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An objective nuclear accident magnitude scale for quantification of severe and catastrophic events

12 December 2011

Introduction and summary

Deficiencies in the existing International Nuclear Event Scale (INES)[1] have become clear in the light of comparisons between the 1986 Chernobyl and 2011 Fukushima Daiichi nuclear power plant accidents.[2–4] First, the scale is essentially a discrete qualitative ranking, not defined beyond event level 7. Second, it was designed as a public relations tool, not an objective scientific scale. Third, its most serious shortcoming is that it conflates magnitude with intensity.

I propose a new quantitative nuclear accident magnitude scale (NAMS). It uses the earthquake magnitude approach to calculate the accident magnitude M = log(20R), where R = off-site atmospheric release of radioactivity, normalized to iodine-131-equivalent terabecquerels. In NAMS the observed frequency-magnitude distribution of 33 well-quantified events over the past 60 years follows an inverse power law, as with earthquakes,[5] but NAMS highlights four exceptional accidents that are greater by 2–3 orders of magnitude than the next largest. These are, in decreasing order of severity, Chernobyl, Three Mile Island, Fukushima Daiichi, and Kyshtym. Such catastrophic accidents can be expected to occur every 12–15 years.

The problem with INES

The International Atomic Energy Agency (IAEA) developed the INES in 1990. It is based in part on a loose analogy with the logarithmic earthquake-magnitude Richter scale, in that one unit difference in event level between 4 and 7 corresponds approximately to a factor of 10 in amplitude. Despite its reference to decade threshold values for off-site radionuclide release for discriminating between levels 4 through 7, the INES is essentially a discrete qualitative ranking. A true location-specific intensity scale measures exposure at a particular time and place due to an accident. Figure 1 shows a popular representation of the scale as a pyramid.

The IAEA, although created under the aegis of the United Nations, is a nuclear industry trade association whose aim is to promote civil nuclear power. Article II of its statute reads, “The Agency shall seek to accelerate and enlarge the contribution of atomic energy to peace, health and prosperity throughout the world.” The IAEA has an inherent conflict of interest when it promotes nuclear power while prescribing its use, by purely voluntary guidelines.

Like any large organization, the IAEA cannot be considered impartial in its consideration of failures and shortcomings of what it seeks to promote. Because the INES “should be used as part of a communications strategy,” it is therefore more of a public relations tool than an objective scientific measure. The INES guidelines for how to inform the “target audience [of] . . . the media and the public” omit any mention of quantified radionuclide releases. The INES level, the nearest we get to quantification, has already been seen to be inadequate in the case of Fukushima Daiichi and Chernobyl, both currently assigned to level 7.

Accident magnitude à la Richter earthquake scale

Soon after Chernobyl but before the INES was first drawn up, Ken Hsü pointed out the likely analogy between the earthquake frequency–magnitude relation and nuclear accidents,[6] but he did not present any data. He suggested that factory-years should be the frequency measure and cost in Swiss francs the magnitude measure (he was for a long time a Swiss resident). Ali Mehmet Celâl Sengör has twice drawn attention to Hsü’s ideas,[7, 8] but they have never been developed, no doubt because both variables are too difficult to estimate in practice.

Seismology employs two distinct but complementary scales, magnitude and intensity. The intensity scale measures the impact of an earthquake on a population at a given place. Seismic intensity is thus analogous to radiological exposure, or dose, at a specific place. Overall, the INES level definitions (requiring a 200-page user manual) more resemble a seismic intensity scale than a magnitude scale. But since intensity varies with location, one level cannot fit a wide region.

The proposed new NAMS simply uses the estimate of the off-site atmospheric release R of radioactive material. The INES already defines approximate order-of-magnitude discrete threshold levels 4 to 7, based on radioactivity released to the atmosphere. The activity of different isotopes is normalized to 131I using the scale of dose equivalence specified in the INES; for example, plutonium-239 is 10 000 times more active for atmospheric release than 131I. Following the INES guideline that boundaries between the INES discrete levels be set every decade (power of 10) of the estimated R, the quantitative relation is M = log(20R), where R is specified in 131I -equivalent terabecquerels (TBq), and M is the NAMS magnitude. M is a real scalar, not an integer as in the INES. The constant 20 in the equation provides the equivalence between the INES and the NAMS.

The INES user’s manual[1] suggests that the boundaries between levels 4 to 7 be drawn approximately at 500, 5000, and 50 000 TBq. So level 4, for example, is in the range of 50–500 TBq. For simplicity, I have set the equivalence between the NAMS and the INES to be at the top of each decade range. So, for example,

500 TBq ≡ upper limit of INES level 4 ≡ NAMS magnitude 4.0.

Near-field and far-field release

A scale reserved for far-field releases of radioactivity, the NAMS is defined only as atmospheric release, like INES levels 4-7. Near field means releases confined within the installation buildings, but it includes direct radiation from a confined source that penetrates walls to the environment immediately outside the installation (see, for example, the Tokaimura 1999 criticality event discussed in the appendix).

With very few exceptions, far-field water contamination is ignored in the present compilation. Exceptions are the evaporation of Lake Karachai near Chelyabinsk in 1967 and the resuspension into the air from contaminated water during the 1955 Sellafield accident. Those can be classed as indirect atmospheric contamination.

Data sources and their compilation

The vast majority of descriptions of nuclear accidents are found only on the Web. My sources also included peer-reviewed journal papers and official government reports. I treated all purported radioactive release estimates with a certain amount of suspicion, and, where possible, I weighed the conflicting claims. This document and its appendix include extensive reference lists with web links, so that readers can view those sources and form their own opinions.

Compilations of international nuclear incidents for the past 60 years are available on many websites, but few descriptions provide a quantitative estimate of radioactive release, and still fewer an estimate of the INES level. Chris Winter has compiled a fairly complete list of nuclear accidents of all types,[9] but his list is probably biased toward military accidents. One INES level-4 French accident is omitted, as are events after mid 2010, including Fukushima Daiichi. He assigns his own INES levels, where none previously existed, to all 985 events. Wikipedia lists civilian and military nuclear accidents separately; a combined nuclear and radiation accident page is incomplete and biased toward US incidents. The civilian list is also incomplete, for example, in its reference to Three Mile Island. In March 2011, the Guardian published a list of nuclear power plant accidents.[10] Wikipedia and other websites (see, for example, references 11 and 12) discuss the more serious individual accidents of INES level 3 and greater.

I have removed medical-related accidents from the new classification and retained only accidents that involve civilian power installations and civilian or military nuclear fuel processing or reprocessing. I omit detonations of nuclear weapons, accidents involving airplanes carrying nuclear weapons, and accidents involving nuclear-powered ships or submarines. In addition I omit long-term leaks to the environment because they are difficult to classify as discrete events. Events involving multiple reactors at one site are treated as one accident.

Table 1, which summarizes the data, is restricted to the 33 events to which an INES level has been assigned or can reasonably be estimated and for which the off-site atmospheric release has been or can be estimated. Calculations of the radiological dose equivalences for atmospheric releases are taken from table 2 of the INES manual.[1] Although only INES levels 4-7 have off-site effects, I have included events classified as INES level 3 to ensure that no possible off-site events have been omitted. I have not examined events of INES level 2 or below. However, several events at INES level 3 turn out to have magnitudes well below 3, so all the graphs shown below include events down to M = 2.0.

The data sources for the individual accidents and their magnitude calculation are given in the appendix. Eight events (two of INES level 5, five level 4, and one level 3, shown at the bottom of table 1) evidently had significant off-site releases, but the releases remain unquantifiable. Those are also discussed in the appendix, together with an anomalous group called criticality accidents.

The data compiled in table 1 are unlikely to have excluded any recorded event of INES level 4 or greater, but some unauthorized discharges, especially from the early era of the nuclear industry, could remain undocumented. Those putative hidden events would most probably have occurred in the UK, the former Soviet Union, the US, and France, given those four countries’ track records of secrecy and official denial.

Magnitude and frequency

It may seem counterintuitive, but 33 events are sufficient for an analysis to see whether any predictions can be made, as opposed to simple description and classification. Taking the earthquake magnitude analogy further, we examine the nuclear accident equivalent of the Gutenberg and Richter frequency–magnitude relation:[5]

logN = a − bM,

where N is the number of earthquakes greater than or equal to magnitude M occurring in a given area over a given time interval, and a and b are constants, usually determined by a best fit to the data, when logN is plotted against M. Such a graph is doubly logarithmic because the magnitude has been defined logarithmically. Note the ingenious part of Gutenberg and Richter’s definition: 'greater than or equal to.' That cumulative frequency enables more information to be extracted from small data sets.

The fact that earthquake data can be fitted well to a straight line with slope b reflects the fact that such earthquake distributions empirically obey an inverse power law between frequency and magnitude; this, in turn, suggests that earthquake distributions are fractal.[13]

Figure 2 plots the frequency-magnitude relationship for nuclear accidents. The white circles show the log-frequency versus INES-level relationship of the 33 events of INES levels 3–7, for which both the INES level and the magnitude can be estimated. The distribution is remarkably linear, with a negative slope (the b value, in seismological parlance) of about 0.3. But a surprising difference emerges when we look at the NAMS magnitudes (the black circles in figure 2) in the same manner. Between 2.0 ≤ M ≤ 5.2 there is still a strong linear relationship, with a b value of about 0.2. But the four biggest events now plot as a group distinct from the lower-magnitude group—a gap in magnitudes exists between 5.2 and 7.3—in which no events have occurred. So the overall frequency-magnitude distribution is clearly bimodal, suggesting that catastrophic accidents behave in a qualitatively different way from lesser events.

There are nine NAMS events of M < 2.0 that are not shown in Figure 2. Their absence explains why the left-hand side of the NAMS trend does not attain the same value of 1.519 (log1033) as does the INES plot (white circles) that uses all 33 events.


What is the predictive power, if any, of the new magnitude classification? I have refrained from fitting straight lines to the log–log graphs shown in figure 2; however, the revised scale suggests that about 10 events of magnitude 4 or greater will occur every 60 years, the period covered by the data. Any prediction assumes that the dataset is statistically stationary, so that the past is indicative of the future. Recurrence times using the two scales are compared in table 2.

The INES defines only two level-7 events in 60 years, and three events of level 6 or greater. INES levels 3 and 4 have shorter recurrence periods than NAMS because of the coarser classification of the former. Level 6 and magnitude 6 cannot be compared because no events between magnitude 5.2 and 7.3 have occurred. INES level 7 overestimates the recurrence time by a factor of 2.5 relative to NAMS and underestimates the severity of some events due to the ceiling at level 7. The comparable recurrence time of 12–15 years between M = 5.2 and 7.3 in figure 2 suggests that if an accident approaches magnitude 5, it is likely to increase by two further orders of magnitude—that is, to a full-blown runaway catastrophe. A magnitude 7 or greater accident is to be expected every 12–15 years.

It may be argued that the past 60 years are not a reliable guide to future accidents, because more modern reactor designs are inherently safer; however, an aging worldwide reactor fleet might be expected to become less reliable as time goes on. Furthermore, the accident record, reviewed by Benjamin Sovacool,[14] implies that new designs are more risky, not less so.

To remove the bimodal distribution on the assumption that it is a data artifact, we would have to reduce the magnitudes of the three biggest events below Chernobyl to about 6.9, 6.3, and 5.7; for example, we could arbitrarily reduce the magnitudes of Three Mile Island by 1.0, Fukushima by 1.2, and Kyshtym by 1.6, as shown in figure 3. I consider this scenario to be highly unlikely, as it requires that each be reduced by than one order of magnitude. Equally unlikely, as discussed above, would be the infill of the current gap in magnitudes by major accidents not yet documented in the public domain.

Given that 20 of the 33 classified events in table 1 are Sellafield accidents,[15], is it possible that Sellafield is distorting the magnitude-frequency relation? To address that question, figure 4 shows the graph for just the 13 non-Sellafield events, but otherwise to the same scale as figure 2. The data are clearly much sparser, but the two-orders-of-magnitude gap remains. The roll-off (the flattening out of the data trend on the left-hand side) below level or magnitude 4 is presumably due to underreporting of non-Sellafield accidents in the INES range 3-4. It would be useful to have detailed reports and analyses, as Webb and coauthors have done for Sellafield,[15] to quantify accidents at the following major nuclear installations:

  • 1.  Chalk River, Ontario (Canada)
  • 2.  Saint Laurent, La Hague, Marcoule, and Tricastin (France)
  • 3.  Idaho National Laboratory, Rocky Flats, Oak Ridge, Los Alamos, Savannah River, and Hanford (USA)
  • 4.  Tokai (Japan)
  • 5.  Hanau (Germany)

Some of the above feature prominently in an IAEA report on significant incidents at nuclear installations.[16] The list is, of course, incomplete. So Sellafield is serving here as a proxy for many other undocumented accidents, which, if they were included in figure 2, would raise and steepen the main sequence part of the NAMS curve.


The NAMS highlights four exceptional accidents that are greater by 2–3 orders of magnitude than the next largest. Those are, in decreasing order of severity, Chernobyl, Three Mile Island, Fukushima Daiichi, and Kyshtym. Such catastrophic accidents can be expected to occur every 12–15 years. The INES, in contrast, fails to reveal that bimodality and underestimates the frequency of severe accidents.

The NAMS makes no predictions about impact and dose, for the same reason that earthquake magnitude does not necessarily indicate damage.

Liquid contamination should in future be included in the NAMS accident quantification. The problem here, which is beyond the scope of the present paper, is how to estimate the radiological equivalences for the various isotopes and to return a value for the magnitude, given also the variety of paths by which activity might eventually be ingested. Alternatively, a separate NAMS scale for liquid contamination could be devised. It would once again be analogous to seismic magnitude scales, where a particular scale is defined by how the magnitude is estimated. The off-site consequences part of the INES (levels 4–7) needs to be completely rewritten. As with seismological scales, decimal real numbers should be reserved for the NAMS magnitude M, and Roman numerals should be used for a proposed revised INES intensity scale based on dose at a specific place and time.

Finally, the NAMS magnitude of an accident having off-site consequences is simple to compute; all that is needed is the estimate of off-site releases of radionuclides. The NAMS can be corrected or updated instantly. It is easily understood by the general public, who are accustomed to earthquake magnitudes, and it is more honest than the INES level. But if the raw release data to enable a calculation are not being made available immediately after an accident, we need to ask why.


David Smythe is Emeritus Professor of Geophysics in the University of Glasgow, but now lives in France. He is currently engaged in persuading the UK government to abandon its plans for a high-level nuclear waste repository in West Cumbria, where, he contends, both the geology and hydrogeology are completely unsuitable. He was a founder and bass player of the Rezillos, Scotland's best-known pop group of the late 1970s, but now prefers to sing the solo tenor classical song repertoire.

Table 1

The 33 accidents of International Nuclear Event Scale (INES) levels 4—7 for which the magnitude M can be quantified. Eight further accidents of level ≥3 are appended, but for which the off-site release, and therefore M, cannot currently be quantified. Sources of the quoted INES level are the reference numbers in the last column (reference 0 = this paper; W = Wikipedia; A = Appendix).

Date INES level Location Release (TBq) M INES Ref.
1949-12-02 Hanford, WA, USA 289 3.8
1955-03-25 Sellafield, UK 1000 4.3 15
1955-07-14 Sellafield, UK 0.0002 -2.4 15
1955-12-08 Sellafield, UK 0.0001 -2.7 15
1957-09-11 Rocky Flats, CO, USA 7800 5.2 0, 9
1957-09-29 Kyshtym (Mayak), Russia 1,000,000 7.3 15, W
1957-10-07 Windscale (Sellafield), UK 1786 4.6 15, W
1961-01-03 SL-1, Idaho Falls, ID, USA 41 2.9
1961-06-19 Sellafield, UK 540 4.0 15
1965-01-20 Lawrence Livermore, CA, USA 259 3.7
1967-04-01 Chelyabinsk; Lake Karachai, Russia 5600 5.0
1968-05-01 Sellafield, UK 550 4.0 15
1969-03-05 Sellafield, UK 2.1 1.6 15
1969-05-11 Rocky Flats, CO, USA 10 2.3
1969-10-12 Sellafield, UK 2.3 15
1970-02-10 Sellafield, UK 2.0 15
1970-03-10 Sellafield, UK 18 2.6 15
1970-08-06 Lawrence Livermore, CA, USA 222 3.6
1970-11-29 Sellafield, UK 16 2.5 15
1971-03-19 Sellafield, UK 4.8 2.0 15
1971-12-15 Sellafield, UK 74 3.2 15
1972-12-06 Sellafield, UK 2.2 1.6 15
1973-09-26 Sellafield, UK 5.4 2.0 15
1979-03-28 Three Mile Island, PA, USA 3,700,000 7.9
1979-07-16 Sellafield, UK 3.7 1.9 15
1979-09-11 Sellafield, UK 130 3.4 15
1980-09-22 Sellafield, UK 2.0 1.6 15
1981-10-01 Sellafield, UK 0.9 1.3 15
1984-07-17 Sellafield, UK 2.9 1.8 15
1986-04-26 Chernobyl, Ukraine 5,200,000 8.0
1993-04-06 Seversk (Tomsk-7), Russia 3500 4.8 W, A26
2003-04-10 Paks, Hungary 360 3.9 W, A27
2011-03-11 Fukushima -Daiichi, Japan 1,592,000 7.5
Not currently quantifiable      
1952-12-12 Chalk River, Ontario, Canada     W, W
1969-10-17 Saint Laurent, France    
1972-12-21 ≥4 Pawling, NY, USA    
1975-11-28 Leningrad, Russia    
1977-02-22 Jaslovské Bohunice, Slovakia    
1980-03-13 Saint Laurent, France     W, W
1982-09-09 Chernobyl, Ukraine    
1983-11-01 Sellafield, UK     15

Table 2

Comparison of recurrence times for the INES and the NAMS magnitude scales.

INES level or NAMS magnitude INES Period (y) NAMS Period (y)
4 / 4.0
5 / 5.0 12
6 / 6.0 20
7 / 7.0 30 12-15
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