In this experiment, vertical profiles of counts at different horizontal locations were averaged together to create a mean profile. The goal was to determine how mean profiles created by averaging profiles from varying horizontal resolutions compare to the mean profile created by averaging profiles at the maximum horizontal resolution of 1 mm (Fig. 1). This procedure will help to determine the minimum horizontal resolution needed to approximate the overall mean profile of a specimen. Note that because vertical profiles from different portions of the specimen with differing structures are averaged together, the results below should not be considered necessarily as machine error. Instead, they indicate the utility of averaging procedures to approximate the mean profile.
As the horizontal spacing between vertical profiles increased, the NRMSE for both the paste and mortar vertical profiles increased (Figs. 9 and 10). Moreover, the NRMSE of mortar scans is more dependent on the horizontal resolution than those for paste scans due to the greater heterogeneity of the mortar mixes caused by the aggregate. The NRMSE for the mortar scans at a horizontal resolution of 10 mm at intensity C is nearly 5 % while that for a 1 mm horizontal resolution is about 0.2 % (Fig. 10c). For these same conditions, the paste NRMSEs vary from about 1 % to 0.2 % (Fig. 9c). For the paste, a horizontal resolution of 10 mm or less at intensity C is sufficient to produce NRMSEs of less than 2 %. For the mortar, a horizontal resolution of 2 mm and counts greater than 4000 are required to obtain a NRMSE of 2 % or less.
In averaging over several horizontal points to compare to a mean profile as is done above, some of the random noise and the microstructural features are averaged out or different microstructural features are measured when different horizontal points are sampled. Therefore, the Poisson estimate does not reliably predict the magnitude of the uncertainty in the horizontal averaging procedure. For the paste, the NRMSEs for all horizontal resolutions are lower than the Poisson estimate (Fig. 9), while for the mortar, some NRMSEs are higher by up to a factor of five and some are lower by up to a factor of four than the Poisson estimate (Fig. 10).
For the paste specimen, the Poisson estimate provides a reasonable approximation to the change in the uncertainty as the number of repetitions or the horizontal resolution changes. For example, at intensity A with one scan, the NRMSE is 4.5 % at a resolution of 10 mm (Fig. 9a). At a horizontal resolution of 5 mm, the NRMSE decreases to 2.8 %, or a factor of 1.6 smaller than that at a resolution of 10 mm. The Poisson estimate of this factor would be 1.4 since the number of counts increased by a factor of two in going from the 10 mm resolution to the 5 mm resolution.
For the mortar specimen, the Poisson estimate provides a reasonable estimate to the change in the uncertainty only for changes in horizontal resolution (Fig. 10). As mentioned above, the increased variability due to the presence of the aggregate masks any Poisson effect when averaging over several repetitions at a given horizontal resolution.
Fig. 9. Normalized Root Mean Square Errors (NRMSE) for the paste specimen in experiment IVP as a function of horizontal scanning resolution and the number of scans performed as indicated in the legend. The true value for the calculation of NRMSE was assumed to be the data in which four scans of the specimen were averaged together at 1 mm horizontal resolution at intensity C. See Table 1 for more information. (a) Intensity A. (b) Intensity B. (c) Intensity C. The numbers at the top-center of each chart indicate the approximate average number of counts per point for the corresponding x-ray source intensity. The values in parentheses below the number of counts are the Poisson estimates of uncertainty or the inverse of the square root of the number of counts.
Fig. 10. Normalized Root Mean Square Errors (NRMSE) for the mortar specimen in experiment IVP as a function of horizontal scanning resolution and the number of scans performed as indicated in the legend. The true value for the calculation of NRMSE was assumed to be the data in which four scans of the specimen were done at 1 mm horizontal resolution at intensity C. See Table 1 for more information. (a) Intensity A. (b) Intensity B. (c) Intensity C. The numbers at the top-center of each chart indicate the approximate average number of counts per point for the corresponding x-ray source intensity. The values in parentheses below the number of counts are the Poisson estimates of uncertainty or the inverse of the square root of the number of counts.