Advances on identification and animated simulations of radioactivity risk levels after Fukushima Nuclear Power Plant accident (with a data bank): A Critical Review

Abstract

This review study has been based on two main foundations as advances on the attainment of the risk radioactive fallouts levels, and the applications of methods for risk assessment to actual data and visual results, which are based on a 3-year study. A risk analysis model is developed with the animated simulations including the isotope distribution based on soil activity data, ¹³¹I measured at 19 stations after the Fukushima accident. Probability distribution functions of the risk levels are obtained in addition to the probability of occurrence (risk) and the probability of non-occurrence (reliability) of the activity risks concerning ¹³¹I. The results are used for prediction of 60-day radioactive fallout subsequence and animated (.mp4) through simulations.

Graphical abstract

Introduction

Three Mile Island, Chernobyl, and the Fukushima Dai-Ichi nuclear power plant (FDNPP) accidents took place in March 2011, which are important spots in the history of nuclear power accidents. The tsunami waves after the Tohoku earthquake of 8.9 magnitude broke out on the Honshu Island openings on March 11, 2011 at 14:46, resulting in the FDNPP reactor accident. After the earthquake, the diesel generators started to supply power to the to the electricity circuit that was automatically interrupted. Tsunami ripples caused electric supply to cease within a short time. Because of energy loss, the cooling systems were shut down and then the explosions came to fruition. Meanwhile, radioactive caesium and iodine emissions were mixed in the atmosphere. Subsequent to the occurrence of the accident, a safety circle of 20 km was built. Approximately 80,000 people were removed from this area, and any non-authorized person was not taken to the area. In this study, radioactive fallout calculations are performed by taking into account the restriction region [1,2,3,4,5,6,7]. The FDNPP accident was classified as level 7 according to the International Nuclear Events Scale (INES) system [1, 8]. After FDNPPA, the effects of radioactive fallout were measured in many places. According to these measurements, almost all of the American and Asian continents were affected by the accident. In some East Asian countries, the dose levels have reached, in places, the limit values announced by the IAEA [9,10,11,12,13,14,15,16,17,18,19,20,21]. Other studies have shown how the accident affected western and northern western regions of Europe [22,23,24,25].

The follow-up radionuclides traces on ecosystems is crucial both scientifically and in terms of viability in the relevant ecosystem [26,27,28]. Radionuclide distribution affects not only humans, but also marine and terrestrial ecosystems [29]. The major reactor accident radionuclides, ¹³⁷Cs and ¹²⁹I, have significant effects on air quality, and to see these effects, researchers have recently started to work on climate models [30,31,32,33,34,35,36,37,38,39,40,41,42]. Concurrently, new models [43,44,45,46], simulation techniques [41, 47,48,49] and risk analyses [50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70] are continuously employed for how to remove radioactive fallout products from nature [71]. In such studies, global radionuclides transport mechanisms [4, 72,73,74,75,76,77,78,79] are generally observed using systems of differential equations or statistical modeling approaches [80,81,82,83].

After the FDNPPA accident in March 2011, a serious radio-nuclear wave was delivered to the neighborhood. It is not possible to instantly monitor (or observe) all of the data related to the radionuclides emitted to the environment, which cause to data deficiencies that are generally a problem for similar investigations. For this reason, atmospheric dispersion model [84,85,86] approaches are used to partially compensate for errors in calculations [87]. The total amount of ¹³¹I released to the atmosphere after the FDNPP accident was measured as 120-380 PBq [10, 83, 87,88,89,90,91,92,93,94,95,96,97] following the accident, the ¹³¹I was first detected in Fukuoka, 1000 km from the FDNPP, 3 days after the accident [16]. Detection of short half-life ¹³¹I is important for short exposures. It causes global atmospheric oscillations despite short half-lives [98]. It was detected in Vietnam at 4500 km from Japan between March 27 and April 22 after the first detection [99]. On March 28, the presence of ¹³¹I was identified in the Republic of Korea about 1000 km from Japan [100]. On the other hand, the precipitation behavior of Iodine’s longer half-life isotope ¹²⁹I (1.6 × 10⁷ years) emissions were characterized as a result of detailed studies [101]. Since, ¹³¹I has a relatively short half-life, it is, therefore, a difficult radionuclide for such long-term monitoring works. This makes it difficult to obtain its transport characterizations. For all these reasons, it was decided to work on ¹³¹I, and moving visual simulations were also made for an effective methodology leading to transport risk analysis and activity intensity maps. Five years after the accident (in 2016), a study by Arai [102] detected the presence of FDNPP-originated radioactive particles in the natural environment. Similar findings were obtained through simulation program developments [21, 103,104,105,106]. This shows the importance of the simulation studies in large-scale areal researches. For example, by simulating Fukushima-derived radioactive fallout, it was possible to obtain meaningful conclusions about the behavior of radionuclides in the oceans [107]. As a result of various works on combining experimental and theoretical studies, the distribution of radioactive fallout was interpreted successfully [108], and hence, visual changes were easily visible. Apart from simulations, activity density maps also provide large-scale analysis [109].

After Fukushima NPPA, the presence of ¹³¹I in Europe has also been identified. The presence of these traces shows that the radioactive clouds move horizontally along with the vertical movement towards the troposphere. This horizontal movement causes ¹³¹I to be present in the air near the ground level [110]. Especially, after the Chernobyl accident, it is possible to see this in the global assessments studies of radioactive contamination [54, 83, 111,112,113,114,115]. It is far more difficult to remove ¹³¹I from the water environment by conventional methods such as coagulation, flocculation and sedimentation methods than ¹³⁴C and ¹³⁷Cs. Its short half-life (t_1/2 = 8.05 days) makes its detection difficult, but increases the importance of its analysis, because it is an important pollutant [95]. After FDNPP, significant quantities of ¹³¹I, ¹³⁴Cs, and ¹³⁷Cs were found to be deposited on the soil surface in Japan on March 21-23, 2011 due to rains in Japan [10,11,12,13,14,15, 116,117,118,119]. Again, according to Unno et al. [10], Xu et al. [11,12,13], Yamaguchi et al. [14, 15] these radionuclides originating from the radioactive fallout had accumulated on the soil surface [120] and adhered to the dust particles resulting from agricultural activities [121] and remixed to the atmosphere. Indeed, the effects of climate change were as a result of anthropogenic radionuclides mixing into the atmosphere in the Asian dust-collecting zone, and radionuclides adhering to dust particles fold into long distances by mixing into the atmosphere [122]. These transports were examined spatio-temporally and the transport characteristics of radionuclides were determined over time [123,124,125].

The Fukushima accident brought to light the issues of public health, economy, international relations and the energy policies re-examination, the release of radioactivity and its distribution [126]. It is scientifically important to see the size of the fallout formed by the radionuclides, which radiate to the atmosphere during and after the FDNPPA. In addition to the long half-life radionuclide exposure that occurs immediately after the reactor accident, short exposure is also important. In this study, the risks and risk scenarios, pollutant and transport characteristics and partially health, sociological and psychological effects of radioactive fallout and especially short half-life ¹³¹I environmental exposures are reviewed, and a new “moving simulation method” is proposed using the spatial analysis method [127]. With the proposed simulation study, it is possible to visualize the behaviours and characteristics of the radioactive fallout clouds. In this study, the proposed simulation method was based on the Kriging methodology [128] and semivariogram concepts [129], which have an important place in geostatistics methods [130, 131]. Somewhat important updates were made on this method and concept and they are used sometimes in different disciplines [127, 132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149].

Risk analysis [150,151,152,153,154], modelling and simulation studies [155,156,157,158,159,160,161] are important in determining the effects of radioactive fallouts. Such studies could make important predictions on the study area at that or for a further time. These estimates are also important for the development of nuclear waste scenarios [162,163,164,165]. The creation of these scenarios is important for society and environmental health [166,167,168]. In this study, previous monitoring, simulation, and modeling studies are evaluated from a broad perspective and motion activity distribution simulations and activity risk analysis for ¹³¹I after FDNPPA is obtained. This article contains two parts, review part (radionuclides and the risk assessments of the radioavtive fallouts, items 1–4) and model simulation part (simulation and risk assessment for the Fukushima Dai-Ichi accident area, item 5). Moreover, significant approaches and interpretations are obtained on the radioactive fallout risk levels. The effects of radioactive fallout in Fukushima are determined for ¹³¹I radionuclide. Both risk analysis and simulation studies are performed to predict the characterization and transport of the radioactive fallout prospectively. The data for the risk analysis is obtained from Japan Ministry of Education, Culture, Sports, Science, and Technology (MEXT). The probability of ¹³¹I contamination (risk) formation up to a distance of 60 km from the site of the reactor accident and the probability of non-occurrence of contamination (confidence) are determined for the next 60 days. The simulation studies for risk and probability calculations in addition to the characterization of contamination are obtained as .mp4 files in motion. These animated simulations provide considerable convenience in visualizing how the contamination evolution takes place by time. Radioactive particles transport simulation is important especially for the characterization of radioactive fallouts and for future predictions.

¹³¹Iodine

Despite the short half-life of ¹³¹I following FDNPP accident, it can be said that global transport [169,170,171] is serious. As a matter of fact, it is estimated that 11,000 km away in the accident region in the USA [172,173,174], in the Pacific Ocean [175, 176], in Canada [177], in Greece [22, 178], in France [179, 180] and in the other regions of Europa [23, 181,182,183,184]. ¹³¹I reached Europe only 7 days after the accident [23, 185,186,187,188]. ¹³¹I and some other fission products were detected at distances from the troposphere layer [189,190,191] whereas, Matsui [192] theoretically calculated ¹³¹I based on the existing information from nuclear reactions and activity densities in the environment.

Radionuclides such as ⁹⁵Zr, ^103,106Ru, and ¹⁴⁰Ba are detected in the Chernobyl reactor accident, and they differ from those in the Fukushima accident. In Fukushima, it is shown that the ones emitting terrestrial broadness are inert gases and volatile radionuclides of which ¹³¹I occupies an important place [19, 193, 194]. Some of the most effective physical mechanisms play a great role in the spread of ¹³¹I, which are wind and rain [195]. In this study, ¹³¹I data for the application of methodologies are taken by MEXT 3 days after the accident. In this period, MEXT reported that the global air circulation was not ineffective [196].

Removal of ¹³¹I from the water environment by coagulation-flocculation-sedimentation methods did not yield the desired results. On the other hand, ¹³⁴Cs and ¹³⁷Cs coagulation could be removed in the same medium [197]. Water purification and filtration systems are recommended for removal of radionuclides from drinking water [197, 198]. “Nano-metallic Ca/PO₄″ has been proposed to remove the above fission products from the surrounding environment or to reduce their mobile capability. This material was used by the ball milling method and reduced the mobilization of the fission products in the soil by about 56% [199]. With the development of these new techniques, it is thought that serious progress can be achieved in the reduction of possible cancer cases [200].

Approximately 80% of ¹³¹I can be stopped at soil depths of 4-6 cm [201, 202]. Apart from soil-depth analyses [203,204,205,206,207], ¹³¹I determination analyses were performed on a large scale on the soil surface [208, 209]. The transport characterization of ¹³¹I is also modeled, which then adheres to the dust particles [14, 80, 210]. This progression and distribution of ¹³¹I in the soil were modeled through numerical simulations [211]. In the atmospheric distribution [92, 212]; the Bayesian method [213], the Monte Carlo technique [214, 215], the time series analysis [12, 216], the mathematical modeling [217, 218] and in particular the inverse modeling methods [43, 55, 80,81,82, 84, 91, 92, 212, 213, 219,220,221,222,223,224,225,226,227,228,229,230,231] have recently become quite popular at atmospheric contaminants [232,233,234] and fallout studies [230, 235]. These modeling techniques have brought a different perspective to the characterization of the atmospheric ¹³¹I transport and some other fission products [90, 92, 93, 219, 226, 236,237,238,239,240]. ¹³¹I and some other fission products have been also reported as significant contamination indicators in the aquatic environments [13, 241, 242]. The environmental contamination of ¹³¹I necessitated the investigation of its effects on the lives of the living creatures [15, 243,244,245,246,247,248,249,250]. Examining the effects of her breastmilk, it was a pleasing situation that no feared results occurred [10, 251]. Apart from cancer research on thyroid [252], improvements and innovations in the field of engineering are also noteworthy. Utilizing the accumulation of ¹³¹I on thyroid glands [253], ultra-sensitive biomonitors were also developed [254]. In a similar vein, from the thesis that “radioactive nuclei could be seen with the naked eye, could be controlled more easily”; high-resolution molecular sensors have been proposed [255].

The answer given by the creature that received the radiation dose is very important [256]. The decontamination map generated as a result of these responses is determined by the atmospheric monitoring results and the dose of radiation calculated from the stations at the first period of the FDNPPA. The decontamination protocol advocates individuals’ reassessment, who wish to live in these areas according to their individual doses and it is an improvement on the response of living things to the dose–response relationship [257, 258]. On the other hand, radiological investigations continue to predict the amount of ¹³¹I activity, which is difficult to detect using experimental methods, unlike the theoretical methodologies in this study [259].

Radioactive fallout risk analyses and risk assessments

Radioactive fallouts risk analysis and studies on nuclear scenarios began in 1992 by Harvey et al. [260], which found practical applications. According to the studies conducted 4–5 years after the Chernobyl accident on the Chelyabinsk-65 population, in human organs and tissues ²³⁸Pu and ^239,240Pu were found three-four times higher than global levels. These results also brought serious health problems like cancer [261,262,263,264,265,266,267]. In 2001, lessons on the radioactive fallout health effects began to be taught [52]. It has led to the development of interesting computational tools such as web-based and GIS-based on the determination of the risk levels of nuclear weapons trials radioactive fallout. The computational techniques development revealed the necessity for eliminating statistical errors and uncertainties [268]. The environmental and health anomalies that appeared even after 50 years of radioactive exposure seem to occupy the present world of science [269,270,271,272]. Indeed, research on radioactive clouds global effects after major reactor accidents is an important step in determining risk levels [83, 111,112,113, 273, 274]. Uncontrolled reactor discharges, which cause to local health and environmental effects, also disrupt atmospheric C-14 equilibrium while Chernobyl’s effects are still discussed [275], although not globally as Chernobyl [276].

Fukushima risk analyses and risk assessments

After the Fukushima Dai-Ichi reactor accident, risk scenarios were established by considering radionuclide fallouts around the area, posing possible health risks for living [3, 277,278,279]. Apart from these scenarios, the radioactive fallout products’ effects on the food sector [280], other than the risks posed to living beings directly in the marine [281, 282] and terrestrial environments, have been studied in a broad perspective [283,284,285]. Communication tools [286, 287] were developed to present Fukushima risk analysis [288,289,290,291] and risk assessment results [238, 292,293,294,295,296,297,298,299,300]. On the other hand, important proposals were made on re-observing the construction of reactors and central buildings on the grounds that nuclear reactors known to be very resistant to earthquakes and they were influenced by tsunami [154, 301,302,303,304,305]. FDNPPA, the sociological and psychological effects of the incident have been on the agenda many times [305,306,307,308,309,310,311,312,313,314]. Particularly, health and more especially the cancer risk and its effects on humans have been one of the serious research topics [237, 238, 277, 315,316,317,318,319,320,321,322,323]. In the time frame from the 2011, Fukushima reactor accident to the present day, risk assessments on the occurrence of reactor accidents and the distribution of radioactive fallouts [9, 83, 99, 111,112,113, 227, 324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340] and determination of the risk of the effects of reactor accidents on health were the main research topics [200, 214, 341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364].

Simulation of radioactive fallout

Ability to simulate the radioactive fallouts propagation has opened a new door in this era [365]. The first simulation study is identified for iodine that belongs to Sorensen [366]. Later, the Japan Atomic Energy Agency obtained dynamic simulations for rice paddy fields and ¹³⁷Cs deposited in rice [367, 368]. Simulation studies have been proposed as a result of the convenience for the spread of radioactive fallouts’ interpretation, and simulation studies for some other hazardous materials [369, 370]. Radioactive fallouts were detected in fallout decay simulations that caused mutation in some flower pollen [371]. For instance, the detection of the ⁹⁰Sr and ¹³⁷Cs fallout products effects on soybean plants play directly a role in the development of the plant physiological development, which is another important consequence [372]. In 2008, Macedonio et al. [373] simulated the fallout, which was formed as a result of the volcanic activity of Vesuvius. Finally, motionless simulations of the reactor accident of Chernobyl were obtained [112]. It is also obtained in this study the propagation of the Fukushima reactor accident product ¹³¹I as moving simulations for a 60-day forward-looking estimation.

On simulation of Fukushima radioactive fallout

Simulation models have been used quite often recently for prospectively predicting the characterization of the amount or behavior of the variables concerned. Simulation techniques [85, 86, 374, 375] applicable from microscale to macro scales include new interpretations [376, 377] and new scientific advances [239, 378,379,380,381,382]. The destructive tsunami effects in the Fukushima reactor caused damage on the accident scenarios as little as possible from similar accidents [383]. In addition, after the reactor accident, alternative solutions to the problems that occurred in electricity generation and related problems came to mind [384] and in parallel with these workings, simulation studies on ¹³⁴Cs and ¹³⁷Cs radionuclides were also made [385,386,387,388,389].

Two-dimensional radioactivity distribution maps were obtained by using numerical simulation techniques for the characterization of radioactive water emitted from FDNPP [156, 215, 390]. These maps give the researcher detailed information about the variable studied in large-scale areas. Takemura et al. [157], Danielache et al. [159] and Behrens et al. [158] modeled global atmospheric radionuclide transport by performing numerical analysis on a global scale. After the Fukushima reactor accident, the radionuclides radiated to atmosphere travel far distances [391] over the ocean and through ocean currents [160, 161, 392,393,394,395,396]. Radioactive particles are under the influence of meteorological variables when transported at these distances [397]. As a result of atmospheric transport, fission products that land on the ground can contaminate groundwater [89].

As mentioned above, the findings obtained by simulating the reactor to investigate the causes of the explosion in the Fukushima reactor and the radioactive fallout are bound to provide important clues for controlling similar accidents in the future [398,399,400,401,402,403,404,405].

Real-time theoretical and practical researches for simulation, risk analysis and modeling of radioactive fallout

FDNPP was established near the Okuma Village in the Futaba district of Fukushima Prefecture in Japan and entered into operation in the 1970s as the first nuclear energy reactors generation. This plant was then transformed into a second-generation nuclear power plant with improvements. The plant has six boiling water reactors operated by the Tokyo Electric Power Company (TEPCO) [2, 8, 406].

In this research, since the risk analysis of ¹³¹I radionuclide is studied, first, the activity value at any time should be known in each station. Since, the measurements taken by MEXT are not synchronous, it is necessary to obtain the activity curves of the short-lived ¹³¹I in each station. For this, whenever the least squares method application to measurements taken from each station, an activity curve can be obtained for each station. Theoretically, once the concurrent activity values are obtained, risk analysis can be performed according to the position. After FDNPPA, soil ¹³¹I activity measurements from 19 stations are taken by MEXT [196]. ¹³¹I was first discovered in nuclear weapons tests, and this radioisotope is a dangerous radionuclide in NPP accidents. The ¹³¹I radioisotope turns into a stable nucleus of ¹³¹Xe after negative beta decay and gamma emissions [407, 408]. Information on the latitude, longitude and FDNPP distances of measurement stations are given in Table 1 [196].

Table 1 Some information for the stations where the measurements are taken

Theory

Least squares method

Gauss (1795) originally proposed the least squares method and in 1801, and used this method to determine the orbit of the Ceres asteroid. This method was published in 1809 as the second edition of Gauss’ collective works [409]. The least squares method is a standard regression method [410,411,412], which is used to write the mathematical relationship between two physical quantities that change in relation with each other.

Measurements taken from some stations by MEXT were not simultaneous. This is an obstacle for simulations and predictions. For this reason, the least squares method (LSM) is recommended for data optimization. The radioactivity values measured at stations in the LSM were determined at equal time intervals. The mathematics of the methodology can be summarized as follows:

In a study, let g(x) be a fitted model curve to the scatter diagram from pairs of x_i − f_i (x; independent variable, f; dependent variable). Here, it is expected that any x_i value corresponds to the values of f_i and g(x_i). The difference between them;

$$\varepsilon_{i} = (g(x_{i} ) - f_{i}^{{}} )$$

(1)

gives the ith error. For every x_i, this error can be positive or negative in the calculation of errors sum, so in order to avoid zero sum each error is squared. Sum of squares errors is given by,

$$E = \mathop \varSigma \limits_{i = 1}^{n} \varepsilon_{i}^{2} = \mathop \varSigma \limits_{i = 1}^{n} \left( {g(x_{i} ) - f_{i} } \right)^{2}$$

(2)

Suppose that the function g(x) depends on x and expressed in terms of coefficients a_i (1 ≤ i ≤ n). Let us choose a function g(x) which yields the following condition.

$$\frac{\partial E}{{\partial a_{i} }} = 0$$

(3)

If Eq. (3) is combined with Eq. (2), then one can obtain,

$$\mathop \varSigma \limits_{j = 1}^{n} g(x_{j} )\frac{{\partial g(x_{j} )}}{{\partial a_{i} }} = \mathop \varSigma \limits_{j = 1}^{n} f_{j} \frac{{\partial g(x_{j} )}}{{\partial a_{i} }}$$

(4)

If i-equation systems are solved as given by equation Eq. (4) and have i-variables, coefficients a_i should be solved for the best fit [411, 413]. Finally, the least squares method can be used to obtain synchronous activity curves, if samples are taken at different times in different stations. Thus, in calculations optimizations are provided.

Least squares method results

The time between the receipt of the ¹³¹I measurements and the date of the reactor accident was taken as parameter t. If the sought dependent parameter is activity, then it is necessary to obtain an exponential curve as in Eq. (5), according to the classical radioactive decay law.

$$A\left( t \right) = A_{0} e^{ - \lambda t}$$

(5)

If natural logarithms are applied on both sides of Eq. (5), a linear equation is obtained as in Eq. (6).

$$\ln \left( {A\left( t \right)} \right) = \ln \left( {A_{0} } \right) - \lambda t$$

(6)

If the equation g(x) in Eq. (4) is written as in Eq. (7), then one can obtain,

$$g\left( x \right) = g\left( t \right) = \ln \left( {A\left( t \right)} \right) = \ln \left( {A_{0} } \right) - \lambda t$$

(7)

Equation (4) can be rewritten as follows,

$$\mathop \varSigma \limits_{j = 1}^{n} g(t_{j} )\frac{{\partial g(t_{j} )}}{{\partial a_{i} }} = \mathop \varSigma \limits_{j = 1}^{n} f_{j} \frac{{\partial g(t_{j} )}}{{\partial a_{i} }}$$

(8)

This is a linear form and has two coefficients as,

$$a_{1} = \lambda \quad {\text{and}}\quad a_{2} = \ln (A_{0} )$$

(9)

After differential operations, Eq. (8) becomes a system of two linear equations with two unknowns (λ and ln(A₀)), in the form of Eq. (10). With the aid of MATLAB^® software program, the system of Eq. (10) is solved to yield activity values given in Table 2 and the other tables in Appendix A (supplementary material) and the coefficients a₁ and a₂ are obtained as in Table 3 for each station. This table shows the station numbers in the first column, the decay constant (λ) in the second column, the initial activity value (t = 0) in the third column and the square of the correlation coefficients (R²) of the curves obtained with these constants in the fourth column.

$$\begin{aligned} & \ln \left( {A_{0} } \right)\sum\limits_{j = 1}^{n} {t_{j}^{2} } + \lambda \sum\limits_{j = 1}^{n} {t_{j} } = - \sum\limits_{j = 1}^{n} {f_{j} t_{j} } \\ & \ln \left( {A_{0} } \right)\sum\limits_{j = 1}^{n} {t_{j} + \lambda n = - \sum\limits_{j = 1}^{n} {f_{j} } } \\ \end{aligned}$$

(10)

Table 2 Activity values measured and calculated versus time for station 1. For example, here “337th term” describes the 337 h after the reactor accident

Table 3 Coefficients and correlation coefficients obtained by least squares method

The half-life of ¹³¹I is t_1/2 = 193.68 ± 0.216 h. Again, the decay constant in hours is theoretically,

$$\lambda \cong 0.00357883{\kern 1pt} \,{\text{h}}^{ - 1}$$

(11)

The differences in activity values in Table 3 are thought to be due to geographical and meteorological situations, which indicate that the activity is stochastic relative to the position. Due to randomness, statistical considerations and calculations should be introduced after this phase. As an example, the experimental (measured) and theoretical (calculated) activity values for Station 1 are given in Table 2. These operations are performed for 19 stations and the theoretical calculations for all other stations are given in Appendix A (supplementary material).

In Table 3, the equation of the curve for Station 1 is.

$$A\left( t \right) = 2 9 9 4 2 7. 4 3 4{\kern 1pt} e^{{{\kern 1pt} {\kern 1pt} {\kern 1pt} - 0. 0 0 3 9 2 7 6 4t}}$$

(12)

The value at the time given by the least squares method in columns three and six is calculated from Eq. (5).

The result of these calculations is shown in Fig. 1, which shows the output of the computer program written in MATLAB^® language to obtain the Activity = f(t) graph for station 1. In this figure, the curve in red line is the most suitable one for station 1 as a result of the LSM. The graphs obtained from all other stations are given in Appendix B (supplementary material).

Fig. 1

Measured (experimental) activity values versus time for station 1. The curve on the graph was obtained by the least squares method. (Color figure online)

Risk analysis and probability distribution functions (PDFs)

Risk can be defined as the percentage of adverse events occurrences. Herein, it is defined as the process of scaling the risks that radioactive fallout will form and the areal determination, where measurements need to be taken. When a risk is determined for an event, the given data sets are considered individually and the states for each data are calculated.

In this part of the research, risk values are obtained with an experimenter approach. If R denotes the probability of occurrence (i.e., the occurrence of activity) and G denotes the probability of non-occurrence of an event, then the sum of the probability and absence of an event (G + R) is always fixed.

$$G + R = s$$

(13)

The constant s is the sum of the probability of occurrence and the probability of non-occurrence of ¹³¹I fallout. By dividing both sides of Eq. (13) by the constant s, the probability of non-occurrence and probability of occurrence ratios are obtained.

$$g + r = 1$$

(14)

At any given moment, the activity values from a location are sorted from small to big values, and hence, any activity value event (A_b) will itself include small activity events (A_s). One can express this event with A,

$$P\left( {A \le A_{b} } \right) = P\left( {A \le A_{s} } \right) + P\left( {A_{s} < A \le A_{b} } \right)$$

(15)

As the expression in Eq. (15) implies the probability of the given activities will increase steadily to one. This means that the probability functions are cumulative [133, 414,415,416].

$$g_{b} = \frac{{m_{b} }}{n}$$

(16)

where g_b is probability of non-occurrence for biggest activity value, m_b is the rank for biggest activity value, and n is the number of all activity events. However, this is not realistic in practice. The greatest activity for this is not 1, it should be very close to 1. Instead of Eq. (16), it is preferable to write,

$$g_{b} = \frac{{m_{b} }}{n + 1} = \frac{n}{n + 1}$$

(17)

Thus, the bigger active area in Eq. (17) is acceptable. The most general form of Eq. (17) for each rank is given as

$$g = \frac{m}{n + 1}$$

(18)

And from Eq. (14) one can obtain,

$$r = 1 - g = 1 - \frac{m}{n + 1}$$

(19)

Equation (19) is the probability of occurrence for each rank. The order given by m is also the order of harm caused by radioactive fallout [133].

This study also considered the harmonization of the important pdfs in the literature with the changes of ¹³¹I in order to be able to make risk analyses and to find out the possibility of the occurrence of ¹³¹I’s activity in the research field and to find out the spatial transformations of these variables with the results obtained later. The calculations took into consideration three pdfs, which have the highest R².

Generalized extreme-value distribution

The generalized extreme-value distribution is based on the combination of Gumbel, Fréchet, and Weibull distributions and the continuous probability distributions developed within the extreme-value theory. The generalized extreme-value distribution can be used as an approach to model the maxima (or minima) of long (end) random sequences. As different from other distribution, it is represented by three parameters, namely, σ; scalar parameter, μ; location parameter and k; shape parameter.

The generalized extreme-value distribution is given by the following expression

$$f(\left. x \right|k,\mu ,\sigma ) = \left( {\frac{1}{\sigma }} \right)\left( {1 + k\frac{{\left( {x - \mu } \right)}}{\sigma }} \right)^{{ - 1 - \frac{1}{k}}} \exp \left( { - \left( {1 + k\frac{{\left( {x - \mu } \right)}}{\sigma }} \right)^{{ - \frac{1}{k}}} } \right)$$

(20)

with k ≠ 0 and $\left( {1 + k\frac{{\left( {x - \mu } \right)}}{\sigma }} \right) > 0$. The cumulative distribution function is then appears as follows.

$$F(\left. x \right|k,\mu ,\sigma ) = \exp \left( { - \left( {1 + k\frac{{\left( {x - \mu } \right)}}{\sigma }} \right)^{{ - \frac{1}{k}}} } \right)$$

(21)

with k ≠0 and $\left( {1 + k\frac{{\left( {x - \mu } \right)}}{\sigma }} \right) > 0$.

The distribution has three alternatives according to the state of the k parameter: type 1 for k = 0, type 2 for k > 0 and type 3 for k < 0, which resemble Gumbel, Fréchet, and Weibull distributions, respectively [417,418,419,420]. The probability of non-occurrence for the generalized extreme-value distribution from Eq. (21) is given as,

$$g_{gev} = F(\left. {x{\kern 1pt} {\kern 1pt} } \right|{\kern 1pt} {\kern 1pt} {\kern 1pt} k,{\kern 1pt} {\kern 1pt} {\kern 1pt} \mu ,{\kern 1pt} {\kern 1pt} {\kern 1pt} \sigma )$$

(22)

Again, from Eq. (14), the probability of occurrence becomes as,

$$r_{gev} = 1 - g_{gev} = 1 - F(\left. {x{\kern 1pt} {\kern 1pt} } \right|{\kern 1pt} {\kern 1pt} k,{\kern 1pt} {\kern 1pt} {\kern 1pt} \mu ,{\kern 1pt} {\kern 1pt} {\kern 1pt} \sigma ) = 1 - \exp \left( { - \left( {1 + k\frac{{\left( {x - \mu } \right)}}{\sigma }} \right)^{{ - \frac{1}{k}}} } \right)$$

(23)

Lognormal probability distribution

The lognormal distribution is a probability distribution for random variables, whose logarithm is normally distributed. If x shows lognormal distribution, then log(x) shows normal distribution. It does not matter what the basis for the logarithm function is. If log_a(x) shows the normal distribution for any two positive numbers a, b ≠ 1, log_b(x) also implies normal distribution. Probability density function for lognormal distribution (μ; position parameter, σ; scale parameter and for x > 0) is

$$f(\left. {x{\kern 1pt} {\kern 1pt} } \right|{\kern 1pt} {\kern 1pt} \mu ,{\kern 1pt} {\kern 1pt} {\kern 1pt} \sigma ) = \left( {\frac{1}{{x\ln (\sigma )\sqrt {2\pi } }}} \right)\exp \left( { - \frac{{\left( {\ln (x) - \ln (\mu )} \right)^{2} }}{{2\ln (\sigma )^{2} }}} \right)$$

(24)

The cumulative distribution function is given as

$$F(\left. {x{\kern 1pt} {\kern 1pt} } \right|{\kern 1pt} {\kern 1pt} {\kern 1pt} \mu ,{\kern 1pt} {\kern 1pt} {\kern 1pt} \sigma ) = \frac{1}{{\ln (\sigma )\sqrt {2\pi } }}\int\limits_{0}^{x} {\frac{{\exp \left( { - \frac{{\left( {\ln (t) - \ln (\mu )} \right)^{2} }}{{2\ln (\sigma )^{2} }}} \right)}}{t}dt}$$

(25)

Lognormal pdf is a distribution function used to model randomly varying states [411].

Weibull probability distribution

The Weibull distribution is a probability distribution for random variables and its mathematical form is as follows.

$$f(\left. {x{\kern 1pt} {\kern 1pt} } \right|{\kern 1pt} {\kern 1pt} {\kern 1pt} \alpha ,{\kern 1pt} {\kern 1pt} {\kern 1pt} \beta ) = \alpha \beta^{ - \alpha } x^{\alpha - 1} \exp \left( { - \left( {\frac{x}{\beta }} \right)^{\alpha } } \right)$$

(26)

where α is the shape parameter, and β is the scale parameter, and x ≥ 0. The cumulative distribution function is then,

$$F(\left. {x{\kern 1pt} {\kern 1pt} } \right|{\kern 1pt} {\kern 1pt} {\kern 1pt} \alpha ,{\kern 1pt} {\kern 1pt} {\kern 1pt} \beta ) = 1 - \exp \left( { - \left( {\frac{x}{\beta }} \right)^{\alpha } } \right)$$

(27)

[411].

Risk analysis and PDFs results

The activity values of the ¹³¹I radioisotope, released after the Fukushima accident are not statistically discrete events; therefore, the probability curve for these activity values is the cumulative probability function. Equation (15) can be applied with the values in Table 4. For example, if the occurrence likelihood of activity value belonging to 10 stations is calculated, this probability value is equal to the sum of the occurrence likelihood of the activity value belonging to station 14 and the probability of occurrence between activity values of 2 stations. If this event is expressed with A, then from Eq. (15) the following expression is obtained.

$$P\left( {A \le \, 12610.98} \right) \, = P\left( {A \le \, 11518.14} \right) \, + P\left( {11518.14 \, < A \le \, 12610.98} \right)$$

(28)

As can be understood from this equation, the probability functions are in cumulative form.

Table 4 The probability of occurrence and probability of non-occurrence at 240th hour

Sample risk calculations are given in Table 4, taking into account the 240th hour after the accident. In this table, the g probability of non-occurrence values are obtained from Eq. 17 and given in column 6, and the risk values from in Eq. 19 in column 7. The probabilities other hours are presented in Appendix C (supplementary material). The data are tested with important pdfs in the literature to see the probability of occurrence values for 19 stations. According to R² values, the most suitable pdfs are Weibull, Lognormal and Generalized Extreme-Value pdfs (Table 5), and pdfs with lower R² values from these three distributions are excluded from the calculations. The variables of the three distributions are calculated as in Table 6 with an illustration graph in Fig. 2 for the 240th hour. At all other times, the risk variance against radioactivity can be seen in Appendix D (supplementary material).

Table 5 Correlation coefficients (R²) of probability distribution functions

Table 6 Distribution functions of radioactivity with respect to the calculated hours and their parameters

Fig. 2

The probability of occurrence (risk) graph against the 240th-hour activity

As for the R² values in Table 5, the generalized extreme-value distribution for all times is the most appropriate one. Another point to note is that the R² value of the generalized extreme-value distribution decreases with time. Furthermore, as time progresses, the risk caused by the fallout decreases.

A graph of the occurrence probability versus activity at 240th hour as an example is given in Fig. 2. One can see that the generalized extreme-value distribution explains the experimental data better than the other pdfs. For this reason, in the advanced risk analysis calculations, operations and interpretations are made by considering the generalized extreme value pdf. In Table 7, the risk, probability of occurrence and probability of non-occurrence values are calculated according to the generalized end-value distribution at the 240th hour and the station numbers are also given.

Table 7 The probability of occurrence (risk) and probability of non-occurrence (confidence) according to the generalized extreme-value distribution at 240th hour

For instance, if the k, μ and σ parameters at 240th hour of generalized extreme-value distribution in Table 5 and the activity value (11,518.14 Bq/kg) at 14th station in Table 7 are substituted into Eq. (23), then r_gev is obtained from Eq. (29) (please refer to 4th column in Table 7). All the changes in other times are given in Appendix E (supplementary material).

$$\begin{aligned} r_{gud} & = 1 - F(\left. {11518.14{\kern 1pt} {\kern 1pt} } \right|{\kern 1pt} {\kern 1pt} {\kern 1pt} 1.552193,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 22590.07,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} 19336.03) \\ r_{gev} & = 1 - \exp \left( { - \left( {1 + 1.552193\frac{{\left( {11518.14 - 22590.07} \right)}}{19336.03}} \right)^{{ - \frac{1}{1.552193}}} } \right) \\ r_{gev} & = 0.983700 \\ \end{aligned}$$

(29)

Kriging methodology

The Kriging is an interpolation method developed by South African mining engineer Krige in the early 1960s and it is a local estimation technique in a region that provides the best linear objective estimate of the unknown characteristics. If one desires to make an estimate at point x₀ for f(x) measurements taken against a value x in a region, it can be expressed as follows.

$$f\left( {x_{0} } \right) = \sum\limits_{i = 1}^{n} {w_{i} \left( {x_{0} } \right)f\left( {x_{i} } \right)}$$

(30)

In this equation, x₀ is the estimation point, f(x₀) is the estimation value, and w_i(x₀) are the weights, which are obtained from one of the covariance or variogram techniques [421]. The reason why Kriging differs from the classical linear regression is that the changes are not independent, and observations for the Kriging technique assume random sampling [422]. In this study, the Kriging method is used to obtain the surface maps of the ¹³¹I radioactivity, probability of occurrence and non-occurrence of the activity.

Kriging method results

The activity distribution map for ¹³¹I after 240 h from the Fukushima accident using the Kriging method [423] is given in Fig. 3. The activity is in exponential decay with time, and hence, the simulation change is better observed at each hour.

Fig. 3

¹³¹I radioactive fallout map at the 240th-hour

In this study the half-circle area at sea is banned. Entrance and exit to this area are closed immediately after the accident. The data from MEXT correspond to the time when this region is forbidden (restricted). After the accident, only dose measurements are made with no activity measurements. The activity prediction maps obtained for all other prospective times are given in Appendix F (supplementary material). In addition, motion simulations of 60 days of activity change in ¹³¹I are given in .mp4 format, Iodine131_Activity_Animated_Simulation_1.mp4 (Appendix G, supplementary material) file. When all of the maps in Fig. 3 and Appendix F (supplementary material) are examined, it is seen that ¹³¹I activity values are high in the north-west and south part of Fukushima. In this case, it can be said that the ¹³¹I core is moved north-west and southward after the accident. The map and animated simulations are available for 1440 h, or 60 days. Due to the short ¹³¹I half-life, the activity value after 60 days has dropped to very low levels and experts have not been able to measure ¹³¹I after the 60th day. As seen in the simulations, the activity values decrease rapidly with time progression.

The probability of occurrence according to the generalized extreme-value distribution, using activity values 240 h after the Fukushima accident is shown in Fig. 4 and the probability of absence in Fig. 5. The maps of these possibilities for all other times are given in Appendices H and I (supplementary material). The motion simulation files, in which the change of the event (activity concentration) and the absence of change after 60 days are estimated, are also given in .mp4 format in the form of Probability of Occurrence.mp4 (Appendix J, supplementary material) and Probability of Nonoccurrence.mp4 (Appendix K, supplementary material) files. These simulations allow seeing radioactive fallout as a whole for given hours.

Fig. 4

Spatial variation of probability of occurrence (risk) according to the generalized extreme-value distribution corresponding to the activity values at 240th-hour

Fig. 5

Spatial variation of the probability of non-occurrence (confidence) values according to the generalized extreme-value distribution corresponding to the activity values at 240th-hour

Looking at Fig. 4 and all of the maps in Appendix H (supplementary material), the probability of occurrence value is close to 1 in the north and south-west part of Fukushima. In other words, it defines the probability of occurrence in regions with low activity and where repetition of these possibilities is the greatest. On the other hand, regions with high activity seem to protect these conditions. This is also radioactive emissions reporter from the reactor on a continuous basis during the period studied. In motion simulations that change the probability of occurrence (Probability of Occurrence.mp4, Appendix J, supplementary material), the areas where activity decreases as time elapses in the accident area can be seen clearly. These simulations allow one to see prospectively whether or not radioactivity is in an environment.

Looking at the whole of the maps in Fig. 5 and Appendix I (supplementary material), the probability of non-occurrence of the activity in the north-eastern and south-western part of Fukushima is close to 0. This is in line with the assessment made for the activity maps. Since the activity values in these regions are close to 0, the probability of non-occurrence values approaches 0. The probability of non-occurrence decreases with time as it can be seen in moving simulation (Probability of Nonoccurrence.mp4, Appendix K, supplementary material).

Conclusions

Nuclear energy is one of the world’s important energy sources, and its production is relatively cheap provided that the energy needs of nuclear power plants are supplied properly. Such a supply reduces the capital spent on energy production and increases purchasing power. This leads to an increase in the welfare level of any country, where nuclear energy investments exist. On the other hand, nuclear energy brings some risks in the event of a possible accident. However, it is possible to reduce the effects and risks that arise as a result of nuclear power plant accidents by employing scientific methodologies. Nonetheless, it can be said that nuclear energy is one of the sources with the lowest risk ratio in terms of both clean energy production and profit/loss balance, compared to other energy alternatives and the risks inherent in people’s daily life. Through “Radioactive Waste Management”, the possible risks that may arise are reduced to a minimum level. Nowadays, “Radioactive Waste Management” enables people to be affected radioactively at the minimum level through audits conducted by many organizations, especially by the International Atomic Energy Agency (IAEA). About 151 countries belonging to IAEA carry out Radioactive Waste Management, published by IAEA. At this point, countries are affected in their daily lives at a lower level of radiation. In addition, it is aimed to minimizing damage for the case of possible accidents. There are many studies for each stage of Radioactive Waste Management. Particularly after the Chernobyl accident, countries have begun to take strict measures, seeking answers to the questions “How should radioactive waste management done in the case of chaos?” Models have been proposed for radioisotope distributions in air, groundwater, sea, river, and soil. A realistic risk analysis model for radioactive fallout to be released after a possible accident will keep radioactive waste harms at the optimum level. On the other hand, this research also contributes considerably to Radioactive Waste Management and Radioactive Pollution Prevention Scenarios.

When FDNPPA was examined, it was observed that the results were generally non-linear, because radioactive materials that are emitted by atmospheres and other environmental systems are influenced in many ways from atmospheric pressure to the humidity of the environment. Artificial intelligence techniques such as artificial neural networks and fuzzy logic can be used in future models for similar studies. In other respects, chaotic calculations provide effective results for non-linear studies and they can be applied as other different steps for this and similar studies. After FDNPPA, the activity of the short half-life ¹³¹I isotope was taken at 19 different stations at different times. The fact that activity measurements are not synchronous in a research area is the main reason why the distribution of radioactivity in that area cannot be determined as a whole. If this research is to be carried out with a long half-life radioisotope, such as ¹³⁴Cs or ¹³⁷Cs, the daily, weekly or even monthly change in activity would be almost constant or would show a linear change. In this case, a single scattering graph coupled with risk assessment is sufficient for long-life radioisotopes. The change in activity values ​​for the short half-life of ¹³¹I (approx. 8 days) is shorter, but at some stations (stations 3, 6, 7, 16, and 19), measurements take long intervals, as simultaneous measurements are not taken from the stations. In this case, model curves are recommended to each station for accurate results in a risk assessment at a given time. The studied parameter is activity and the variation is visibly exponential, and hence, this model is obtained by the least squares method. In this study, a risk analysis method is developed to determine the risk levels of non-asynchronous radioactivity data. The power of Least Squares Method in obtaining simultaneous data is percieved. After getting the risk values of the simultaneous data, it is decided that the pdf, which best describes these risk values, is the Generalized Extreme-Value Distribution (GEVD). In the literature, GEVD is a type of distribution proposed for the data under a sudden development of chaotic conditions. The activity a value of the ¹³¹I radioisotope is released after the FDNPP accident, which is a chaotic situation, and they are expected to comply with GEVD. As the time progresses, the activity values of the stations change, but the suitability of GEVD remains unchanged. Hence, the activity risk values in the GEVD are equivalent to saying that the GEVD is a characteristic of the activities in this region. It is then expected that as the value of activity decreases, the appropriateness of the GEVD decreases. As a result, for Fukushima, it can be said, “the risk distribution according to the position of ¹³¹I corresponds to the GEVD”.

Changes in the ¹³¹I contamination probability to occur and not to arrive are determined for the next 60 days after the accident. These possibilities are shown on maps and as simulation animations, which prospectively made it possible to glimpse the possibility of radioactivity being in an environment or not. The simulation method gives instant information for the risk levels that the corresponding radionuclide generates leading to a strong belief that it can be easily used in other similar studies.

With the methodologies proposed in this study, if the examined radionuclide has a long half-life, it can display moving or still simulation images; sampling areas can be expanded to reach global measurements in terms of both distance and duration. Especially, moving simulations help to facilitate the interpretation of the case/event of the scientists, who can work on the advanced subject. With this simulation method, it is possible to observe the spread of radioactivity over ground layers, as well as the propagation of atmospheric radioactive fallout.