Spatial Statistics and Interpolation Methods for TOF SIMS Imaging Tammy M. Milillo and Joseph A. Gardella, Jr. Department of Chemistry University at Buffalo, State University of New York Buffalo, NY 14260-3000 USA [email protected]

Abstract Multivariate statistical methods such as principal components analysis (PCA) and factor analysis (FA) have been applied to mass spectral data to extract higher quality information from ion intensities in the mass spectrum. This often leads to better image quality in the resulting image analysis of principal components or factors. This paper presents a second multivariate statistical approach by examining the spatial statistics of the two dimensional image data. Geographic information is analyzed using two and three dimensional spatial statistical methods focused on interpolating spatial distributions. Methods such as Kriging and inverse squared distance weighting are often used to develop spatial distributions of common surface features distributed over geographic distances of meters, kilometers, miles, etc. Geospatial statistics have not been widely applied to spatial chemical distributions of microscopic dimensions. In this paper, we compare ordinary Kriging and inverse squared distance weighting for the analysis of ToF-SIMS image data. . By selectively eliminating pixels from the original image, we evaluate the accuracy of images reconstructed from 50 to 0.5% of the original dataset. Accurate image reconstruction from small data sets can provide added speed to ToF-SIMS image collection and analysis, a potential advantage for on-line ToF-SIMS analysis. Keywords: ToF-SIMS, Imaging, Multivariate Statistics, Geospatial Statistics

1. Introduction 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 of detectable species in the spectrum [13]. In combination with the ToF-SIMS ability to quantify trace elements of surface species, depth profile and the very high mass resolution (to distinguish different signals) makes 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 sources[4] to improve signal intensity [4-10],image depth profiling limit sampling depth [9,10], improve image quality [11- 20] and improve depth profiling [21-24] has been reviewed by Winograd [25]. 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. Organic films, materials and polymers have often been the topic of ToF-SIMS imaging studies [2628], generally suffering from a poor signal to noise ratio. A major focus on new cluster ion imaging and depth profiling has had a huge impact on proving the three dimensional surfaces and in-depth characterization of organics and polymers. 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 [29-40]. There have been multiple attempts to improve the clarity of the spectra by reducing the number of peaks, and thus the

clarity of the image by implementing different forms of multivariate statistics. In these studies it was evident that pre-treatment 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). Two previous studies have been done that utilize different methods in order to reduce Poisson noise, thereby, improving the accuracy of PCA. Improving the accuracy of PCA would mean that the PCA analysis would result in fewer PCs. The resulting PCs would incorporate a greater percentage of the variance in the dataset. The resulting eigenvectors that would 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, box car, and wavelet filtering were compared in order to see which algorithm successfully filtered out the most noise, which resulted in the most accurate principal component analysis[38]. In a separate study a weighted scheme was derived in an attempt to account for the Poisson noise present in a ToF-SIMS image [33]. There was no study of how the spatial resolution factor affected the noise present in the image. In a recent paper, McIntyre, et al. [37] used PCA to improve contrast and resolution of the ToF-SIMS images over that of regular TOF-SIMS images when static conditions are used. 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 auto correlation factors are two examples of such techniques. These techniques use different statistical means to try to reduce the amount of peaks present in the spectra, thereby, displaying the variance present in a more simplified form[38, 39, 40]. There are different adjustments that can be made to re-scale the spectrum in an attempt to equilibrate the intensity of the peaks. Three such techniques are normalization, mean-centered and autoscaling[34,46]. Pre-treatment of the spectral data before statistical analysis using PCA is an area of great interest. If there was a technique that could reduce the amount of noise without creating the biases associated with current pre-treatment processes, this would increase the use of PCA in the analysis of ToF-SIMS data. A bias results when a particular portion of the dataset is given unfair or unequal weighting or importance to that portion. This results in changing the spectrum and the number of peaks seen to be important in the characterization of the sample. The present paper investigates alternative multivariate methods based on geospatial statistical methods commonly used in geographic information analysis [41,42,47]. 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 contrasts with 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 can be evaluated, thus increasing speed in ToF-SIMS image collection. In addition, using this technique may allow for the visualization of spatial distributions of different species. Two methods investigated in the present study are ordinary Kriging and inverse distance squared weighting (IDW) methods [47]. 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

Vˆ ( x 0 )   wi  V ( xi ) i 1

^

V (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 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 variogram analysis of the distances between points (random sampling and then variogram). A semivariogram or variogram model is a graphic representation of the similarity or autocorrelation present in a dataset as a function of distance. Inverse Distance squared Weighting [47], or IDW, is an additional interpolation method used in geostatistics. IDW assigns weights inversely proportional to the distance that a 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:  vˆ 1 



n i1

1 d

n i1

p i

d

vi

1 p i

In this equation, 1/d is the inverse distance of separation between points and νi are the individual intensity values at specific pixels. This approach is the most effective for accurately modeling the spatial autocorrelation because only the points that are closest to the point of estimation will exert influence in the prediction. Using 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. In the present study, data from ToF-SIMS secondary ion images obtained from a surface which presents a clear image and features, a 2 Euro coin, is utilized to explore the limits of reduction of the data set size on the reconstruction of an image. We compare results from ordinary Kriging and inverse squared distance weighting for the analysis of the ToF-SIMS image data. By selectively eliminating pixels from the original image, we evaluate the accuracy of images reconstructed from 50 to 0.5% of the original dataset. Accurate image reconstruction from small data sets can provide added speed to ToF-SIMS image collection and analysis, a potential advantage for on-line ToFSIMS analysis. 2. Experimental 2.1 ToF-SIMS Data Images of 2 Euro coin were collected using an ION-TOF GmbH (Munster, Germany) TOF SIMS 4 instrument, with a LMIG gun equipped to produce Gold primary ions. The Au+ ions were accelerated to 25kV, at a current of 0.80 pAmp for a total acquisition time of 6710 seconds, cycle time of 200 usec and a total ion beam time of 306 seconds, yielding a total dosage of 4.3 x 109 primary ions/cm2 to collect the images. Images were collected from 256 x 256 pixels (65,536

points), separated by 1048 um step size, for an image size of 26200um x 26200um or 2.62cm x 2.62cm. Data was selected from four positive secondary ions (H, CH3, Na, K). Data was generously provided by Nathan Havercroft of ION TOF USA and Markus Terhorst of ION TOF GmbH. 2.2 Software and Image Analysis Data was received as text files of signal intensity versus pixel position for a particular ion and converted to x-y coordinates using Microsoft Excel. Data points with intensity values of zero were removed and not considered as part of the map data. The typical data size of 65,536 data points was

Figure 1: ToF-SIMS Images from 2 Euro coin sample

reduced to a data set of approximately 42,000 data points. (See results and discussion for justification of removal of zeros.) The resulting x,y spreadsheets were imported into ARCMap (ESRI Associates, Redlands, CA) version 9.0 and Matlab for Students (The Mathworks, Natick, MA), version 6, with PLS Toolbox [43] (Eigenvector Research, Manson, WA). Data was then replotted using a standard 7 level color intensity scale from ColorBrewer.com, ranging from dark blue (low intensity), through light blue, tan, orange and red for high intensity. Data sets were reduced to subsets of 99%, 50%, 20%, 10%, 1% and 0.5% of the original points. Fewer than 200 total points (0.5%) yields autocorrelation and validity results that are not acceptable [42]. Data sets were reduced by randomly eliminating data points using functions in Geostatistical Analyst extension of ARCMap. Ordinary Kriging and Inverse Distance Squared Weighting processing methods were available as part of the Geostatistical Analyst extension of ARCMap. 3. Results and Discussion 3.1 Data Analysis Figure 1 shows the resultant ToF-SIMS positive ion images from the 2 Euro coin. Surface contamination from organics (H, CH3) and Sodium and Potassium yielded high signal to noise data defining the surface image. This data set was determined to be useful because of the clearly recognized 2 on the surface feature. Figure 2 shows the results from data conversion from Ascii text into data files represented by color coded intensity levels using the Colorbrewer color ranking system, a standard in geographic representations of intensity scales (www.colorbrewer.com). This resulted in an inversion of the image from the upper left corner designation of pixel number of 1,1 to lower left corner. Data was first analyzed including all data points, with values of zero intensity included and then following the results of removal of the zeros outside of the perimeter of the image. Removing the zeros improved the characteristics of the distribution so that it approached a normal

distribution, a requirement for Kriging. A large fraction of zero intensity values created essentially a bimodal distribution, with one mode at zero, and then followed by a normal distribution of data points. Other techniques were considered for dealing with zero intensity points, but elimination appeared to be most relevant for this dataset. Data was then analyzed by both ordinary Kriging and inverse distance squared weighted (IDW) methods using the Geospatial analyst, an extension of ARCMap, found in the software suite of ARCGIS. The procedure involved selecting a randomly distributed (across the image) subset of datapoints and then “reconstructing” the image data using both interpolation methods. Figure 3 shows a series of results from reconstructing the image data Figure 2: ToF-SIMS Images from 2 Euro coin sample replotted in + using successively smaller data Matlab and ARCMap Left image is CH3 ion image all data points; + sets, and illustrates the strengths Right image is scaled Na ion image after removal of zero values outside of coin perimeter. Seven level colorbrewer scale, four and weaknesses of Kriging, a shades of blue to light blue, followed by yellow, orange and red for method which focuses on highest intensities, scele from 0-512 counts minimizing large variations in data. Within Figure 3, it is obvious from visual inspection that a recognizable image can be reconstructed using as few as 10 percent of the data points (90% data removal), with still recognizable features at the 5% level, but at 1% and 0.5% (the minimum size data set recommended for Kriging by Webster [42]). Despite the loss of visually recognizable features, the resulting concentration distribution does have high statistical validity (as discussed below, section 3.2, Criteria for Validity). In comparison to ordinary Kriging, IDW geospatial analysis data generally results in sharper concentration contours. For image reconstruction, this yielded recognizable features from the original images with fewer data points. Figure 4 shows a comparison of data from Na ion images reconstructed with both methods. IDW results clearly show features of the image recognizable with a smaller size data set, down to 1%. IDW of the 99.5% removal Figure 3 CH3+ image data with ordinary Kriging analysis of successively smaller datasets.

did not show a visually recognizable image feature for either Na+ or CH3+ datasets.

Figure 4 Comparison of ordinary Kriging and IDW analysis of Na+ image data with 90% and 99% removal of data.

3.2 Validation of Geospatial Analysis Aside from visual comparisons of recovered image features, Kriging and IDW are commonly evaluated by multivariate statistics of the data distributions and the predicted data. A standard comparison of the advantages of ordinary Kriging and IDW [47] provides the following conclusions. Kriging provides a built-in error analysis; it adjusts for subtle differences in value over short distances, in order to smooth plots and takes care of redundancies in the covariance matrix. Kriging allows for the calculations of weights. The resulting weights are associated with each sample which is proportional to how many points are in the area of estimation. The weighting system alleviates the biases of having clusters present in the data set. Clusters have an adverse effect on estimations often skewing the weighting system calculated by the linear combination of variable results. IDW yields, for the present analysis, a sharper resolution of the recovered image because it does not smooth intensity contours, but does not give an estimation of error, so validation or cross validation of the data points has to be done separately. Cross validation is a technique used in both IDW and ordinary Kriging. The technique is used for checking the accuracy of the model produced in both situations. The basic principle of the technique involves removing points one at a time and checking the estimate given at the point that is removed with the actual value. Theoretically, cross validation should give you the closest approximation with the least error because it uses all of the data set with the exception of the one missing point to produce the prediction. The greater number of points involved in a prediction the more accurate a prediction will be.

The cross validation produces parameters that are useful in assessing the error involved in the prediction. These parameters consist of the average standard error, root mean squared prediction error, root mean squared standardized error, and the mean error. Ideally, the mean should be close to zero, which would indicate an unbiased prediction. The mean is therefore, the mean error associated with the interpolation surface created. The standardized prediction errors are given by dividing the errors given with the individual prediction by a standardized prediction error. To properly assess the variability in a kriging model it is important that the average standard errors are close to the root mean squared prediction errors. If the average standard errors are greater than the root mean squared prediction error there is an overestimating of the variability of the prediction. If the average standard errors are less than the root mean squared prediction errors then there is an underestimation of the variability in the predictions given by the models. Another way to calculate the variability in the prediction is to divide each prediction error with the estimated prediction standard error. This calculation should produce a standardized root mean squared value that is close to one. If the root mean squared standardized, RMS standardized, value is less than one the model overestimates while a value greater than one indicates an underestimation [42]. Table 1 compares the statistics from the series of image reconstructions from the CH3+ ion images shown in Figure 3, along with IDW results. Table 1 Statistics from Geospatial Analysis of CH3+ Ion Images with Data Removed Statistic/Dataset Kriging Analysis Mean Root Mean Square (RMS) Average Standard Error Mean Standardized RMS Standardized Number of Data Points IDW Analysis Mean RMS Number of Data Points

Percentage of Datapoints 80% 90% 0.01988 0.1827 30.28 30.28 38.54 38.22 0.00486 0.009528 0.7855 .8555 8334 4167

Removed 95% 0.4079 34.21 39.08 0.009747 0.8749 2500

99% 0.6672 34.98 34.84 0.01694 1.00 416

99.5% 0.6101 34.24 33.48 0.01528 1.017 208

0.4782 26.00 8334

1.145 30.85 2500

1.806 33.98 416

0.8764 34.44 208

0.7981 28.69 4167

These results show good data for all recovered images, even those without recognizable visual features, such as the small data set ordinary Kriging results. As stated previously kriging is meant to smooth out boundaries contained within the area of interpolation. By retaining at least 200 points the variogram model can be trusted to predict concentrations that are in an acceptable range of error. For this reason the statistics for the smaller data-set is reliable for concentration predictions but visually the interpolation has been smoothed to the point where there are no recognizable boundaries. It therefore, becomes a decision on how to balance the sample number with the visual consequences, and what is desirable in the analysis. These data analyses compare the data predicted to the dataset used; they do not evaluate the predicted distributions versus the FULL data set, including the portion eliminated. Further work developing an algorithm that accomplishes a comparison of all predicted data points in the reconstructed images to the original datapoints in the full image is underway. A third validation method proposed for this work is to use resolution changes across a line of datapoints that includes a large change (increase or decrease) in the intensity distribution. We propose that we would measure the change in resolution from the 84-16%, adapting the ASTM E673-90 Depth Profile resolution standard [44], to an image resolution evaluation, documenting the loss of spatial resolution as a function of image processing technique and level. Both the latter proposals for image validation are underway. 4. Conclusions Two dimensional geospatial multivariate statistical methods can be used to reduce the size of datasets for ToF-SIMS images, by (re) constructing images using interpolation techniques. Both

Kriging and Inverse Distance Squared Weighting (IDW) methods can reconstruct accurate images from a small subset of datapoints, and these can be validated using standard geospatial statistical methods. IDW methods result in sharper edges for boundaries in a spatial distribution as appraised by visual comparison of images constructed from the same number of data points in the subset. Kriging offers a better set of statistical figures of merit to compare the accuracy of the interpolated data set to real data. A minimum of 200 datapoints for an image is needed for appropriate statistics. High resolution spatial analysis with geospatial statistics will require additional methods to evaluate accuracy. Further work using three dimensional data sets, including depth profiling, will require additional geospatial methods, but these exist. At present, the gain in speed is limited to a factor of 200 times; 0.5% of the original dataset allows reconstruction of an accurate image as evaluated by statistical figures of merit. Acknowledgements The authors acknowledge support from the US National Science Foundation Chemistry Division (Grant CHE 0316735). The authors are grateful for continued support of Nathan Havercroft of IONTOF USA for facilitating the provision of data on the samples reported on in this paper. References 1)

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