3.1 Utility of the Poisson Uncertainty Analysis

The utility of the Poisson approach to x-ray absorption measurement uncertainty was tested by comparing the Poisson estimate to the NRMSE and the NSD of measurements for water, paste, and mortar specimens. In experiment OP (Table 1), one point in the water specimen was sampled 100 times at three different intensities. In hindsight, performing this same task for the paste and mortar specimen may have been enlightening, but these measurements were not done.

Because many studies using x-ray absorption have involved count profiles, vertical count profiles for the water, paste, and mortar specimens (Fig. 1) were measured four different times at a fixed horizontal location for three different intensities in experiments WVP and IVP (Table 1). For the vertical profiles, the NSD can only be used as a measure of uncertainty for the water specimen (or other uniform specimen). As will be shown below, the NSD for the paste and mortar specimens provides information about the density structure and composition of these specimens that is not measurement error.

As the x-ray intensity increased, the number of counts from the water specimen point sampled 100 times increased (Table 4). All of the uncertainty indices decreased as the number of counts increased. For example, the Poisson estimate of uncertainty decreased accordingly from 2.7 % at intensity A to 0.3 % at intensity C while the NSD decreased from 3.6 % to 0.5 % over the intensity range. When sampling one point repeatedly, the Poisson estimate slightly underestimated the uncertainties estimated by the NRMSE and the NSD. Small variations in the thickness of the specimen or the container housing it may increase the uncertainty estimates from the NRMSE and NSD over what might be expected from the random noise indicated by the Poisson estimate. In general, the Poisson estimate provides a reasonable approximation to the uncertainty of point estimates.

The uncertainties of the vertical profiles from the water, paste, and mortar specimens (experiment IVP; Table 1) also decreased as the x-ray source intensity increased (Table 5). In these experiments, four vertical profiles at a fixed horizontal location were measured in each specimen. The values for NSD and NRMSE in Table 5 were computed in two ways, with the average profile at the highest intensity being used as the true value for the NRMSE calculation. First, the NRMSE and NSD of each individual profile for each intensity were averaged together to determine mean values (NRMSEa and NSDa in Table 5). Second, an average vertical profile for each intensity was determined and then compared to the true profile (NRMSb and NSDb in Table 5). These two methods were used to demonstrate how constructing a mean profile from several vertical profiles over the same points can reduce the uncertainty. Strictly speaking, the use of the Poisson estimate for several different points of heterogeneous specimens like pastes and mortars is not warranted. The Poisson values, however, were computed to examine how the Poisson estimate would perform with simple vertical profiles.

For the water specimen, creating a mean profile reduced the NRMSE and the NSD by a factor of two, as might be expected for a Poisson process. For intensity B, NRMSEa is 1.2 % and for NRMSEb is 0.6 % (Table 5). The NSDa is 1.2 % and the NSDb is 0.5 %. In fact, the averaging reduced the NRMSEs and NSDs below the Poisson uncertainty estimate (0.8 %). The reduction in the paste’s NRMSE is 45 % and in the NSD is less than 20 %. For the mortar, the NRMSE and NSD are reduced by 33 % and 2 % or less, respectively. The reason that the NRMSE is reduced more by averaging than the NSD for the paste and mortar is that random noise is averaged out with the NRMSE calculation while the NSD is indicating some physical density or composition variations in the specimens (Fig. 2). The NRMSE and NSD of the water specimen are reduced approximately the same amount as might be expected for a uniform specimen. Moreover, the NSD and NRMSE of a uniform specimen should approach zero when the random noise is averaged out, as is the case with the water specimen.

This comparison of NRMSE and NSD may be a convenient way with which to distinguish between random noise from the measurements and physical detail in the specimen (Table 5; Fig. 2). At higher counts, the NRMSE and NSD for the paste and mortars are considerably different. The NRMSE of the mortar at intensity C approaches zero (1.5 %), as random noise is averaged out of the measurements. At the same time, NSD converges to a high variability (8.3 %), indicating more detail in the profile.

Fig. 2. Vertical profiles of normalized counts for water, paste, and mortar specimens from experiment IVP. Each mean profile was determined by averaging together four vertical scans at one horizontal location.