 99.13 99.25 98.06 98.33 98.5 98.66 98.87 98.89

7

100.00 99.35 99.24 97.71 98.24 98.13 98.77 98.63 98.57

8

100.00 99.03 97.45 97.99 97.82 98.58 98.41 98.34

9

100.00 97.76 98.21 98.26 98.60 98.66 98.59

10

100.00 99.29 99.31 99.01 99.16 99.08

11

100.00 99,26 99.29 99.20 99.11

12

100.00 99.14 99.20 99.18

13

100.00 99.27 99.18

14

100.00 99.14

15

100.00

16

Table 1 Normalized correlation coefficients (NCC) for 16 replicate spectra of the sample AGV (Andesite). Spectra are identified by the numbers 1-16. Pairwise correlation coefficients form a triangular matrix. All spectra were obtained with the X-ray generator operating at 45 kV, Spectra numbers 3 through 8 have differing anode currents and accumulation times such that the total counts vary by as much as a factor of 4; the remaining spectra were taken under identical conditions to assess the statistical spread of the correlation coefficients. The coefficients are multiplied by 100 to appear as percentages.

t-,

q~

R-

"

t.o t,o

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Lo I Yin et al. / Pattern recognition approach in x-ray fluorescence analysis

Table 2 Normalized correlation coefficients, in percent, for the spectra shown in fig. 2

AGV Mars BIR G2A GSP

AGV

Mars

BIR

G2A

GSP

100.00 85.54 91.13 90.35 89.79

100.00 82.44 81.23 77.93

100.00 74.39 71.02

100.00 97.98

100.00

of the NCC varied from 24.57 to 99.87. Based on the results of table 1, a discrimination threshold of 97.0 was applied to these NCCs to select those spectra which were judged similar in chemical composition by the pattern recognition technique. First of all, without exception, different spectra from the same sample were always correctly identified as having an NCC larger than 97.0. Spectra with obviously different structures by visual inspection indeed have NCC values considerably below this threshold. Furthermore, the technique has identified some samples as having similar structures in the region of interest. Two such samples, G2A and GSP, can be seen in fig. 2. The NCC values for the five spectra in fig. 2 are tabulated in table 2. The quoted chemical compositions for four of the samples in fig. 2 are listed in table 3. It can be seen that, as expected, G2A and GSP do have rather similar chemical compositions. The calculations of the NCC were carried out on a 8-MHz 80286/287-based personal computer. For the examples given above, using the raw XRF data, the computation of the pairwise correlation values of up to 10 spectra such as those shown in tables 1 and 2 took about 10 s. 3.2. Alloy samples

We have studied the XRF spectra of 23 alloy sampies, including some common laboratory items such as machine screws and drill bits whose quantitative chemical compositions are largely unknown. For illustration, we have selected the spectra of two common alloy groups: stainless steel and brass. Fig. 3 shows the de-

%

.17-4PH

%

i

,

i

i

t5

i

i

i

i

L

i

,

10 Energy, keY

0

,

,

i

i

i

15

i

i

20

Fig. 3. XRF spectrum of precipitation-hardened stainless steel sample 17-4PH. The W L lines are mostly overwhelmed by the strong lines of Ni and Cu. The 3 regions of interest chosen for pattern-recognition analysis are: 4.089-6.127 keV; 6.734-9.432 keV; and 15.494-19.935 keV.

tailed spectrum of the precipitation-hardened stainless steel sample 17-4PH, and fig. 4 shows the composite spectra of five stainless steel samples: 17-4PH, 304, 303, 300 *, and NBS standard stainless steel filings. Fig. 5 shows the composite spectra of three brass samples of unknown composition. In general, the spectral intensity from the alloy sampies were much higher than those of the geological samples. In order not to have an excessive deadtime in the multichannel analyzer, the X-ray generator was operated at 45 kv and 10 ~A, instead of 15 ~A, and the accumulation time per spectrum was also decreased from 10 rain to 5 min. In the region between 7.3 and 10 keV, the coherently scattered W L lines from the X-ray anode~ which were quite prominent in figs. 1 and 2, have now been mostly overwhelmed by the strong K~ and Kp lines of Ni, Cu, and Zn (see figs. 3-5). Further-

* The sample "300" was analyzed by the Materials Branch of Goddard Space Flight Center, and certified to be a stainless steel in the 300 series. The nominal composition is given in table 5.

Table 3 Chemical composition quoted for the sample used in fig. 2 and table 2. AGV denotes Andesite; BIR, basalt; G2A, granite, GSP, granodiorite Sample AGV BIR G2A GSP

K 2° [%]

CaO [%]

2.92 0.03 4.46 5.51

4.94 13.31 1.96 2.03

[%]

Fe203 [%]

Rb [ppm]

Sr [ppm]

Y [ppm]

Zr [ppm]

1.06 0.94 0.48 0.66

6.78 11.25 2.69 4.30

67 <2 168 254

657 106 479 233

21 19 12 30

225 35 300 500

T i O

2

624

Lo I Yin et al. / Pattern recognition approach in X-ray fluorescence analysis

¢O

0

5

10

15

20

pairwise NCC values analogous to table 1. Because of the higher intensities, and hence better counting statistics of these spectra as compared to the geological samples, the minimum value of NCC was 99.54 in ROI 1, 99.31 in ROI 2 for the NBS sample, and 99.42 in ROI 1, 99.45 in ROI 2 for the 300 sample. In ROI 3, the minimum value of NCC was 98.08 for the NBS sample and 92.83 for the 300 sample due to poorer statistics. The NCC values for the five stainless steel spectra of fig. 4 are tabulated in table 4 for the three regions of interest. Using the statistical spread of NCC values mentioned above as a guide, we can see that while all five samples have highly correlated structures in ROI 1, with NCC values above 99%, only samples 304, 303 and 300 have mutual NCC values above 99% in ROI 2, and above 90% in ROI 3. These results correlate well with the quoted nominal compositions of these samples, given in table 5. From table 5 it is seen that among the major constituents, although the Cr and Fe contents of all five samples do not differ significantly (ROI 1), the Ni content (ROI 2) of 304, 303, and 300 are more similar to each other than to either 17-4PH or NBS. Similarly, in ROI 3 17-4PH contains Nb in addition to Mo, and the NBS sample has a much higher Mo concentrations

Energy, keY Fig. 4. Composite XRF spectra from five stainless steel sampies: 17-4PH, 304, 303, 300, and NBS standard stainless steel filings. Pattern-recognition analysis show high correlations among 304, 303 and 300.

more, in those spectra where the counting rates were very high, such as the two brass samples in fig. 5, sum peaks began to appear at twice the energy of the main peaks due to the near simultaneous absorption of two X-ray photons in the detector within the resolving time of the pulse circuitry. Sum-peak structures can also be seen in the stainless steel spectra of figs. 3 and 4. Looking at the stainless steel spectra of figs. 3 and 4 in more detail, we have partitioned the spectra into three regions of interest (ROI): (1) 4.089 keV to 6.127 keV; (2) 6.734 keV to 9.943 keV; and (3) 15.494 keV to 19.935 keV. The rationale for this partitioning is as follows. Similar to the geological samples, the Fe K~ peak intensity is many times that of the elements on either side. In order to bring out the spectral differences due to the elements below Fe in the evaluation of the NCC, we have included in ROI 1 only the peaks due to Ti, V, and Cr. Similarly, in ROI 2 we have excluded the Fe K<, peak, but included the Fe Kp peak together with the K,~ and Kp peaks of Co, Ni, Cu, and Zn. ROI 3 contains the K~ and K~ peaks of Zr, Nb, and Mo. To establish the statistical spread of the NCC values in the three regions, we collected 7 spectra from the NBS sample and 10 spectra from the 300 sample under identical excitation conditions, and evaluated all the

sample 2

Ill .o..

o

-

xl

0

hex nut

5

10

15

20

Energy, keV Fig. 5. Composite XRF spectra from three brass samples of unknown composition. Although all three spectra appear quite similar, pattern-recognition analysis shows that samples 1 and 2 are more highly correlated which each other than with the hex nut.

Lo I Yin et al. / Pattern recognition approach in x-ray fluorescence analysis Table 4 Normalized correlation coefficients, in percent, for the five XRF spectra from stainless steel samples shown in fig. 4 Region-Of-Interest 1 (4.089-6.127 keV)

17-4PH 304 303 300 NBS

17-4PH

304

100.00 99.86 99.87 99.36 99.73

100.00 99.95 99.37 99.89

303

100.00 99.38 99.90

300

Sample 1 Sample 2 Hex nut 100.00

Region-Of-Interest 2 (6. 734- 9.943 keV) 17-4PH 304 303 300 NBS

17-4PH

304

303

300

NBS

100.00 93.66 96.03 95.29 86.93

100.00 99.48 99.11 98.28

100.00 99.47 96.44

100.00 95.35

100.00

300

NBS

Region.Of-Interest 3 (15.494-19.935 keV) 17-4PH 304 303 300 NBS

17-4PH

304

303

100.00 72.12 76.27 71.54 54.71

100.00 93.18 94.88 89.01

100.00 93.18 83.00

Table 6 Normalized correlation coefficients, in percent, for the three XRF spectra from brass samples shown in fig. 5 Region-Of-Interest 6.734- 9.943 keV

NBS

100.0 99.28

625

Sample I

Sample 2

Nex nut

100.00 99.87 98.33

100.00 98.96

100.00

with the hex nut. Closer examination of the spectra reveals that the hex nut most likely containes a higher concentration of Zn than the other two samples. Because of the wide variety of compositions among the alloy samples, their N C C values varied over ranges wider than those for the geological samples. A m o n g all 23 alloy samples, N C C s were obtained with values from 2.29 to 99.95 in R O I 1, 2.15 to 99.90 in R O I 2, and 7.44 to 9.32 in R O I 3.

4. Conclusions 100.00 88.37

100.0

than any of the other samples. Thus, while 304, 303 and 300 correlate well in R O I 3, with mutual N C C values above 90%, they correlate less well with 17-4PH and NBS, with N C C values below 90%. The three X R F spectra of brass samples shown in fig. 5 appear to be quite similar. Samples 1 and 2 were in sheet form; sample 3 was a hexagonal machine nut. The nominal handbook composition for yellow brass is about 67% Cu and 33% Zn. Indeed in fig. 5 the highest peak is the K~ peak of Cu, and the second highest is that of Zn K~. The shoulder on the right of the Zn K , peak is from Cu Kt~; the third peak is Zn K s. The mutual N C C values for these three samples are shown in table 6. Despite the similarity in appearance among all three spectra, the N C C values indicate that the brass samples 1 and 2 correlate with each other better than

We have attempted to illustrate, with some typical experimental examples, that the pattern recognition technique can be a very simple and efficient method in the monitoring and classifying of X R F spectra. The analysis is m a d e by means of a single parameter, the normalized correlation coefficient, which can be computed easily and quickly from the raw data on a personal computer. The pairwise comparison of ten samples (55 pairs) takes only about 10 s on our personal computer. Therefore, in applications where quantitative compositional information is not of primary concern, pattern recognition may be an attractive alternative in the analysis of X R F spectra. We should point out that while the N C C is shown to be an excellent pattern discriminant function, independent of spectra intensity, it is nevertheless affects by counting statistics. F r o m our calculations of the N C C in the various regions of interest for replicate spectra from both the geological and alloy samples, we find the following. F o r peak intensities above 1000 counts (i.e.,

Table 5 Nominal chemical composition for the five stainless steel samples used in fig. 4 and table 4. The quoted compositions are from the analysis provided either by the manufacturer or by the Materials Branch at Goddard Space Flight Center

17-4PH 304 303 300 NBS

Cr [%]

Mn [%]

Fe [%]

Ni [%]

Cu [%1

Nb [%]

Mo [%]

15.76 18.51 17.96 18.40 18.00

0.50 1.01 1.55 1.30 -

75.45 69.21 69.63 71.10 65.00

4.17 9.60 9.01 8.70 14.00

3.39 0.49 0.50 -

_ a) -

0.22 0.50 0.30 3.00

a) Industry standards specify the N b + T a content of 17-4PH steel to be 0.15-0.45%.

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Lo I Yin et al. / Pattern recognition approach in X-ray fluorescence analysis

statistical fluctuations below about 3%), the NCC values are greater than 99.3; for 300-500 counts (4-6% statistics), the NCC values are greater than 97-98; and for 70-80 counts (10-12% statistics), the NCC values are greater than 92-93. As expected, the poorer the counting statistics in the region of interest, the lower the NCC for spectra from the same material. However, as we have seen in the stainless steel case, even threshold NCC values in the 92-93 range can still be useful as a discriminant function for certain tasks.

Acknowledgements We gratefully acknowledge the support for this research through NASA RTOP 682-157-03-50. We thank

Robert G. Johnson of the US Geological survey for providing us with the geological samples a well as useful discussions and advice. We are also indebted to Diane M. Kolos of the Materials Branch of the Goddard Space Flight Center for supplying the alloy samples.

References [1] J.T. Tou and R.C. Gonzales, Pattern Recognition Principles, (Addison-Wesley, Reading, MA, 1974) chap. 3. [2] L. Yin, J. Trombka, L. Evans and S. Squyres, in: Mars Sample Return Science Workshop, Lunar and Planetary Institute, Houston, TX, Report 88-07 (1988) p. 182. [3] L. I Yin, J.I. Trombka, S.M. Seltzer, R.G. Johnson and J.A. Philpotts, J. Geophys. Res., submitted.