## 1. Introduction

The radioactivity concentration data of radionuclides are important in radioactive waste management from the beginning of waste generation to the final disposal. The radioactivity concentration should be accurate and reliable for decision-making at each step of waste management. Among the four methods used to determine the radionuclides present in radioactive wastes—accountability of material, classification by source, measurement of gross radioactivity, and direct measurement of individual radionuclides— the direct measurement method has two ways to determine the radioactivity concentration of individual radionuclides: the individual radionuclides are directly measured, or they are determined through the correlation between two radionuclide concentration data obtained by a *priori* direct measurements [1].

Direct measurements are labor-intensive and timeconsuming radiochemical procedures; therefore, indirect methods such as the scaling factor (SF) method are applied as an efficient concentration determination method based on direct measurement data. The SF method is the most widely adopted indirect method worldwide for the waste of a specific facility and waste stream [2, 3]. The SF method is based on the correlation between the radioactivity concentration of an easy-to-measure (ETM) radionuclide and a difficult-to-measure (DTM) radionuclide. Radioactivity concentrations of both radionuclides are determined using radiochemical analysis. The radioactivity concentration of the DTM can be obtained from the radioactivity concentration measurement of the ETM as a key nuclide and using various statistical models.

Thus, the accuracy of the DTM concentration depends on the uncertainties associated with sampling and radiochemical procedures, including measurements, modeling, and predictions. The uncertainties in the modeling and implementation of SF mainly result from the underestimation and overestimation of the radioactivity concentrations. Both underestimation and overestimation lead to poor waste management practices. Nevertheless, the underestimation has been regarded as being much more severe in terms of public health and safety than overestimation, leading to the underutilization of disposal facilities and a significant increase in costs.

The determination of radioactivity concentration in radioactive wastes and the issues regarding overestimation and underestimation are important from two perspectives: qualitative waste classification and quantitative inventory assessment. The overestimation of radioactivity concentration fundamentally results from the use of data biased from the trueness. One fundamental source of bias is the use of the minimum detectable activity (MDA) values in lieu of true activity values. Disposal concentration limits for lowlevel waste (LLW) and very-low-level waste (VLLW) disposal is extremely low, and the radioactivity concentration of some long-lived radionuclides in the LLW and VLLW is present at the ultra-trace level; therefore, it is inevitable to use the MDA values in place of true activity values as an alternative countermeasure option in the waste characterization strategy.

However, the magnitude and aspect of potential bias due to the use of MDA values must be carefully examined from the perspectives of waste classification and inventory assessment, as it may result in serious overestimations. Because the true activity data can always be obtained when the MDA of an analytical method is sufficiently low, the current status and future prospects of measurement techniques are briefly described in this study to help establish a waste management strategy.

Currently, Korea has only one disposal facility for lowand intermediate-level radioactive waste (LILW) as Asia's first underground radioactive waste disposal facility, which can hold up to 100,000 drums of waste in the first phase of disposal [4]. Efficient operation of the facility is no less critical than public health and safety in the future of radioactive waste management in Korea. This study investigated the probable overestimation or underestimation of radioactivity concentrations resulting from modeling and prediction using SF. However, the discussion is currently limited to iodine-129 (I-129) and technetium-99 (Tc-99) existing at ultra-trace concentration levels in the LLW/ VLLW and clearance waste (CW), where the possibility and impact of overestimation is relatively large when MDA values are used as true radioactivity concentration values. I-129 and Tc-99 have drawn much attention from the perspectives of radioactive waste disposal and radiation safety.

The characterization and safety assessment of I-129 and Tc-99 regarding disposal facilities are essential, as I-129 and Tc-99 have high environmental mobility, high solubility, and long half-life [5-7]. Their chemical and physical properties enable their transport in the groundwater environment around disposal facilities [8]. This characteristic is also applied to the regulatory framework of radioactive waste. According to the U.S. Nuclear Regulatory Commission (NRC) [9, 10], and the Nuclear Safety and Security Commission (NSSC) of Korea [11, 12], radioactivity concentrations of I-129 and Tc-99 must be quantified in radioactive waste. However, the radioactivity concentrations of I-129 and Tc-99 have been reported based on overestimated data quantified using the SF method based on the combination of direct and indirect measurements because it is highly probable that I-129 and Tc-99 concentrations were below MDA, but MDA values were used as true activity values [13, 14].

Overestimated DTM concentrations result in an overestimation of SF values in the construction of the SF model. The impact of overestimated SF values is huge, which induces the underutilization of radioactive waste disposal facilities. The overestimation of SF results from statistical modeling and the prediction of the radioactivity concentration. Concentration overestimation leads to an overestimation of SF, and vice versa. Overestimation or underestimation of the inferred radioactivity concentration occurs depending on the model and the region of interest. In this study, simple univariate linear models are investigated to determine the conditions under which overestimations or underestimations occur.

## 2. Statistical Models for the Determination of SF

As described in previous reports [15] and Eq. (1), the correlation between radioactivity concentration ratios (*SF _{i}*) between DTM and ETM is qualified by t-test with the correlation coefficient, and the simplest linear model with a slope and the intercept passing through the origin can be used for the determination of SFs.

where *SF* is a proportionality constant. There are two ways to determine the SF: the arithmetic mean (*SF _{AM}*) and the geometric mean (

*SF*) value of radioactivity concentration ratios between the DTM and ETM using Eq. (2) and Eq. (3), respectively.

_{GM}

If the radioactivity concentration ratios between the DTM and ETM are normally distributed, the arithmetic mean is representative to report the central tendency. The geometric mean is more appropriate when the radioactivity concentration ratios between the DTM and ETM show a log-normal distribution. One advantage of the geometric mean over the arithmetic mean is its robustness to extreme outliers. The geometric mean is less affected by the existence of a large extreme value.

A simple linear regression analysis with two parameters is more advanced than the simple linear model with only one parameter in Eq. (1). Regression analysis is the most widely used method in modeling studies.

A simple linear relationship is expressed as a linear regression model with the regression coefficient (*β*), intercept (*β*_{0}), and random error term (*ϵ*) of the ETM vs. DTM concentration curve in Eq. (4). The intercept of the simple linear regression is not always zero. The slope and the intercept parameters are determined through various computational algorithms, such as the ordinary least-squares method, maximum likelihood method, and Bayesian regression method [16].

Interestingly, the linear regression model has never been used as a linear model in the general SF community. Instead, the log-linear regression in Eq. (5) has been suggested to represent the non-linear relationship of the data when the DTM concentration does not linearly depend on the ETM concentration [2].

If more complicated models are adopted, the parameter estimation and interpretation of the data are not simple. However, even in simple linear models, abuse and misuse of SF methods are frequently encountered.

## 3. Overestimation of SF in a Simple Linear Model: *y* = *ɑx*

Eq. (1) is the simplest linear model form with only one parameter (parameter *ɑ*). The DTM radioactivity concentration (*y*) can be calculated by using the arithmetic mean or geometric mean of the concentration ratios between the DTM and ETM as a SF. Depending on the shape of the concentration ratio distributions, various measures can be used to represent the central tendency of the distributions. Three representative measures of central tendency are: mean, median, and mode [17]. In the SF methodology, the arithmetic mean and geometric mean are the two most widely used measures of central tendency in parametric statistics. If the concentration ratios are normally distributed, the DTM concentrations calculated using the geometric mean and the arithmetic mean would be equal because the three classical Pythagorean means would be equal for normal distribution, as described in our previous report [15]. However, if the ratio shows a perfect standard log-normal distribution, as shown in Eq. (6), with two parameters *μ _{SF}* and

*σ*(where the location parameter is ignored), the arithmetic mean is always greater than the geometric mean, as shown in Eq. (7), which implies the overestimation of the SF value when using the arithmetic mean as an SF for log-normally distributed concentration ratio data.

_{SF}

The extent of overestimation is the difference between the AM and GM of the SFs in Eq. (7), where the extent of overestimation is a function of *μ _{SF}* and

*σ*. As

_{SF}*μ*and

_{SF}*σ*are the mean and standard deviation of the log of the

_{SF}*SF*distribution, and

_{i}*μ*and

_{SF}*σ*are the log of GM and two sigma log-mean dispersion (LMD), LMD is a measure of the dispersion of the log-normal distribution. As LMD increases, the distribution of

_{SF}*SF*is increasingly right-skewed, as shown in Fig. 1. As

_{i}*μ*and

_{SF}*σ*increases, the log of GM or LMD increases, and the GM value becomes increasingly higher than the AM value; that is, the AM-GM difference increases, and the extent of overestimation becomes severe. If the GM is used despite the

_{SF}*SF*data deviating significantly from the normal distribution, the overestimation of SF will become more severe. The normality test is a good criterion for the fundamental step of decision-making, but it cannot decide whether the distribution is normal or not; it should solely act as a guide because most

_{i}*SF*distributions do not follow either normal or log-normal distributions. Decisionmakers, policymakers, and any stakeholders should check the extent of overestimation by comparing the results between the geometric mean and arithmetic mean carefully. If the GM or LMD is extremely high, it should be cautiously evaluated to choose the best model representing the actual data sets of

_{i}*SF*more accurately. If the

_{i}*SF*distribution is perfectly log-normal, then the GM is exactly the median, which is a useful non-parametric measure of the central tendency because it does not depend on the existence of outliers and large values of the concentration ratios.

_{i}The presence of outliers is a well-known problem in statistical analyses. In our case study, the presence of a single data point deemed to be an outlier was examined, as shown in Fig. 2. Red and black solid lines in Fig. 2(a) represent *SF _{GM}* with and without the outlier, respectively, but they are too close to each other to distinguish. We added a point (a red dot) arbitrarily so that the suspicious point was just above the common outlier criterion of the critical standardized residual, which is frequently used as a model diagnosis in regression analysis, as shown in Fig. 2(b). The best practice is to only remove outliers after a careful review, especially in cases where the suspicious points are not far away from the critical level of the diagnosis, although the outlier results in the overestimation. As expected from the basic principles of statistics,

*SF*is much more sensitive to outliers than

_{AM}*SF*. Since the

_{GM}*SF*is a parametric version of a non-parametric measure of the central tendency, it is as robust as the median, regardless of distributions. Owing to the use of the geometric mean, instead of removing the point, the point can remain in the database for the future update of SFs.

_{GM}The number of samples (*n*) may have an effect on the overestimation. We prepared two sets of data generated using the random number generator function and the inverse of the log-normal distribution function, where each data set has n = 30 and 200, respectively. Both datasets have the same parameters, *μ _{SF}* =0 and

*σ*= 1 in Eq. (6).

_{SF}*σ*= 1 corresponds to roughly LMD < 10, which is an accuracy tolerance of a factor of 10 [15]. Four hundred sets of random data composed of n = 30 and n = 200 were generated to have the same parameters of

_{SF}*μ*= 0 and

_{SF}*σ*= 1, and then the four hundred LMD values could be also obtained. The number of data sets with LMD < 10 for n = 30 and n = 200 was 43 (c.a. 89%) and 1 (c.a. 99.8%) out of 400 data sets, respectively. In addition, maximum value of

_{SF}*SF*and

_{AM}*SF*in the 400 data sets was 4.4 and 1.6, respectively, for n = 30 while 2.5 and 1.3, respectively, for n = 200. Since the higher LMD and

_{GM}*SF*have the higher chance of overestimation, the number of samples should be considered with great care. However, as in the previous case with the presence of outliers, the effect of the number of samples was not significant for

*SF*.

_{GM}Predicted DTM concentrations can also be overestimated when MDA values are used in lieu of true activity values because some long-lived radionuclides in the LLW and VLLW at the ultra-trace level are present below MDA. If MDA values are used, they may have a significant influence on SF values, particularly in the range of high ETM concentrations. The mean activity method is recommended to reduce the influence of overestimation by using the MDA values because the correlation test cannot be passed when the majority of data are below the MDAs. Nevertheless, if the SF method might be applied even in such cases using MDA values as true activity values, careful consideration should be taken not to severely overestimate the SF. The extent of overestimation can be understood by taking an example with I-129 or Tc-99 concentration data, which is primarily composed of MDA values as reported previously [13, 14, 18-23]. Fig. 3 shows randomly generated simulation data produced based on our field experience and the assumption that the wide concentration range of Cs- 137 spans roughly 10^{−2}–10^{6} Bq‧g^{−1}, whereas the minimum detectable concentration (MDC) range of I-129 spans only 10^{−3}–10^{−2} Bq‧g^{−1} because the MDC values do not vary significantly compared with the actual concentration variation. Compared with the estimated mean concentrations of DTM determined by the mean activity method (red dashed line), the estimated concentration values by the SF method (black dashed line) are underestimated below the crossing point between two lines and severely overestimated above the point, especially at the high concentration range of ETM. Again, when there is no choice but to use MDA values as true activity values, extra care is required.

## 4. Overestimation of SF in a Simple Linear Regression Analysis: *y* = *ax* + *b* + *ϵ*

In principle, neither the slope nor the intercept of Eq. (4) and (5) tends to be systematically overestimated or underestimated. However, the estimation of radioactivity can be overestimated or underestimated because of the existence of outliers or “leverage,” which is a measure of how far away the ETM concentration values are from the average of the ETM concentrations, and a consequent overestimation and underestimation of the slope and intercept. Outliers and leverage are unusual data points, and both could be influential in the modeling of regression analysis. Although an influential point typically has high leverage, a high-leverage point is not necessarily an influential point [24]. The effect of leverage values has never been considered in the determination of radioactivity concentrations in SF methodologies. At the same time, outliers are extreme values of DTM concentration relative to the fitted regression values of DTM concentration. Hence, both outliers and high-leverage points have the potential to be influential points, that is, to cause large changes in the parameter estimates when they are deleted. The effect of leverage values on the concentration of DTM can be minimized for a simple linear model with only one parameter, as shown in the previous section of this study and Eq. (1). If the predicted DTM concentration based on a model is extremely different from an observed DTM concentration value, then the observed value is regarded as an outlier. In contrast, if an ETM concentration value is extremely different compared with all the other ETM concentration values, then the ETM concentration value is said to be a high leverage point. Thus, outliers and high leverages can impact the regression analyses in a different manner. An observation can be both an outlier and a high leverage, and the impact of outliers and high-leverage data points should be examined carefully, once outliers and high-leverage data points are identified.

The extent of influence of a specific data point on the model parameters can be evaluated using two measures: Cook’s distance (*D _{i}*) and difference-in-fits (

*DFFITS*) as shown in Eq. (8) and Eq. (9), respectively [25]. Basic fundamentals behind the two measures is identical. One data point is omitted at a time, and each time repetitive model fit is performed with the remaining n – 1 data points. Then, the regression analysis results using all n data points is compared to the results with the i

_{i}^{th}data point deleted to evaluate the influence of this data point on the regression analysis.

where

k is the number of regression parameters excluding the intercept, and *h _{i}* is the leverage defined as the diagonal elements of the hat matrix.

*MSE*and

*MSE*− are the mean squared error with all data points and without the i

_{i}^{th}data point, respectively, whereas

*ŷ*and

_{i}*ŷ*− are the i

_{i}^{th}predicted DTM concentration by the model with all data points and without the i

^{th}data point, respectively.

*x*, and

_{i}*x*are the i

^{th}ETM concentration and average ETM concentration, respectively.

*D*and

_{i}*DFFITS*can also be easily obtained by using the leverage in Eq. (5).

where

A common rule is to identify any data points as influential data if its leverage value *hi _{i}* is more than three times larger than the mean leverage value. The recommended guideline for Cook’s distance is that the data point

*D*of 1.0 or greater is identified as an influential point, whereas the guideline for

_{i}*DFFITS*is that the

*DFFITS*values of greater than 2$\sqrt{k/n}$ are identified as influential points. Suppose the measured DTM concentration of the influential data is higher than the predicted one. In that case, overestimation will occur near the influential data point, while underestimation will occur if it is lower. Careful consideration is needed once any data points are identified as influential and further evaluation procedures are required. Data points should be re-measured, or more data points near the influential data points should be acquired for validation.

_{i}The effect of an influential point on the overestimation was depicted in Fig. 4, where the same data set including the suspicious outlier used in Fig. 2 were taken to calculate *D _{i}* and

*DFFITS*values. The outlier designated with red open circle is actually an influential point with

_{i}*D*= 1.6 and

_{i}*DFFITS*= 2.0, since both of them are higher than the decision criteria for

_{i}*D*= 1.0 and

_{i,critical}*DFFITS*= 0.35, respectively. The regression line has a supporting point, which can be regarded as a fulcrum, as shown in Fig. 4, where The predicted DTM concentration on either side of the fulcrum of the fitted regression line is overestimated or underestimated, respectively, depending on the slope and position of the influential data points. For example, suppose an influential data point is on the right-hand side of the fulcrum of the linear curve with a positive slope and is higher than the fitted line. In that case, the predicted DTM concentration data are overestimated. Conversely, the predicted DTM concentration data are underestimated if an influential data point is on the left-hand side of the fulcrum and is lower than the fitted line.

_{i,critical}The second case of overestimation occurs when using MDC values as true DTM concentration values. As in the previous section, the use of MDC as a true DTM concentration significantly influences the regression coefficient, the intercept of the model, and, consequently, the predicted DTM concentration. If both DTM and ETM concentrations are below the MDC, then the use of that data for the SF method is not recommended. The best solution is to reduce the MDC by using methods, such as neutron activation and mass spectrometry, which are discussed later in the paper. In addition, decision-makers, policymakers, and stakeholders must consider that a post-disposal re-evaluation of DTM concentration based on the revised SF updated with true radioactivity values is inevitable for the best implementation of the SF methodology for radioactive waste management to save our valuable tangible and intangible resources.

## 5. Current Status of the United States and Korea From the Perspective of Overestimation of Radioactivity Concentration in SF Methodology

The U.S. NRC Notice in 1986 remarked that “while underestimation is clearly a problem, overestimation of the gross radioactivity concentration is also of concern” [26]. Overestimation of gross radioactivity concentration results in the underutilization of the disposal facility. Underutilization is not trivial in waste management; it has been reported in multiple studies [14, 21, 22, 27]. A major source of overestimation is the use of MDA values as true activity values. The use of MDA values as true activity values has a long history. In the early days of SF implementation, some countries, including the United States, lacked data for statistical modeling. For instance, between 1992 and 1995, France used the SF values given in the literature published by other countries [3]. In the United States, if the number of samples is not sufficient for SF implementation, generic SFs are used regardless of waste streams [26]. The NRC also noticed a bad practice of combining the generic data with plant-specific data to produce plant-specific SF. The concentrations determined using the combined SFs are underestimated by a factor of ten, which does not comply with the regulation of a factor of ten by 10 CFR 61.55. Thus, the NRC recommended not combining generic data with plantspecific data. The United States also had a problem with the lack of true activity concentration data for DTM because most of the concentration values of DTM were below MDC values, especially for I-129 and Tc-99. As a last resort, MDC values were used as the true activity values. Consequently, the radioactivity concentrations of I-129 and Tc-99 were overestimated because of the overly high SF values. The early days of the SF database in the 1980s and the use of MDA data as true activity values have been well documented in previous publications [13, 14, 18-23]. As recently as 2012, MDA values have been extensively adopted as true activity values for LILW waste [13, 14]. The Electric Power Research Institute (EPRI) recognized the problems of databases in the United States using MDA values as true activity values for I-129 and Tc-99 in LILW and the consequent biases [21]. Hence, high-accuracy mass spectrometry could be used for trace levels of I-129 and Tc-99. Mass spectrometric analysis performed by Pacific Northwest National Laboratory (PNNL) enables the detection of true activity, which is much lower than the previous MDA of radiometric methods. SFs obtained using the conventional radiometric methods were overestimated by three to five orders of magnitude higher than those determined by mass spectrometry, as described in various U.S. organizational reports [14, 20, 22, 23, 27, 28]. PNNL’s SFs [18, 19] were compared with the old version of SFs [14, 28] as summarized in Table 1. PNNL’s SFs were found to be much lower than the old SFs, indicating that the overestimation can be avoided successfully. In 2013, the EPRI recommended a further reduction in MDC to 1% of the concentration limit listed in Class A of Table 2 [22], which is ten times lower than the previous requirement by the NRC [1]. According to the U.S. guidelines for the direct measurement of a particular radionuclide, MDC should be no more than 0.01 times the concentration limit for long-lived Class C radionuclides and 0.01 times the smallest concentration limit for short-lived radionuclides in Table 2 [1]. The use of MDC is appropriate in decision-making for the classification of radioactive wastes; however, MDC should not be used for inventory assessment in SF implementations. This is a representative example of misconceptions regarding the use of MDCs.

Periodic verification of SF should be conducted at least once every two years for Class A and at least once a year for Class B and Class C. If there is a significant possibility of changes occurring in the waste stream by a factor of ten, the SF should be re-evaluated for the update of SFs [1]. According to the NRC Notice in 1986, SFs can be updated based on recent data or a combination of the latest data with old data [26]. However, reportedly, no detailed descriptions of the process of conducting SF updates have been made available to the public.

In this section, the current status of Korea is discussed. The Gyeongju LILW disposal facility was launched in 1986 after 20 years of intense disputes over the first radioactive waste repository. The Gyeongju LILW facility has six underground silos and began operation in 2015 [4]. Although the facility site is expected to accommodate a total of 800,000 drums of waste in the long run, additional construction of underground silos for LILW is improbable within the next several decades. Thus, a total of 100,000 drums of LILW that are expected to be disposed in the underground silos must be handled with extreme care. This is the main reason why underutilization is key to successful LILW waste management in Korea. According to the NSSC notice [12], radionuclides present in various radioactive wastes such as H-3, C-14, Fe-55, Co-58, Co-60, Ni-59, Ni-63, Sr-90, Nb- 94, Tc-99, I-129, Cs-137, Ce-144, and gross-alpha nuclides should be quantitatively characterized so as to make them up for 95% of the total sum of radioactivity concentrations of radionuclides present in the waste. Underestimation of radioactivity concentrations is strictly prohibited by law in Korea. At the same time, presumably, there are no reports, governmental guidance, and regulations regarding the overestimation related to the implementation of the SF methodology. The ignorance of the overestimation issue reflects public sentiment on nuclear safety issues, not based on scientific considerations.

NSSC Notice defines the maximum radioactivity concentration limit of waste expressed in Bq‧g^{−1} that can be disposed of as LILWs in disposal facilities [11], as summarized in Table 3. MDCs for the direct measurement of the radioactivity concentration of individual radionuclides should be lower than 1% of LLW and CW concentration limits, respectively, which is a guideline for waste classification and inventory assessment. When the concentration is expressed in Bq‧g^{−1}, care must be taken in the waste classification to avoid overestimating the gross radioactivity per drum because the actual radioactivity per drum of the lower-density waste is lower than the radioactivity limit per drum estimated for the Gyeongju disposal facility. Gyeongju disposal facility for LILW has the total radioactivity capacity of 9.99 × 10^{8} Bq and 7.85 × 10^{11} Bq for I-129 and Tc-99, respectively [29], while the total volume capacity is 100,000 drums based on the assumption that all 200-L-drums store 168.9 kg (= 0.8 g∙cm^{−3} × 211,180 cm^{3}) of dry active wastes with a uniform radioactivity concentration [30]. Based on a similar simple assumption of uniform distribution of I-129 and Tc-99 all throughout the drums at full capacity of the facility, the average radioactivity concentration of I-129 and Tc-99 in an individual drum is presumed to be 0.059 Bq‧g^{−1} of I-129 and 46.5 Bq‧g^{−1} of Tc-99 by a simple arithmetic calculation. Both these values are between the I-129 and Tc-99 concentration limits of CW and VLLW in Table 3. Current radiometric method such as gamma spectrometry and gas proportional counter, which are the common practice for direct measurement of I-129 and Tc-99 in Korea [31], seems to be adequate for classifying the VLLW as the typical MDC for I-129 and Tc-99 is varied from 0.01 Bq‧g^{−1} to 0.002 Bq‧g^{−1}. However, the actual I-129 and Tc-99 concentrations in a considerable number of LILW drums could be well below the average concentrations of 0.059 Bq‧g^{−1} and 46.5 Bq‧g^{−1} for I-129 and Tc-99, respectively. For CW, the MDCs required by the NSSC are below the typical radiometric MDC levels. Moreover, the lower MDC, the better is the perspective of the concentration margin for the facility. The disposal inspection report of the Korea Institute of Nuclear Safety (KINS) [32] reports that the current average I-129 activity is 882 Bq per package for 21,374 packages disposed, as of May 2021, which is sufficiently below the calculated average activity of 9,990 Bq per 200 L-drum based on the volume capacity of the disposal facility.

Korea Hydro and Nuclear Power has used Cs-137 as a key nuclide to determine I-129 and Tc-99 concentration [33-35]. The PWR SFs of the multiple nuclear reactors at the six sites are summarized in Table 4, together with the SFs calculated using the ORIGEN-S code [36]. The SF value for I-129 was of the same order as the U.S. generic SF in the 1980s, as summarized in Table 1, whereas Tc-99 was three orders less than the U.S. generic SF. Nuclear fission and neutron activation for Hanul nuclear power plant unit 1 are considered in the theoretical SF calculation. Interestingly, SF values calculated using the code are three to four orders lower than the current SF values, which is evidence of overestimation and good motivation for lowering the MDC. As for the U.S., Korea needs to eliminate the overestimation of the SFs using more accurate and precise measurement techniques.

Inventory assessment and classification of radioactive waste, where SF plays a crucial role, has a different aspect of SF implementation. For waste classification, the use of MDC is not a serious problem only if the MDC is lower than 1% of the concentration limit. However, the impact of overestimation on inventory assessment is huge due to the use of MDC, even if the 1% criterion is met. As summarized in Table 1, SFs in the U.S. were reduced by approximately three orders of magnitude. Based on theoretical calculations, the same order of reduction can be expected for Korea, as summarized in Table 4. The impact of this level of reduction is huge in terms of facility assessment, especially for I-129 in Korea, since the maximum radioactivity capacity of I-129 for the Gyeongju disposal facility is much smaller than that of any other radionuclides. The overestimation may distort the facility assessment results seriously and lead to erroneous decisions [8]. The true quantities of I-129 and Tc-99 should be used for rational and accurate performance assessment of the disposal facility because they are the limiting radionuclides affecting the permitted total quantities of LILW [21, 28]. In addition, I-129 and Tc-99 need a special requirement for the long-term safety of disposal facilities [10, 37], and this leads to additional costs for extra care to satisfy the requirement [21].

## 6. Overestimation-Preventable Instrumental Techniques for Trace Level Detection

Among the many radionuclides of interest in waste classification and inventory assessment, beta-emitting radionuclides such as I-129 and Tc-99 are difficult to measure at trace levels. Selection of proper instrumentation and reduction of MDA values for each measurement technique is key to the prevention of overestimation. The typical MDA values reported previously are compared in Table 5 [28, 38-43]. Since Tc-99 is a beta-emitting radionuclide, a gas proportional counter (GPC) or liquid scintillation counter (LSC) can be used to quantify Tc-99 in waste. The typical MDA range of GPC and LSC is of the order of 10^{−2} Bq [21, 44].

I-129 is also a beta-emitting radionuclide; however, photon emission of its daughter nuclide Xe-129 can be indirectly detected by the gamma-spectrometric method. Advanced gamma spectrometry equipped with a well-type detector and analyzed using the anti-coincidence method can improve the MDA to be of the order of 10^{−8} Bq [39, 45]. The anti-coincidence method utilizes shield detectors around the main detector to remove the background signal from the main detector. Background reduction and improved efficiency make MDA much lower and improve the accuracy [46].

In contrast, mass spectrometry is more sensitive and accurate than radiometric methods because the mass spectrometer directly detects the radionuclide particles in the sample. In contrast, radiation measurement detects the decay of radionuclides. It is difficult for radiometric spectrometers to detect long-lived radionuclides that do not emit sufficient radiation on the radiation detector. Consequently, mass spectrometric methods such as inductively coupled plasma-mass spectrometry (ICP-MS), thermal ionization mass spectrometry (TIMS), and accelerator mass spectrometry (AMS) have been used to determine I-129 [39, 47-54] and Tc-99 [55-60]. TIMS and AMS are powerful tools for decreasing MDAs. However, only a few research groups have experimented with TIMS and AMS due to their cost prohibition from the perspective of SF implementation for I-129 and Tc-99 [28, 39, 40, 44, 48, 57]. Neutron activation analysis (NAA) is also a good detection method for I-129, but the MDA of Tc-99 using NAA is three orders of magnitude lower than that obtained using TIMS or AMS. For Tc- 99, the MDA of AMS was comparable with that of TIMS. As the radioactivity concentration of I-129 is generally two orders of magnitude lower than that of Tc-99, TIMS is preferred to ICP-MS for I-129. TIMS requires a more sophisticated procedure; therefore, TIMS is not cost-effective [52, 53]. Nevertheless, ICP-MS is a reasonable choice for Tc-99 because the required concentration level of Tc-99 for waste characterization is much higher than that of I-129.

MDAs shown in Table 5 are instrument detection limits, which are figures of merit that represents instrumental performance. MDAs of five instruments for I-129 reported by PNNL were determined through the same sample preparation procedures of drying, combustion and collection of iodine in an aqueous ammonia solution including an additional purification using a micro-extraction procedure if needed [45]. For gamma spectrometry and ICP-MS, the solution sample is directly used whereas the sample was placed in a quartz ampoule to be irradiated for NAA [48]. In case of TIMS and AMS, the sample in the form of silver iodide is placed on the filament and cathode, respectively [52, 53]. Similarly with the sample preparation of I-129, the samples for Tc-99 undergo well-known radiochemical procedures [40, 42, 43, 57, 61, 62]. MDAs of Tc-99 can be further reduced if more advanced high-performance instrumentations are applied [40, 63, 64].

In contrast, MDC based on the method detection limit is associated with all analytical procedures, from sampling to the final data assessment. The amount of sample, sample preparation techniques, and measurement conditions are important factors regarding the determination of MDC: increase in the sample amount, preconcentration of the analytes, and ultra-clean environment can lower the MDC. However, a large amount of samples and long counting time are not practical to decrease MDC in the radiometric method. In this regard, the MDA of I-129 with AMS is the lowest, which simply means that the smallest amount of sample is required to determine the same level of radioactivity concentration. In other words, compared with the conventional radiometric methods, AMS can reduce the I-129’s MDC by seven orders of magnitude under the same conditions such as sample amount and measurement time so that the overestimation can be avoided effectively.

## 7. Conclusions

The SF method may produce overestimated radioactivity values of DTM nuclides. Underutilization caused by the overestimation of SF is a problem that must be solved for the efficient management of radioactive waste. Some issues on overestimation of SF and consequent erroneous DTM concentration overestimation were investigated through careful case studies. The overestimation problems were caused by the statistical misinterpretation of a measure of the central tendency, disregard of the impact of influential points such as outliers and leverage, and the inappropriate use of MDA value as a true activity value, all of which have not been seriously examined for LILW disposal in Korea. I-129 and Tc-99 are good examples of easily overestimated DTMs in this regard.

These two radionuclides are of significant interest in radioactive waste management and safety assessment of disposal facilities; however, the overestimation induces underutilization of disposal facilities and extra costs for safety assessment. Both Tc-99 and I-129 can be measured using various detection methods such as gamma spectrometry, LSC, inductively coupled plasma-mass spectrometry (ICP-MS), NAA, TIMS, and AMS. As the MDAs are reduced, the benefit from the reduction becomes much larger in the inventory assessment. Mass spectrometric methods such as AMS and TIMS can reduce MDA and detect the true activities of I-129 and Tc-99. Other analytical methods, such as the advanced radiometric method and NAA, can also be improved to produce true radioactivity data at trace levels. Other overestimation issues related to sampling, radiochemical procedures, and measurement itself will be examined in a future study.

Existing literature has demonstrated that the U.S. has been leading the SF implementation since the 1980s. A famous aphorism, “all models are wrong, but some are useful” has many aspects of empirical models based on the measurement data. For heavily scattered data reflecting real-life truth, it is extremely difficult to build and realize a precisely accurate model. Overestimation and underestimation are inevitable because the inevitability is caused by the inevitable “wrong model” which will provide “wrong data.” Nevertheless, if flexibility and rationality are taken to heart, as we mentioned in our previous paper, “useful models” will be established so that the overestimation and underestimation are minimized. The best solution to avoid the overestimation of SFs is to produce sufficient radioactivity data above the MDA. In reality, however, it is improbable that the current measurement technique adopted in Korea cannot characterize the wastes at trace concentration levels of some radionuclides of interest.

The use of MDA values as an alternative countermeasure option is inevitable until the development of more advanced radiochemical techniques is completed to reduce the MDA to a sufficiently low level and to identify the existence of the radionuclides of interest. Considering the experiences of other countries with the implementation of the SF method, including the periodic verification of SFs, it is essential for decision-makers, policymakers, and stakeholders that interim disposal is required at the final disposal site until the re-evaluation of SFs. The waste deployment in the disposal facility based on the re-assessment may be possible after the new and more advanced methodologies are ready to be used to produce useful data. There is a need for wise and reasonable policies for determining the best practices for the operation of the first waste disposal facility in Korea. The advent of recent new tools of big data science worldwide will improve the models by evolving them over time as data is generated. In Korea, the SF methodology is still halfway through the final destination of perfect modeling as an indirect method via trial-anderror problem-solving. To achieve this goal, a collaboration between data scientists and radiochemists is important to connect useful data with a useful model. In this study, the importance and impact of reducing the MDAs was verified. It was shown that the need to improve the performance of radiochemical instruments and procedures is urgent for the safe and efficient operation of radioactive waste disposal facilities in Korea. SFs should be validated every two years and at the time of change in waste composition. Therefore, re-evaluation and updating of SF should not only be limited to the update of the data but should also pay attention to more advanced statistical models required for rational and reliable decision-making for the safe and efficient operation of the disposal facility.

In conclusion, we recommend several countermeasures for the overestimation including the use of geometric mean rather than the arithmetic mean, if possible, as a distributionfree robust measure of central tendency against outliers and leverages. The impact of outlier-like and leverage-like data points on the model parameters must not be disregarded and removed because they may be important for the data interpretation and the update of SFs in the future. We also propose that the MDA values should not be used as true activity values, but only as interim SFs when the data is not sufficient to implement the statistical SF methodology. The use of advanced data science, artificial intelligence and big data analysis will further improve the quality of information and optimize the issues that we currently face.