Spatial Analysis of Time of FlightâSecondary Ion Mass Spectrometric

Anal. Chem. 2008, 80, 4896–4905

Spatial Analysis of Time of Flight-Secondary Ion Mass Spectrometric Images by Ordinary Kriging and Inverse Distance Weighted Interpolation Techniques Tammy M. Milillo and Joseph A. Gardella, Jr.* Department of Chemistry, University at Buffalo, State University of New York, Buffalo, New York 14260 Ordinary kriging and inverse distance weighted (IDW) are two interpolation methods for spatial analysis of data and are commonly used to analyze macroscopic spatial data in the fields of remote sensing, geography, and geology. In this study, these two interpolation techniques were compared and used to analyze microscopic chemical images created from time of flight-secondary ion mass spectrometry images from a patterned polymer sample of fluorocarbon (CxFy) and poly(aminopropyl siloxane) (APS, a.k.a. siloxane). Data was eliminated from the original high-resolution data set by successive random removal, and the image file was interpolated and reconstructed with a random subset of points using both methods. The statistical validity of the reconstructed image was determined by both standard geographic information system (GIS) validation statistics and evaluating the resolution across an image boundary using ASTM depth and image resolution methodology. The results show that both ordinary kriging and IDW techniques can be used to accurately reconstruct an image using substantially fewer sample points than the original data set. Ordinary kriging performed better than the IDW technique, resulting in fewer errors in predicted intensities and greater retention of original image features. The size of the data set required for the most accurate reconstruction of the original image is directly related to the autocorrelation present within the data set. When 10% of the original siloxane data set was used for an ordinary kriging interpolation, the resulting image still retained the characteristic gridlike pattern. The CxFy data set exhibited stronger spatial correlation, resulting in reconstruction of the image with only 1% of the original data set. The removal of data points does result in a loss of image resolution; however, the resolution loss is not directly related to the percentage of sample points removed. Time of flight-secondary ion mass spectrometry (TOF-SIMS) with either secondary ion focusing (microscopy) or primary ion focusing (microprobe) has the ability to visually display information about the morphology, topology, and chemical composition * Corresponding author. E-mail: [email protected]. Fax: 716-645-6963.

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of detectable species in the spectrum.1–3 In combination with the TOF-SIMS ability to quantify trace elements of surface species, the depth profile and the very high mass resolution (to distinguish different signals) make it a valuable tool for the analysis of surfaces in three dimensions.3 Recently, there has been increased attention regarding the use of cluster primary ion sources4 to improve signal intensity,4–10 image depth profiling, limit sampling depth,9,10 improve image quality,11–19 and improve depth profiling,20–23 as reviewed by Winograd.24 Organic films, materials, and polymers (1) Luc Van Vaeck, A. A.; Gijbels, R. Mass Spectrom. Rev. 1999, 18, 1–47. (2) Luc Van Vaeck, A. A.; Adams, F. Mass Spectrom. Rev. 1999, 18, 48–81. (3) Vickerman, J. C., Ed.; Surface Analysis: The Principal Techniques; John Wiley & Sons: New York, 1997. (4) Gillen, G.; Roberson, S. Rapid Commun. Mass Spectrom. 1998, 12, 1303– 1312. (5) Weibel, D.; Wong, S.; Lockyer, N.; Blenkinsopp, P.; Hill, R.; Vickerman, J. C. Anal. Chem. 2003, 75, 1754–1764. (6) Kersting, R.; Hagenhoff, B.; Killmer, F.; Mo ¨llers, R.; Nichuis, E. Appl. Surf. Sci. 2004, 231-232, 261–264. (7) Weibel, D. E.; Lockyer, N.; Vickerman, J. C. Appl. Surf. Sci. 2004, 231232, 146–152. (8) Kollmer, F. Appl. Surf. Sci. 2004, 231-232, 153–158. (9) Postawa, Z.; Czerwinski, B.; Szewczyk, M.; Smiley, E. J.; Winograd, N.; Garrison, B. J. Anal. Chem. 2003, 75 (17), 4402–4407. (10) Postawa, Z.; Czerwinski, B.; Szewczyk, M.; Smiley, E. J.; Winograd, N.; Garrison, B. J. J. Phys. Chem. B 2004, 108 (23), 7831–7838. (11) Roddy, T. P.; Cannon, D. M., Jr.; Ostrowski, S. G.; Winograd, N.; Ewing, A. G. Anal. Chem. 2002, 74 (16), 4020–4026. (12) Roddy, T. P.; Cannon, D. M., Jr.; Meserole, C. A.; Winograd, N.; Ewing, A. G. Anal. Chem. 2002, 74 (16), 4011–4019. (13) McQuaw, C. M.; Sostarecz, A. G.; Zheng, L.; Ewing, A. G.; Winograd, N. Langmuir 2005, 21 (3), 807–813. (14) Xu, J. Y.; Braun, R. M.; Winograd, N. Anal. Chem. 2003, 75 (22), 6155– 6162. (15) Walker, A. V.; Winograd, N. Appl. Surf. Sci. 2003, 203-204, 198–200. (16) Touboul, D.; Halgand, F.; Brunelle, A.; Kersting, R.; Tallarek, E.; Hagenhoff, B.; Laprevote, O. Anal. Chem. 2004, 76, 1550–1559. (17) Xu, J.; Ostrowski, S.; Szakal, C.; Ewing, A. G.; Winograd, N. Appl. Surf. Sci. 2004, 231-232, 159–163. (18) Xu, J.; Szakal, C. W.; Martin, S. E.; Peterson, B. R.; Wucher, A.; Winograd, N. J. Am. Chem. Soc. 2004, 126 (12), 3902–3909. (19) Sostarecz, A. G.; McQuaw, C. M.; Ewing, A. G.; Winograd, N. J. Am. Chem. Soc. 2004, 126 (43), 13882–13883. (20) Sostarecz, A. G.; McQuaw, C. M.; Wucher, A.; Winograd, N. Anal. Chem. 2004, 76 (22), 6651–6658. (21) Wucher, A.; Sun, S.; Szakal, C.; Winograd, N. Anal. Chem. 2004, 76 (24), 7234–7242. (22) Mahoney, C. M.; Roberson, S. V.; Gillen, G. Anal. Chem. 2004, 76, 3199– 3207. (23) Mahoney, C. M.; Yu, J.-X.; Gardella, J. A., Jr Anal. Chem. 2005, 77, 3570– 3578. (24) Winograd, N. Anal. Chem. 2005, 77, 143A–149A. 10.1021/ac702640v CCC: $40.75  2008 American Chemical Society Published on Web 06/07/2008

have often been the topic of TOF-SIMS imaging studies,25–27 generally suffering from a poor signal-to-noise ratio. The development of cluster ion beams for imaging and depth profiling is projected to have a great impact on the imaging of organic materials and surfaces. A second area of great activity is the use of multivariate statistics to improve the quality of data and signal-to-noise in TOF-SIMS spectra of organic and polymeric materials.28–39 There have been multiple attempts to improve the clarity of the spectra by implementing different forms of multivariate statistics. In these studies, it was evident that pretreatment steps must be performed on the data before any multivariate statistical analysis could be performed. Of particular interest is a multivariate statistical technique known as principle component analysis (PCA). PCA is a multivariate statistic technique which produces a smaller number of useful spectra (a.k.a. loadings) by taking a combination of peaks, which typically improves the contrast of the image. Two previous studies have been done that utilize different methods in order to reduce Poisson noise, thereby improving the accuracy of PCA.40,41 Improving the accuracy of the PCA analysis would result in fewer PCs. The resulting PCs would incorporate a greater percentage of the variance in the data set. The resulting eigenvectors that correspond to these individual PCs would be easier to distinguish from those eigenvectors which contain mostly noise. In one study, common algorithms such as down binning, boxcar, and wavelet filtering were compared in order to see which algorithm successfully filtered out the most noise, resulting in the most accurate PCA.37 PCA has been applied to improve TOF-SIMS image data. Keenan and Kotula32 developed a weighted scheme to account for the Poisson noise present in a given TOF-SIMS image. There was no study of how the spatial resolution factor affected the noise present in the image. In a recent paper, Biesinger et al.36 used PCA to improve contrast and resolution of the images over that of regular TOF-SIMS images when static conditions are used. (25) Vargo, T. G.; Thompson, P. M.; Gerenser, L. J.; Valentini, R. F.; Aebischer, P.; Hook, D. J.; Gardella, J. A. Langmuir 1992, 8, 130–134. (26) Li, L.; Ng, K.-M.; Chan, C.-M.; Feng, J.-Y.; Zeng, X.-M.; Weng, L.-T. Macromolecules 2000, 33, 5588–5592. (27) Satriano, C.; Spinella, N.; Manso, M.; Licciardello, A.; Rossi, F.; Marletta, G. Mater. Sci. Eng. 2003, 23, 779–786. (28) Raczkowska, J.; Rysz, J.; Budkowski, A.; Lekki, J.; Lekka, M.; Bernasik, A.; Kowalski, K.; Czuba, P. Macromolecules 2003, 36, 2419–2427. (29) von Gradowski, M.; Wahl, M.; Forch, R.; Hilgers, H. Surf. Interface Anal. 2004, 36, 1114–1118. (30) Eynde, X. V.; Bertrand, P. Surf. Interface Anal. 1997, 25, 878–888. (31) Pisciotti, F.; Lausmaa, J.; Boldizar, A.; Rigdahl, M. Polym. Eng. Sci. 2003, 43, 1289–1297. (32) Keenan, M. R.; Kotula, P. G. Surf. Interface Anal. 2004, 36, 203–212. (33) Keenan, M. R.; Kotula, P. G. Appl. Surf. Sci. 2004, 231-232, 240–244. (34) Wagner, M. S.; Graham, D. J.; Ratner, B. D.; Castner, D. G. Surf. Sci. 2004, 570, 78–97. (35) Thomas, C. G.; Harshman, R. A.; Menon, R. S. NeuroImage 2002, 17, 1521– 1537. (36) Biesinger, M. C. P.; Paepegaey, P-V.; McIntyre, N.; Stewart, N.; Harbottle, R. R.; Petersen, N. O. Anal. Chem. 2002, 74, 5711–5716. (37) Smentkowski, V. S.; Keenan, M. R.; Ohlhausen, J. A.; Kotula, P. G. Anal. Chem. 2005, 77, 1530–1536. (38) Larsen, R. Chemometrics 2002, 16, 427–435. (39) Engrand, C.; Lespangnol, J.; Martin, P.; Thirkell, L.; Thomas, R. Appl. Surf. Sci. 2004, 231-232, 883–887. (40) Keenan, M. R.; Kotula, P. G. Surf. Interface Anal. 2004, 36, 203–212. (41) Wickes, B. T.; Kim, Y.; Castner, D. G. Surf. Interface Anal. 2003, 35, 640– 648.

Topographical and ion yield (matrix) effects can be removed effectively, allowing the analyst to concentrate on chemical variations across the surface within the system under study. There have been additional TOF-SIMS studies that have attempted to eliminate noise by other means. In particular, there have been a variety of multivariate techniques using different principles. Multivariate curve resolution (MCR) and maximum autocorrelation factors are two examples of such techniques. These techniques use different statistical means in order to reduce the amount of peaks present in the spectra, thereby displaying the variance present in a more simplified form.37–39 There are different methods that can be used to rescale the spectrum in an attempt to equilibrate the intensity of the peaks. Three common methods are normalization, mean-centering, and autoscaling.33,42 Pretreatment of the spectral data before applying PCA is often necessary, making this an area of great interest. If there was a technique that could reduce the amount of noise without creating biases common to the current pretreatment processes, this would increase the use of PCA in the analysis of TOF-SIMS data. The resulting TOF-SIMS image would display the spatial distribution of ion species of interest contained within the image, with clearly defined boundaries. A bias refers to the unequal weighting that is given to a particular portion of the data set. This phenomenon results in changing the spectrum and the number of peaks that are important in the characterization of the sample. The resulting image often has decreased spatial resolution and poorly defined regions for the ions of interest.36,43 The present paper investigates alternative geospatial statistical methods that are commonly used in geographic information analysis.44–46 Geospatial statistical methods are a form of multivariate statistics that take into consideration the geographic (or spatial) location within an individual sample distribution associated with the individual data points. This is in contrast to PCA and other more common chemometric multivariate methods that examine multiple variables among a series of signals. These methods are commonly used to construct interpolated images or concentration contours from small data sets. By using these methods, it is hoped that an evaluation of the use of small data sets to construct accurate images of surface concentrations will result in more effective, accurate modeling with fewer sample points needed, thus increasing speed in TOF-SIMS image collection. In addition, using this technique may allow for the visualization of spatial distributions of different species. A smaller data set could allow for faster analysis of large area samples, such as disk drives,47 or may allow increases in spatial resolution. In the present study, TOF-SIMS secondary ion images were obtained from the surface of lithographically modified fluoropolymer samples prepared as previously described by Vargo et al.25 Transmission electron microscopy (TEM) grids were used as masks, and the modified regions of the fluoropolymer were (42) (43) (44) (45)

Pachuta, S. J. Appl. Surf. Sci. 2004, 231-232, 217–223. Wagner, M. S.; Castner, D. G. Appl. Surf. Sci. 2003, 698–703. Geostatistical Analyst, A help file in ArcGIS 8.1, 2002. Webster, R.; Oliver, M. Geostatistics for Environmental Scientist; John Wiley & Sons: New York, 2001. (46) Isaak, E. H.; Srivastava, R. M. An Introduction to Applied Geostatistics; Oxford University Press: New York, 1989. (47) Spool, A. M.; Forrest, J. Appl. Surf. Sci., in press; Proceedings of the 16th International Symposium on Secondary Ion Mass Spectrometry (SIMS XVI), Kanazawa, Japan, 2007.


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model consist of having a mean error equal to zero and minimal error variance. These two criteria make ordinary kriging an unbiased model. Kriging requires a semivariogram or variogram model, which is a graphic representation of the similarity or autocorrelation present in a data set as a function of distance. This acts as the basis for the interpolated surface. The more accurate the base model, the more accurate the interpolation. IDW46 is another interpolation method used in geostatistics. IDW assigns weights to sample points, which are inversely proportional to the distance that the particular sample is separated from the point of estimation. Typically, inverse squared distance is used, meaning that the weight assigned to a particular point diminishes by the square of the distance. The equation for the inverse squared distance weighted interpolation is n

^

v1 )

∑ d1 v

p i

i)1 n

∑ i)1

Figure 1. Results of sample analysis. (a) Large area (4200 × 4200 µm2) with color composite of pseudocolor intensity scales for fluorocarbons (red), sodium from sodium chloride (yellow/green), and a sum of ions related to siloxane functionality (blue). (b) Scale reduced to 196 × 196 µm2 region closeup.

refunctionalized with poly(aminopropyl siloxane) (APS). Imaging TOF-SIMS data was collected (Figure 1). The two interpolation methods of ordinary kriging and inverse distance weighted (IDW) were applied to the original TOF-SIMS image, in order to compare the capabilities of both techniques to reconstruct the image. Multiple interpolations were performed (see the Supporting Information), in order to evaluate the effectiveness of the methods as the percentage of data points removed was increased. The percentages of points removed resulted in a series of seven interpolations, which contained amounts ranging from 99% to 0.5% of the original image data. THEORY Two methods investigated in the present study are ordinary kriging and IDW methods, with detailed theoretical background available elsewhere.46 Kriging is an interpolation method that can be applied to estimate concentration values at points where no sampling has taken place. The values produced are a weighted linear combination of the available sample points. The equation for this technique is as follows: n

ˆ (x0) ) V

∑ w V(x ) i

i

(1)

i)1

In eq 1, Vˆ(x0) is the value predicted at a certain point given the summation of all the values (V(xi)) multiplied by the appropriate weight (wi). The optimal characteristics of an ordinary kriging 4898


i

(2)

1 dip

In eq 2, 1/d is the inverse of the separation distance between points and i represents the individual intensity values of specific pixels. This approach is the most effective for accurately modeling the spatial autocorrelation because only points closest to the point of estimation will exert influence in the prediction. The use of the squared distance also reduces the number of calculations, making this approach effective for reducing the amount of computation needed to produce an estimate. It is also an exact interpolator, which means theoretically it should produce the exact value given at a sample point. The IDW method does not rely on a variogram model. It uses only the values of the known sample points to estimate unknown points of interest. EXPERIMENTAL SECTION TOF-SIMS Imaging. The positive and negative spectra were obtained for each of these ions of interest, along with the corresponding images, using a SIMS 5-100 by IonTof GmbH (Muenster, Germany) with static mode conditions. This instrument utilized a bismuth liquid metal ion gun, or LMIG, of Bi3+ at 25 keV. The instrument was also equipped with a dual source column for cesium and C60 ion sources. The ion source was bunched using a target current of 0.02 pA. Bunching caused an increased acceleration resulting in a larger amount of ions reaching the detector. The stronger signal that resulted improved both the image and the spectral data, by obtaining better mass resolution within a given spectra. The analyzed area was equal to 196 × 196 µm2, and the amount of time taken to collect the sample was 100 s. Lithographic Polymer Surface Modification. The sample consisted of a silicon wafer that had a layer of fluorocarbon polymer deposited onto it. The TEM grids (7 µm which are spaced 15 µm apart) were then placed on the masked sample and went through the rf plasma modification,25 which involved methanol and H2 mixture for defluorination. The modification results in the replacement of the fluorine at the surface with highly reactive hydroxyl functionality. The sample is then dipped in dilute aminopropyltriethoxysilane, which results in an APS layer in the regions that were modified.25

Figure 2. (a) Reconstructed image, utilizing 100% of the original siloxane data set. As the intensity increases, the color interval lightens. (b) Reconstructed image, utilizing 100% of the original CxFy data set. As the intensity increases, the color interval lightens.

Interpolation Techniques. The original data was exported as ASCII files from the TOF-SIMS image and converted and imported into ArcMap, a software suite produced by Environmental Systems Research Institute, Inc. in Redlands, California.33 The geostatistical analyst extension of this program facilitates the exploration of data sets and the creation of interpolated surfaces. This extension has the ability to generate simple univariate statistics pertaining to the data set, which are often the starting point for the characterization of the distribution of data. It allows for the creation of histograms and normal Q-Q plots. These provide a graphic representation of the distribution of points. The software provides the ability for parameters involved in the creation of the variogram model to be altered as needed. In addition, multiple algorithms are preprogrammed into the software to allow for multiple interpolation techniques, based on the current data sets and the concentration information desired. This study utilized the interpolation methods of ordinary kriging and IDW. ColorBrewer.org is a Web site used to develop the color scheme that depicts the given signal intensities.48 The scale for these images was determined by the characterization of the data set produced by the geostatistical analyst. Validation. Three types of validation procedures were performed to test the accuracy of the interpolated surfaces created. Cross-validation removes a single point from the data set, and that value is estimated from the remaining points. This process is repeated for every point contained within the data set, and five summary statistics are then calculated: mean, mean standardized (48) Brewer, C. ColorBrewer. http://www.colorbrewer.org (accessed Dec 10, 2007).

(MStd), average standard error (ASE), root mean squared (rms), and root mean squared standardized (rms Std). The mean is the average value calculated for a given distribution. The mean standardized value is an average value calculated by removal of outliers found within the distribution, accounting for biases found within the distribution. ASE is the average standard error, which is calculated by comparing the predicted value with the actual value for the points contained within a distribution and then calculating an average error estimate that has been standardized by the standard deviation of the error distribution within the data set. rms is the root mean squared error, calculated by squaring the difference between the predicted and actual values, averaging them, and taking the square root. Lastly, the rms Std (root mean squared standardized) is the rms calculation standardized by the standard deviation of the error. An ideal interpolation is one that has a mean close to zero, and rms and ASE values that are close to one another, indicating accurate point-to-point variability (or the ability of the model to predict subtle changes between two sample points in close proximity to one another). The MStd value is used to check for bias in the data, as the mean is an easily biased statistic. This MStd calculation can provide insight into the effect that outliers, or points not characteristic of the data set, have on the calculation of the mean. The rms Std calculation is the mathematical measure of the accuracy of the model as a whole in its ability to accurately predict intensities. An rms Std value of 1 is the ideal value. A number less than one indicates an overestimation of intensities Analytical Chemistry, Vol. 80, No. 13, July 1, 2008

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Figure 3. (a) 1% of the CxFy data set; this is the data set used for the most representative interpolation given by the ordinary kriging technique. Panel b is the complete original CxFy image shown in Figure 2b. The green dots indicate the validation points that were used to evaluate resolution loss.

Figure 4. (a) Reconstructed image, utilizing 10% of the original siloxane data set and the ordinary kriging interpolation method, with a spherical base variogram model. The predicted intensities range from 0 to 19. The higher the intensity, the darker the color interval. (b) Reconstructed image, utilizing 0.5% of the original siloxane data set and the ordinary kriging interpolation method, with a circular base variogram model. The predicted intensities range from 0 to 17. The higher the intensity, the darker the color interval. 4900


the five parallel sample points within the validation set. The intensity profile allows for the examination of the successive resolution change of intensities over a given feature present within the image. The equation is49 ∆Z ) (0.16 - 0.84) ) 2σ

(3)

∆Z is the interface width of the sample, the 0.84 is the intensity obtained when the signal was at 84%, and 0.16 is the intensity when the signal had dropped down to 16%. The ratio is equal to two standard deviations of the interface width.

Figure 5. Reconstructed image utilizing 1% of the original CxFy data set and the ordinary kriging interpolation method, with a spherical base variogram model. The predicted intensities range from 0 to 19. The higher the intensity, the darker the color interval.

for the entire surface, whereas a value greater than 1 indicates an underestimation of intensities. Cross-validation technique was used for kriged surfaces only. Alternately, validation is a technique that requires the formation of a subset of data, allowing for the comparison of predicted values with the actual values of the subset. rms and mean error are then calculated for each point contained within the subset, which allows for an alternate method for testing the accuracy of an interpolated surface. This validation technique is used for both kriging and IDW interpolations.44,46 In order to have a quantitative measure of the resolution lost within a given image with each successive removal of points and the resulting interpolation, an image intensity resolution profile was created.49 The subset used in the intensity profile consisted of 5 parallel rows of data, each containing 154 points, for a total of 770 points. These data encompassed an area in the image with both CxFy-siloxane interface and a siloxane-NaCl interface. Each value on the intensity profile was an average value calculated from

RESULTS AND DISCUSSION Figure 2 displays the original TOF-SIMS images after being converted to an ArcMap-compatible format. Figure 2a depicts the intensity and spatial distribution of the siloxane ions. Figure 2b shows the intensity and spatial distribution for the fluorinecontaining species. The color scheme used in these images was suggested by ColorBrewer.org. This Web site, created and maintained by Dr. Cindy Brewer at Pennsylvania State University, gives color schema and other useful visual display techniques commonly used with ArcMap.48 With the use of suggestions given on this Web site and the knowledge learned from the characterization of the data set, a reverse continuous red-tone color scheme was used that characterized the data into five intervals. The unaltered data sets were displayed with intervals ranging from 0 to 20 and 0 to 19, respectively. The higher the intensity, the lighter the color used to display the value. The interval boundaries were chosen based on characteristics of the distribution of the intensity values found within the data set. For this reason, the interval size changes in order to display the image with all characteristic features of the distribution. A series of 28 interpolations were performed for each species, which utilized the intensity values for the respective ions. Images were created using both the kriging and IDW interpolation techniques. (For a complete table, interpolations, and statistics, see the Supporting Information.) A series of ordinary kriging maps were created to evaluate the effects of changing certain parameters. These parameters consisted of variogram model type and percentage of sample points removed. The three variogram types used were spherical, circular, and Gaussian base models. For each one of these base models, seven percentages of sample point removal were used. The percentage intervals were 1%, 50%, 80%, 90%, 95%, 99%, and 99.5% removed, corresponding to 99%, 50%, 20%, 10%, 5%, 1%, and 0.5% of the original data remaining. Figure 3a is an example of the CxFy data set with only 1% of the data remaining that was used to create the interpolation

Table 1. Summary Table of Ordinary Kriging Results for Siloxane and Fluorine Components ion

% data remaining, interpolation

sample size

mean

mean std

ASE

rms

rms Std

siloxane

0.5% CR 10% SP 1% SP 1% GA 10% CR 5% GA 5% SP 1% SP 0.5% SP

328 6553 656 656 6553 3276 3276 655 327

0.01136 0.000205 -0.00059 0.000523 0.002194 0.002151 0.001802 0.008291 0.009624

0.004847 0.0000718 0.0001225 0.0004726 0.0009374 0.0009379 0.0007663 0.001797 0.001394

2.666 2.596 2.679 2.68 2.215 2.229 2.173 2.195 2.4

2.67 2.55 2.68 2.675 2.235 2.228 2.24 2.187 2.375

1.001 0.9823 1.001 0.9984 1.009 0.9995 1.03 0.9959 0.9884

CxFy


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Table 2. IDW Results/Summary Table of IDW for Fluorine and Siloxane ion

% data points remaining

mean

rms

siloxane

20 10 5 10 5 1

0.00487 0.00256 -0.037 0.01269 0.01468 0.08057

2.66 2.722 2.709 2.389 2.399 2.348

CxFy

displayed in Figure 5, compared to the original complete image found in Figure 2b, redisplayed in Figure 3b for comparison purposes. For the removal of data points, the sample of data that is removed is random, as can be seen by Figure 3. Since each percentage of removal is taken from the original data set, error is minimized by interpolation technique, and the error distribution map created ensures that the errors within the map are evenly distributed, and a standardized error measure is given for each interpolation. All of the error analysis including calculated statistical values and maps of the distributions can be found in the Supporting Information. Table 1 is a summary table showing the values calculated for the ordinary kriging surfaces that are most representative of the original samples. From an initial inspection of the table it is evident that changing the model type has little effect on the overall estimation of the given surface. Figure 4a is the interpolation that results when only 10% of the original siloxane data remains and a spherical variogram base model is used. The image in Figure 4a uses only 6556 out of the original 65 538 sample points. The corresponding statistics are highlighted in Table 1. The yellow-to-brown color scale corresponds to the predicted intensity at a given location; the darker color intervals correspond to higher predicted intensities. The intervals predict intensities ranging from 0 to 19. Figure 4a clearly displays the characteristic lines of the grid. The fairly consistent intensities found within the region contained within the grid allows for the visual detection of a distinct low-intensity region where the NaCl contamination is located. This interpolation provided the most appropriate balance between statistical accuracy and image retention. When the statistical results are examined (see Table 1), there are two important observations: First, the spherical and circular variogram models only produce small visual changes between specific interpolations. The Gaussian variogram model, however, produced the most estimation error, with consistently higher image degradation. Second, there must be a balance between the accuracy of the interpolation and the amount of image degradation resulting from the removal of the data points. Figure 4a is a perfect example of this. When comparing the statistics included in Table 1, the rms Std value for the circular model with 0.5% of the original data set remaining suggests that the resulting surface is the most accurate. However, there are no recognizable sample features retained in this image (see Figure 4b). Figure 4b displays a phenomenon commonly exhibited when the ordinary kriging method is applied. The subset of siloxane data that was used to create this image contained too few points (49) ASTM E673-90 Depth Profile Resolution Standard. Surf. Interface Anal. 1991, 17, 951.

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to allow this interpolation technique to accurately display the image. When too few data points are used, the smoothing effect caused by the ordinary kriging technique is very pronounced. Although this characteristic is useful for dealing with small changes in elevation in geographic data sets, this effect negatively impacts the interpolation’s ability to reconstruct the image. For this reason Figure 4a, the interpolation that utilized only 10% of the original sample points and the spherical variogram model, produced the best image that retained most of the original image integrity while minimizing the errors given to the predicted intensities. Figure 5 is the interpolated surface that resulted when 1% of the CxFy data set was used, along with the spherical variogram model type. The series of interpolations created from the CxFy data utilized the same variogram model types and percentages of data point removal as the siloxane images. The color scheme for the predicted intervals is also the same. It is important to note that the siloxane and CxFy images are complementary images of one another, which means that where the grid lines were represented by low-intensity values (yellow shades) in the siloxane images, the same areas are now characterized by the higherintensity values (brown shades) present in the CxFy images. The object of this paper is to introduce this technique as a way to reduce the number of sample points needed, thereby increasing analysis time and making TOF-SIMS imaging techniques more practical for a wider range of applications. In addition, examining statistical relevance is appropriate and effective when calculations are being done with relevant data samples, not copious amounts of “extraneous samples” that make spectral and image analysis convoluted and tedious. The techniques of ordinary kriging and IDW are interpolation techniques that take into account the geographic location in the statistical calculation. At some level, there is a structure to the correlation between intensity and pixel location. For this reason, it is possible to create an “optimal” image from a given reduced set of data, because the separation distance between given pixels has become characteristic of the autocorrelation structure exhibited by the sample. When all the points are included, it is possible to see the smallest number of pixels, but when autocorrelation is correctly depicted within an interpolation, it is possible to determine the separation distance necessary to show correlation between species within the sample. When this happens, the number of pixels across the feature increases; when these factors have been optimized the resolution is nearly that of the unaltered image. The optimal image for the CxFy is the spherical interpolation with 1% of the original data set remaining. Only 656 of the original data points remain, and it is apparent in Figure 5 that the grid pattern is still distinguishable. This is not solely a function of signal intensity as much as a combination of signal intensity and geographic location, since these are the variables of interest displayed on the map. Furthermore, it is evident from the calculated statistics in Table 1 that this data set performed much better in the interpolation process. The rms Std value of 0.9959 indicates a slight overestimation of intensity for the image as a whole. The ASE value of 2.195, when compared to the rms value of 2.187, indicates that the model performs consistently when estimating point-to-point variation within the image. The statistics

Figure 6. (a) Siloxane image with 10% of original data set remaining, utilizing IDW interpolation. Intensities range from 0 to 19. The higher the intensity, the darker the color interval. (b) CxFy image with 1% of original data set remaining, utilizing IDW interpolation. Intensities range from 0 to 15. The higher the intensity, the darker the color interval.

are much closer to that of an ideal interpolation. The CxFy data set is able to reproduce with greater accuracy and integrity when compared to the siloxane image, because there is stronger correlation between signal intensity and spatial distribution. The stronger the correlation, the fewer the points needed for image reconstruction. The IDW technique is a type of interpolation that is dependent only on the distance between a given sample point and the point of estimation. As this distance increases, the weight given to the sample value decreases; thus, a point located furthest from the point of estimation will have the least influence on the predicted intensity. The IDW technique does not have an error measure associated with it. For this reason a validation set of approximately 770 points was selected from the image, and a mean and rms were calculated, comparing predicted and actual values for each percentage of data point removal (see Table 2). The rms calculated for the siloxane images indicate that our model is well-suited for our data set, due to the small range of rms values and how close the mean values are to zero. When comparing both ordinary kriging and IDW techniques, it is apparent that the most representative IDW models had the same amount of data points removed as the most representative kriging interpolations. Parts a and b of Figure 6 are the most representative models for both species using the IDW interpolation method. The same color gradient scale is used to display the predicted intensities given by the interpolation. The IDW predicted intensities range from 0 to 19 and 0 to 15, respectively. When the IDW images are visually compared to the kriging results, it is apparent that the IDW technique produces a grainier image. This is to be expected with this type of interpolation: because there is

no variogram base model, fewer assumptions are made to produce the interpolated surface. Figure 7 is a graph that displays the average predicted intensity for a series of points contained within the sample. In order to quantitatively measure loss of spatial resolution, values calculated from eq 3 (in the introductory text) were used to compare the different percentages of data point removal. The average intensity values were calculated by taking a subset that contained 770 points, equally distributed over five rows. Those values were then averaged together to obtain one intensity value for a particular pixel. The validation set crossed vertically over both edges of the grid, while passing through the characteristic feature caused by the NaCl. The resulting graph of the siloxane images (see Figure 7) showed two characteristic peaks. The edge feature analyzed was found between pixels 78 and 88. The sodium contamination feature produced a peak between pixels 149 and 168. The latter peak is useful in the resolution analysis. The change of signal intensity between the NaCl and edge of the grid could be detected continuously throughout the series of point removals. As the percentage of data points removed increases, the peaks broaden and the height of the peaks decreases. This would be the expected result with loss of resolution. However, it is important to point out that when looking at Table 3, which shows the relative number of pixels covered by a given feature, that the loss of resolution is not linearly dependent on the size of the data set. In fact, it is possible to optimize the percentage of point removal with loss of resolution for a given feature, as indicated in Table 3. With 99% of data remaining, the resolution across both features was 4 and 8 pixels, respectively. With 50% of the data remaining, resolution across the edge feature increased to 8 pixels, and with Analytical Chemistry, Vol. 80, No. 13, July 1, 2008

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Figure 7. Intensity profile comparing loss of resolution and percentage of data points. The 99% remaining line is green, the 10% remaining line is pink, the 5% remaining line is red, the 1% remaining line is blue, and the 0.5% remaining line is purple. Table 3. Relative Number of Pixels across a Given Feature for Siloxane Image edge feature

sodium crystal

% data remaining

no. of pixels

% data remaining

no. of pixels

99 50 20 10 5 1 0.50

4 8 6 5 8 7 NA

99 50 20 10 5 1 0.50

8 10 12 10 20 18 21

20% of the data remaining, the sodium peak increased to 12 pixels. With 10% of the data remaining, loss of resolution is minimized for both features: resolution across the edge feature is 5 pixels, and the sodium peak is 10 pixels across. Assuming that 1 h is needed to obtain the original image that has 156 × 156 pixels resulting in 65 635 individual sample points, for a sample like siloxane, which needs 10% of the original data, the analysis would take approximately 6 min. For a sample of CxFy component, which would also take 1 h for the original image, the 1% of the data necessary to interpolate could be gathered in under 1 min. Obviously, time would vary depending on the number of pixels, the detail level obtained in the image, and the amount of correlation present. To the authors’ knowledge, this technique has not been applied to the analysis of TOF-SIMS images prior to this study. For this reason, there was a significant learning curve in learning the techniques, software, and programming necessary for the presented interpolations. The time necessary for setting parameters decreased significantly as methods were established. The authors foresee that with changing the sample (and the components within the sample), there will be changes necessary for adequate analysis. Assuming that the researcher has sufficient knowledge of software, sample composition, and statistics, analysis time for a given 4904


sample could range from a couple of minutes to a couple of hours, depending on the sample’s complexity. If there are multiple samples using similar or identical parameters, the preliminary analysis need only be done once, meaning the whole process could be completed in about 15 min. CONCLUSIONS The common interpolation methods of ordinary kriging and IDW can be utilized in the analysis of a given TOF-SIMS image to decrease data analysis time by limiting the number of sample points taken. In fact, it is possible to obtain an optimal sample size indicating the amount of sample points needed for a given image by calculating the autocorrelation present between different chemical species contained within a given sample. For the samples discussed here, it is apparent that for the siloxane ion, only 10%, or 6553 of the original 65 635 sample points were necessary. For the CxFy ions, the correlation between sample intensity and sample location is even stronger, thus resulting in fewer points needed for an accurate interpolated surface to be created. In this image, 655 of the points from the original sample needed to be retained. Image resolution for a given image was not directly related to the amount of points removed from the data set. In fact, it was possible to obtain an image that had only 10% of the original points remaining, yet had nearly the same resolution as the image with 99% of the original points remaining. These results indicate that the ordinary kriging technique produced better results than the IDW technique for this type of sample analysis and that the ordinary kriging interpolation technique could be instrumental in substantially increasing the speed of image analysis for images created with the use of the TOF-SIMS instrument. The samples used in this study were uniform; however, this technique could be used on samples that were not uniform. A basic requirement is that there would have to be correlation between species present in the sample with respect to geographic location.

The number of sample points (pixels) needed to reproduce an image is dependent on the correlation present in the data set. The intensity value was used to measure the autocorrelation present in the study. If autocorrelation is determined to be present, there would not have to be drastic contrast between the intensities found within a sample to utilize this technique. Therefore, nonhomogeneous samples could be used that exhibited less visual contrast. If one was working with multiple samples of the same components, the analysis could be done once. If chemical composition changed on the sample, causing the autocorrelation to change, the number of points necessary would vary. It is

necessary to point out that as one becomes familiar with the interpolation and the techniques, the process of experimentally determining experimental size becomes more efficient. SUPPORTING INFORMATION AVAILABLE Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.

Received for review December 31, 2007. Accepted April 28, 2008. AC702640V


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Spatial Analysis of Time of FlightâSecondary Ion Mass Spectrometric

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