Jeffrey LewisMaking Yield Estimates

Update | James says I misrepesented his method, which I may well have done.  He is going to post a comment that explains his method in better detail than I have apparently done.

Apologies that it took so long to post the companion piece on yield estimation to my Foreign Policy column, North Korea’s Big Bang.

The good news is that there are now several yield estimates that help illustrate the point I want to make: Early yield estimates are very rough approximations.

Source Equation Mb Best Min Max
Acton 5.1 4 15
Aster Mb=4.05+0.75 log (W) 5.1 20 15 40
NORSAR 5.0 10
BGR 5.2 40
ROK MND 5.0 6-7

You will undoubtedly notice the very large range.  I still think the best thing is to say the event was “several kilotons” or “on the order of ten kilotons” (as opposed to 1 or 100.) But the most important information to convey is that these are approximate yields based on a well-understood but rough method.

Some of the difference is explained by differences in estimating the magnitude of the body wave generated by the explosion, which ranges from 5.0 to 5.2.

But a much greater discrepancy arises from the equation that is used to model the geology of the test site and the resulting relationship between the size of the explosion (yield) and body wave (Mb).  This equation is normally given in the form of Mb = A + B Log (Yield).

This equation differs from test site to test site, and while a good fit can be achieved, it is far from perfect.  Here, for example, is some data from the then-Nevada Test Site (now Nevada National Security Site) that is used in the foundational study by Nuttli (1986) to give us the canonical equation: Mb = 4.05 + 0.75 log (W), where W is yield.

Reported Yield (W) (in kt) Reported Mb Expected Yield (from reported Mb) Expected Mb (from reported W)
35 4.8 10 5.2
16 5 18 5.0
21 5 18 5.0
65 5.4 63 5.4
70 5.4 63 5.4
71 5.4 63 5.4
105 5.5 86 5.6
80 5.5 86 5.5
110 5.6 117 5.6
85 5.6 117 5.5
150 5.7 158 5.7
250 5.8 215 5.8
220 5.8 215 5.8
300 6.1 541 5.9
1300 6.2 736 6.4
825 6.3 1000 6.2

Reported yield and reported Mb are the actual data.  “Expected yield” is what one would expect based on the equation given the reported Mb. “Expected Mb” is what one would expect based on the equation given the reported yield.

So, for example, the 4.8 Mb event had a reported yield of 35 kilotons — much greater than the 10 kilotons that would expect based on the equation.  Several events with different yields have the same Mb.  There is nothing wrong here, it’s just that the physical relationship is an approximate one. A usefully approximate one.

Now, Nevada is not a perfect example — seismic waves don’t propagate all that well at the Nevada National Security Site.  As a result, the yields for a given Mb in Nevada are a lot higher than in North Korea, which may lead to some head-scratching.  (Blame the terroir!)  Also, the data is noisier than it would be if we could make the model out of reliably reported North Korean events.

The major take-away is that while it’s a good fit, the standard method is not all that precise.  In the Nuttli paper, he had a standard deviation of 0.2 Mb units.  That’s quite a lot.  (Later work seems to suggest that corrections for bias can get that number down to about 0.1 Mb unit.)

It is very hard to know what the proper relationship is for the North Korean test site.  James Acton did something rather clever, assuming that the 2006 and 2009 tests were at the same depth, which allows him to assume that B=1.  Fitting a line to two data points does not inspire huge confidence, especially when we don’t know the yield, location, and depth of burial, but it is better than not doing it at all and probably more appropriate than using Nuttli’s equation for Nevada or Murphy’s equation for Semipalatinsk, Lop Nor, and Pokhran.  The Chinese seismological community takes a similar approach, using Mb = 4.25 + 0.75 Log W for events of more than 1 kiloton — the Bowers et al equation for Novaya Zemlya — based on three chemical explosions detonated in 1998.

As James is careful to point out that, these are back-of-the-envelope calculations that are used to help us start sorting through what this might mean.  So, for example, we have a general sense of whether the test was successful (seems so) and whether it was a thermonuclear device (seems not, with some qualifications).

Competent seismologists — i.e., not policy hacks like myself or lapsed physicists like James — will come along and make much more sophisticated estimates. This will take some time and, I am sorry to tell you, almost certainly fail to produce a definitive result.  Debate still exists about the yield of the 2009 event.  The relevant literature illustrates a number of more sophisticated ways to estimate yield, none of which agree.  While the US intelligence community placed the size of the 2009 test at approximately 2 kilotons, open-source estimates — including work done at Livermore and Los Alamos — range from 2 kilotons using wave-form coda to more than 5.7 kilotons using hydrodynamic calculations and satellite images.  Nice to see Livermore and Los Alamos can still disagree.  I would write a literature review, but I am not a seismologist and I find this stuff very, very dense.  Feel free, in the comments, to do so, and to add any corrections.

So, for the DPRK test, my advice is for everyone to take a deep breath.  The explosion was clearly bigger than the last.  There is no harm in making rough yield estimates, provided we are all as careful as James.  But let’s not take any of this too seriously just yet.


  1. James Acton (History)

    For the record, I did NOT assume that the 2006 and 2009 tests were at the same depths. Here’s what I did do…

    For testing at fixed depth mb = A + log Y (i)

    Prior to the test, satellite imagery indicated that either North Korea would test in the same tunnel as in 2009 or in a new tunnel that was somewhere between the 2006 and 2009 tunnels in depth.

    I therefore obtained an upper bound for the yield of the 2013 test by assuming it was at the same depth as the 2009 test. The yield of the 2009 test (mb = 4.7 USGS) appears to have been between 2 kt and 7 kt. This implies the constant A from formula (i) is between 3.9 and 4.4 for the 2009 tunnel.

    I obtained a lower bound for the yield of the 2013 test by assuming it was at the same depth as the 2006 test. The 2006 test (mb = 4.3 USGS) probably had a yield between 0.7 kt and 1 kt. For the 2006 tunnel, A is, therefore, between 4.3 and 4.5.

    A very conservative approach to estimating the yield of the 2013 test is, therefore, to use mb = 3.9 + log Y for an upper bound and mb = 4.5 + log Y for the lower bound. Given mb = 5.1 (USGS) for the 2013 test, I obtain a yield range of between 4 kt and 16 kt.

    Seismological evidence released this morning indicates that, in fact, North Korea re-used the 2009 tunnel. Assuming the 2009 and 2013 tests were at about the same depth, I now reduce my yield range for the 2013 test very slightly to between 5 kt and 16 kt.

    Significantly, however, regardless of the uncertainty in absolute values, it now appears that the 2013 test produced a yield 2.5 times bigger than the 2009 test.

    Finally, I note that any formula of the form mb = A + 0.75 log Y is unlikely to give a good estimate as North Korea appears to be much bigger overburdens (given the different rock type) than the United States or Russia did.

    James Acton

    • rwendland (History)

      Some US and UK weapons had primaries in the 0.5 kt range, possibly down to 0.33 kt. So I suppose it is just about conceivable that the NK 2006 test was completely successful, rather than a fizzle, and 2009 was boosted a bit – but that would have been very ambitious for a first weapon.

      UK declassified docs show the WE.177A, as designed, had a 0.5 kt primary, boosted to 10 kt. But a later doc considering the replacement of WE.177A has a hand-written note that it was 1/3 (0.33) kt, boosted to 10 kt. Take your choice on which is more accurate. Another UK document says Lance had “three yields between .5 and 50 kt”.

    • rwendland (History)

      … whoops, my reply above is intended to reply to comment @Cthippo below, not James above. Clicked wrong reply link!

  2. Cthippo (History)

    The responsible reporters, including you, Jeffrey, are saying “This is preliminary, but I’m wondering preliminary to what?

    North Korea isn’t going to release yield data that anyone is going to believe, even if we get noble gas measurements that reveal what the bomb was fueled with, they won’t say anything about the yield (unless it was a fizzled two-stage device).

    I think we have to assume that the North Koreans are working on a primary for a two or three stage weapon because, well, why wouldn’t you? The primary is the hard part, get that to work and the secondary and radiation case are easy. Even the low end estimates in your table above are probably enough to ignite the secondary, but the more the merrier in terms of safety factor. My understanding is that the US primaries are in the 1-5 kT range, but have been extensively tested so the weaponeers know that they’ll make that yield every time.

    Something else to think about in terms of materials inventory. We always talk about the 2-6 KG of Pu or U235 for the pit, but how much does the spark plug for the secondary require and can you use U235 for that? I’ve always heard it referred to as Pu, but I don’t know if it has to be.

    • Jeffrey (History)

      I am an academic with a blog. I imagine reporters would be offended by the comparison.

    • Cthippo (History)

      Perhaps they would, but having seen some of the reporting on this I value the fact that when you say something it’s usually correct.

    • kme (History)

      Sidebar: If you’re talking about ensuring the bomb goes boom, isn’t it really an “unsafety factor”?

  3. Bradley Laing (History)

    —If I were writing a fictional version of this, I’d have the Evil Supreme Leader say: “The weapons must be the same yield as those the Americans used in 1953 to force the cease fire on us! Nothing else will do!”

    —But that wouldn’t work to determine how big the design the North Korean government is working on, in reality.

    —Would it?

    • Sineva (History)

      Bradley you could be the next tom clancy,and I`m sure the cold war era “experts” guestimated using far flimsier and crazier “data”

  4. David Watson (History)

    Any ArmsControlWonk thoughts on North Korea’s ongoing renovation of the Tonghae launch centre? 2 space launch centres seems a bit much for one poor country…

    • Jeffrey (History)

      The North Koreans have been pretty clear that we’re in for another launch in the near future.

    • rwendland (History)

      My guess is that they want 2 to support both polar and near equatorial orbits. They probably do not want to launch east over their land from Sohae. eg Yongbyon is 91km away on 80 degrees, which is near the trajectory to overfly the almost-gap between Japan’s main island and Hokkaido, which I think they aimed to overfly on past eastern launches from Tonghae.

  5. Stephen (History)

    Would China be thinking about a change of management right now?

    Or might China take the view that NK with nukes will be less paranoid and therefore more stable?

    • Cthippo (History)

      China finds the thought of a stable, if nuclear armed, North Korea much more comforting than one which may implode and result in US or South Korean troops on it’s border.

      I think we need to keep this in perspective. A nuclear weapons state tested a bomb. It’s really no different than if any of the other NWSs had tested, and really doesn’t change the political calculus. It keeps us wonks interested arguing about technical issues, but in the grand scheme of things changes nothing.

  6. Peter Hayes (History)

    “If I were writing a fictional version of this, I’d have the Evil Supreme Leader say: “The weapons must be the same yield as those the Americans used in 1953 to force the cease fire on us!”

    Myth: we used nuclear threat to compel the North Koreans and the Chinese to stop the fighting. These threats failed. In fact, Kim Il Sung took extra time to build the Armistice signing building (now just north of Panmunjon) to make the point he wasnt being compelled to do anything he didn’t want to do. In that extra time, scores of thousands died as the war raged on.

  7. Rob Goldston (History)

    4.2 Mb -> 4.7 Mb -> 5.2 Mb
    1 kt -> 6 kt -> ?

    • Jeffrey (History)

      Assuming we have any confidence in your estimates of 1 and 6 kt, which we don’t.

    • Rob Goldston (History)

      There seems to be some consensus around the 1 and LLNL says > 5.7. Some of the estimates you mention include 40 for 2013. Just sayin’ we shouldn’t exclude the obvious, which is that this test worked quite well, and delivered what NK promised… “a higher level”.

    • Jeffrey (History)

      It is worth noting that the official US estimates are less than kiloton for 2006 and 1-2 kilotons for 2009.

      Having said that, I don’t exclude a higher yield for 2009 and a much higher yield for the 2013. I continue to think it is premature to exclude possibilities, given the uncertainties. I think it probably wasn’t a Teller-Ulam device, though with the important caveat that the Indians claim to fired one and gotten 43 kt despite seismic data that would suggest the yield was much lower.

    • George William Herbert (History)

      The problem is, the uncertainty bounds change the potential risk and military utility quite significantly.

      Some might indicate, for example, testing of a Teller-Ulam device.

      So … the uncertainty is rather annoying.

    • OT (History)

      Uncertainity is also cheapest deterrence. They do their best to maximize it. Why tinker with sophisticated design, when you can use surest method to produce seismic waves, and let the wonks stir the pot until they see missile ready uranium primary in the hole.

  8. FOARP (History)

    I propose asking XKCD for an estimate. . .

  9. ulan (History)

    according to CAS-IGG, the yield is 8.08 kt.

  10. ulan (History)

    personally I think there are basically 2 possibilities:

    1. the test failed, if the goal is a TN-device, with 6-8kg HEU primary and 100kt expected yield.

    2. the test succeed, if the goal is similar to CHIC-4

  11. Ara Barsamian (History)

    If it would have been HEU, you could be sure DPRK would trumpet it very loudly.

    I guess they did not have enough HEU material for this test, or they’re hoarding it for another test.

    As I mentioned it before, imploding HEU is much easier than Plutonium, and any retard should be pretty confident of getting a big bang out of it; just pack it with enough (100 to 200kg)high explosive.

    The yield “smells” of something like 8 to 16 kT, which again, if using Plutonium, would be challenging because of pre-detonation. Miniaturizing it for an ICBM warhead would also be an engineering feat because of heat constraints in addition to pre-detonation, and fitting everything in a RV…

    If they are thinking about a primary in a TN, you are talking about a minimum of about 10kT…that’s another story, with its own hassles and challenges.

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