climateetc--2020-06-21--Did lockdowns really save 3 million COVID-19 deaths, as Flaxman et al. claim?

Key points about the recent Nature paper by Flaxman and other Imperial College modellers 1 ) The transition from rising to declining recorded COVID-19 deaths in the in 11 European countries that they studied imply that transmission of COVID-19 must have reduced substantially .
The study was bound to find that together the five government non-pharmaceutical interventions ( NPI ) they considered contributed essentially 100 % of the reduction in COVID-19 transmission , since in their model there is nothing else that could cause it .
2 ) The prior distribution they used for the effects of NPIs on transmission @ @ @ @ @ @ @ favours finding that almost all the reduction in transmission is due to one , or possibly two , NPIs with all the others having a negligible effect .
The probability density of the prior distribution at their median estimates of the effect on transmission of each type of NPI , which allocate essentially all the reduction in transmission to lockdowns , was many billion times greater than it would have been if the same total estimated reduction had been spread evenly across the types of NPI .
3 ) Which intervention ( s @ @ @ @ @ @ @ critically on the assumptions regarding the delay from infection to death .
When using their probabilistic assumptions regarding the delay from infection to death , a huge ( and highly improbable given other assumptions they made ) country-specific effect is required to explain the reduction in transmission in Sweden , where no lockdown occurred .
If delays from infection to death are increased by just three days , their model no longer finds lockdowns to have the largest effect , and a more moderate country-specific effect is required to explain the reduction in @ @ @ @ @ @ @ estimated relative strengths of different NPIs are also considerably affected by the use of an alternative prior distribution for their effects on transmission that does not strongly bias the estimation of most of them towards a negligible level .
They are also considerably affected by phasing in over a few days the effects of the two NPIs that seem unlikely to have had their full effect on their date of implementation .
5 ) It follows from the above that that study provides no information whatsoever as to the actual contribution from all @ @ @ @ @ @ @ , and nor does it provide robust estimates of relative effects of different NPI .
On 8 June 2020 , Nature published a paper ( Flaxman et al .
2020 [ 1 ] ) by modellers in the Imperial College OCIVD-19 response team .
Its abstract ends with : Using a counterfactual model , the paper also estimated the impact of interventions on deaths from COVID-19 in the 11 European countries studied , saying : The mainstream media publicised the ‘ 3 million deaths saved ’ claim , without critically appraising the paper @ @ @ @ @ @ @ caveat in the paper : In Imperial College ’ s press release Dr Flaxman ignored his own caveat , saying In this article I examine the main claim – that major non-pharmaceutical interventions ( NPI ) have had a large effect on reducing transmission of COVID-19 , to which the inferred reduction in deaths is attributable , with almost all the reduction due to lockdowns .
I show that this claim is strongly dependent on the assumptions made and is highly dubious .
The case of Sweden , where the authors find the @ @ @ @ @ @ @ moderately weaker than in other countries despite no lockdown having occurred , is prima facie evidence against the paper ’ s main claim .
How the effects of lockdowns and other interventions were estimated Flaxman et al .
employ a ‘ hierarchical Bayesian ’ statistical model .
It uses data on daily deaths ( up to 5 May 2020 , when two countries relaxed their lockdowns ) , the dates of imposition of five types of NPI ( school or university closure , case-based self isolation , public events banned , lockdown ordered @ @ @ @ @ @ @ estimates of the infection fatality rate , for each of 11 European countries .
[ 2 ] Using these data , the model infers what time profiles of the effective reproduction number ( R , the number of people whom an infected person in turn infects ) – and hence of new infections – would produce the best match between projected and recorded deaths for each country .
To do so it uses a simple model of epidemic growth and probabilistic estimates , common to all countries , of the time from infection @ @ @ @ @ @ @ ( that from a person becoming infected to them infecting others ) .
The assumed infection fatality rate ( IFR ) is common between countries for each age band , but reflects the age-structure of each country ’ s population .
It averages slightly over 1 % .
A separate initial value , R ( the basic reproduction number ) , of the reproduction number R is inferred for each country .
R then changes from R in stepwise fashion at the date of each NPI , which act multiplicatively with an equally @ @ @ @ @ @ @ Each country ’ s epidemic is seeded by a series of infections starting 30 days prior to a total of 10 recorded deaths .
[ 3 ] The model is described in more detail here , and is illustrated in Figure 1 , taken from Flaxman et al .
Fig .
1 .
Reproduction of Flaxman et al .
Extended Data Fig .
3 : Summary of model components The model uses no information on NPI ’ s except their type and their implementation date in each country .
NPI of each type @ @ @ @ @ @ @ multiplicative ) effect on R in each country .
Each type of NPI is treated identically .
As well as the five types of actual interventions , all first interventions ( whatever type ) are treated as an extra type of intervention , for each country occurring on the date of implementation of its first actual NPI ( almost always either self isolation or public events ban , and never lockdown ) .
Hence there are six NPIs with shared values for all countries .
In addition , a pseudo-NPI with a strength @ @ @ @ @ @ @ is treated as taking place on the same date as the last actual NPI .
These country-specific pseudo-NPIs allow for variation between countries in the effectiveness of the implementation of their NPI .
They are probabilistically constrained to be relatively small , making a country-specific effect large enough to cause a halving of R exceedingly improbable .
In all 11 countries the exponential growth in infections and deaths experienced early in the epidemics slowed and then turned negative , with infections and deaths decreasing .
This implies that in all 11 countries R @ @ @ @ @ @ @ , since the start of their epidemics .
In the Flaxman et al .
model the only factor that can cause R to decrease significantly is the effect of each NPI .
Therefore , the estimated overall effect of the NPIs in reducing R , and hence deaths resulting from COVID-19 disease , is bound to be very strong .
The only non-NPI factor that affects R in the Flaxman et al .
model is the reduction arising from the proportion of the population susceptible to infection ( set at 100 % initially @ @ @ @ @ @ @ individuals already infected by COVID-19 becoming immune to it .
This reduction is very small in their model , for two reasons : • they make the very unrealistic assumption that all individuals in a country are equally susceptible to COVID-19 and , if infected , are equally likely to infect others .
• the relatively high infection fatality rates they assume result in only very small proportions of countries ’ populations becoming infected in their model .
Therefore , their model has to attribute almost all the overall reduction in R to @ @ @ @ @ @ @ Flaxman et al.
, all of which are highly likely to have caused some reduction in COVID-19 transmission , and which between them may well have caused substantial reductions in R in all 11 countries , include : • population heterogeneity in social connectivity – which generates highly correlated heterogeneity in both susceptibility and infectivity – and in other factors determining susceptibility to COVID-19 • unforced changes in the behaviour of individuals as they adjust it to reflect COVID-19 risk • seasonal factors : infections by common coronaviruses peak in the winter and @ @ @ @ @ @ @ is well known by competent epidemiologists , the first of the above-mentioned factors causes R to diminish faster , potentially much faster , with the number of people who have been infected than if it were proportional to the number of people remaining uninfected , as assumed by Flaxman et al .
The other factors directly reduce R .
If follows that Flaxman et al.
’ s counterfactual case , which predicts ~3,200,000 deaths in the absence of any NPIs ( their ‘ counterfactual model ’ ) , is completely unrealistic , as @ @ @ @ @ @ @ saved by interventions .
It also follows that Flaxman et al.
’ s claim : may be invalid .
As shown here , due to population heterogeneity in susceptibility and infectivity the herd immunity threshold it is bound to be lower – quite possibly very substantially so – than if , as required for it to be ~70 % at an R of 3.8 , populations are homogeneous .
Flaxman et al.
’ s assertion that all the reduction in transmission ( i.e.
, the reduction in R ) was due to NPIs @ @ @ @ @ @ @ more people have been infected and become immune , is unsound .
Nevertheless , it seems quite likely that NPIs have had a significant , perhaps substantial , effect on R .
However , given the confounding effects of the other factors mentioned it is impossible reliably to estimate the total effect of NPIs on R and hence on deaths .
Even when making the unrealistic assumption that almost all the reduction in R was due to interventions , any allocation of that reduction between the NPIs is very fragile .
Flaxman et @ @ @ @ @ @ @ NPIs other than lockdown , writing : On their median estimates , lockdown caused an 82 % reduction in R , whereas no other NPI caused as much as a 1 % reduction in R .
While it would not be particularly surprising if such a drastic intervention as lockdown had had stronger effects than other NPIs , even if lockdown had a strong effect one would expect some other NPIs to have had a significant effect .
So how did Flaxman et al .
find that , remarkably , almost the entire @ @ @ @ @ @ @ ?
The answer , which turns out to be two-fold , shows that their finding is not credible .
Why Flaxman et al .
found almost all reduction in COVID-19 transmission to be attributable to a single intervention Flaxman et al .
use a subjective Bayesian statistical method .
I have repeatedly criticised this type of Bayesian method in the climate science field , but – probably due to its ease of use – it remains standard practice there and in many other fields .
A subjective Bayesian method requires prior probability distributions @ @ @ @ @ @ @ whose value is to be inferred .
These prior distributions are then modified by the likelihood function , which reflects how well the modelled deaths fit the daily deaths data at varying values of the parameters , in order to arrive at a ‘ posterior ’ probability distribution for the parameter values .
They use a common method of achieving this that results in a large number of quasi-random draws ( ‘ posterior draws ’ ) from the derived posterior probability distribution .
They represent the strength of interventions by a six dimensional @ @ @ @ @ @ @ the synthetic first intervention NPI ) , with the corresponding effect of intervention i ( i being 1 , 2,3 , 4 , 5 or 6 ) [ 4 ] on R being to multiply it by exp ( -alpha [ i ] ) .
The combined effect of all interventions is then to multiply R by exp [ - ( alpha [ 1 ] + alpha [ 2 ] + alpha [ 3 ] + alpha [ 4 ] + alpha [ 5 ] + alpha [ 6 ] ) ] [ @ @ @ @ @ @ @ the sum of the individual alpha values .
Their own posterior draws show a median value of the sum of the alphas of 1.75 , which corresponds to an 83 % reduction in transmission ( 1 – e−1.75 = 0.83 ) .
The prior distribution assigned by the authors to the strength of the reduction in R caused by each intervention is of particular concern .
Each of the six alpha values is assigned a gamma-distributed prior probability distribution ; a small offset is applied , so that the gamma-distributed values inferred initially @ @ @ @ @ @ @ a cosmetic feature .
[ 6 ] The authors write : That is not in fact true .
As the left hand panel of Figure 2 shows , their prior allows each intervention to decrease R by up to 100 % , but only to increase it by less than 1 % .
And the combined effect on transmission of all interventions ( right hand panel ) can only vary between –100 % and + 5 % .
However , since the trajectory of the deaths data is , on their assumptions , @ @ @ @ @ @ @ being found to strongly reduce transmission , the +5 % limit is of no real consequence .
Fig .
2 .
Reproduction of the upper panels of Flaxman et al .
Supplementary Fig .
3 : Cumulative distribution function F ( x ) of the prior for one intervention ’ s multiplicative effect x ( = e–α ) on transmission ( left ) or for the effect of all interventions combined ( = e–Σα ) ( right ) .
On the face of it , the combined effect of the six-dimensional joint alpha @ @ @ @ @ @ @ range in which the estimated reduction in R could fall ; it assigns a similar probability to a reduction in the range 40 % to 50 % and in the range 80 % to 90 % , for example .
However , that only looks at one aspect of the six-dimensional prior distribution .
If I take the sum of the six alphas to be 1.75 ( the median sum from their posterior draws ) and set them to be all equal , at 1.75/6 , their joint prior probability density is 0.0023 @ @ @ @ @ @ @ the alpha values to 1.70 and the remaining five to 0.01 , giving the same overall reduction in transmission , the prior probability density is 64.3 .
That means their prior distribution assigns a 28,000 times higher prior probability assumption to this case , where one type of intervention has a completely dominating effect relative to all the others , than to a case where the same overall reduction in transmission is caused equally by all types of intervention .
The reason is that the offset-gamma distribution used assigns a strongly increasing probability @ @ @ @ @ @ @ −0.008 , its lowest permitted level , favouring cases where the effect of all but one or two NPIs is estimated to be almost zero .
So it is unsurprising that they found a single intervention to be totally dominant .
The median individual alpha values in their 2,000 archived posterior draws are −0.007 , −0.007 , −0.007 , −0.007 , 1.699 and −0.006 .
So all interventions except lockdown were estimated to have a completely negligible effect .
The median ratio , across their own posterior draws for alpha , of the @ @ @ @ @ @ @ have been if in each draw the total effect of the intervention had been spread evenly across them , was in fact 392 billion to one !
It is not clear that the authors realised that the prior distribution they used very strongly favoured finding that most interventions had a negligible effect , and I very much doubt that any of the peer reviewers appreciated that this was the case .
Using the code and data accompanying the Nature paper as is , except with the 8,000 draws split between 4 not 5 @ @ @ @ @ @ @ I can accurately replicate Flaxman et al.
’ s findings , with lockdown accounting for almost the entire reduction in R ( Figure 3 ) .
Fig .
3 .
Effect of interventions on R in the base case , with all aspects of the model as per the original version ( that archived for the Nature paper ) .
The red First intervention estimate includes the effect of the synthetic first intervention NPI and so only applies for countries where the NPI concerned was the first to be implemented ; it should @ @ @ @ @ @ @ Mean relative percentage reduction in R is shown for each NPI ( filled circle ) together with the 95 % posterior credible intervals ( line ) .
If 100 % reduction is achieved , R = 0 and there is no more transmission of COVID-19 .
Sweden did not have a lockdown , but it still had a large reduction in R , albeit one not quite as large as the average for other countries .
So how did the model account for that ?
This is where the country specific factors , @ @ @ @ @ @ @ date of the last actual intervention and in effect are an addition to its alpha , come in .
The country specific factors are given an apparently small influence , being zero-mean normally distributed with a standard deviation that is itself zero mean normal+ distributed [ 7 ] with a standard deviation of 0.2 .
But for Sweden a value of 1.27 , in the far tail of the resulting distribution , was inferred .
The probability of such a large country factor arising by chance appears to be about 1 in 2,000 @ @ @ @ @ @ @ model does not adequately represent reality .
I investigated use of a prior distribution for the six alpha parameters that was essentially flat over the alpha parameter range relevant for NPI , both for each parameter separately and for the six-dimensional joint alpha parameter .
For technical reasons , rather than using a uniform distribution I chose an independent zero mean normal distribution with a standard deviation of 10 as the prior distribution for each parameter .
I hereafter refer to this as the ‘ flat prior distribution ’ , even though it @ @ @ @ @ @ @ range of interest ( approximately 0 to 2 ) .
I then ran the model using the same assumptions , but using the flat prior distribution rather than the original offset-gamma prior distribution .
Doing so should eliminate the previous strong bias towards finding that most interventions had almost no effect .
The resulting estimates of the effect of each intervention were as shown in Figure 4 .
The estimated effects of NPI other than lockdown all increase markedly from their near zero values when using the original prior , but the contribution @ @ @ @ @ @ @ 4 .
Effect of interventions on R : as in Fig .
3 , but with the flat prior distribution for alpha substituted for the offset-gamma prior distribution in the original model..
The country specific factor for Sweden was slightly less high than before , at 1.12 .
The probability of such a large country factor arising by chance appears to be about 1 in 900 ; still minute .
So , even when using the flat prior , the Flaxman et al .
model does not adequately fit reality .
The problem @ @ @ @ @ @ @ lockdown to account for the vast bulk of the total reduction in R , it can not adequately account for the reduction in R that occurred in Sweden , where there was no lockdown .
Why Flaxman et al .
found lockdown was the intervention that dominated the reduction in COVID-19 transmission I have explained why it to be expected , given Flaxman et al.
’ s choice of prior distribution for the effect of interventions on the transmission of COVID-19 , that a single type of intervention ( or at most two @ @ @ @ @ @ @ bulk of the reduction in R .
But why lockdown ?
The key here seems to be that lockdown was , other than in Sweden , on average imposed at a point in time that , allowing for the assumed probabilistic delay between infection and death , would result in deaths peaking at about the time that they actually peaked .
Also , the timing of lockdown , relative to the peak in recorded deaths , differed slightly less between countries that locked-down than was the case for most other interventions .
Flaxman @ @ @ @ @ @ @ the delay from infection to symptoms appearing and from symptoms appearing until death , with assumed mean values of 5.1 and 17.8 days respectively , and added them to obtain the infection to death delay values .
The 5.1 day delay from infection to onset of symptoms seems reasonable .
But the 17.8 days mean from onset of symptoms until death looks as if it may be on the short side for European countries .
Ideally , a separate onset of symptoms to death delay distribution would have been estimated for each country @ @ @ @ @ @ @ have been unable to find suitable European data .
They actually used a value estimated by Verity et al .
[ 8 ] ( also members of the Imperial College COVID-19 modelling team ) from just 24 cases in mainland China .
One of the peer reviewers suggested that the value Flaxman et al .
were using for the delay from onset of symptoms until death of ( in the originally-submitted manuscript [ 9 ] being reviewed ) 18.8 days , not 17.8 days , was rather short , writing : I therefore @ @ @ @ @ @ @ symptoms to death slightly .
I also took the opportunity to correct the dates used in the model inputs for school/university closure in Sweden and for self-isolation in Spain to those given in Flaxman et al .
Extended Data Figure 4 , which agree to those in their Supplementary Table 2 .
I found that adding 3 days to the infection to death delay , bringing the average onset of symptoms to death delay to ~21 days ( median 19.6 days ) – which is fully consistent with the peer reviewer ’ s @ @ @ @ @ @ @ problem of Sweden needing a very large country-specific factor .
That factor was then estimated at ~0.4 , to match the reduction in transmission in Sweden – still over twice as large as for any other country , but no longer statistically-inconsistent with their assumptions .
The resulting estimated effectiveness of the various interventions , using the authors ’ original prior distribution for alpha , is shown in Figure 5 .
Fig .
5 .
Effect of interventions on R : as in Fig .
3 ( original prior ) but with the @ @ @ @ @ @ @ days , and one intervention date corrected for each of Spain and Sweden ( see text ) .
School closure is now found to have a slightly stronger effect on transmission than lockdown .
This may seem rather unlikely in reality , but the model has no information to go on regarding the likely relative strengths of each type of intervention – it just knows when they were implemented in each country .
Other interventions are found to have almost zero mean effect , as is to be expected given the nature of @ @ @ @ @ @ @ the flat prior gives slightly different estimates of the effectiveness of the various interventions ( Figure 6 ) , with school closure not having quite as strong an effect as when using the original prior .
The effects of social distancing , and to a slightly lesser extent public events ban and self isolation ( one of which is generally the first intervention , so the red line applies to it ) , all cease to be negligible .
Fig .
6 .
Effect of interventions on R : as in Fig .
@ @ @ @ @ @ @ delay increased by 3 days , but using the flat prior distribution instead of the original prior distribution .
If the infection to death delay is increased by 5 rather than 3 days from Flaxman et al.
’ s assumed probabilistic magnitude – arguably still as reasonable as Flaxman et al.
’ s assumption – and the original prior used , the changes in the relative effectiveness of different interventions become even more marked ( Figure 7 ) .
Lockdown is now estimated to have far less effect than school closure , while @ @ @ @ @ @ @ .
The country-specific factor for Sweden becomes small .
Fig .
7 .
Effect of interventions on R : as in Fig .
5 ( original prior ) but with the infection to death delay increased by 5 days not 3 days .
When the flat prior is used instead , the estimated effect of school closure reduces while that of all other interventions increases ( Figure 8 ) .
Fig .
8 .
Effect of interventions on R : as in Fig .
6 ( flat prior ) but with the infection @ @ @ @ @ @ @ not 3 days .
Finally , I investigated the effects of phasing in certain of the interventions .
Flaxman et al.
’ s assumption that all interventions immediately have their full effect on their date of implementation is questionable .
It may not be too unrealistic for closing schools , banning public events and decreeing a lockdown , all of which it is feasible to enforce .
However , responses to self isolation advice and social distancing encouragement ( which both generally preceded a lockdown ) are more within the discretion of the @ @ @ @ @ @ @ take a little time to reach their final strength .
I examined phasing in over four days the effects of just those two NPIs , with their strength increasing evenly from 25 % on the date of implementation to 100 % three days later .
The result , using the original prior distribution for alpha and making a ~3 day increase in the delay from symptoms to death , is shown in Figure 9 .
The strength of the reduction in transmission attributed to lockdown reduces slightly compared with the no phase-in case @ @ @ @ @ @ @ increases .
Fig .
9 .
Effect of interventions on R : as in Fig .
5 ( original prior ) , but with the effects of self isolation and social distancing phased in over 4 days and the infection to death delay increased by 3.2 days .
Finally , I repeated this experiment using the flat prior ( Figure 10 ) .
The strength of the reduction in transmission attributed to lockdown reduces noticeably compared with the no phase-in case , although it is still larger than that of school closure ( @ @ @ @ @ @ @ marginally ) , while the estimated effects of banning public events and ( particularly ) social distancing increase markedly .
Fig.10 .
Effect of interventions on R : as in Fig .
6 ( flat prior ) , but with the effects of self isolation and social distancing phased in over 4 days and the infection to death delay increased by 3.2 days .
First and foremost , the failure of Flaxman et al.
’ s model to consider other possible causes apart from NPI of the large reductions in COVID-19 transmission that @ @ @ @ @ @ @ the overall effect of NPI unscientific and unsupportable .
That is because the model is bound to find that NPI together account for the entire reduction in transmission that has evidently occurred .
Secondly , their finding that almost all the large reductions in transmission that the model infers occurred were due to lockdowns , with other interventions having almost no effect , has been shown to be unsupportable , for two reasons : • the prior distribution that they used for the strength of NPI effects is hugely biased towards finding that @ @ @ @ @ @ @ transmission , with almost the entire reduction being caused by just one or two NPI .
• the relative strength of different interventions inferred by the model is extremely sensitive to the assumptions made regarding the average delay from infection to death , and to a lesser extent to whether self isolation and social distancing are taken to exert their full strength immediately upon implementation or are phased in over a few days .
It seems likely that the inferred relative strengths of the various NPIs are also highly sensitive to other assumptions @ @ @ @ @ @ @ to structural features of their model .
For instance , their assumption that the effect of different interventions on transmission is multiplicative rather than additive will have affected the estimated relative strengths of different types of NPI , maybe substantially so .
The basic problem is that simply knowing the dates of implementation of the various NPI in each country does not provide sufficient information to enable robust estimation of their relative effects on transmission , given the many sources of uncertainty and the differences in multiple regards between the various countries .
@ @ @ @ @ @ @ Mishra , S. , Gandy , A. et al .
Estimating the effects of non-pharmaceutical interventions on COVID-19 in Europe .
Nature ( 2020 ) .
https : //doi.org/10.1038/s41586-020-2405-7 [ 3 ] The seeding continues for 6 days , with the average number of seed infections per day being inferred by the model .
[ 4 ] The numbering of interventions used in their code is 1. school ( and/or university ) closure ordered ; 2. case-based self isolation mandated ; 3. public events banned ; 4. first intervention ; 5. lockdown ordered @ @ @ @ @ @ @ [ 6 ] The alpha distributions are defined by α ~ Gamma ( shape=1/6 , scale=1 ) − log ( 1.05 ) /6 .
Hence alpha can range between −log ( 1.05 ) /6 ( approximately −0.008 ) and plus infinity .
[ 7 ] “ Normal+ ” means a normal distribution with the negative part of the distribution excluded .
[ 8 ] Verity R , Okell LC , Dorigatti I , et al .
Estimates of the severity of COVID-19 disease .
medRxiv 13 March 2020 ; https : //www.medrxiv.org/content/10.1101/2020.03.09.20033357v1 .
@ @ @ @ @ @ @ al .
manuscript was submitted on 30 March 2020 , the same date as Imperial College published “ Report 13 : Estimating the number of infections and the impact of non-pharmaceutical interventions on COVID-19 in 11 European countries.
” , by the same ( or almost the same ) authors : https : //spiral.imperial.ac.uk/bitstream/10044/1/77731/9/2020-03-30-COVID19-Report-13.pdf .
From the referencing of comments in the Nature peer review file , it appears that the original Flaxman et al .
manuscript was almost identical to Report 13 .
Originally posted here , where a pdf copy is @ @

By Nic Lewis
Key points about the recent Nature paper by Flaxman and other Imperial College modellers
1) The transition from rising to declining recorded COVID-19 deaths in the in 11 European countries that they studied imply that transmission of COVID-19 must have reduced substantially.
The study was bound to find that together the five government non-pharmaceutical interventions (NPI) they considered contributed essentially 100% of the reduction in COVID-19 transmission, since in their model there is nothing else that could cause it.
2) The prior distribution they used for the effects of NPIs on transmission in their subjective Bayesian statistical method hugely favours finding that almost all the reduction in transmission is due to one, or possibly two, NPIs with all the others having a negligible effect.
The probability density of the prior distribution at their median estimates of the effect on transmission of each type of NPI, which allocate essentially all the reduction in transmission to lockdowns, was many billion times greater than it would have been if the same total estimated reduction had been spread evenly across the types of NPI.
3) Which intervention(s) is/are found to be important depends critically on the assumptions regarding the delay from infection to death. When using their probabilistic assumptions regarding the delay from infection to death, a huge (and highly improbable given other assumptions they made) country-specific effect is required to explain the reduction in transmission in Sweden, where no lockdown occurred. If delays from infection to death are increased by just three days, their model no longer finds lockdowns to have the largest effect, and a more moderate country-specific effect is required to explain the reduction in transmission in Sweden.
4)The estimated relative strengths of different NPIs are also considerably affected by the use of an alternative prior distribution for their effects on transmission that does not strongly bias the estimation of most of them towards a negligible level. They are also considerably affected by phasing in over a few days the effects of the two NPIs that seem unlikely to have had their full effect on their date of implementation.
5) It follows from the above that that study provides no information whatsoever as to the actual contribution from all NPI combined to the reduction in transmission, and nor does it provide robust estimates of relative effects of different NPI.
Introduction
On 8 June 2020, Nature published a paper (Flaxman et al. 2020[1]) by modellers in the Imperial College OCIVD-19 response team. Its abstract ends with:
Our results show that major non-pharmaceutical interventions and lockdown in particular have had a large effect on reducing transmission. Continued intervention should be considered to keep transmission of SARS-CoV-2 under control.
Using a counterfactual model, the paper also estimated the impact of interventions on deaths from COVID-19 in the 11 European countries studied, saying:
We find that, across 11 countries, since the beginning of the epidemic, 3,100,000 [2,800,000 – 3,500,000] deaths have been averted due to interventions.
The mainstream media publicised the ‘3 million deaths saved’ claim, without critically appraising the paper or, generally, mentioning the relevant caveat in the paper:
The counterfactual model without interventions is illustrative only and reflects our model assumptions.
In Imperial College’s press release Dr Flaxman ignored his own caveat, saying
Using a model based on data from the number of deaths in 11 European countries, it is clear to us that non-pharmaceutical interventions– such as lockdown and school closures, have saved about 3.1 million lives in these countries
In this article I examine the main claim – that major non-pharmaceutical interventions (NPI) have had a large effect on reducing transmission of COVID-19, to which the inferred reduction in deaths is attributable, with almost all the reduction due to lockdowns. I show that this claim is strongly dependent on the assumptions made and is highly dubious.
The case of Sweden, where the authors find the reduction in transmission to have been only moderately weaker than in other countries despite no lockdown having occurred, is prima facie evidence against the paper’s main claim.
How the effects of lockdowns and other interventions were estimated
Flaxman et al. employ a ‘hierarchical Bayesian’ statistical model. It uses data on daily deaths (up to 5 May 2020, when two countries relaxed their lockdowns), the dates of imposition of five types of NPI (school or university closure, case-based self isolation, public events banned, lockdown ordered and social distancing encouraged), and estimates of the infection fatality rate, for each of 11 European countries.[2] Using these data, the model infers what time profiles of the effective reproduction number (R t , the number of people whom an infected person in turn infects) – and hence of new infections – would produce the best match between projected and recorded deaths for each country. To do so it uses a simple model of epidemic growth and probabilistic estimates, common to all countries, of the time from infection to death and of the generation time (that from a person becoming infected to them infecting others). The assumed infection fatality rate (IFR) is common between countries for each age band, but reflects the age-structure of each country’s population. It averages slightly over 1%.
A separate initial value, R 0 (the basic reproduction number), of the reproduction number R t is inferred for each country. R t then changes from R 0 in stepwise fashion at the date of each NPI, which act multiplicatively with an equally strong inferred effect for all countries. Each country’s epidemic is seeded by a series of infections starting 30 days prior to a total of 10 recorded deaths.[3]
The model is described in more detail here, and is illustrated in Figure 1, taken from Flaxman et al. Fig. 1. Reproduction of Flaxman et al. Extended Data Fig. 3: Summary of model components
The treatment of interventions
The model uses no information on NPI’s except their type and their implementation date in each country. NPI of each type are treated as having the same (multiplicative) effect on R t in each country. Each type of NPI is treated identically. As well as the five types of actual interventions, all first interventions (whatever type) are treated as an extra type of intervention, for each country occurring on the date of implementation of its first actual NPI (almost always either self isolation or public events ban, and never lockdown). Hence there are six NPIs with shared values for all countries.
In addition, a pseudo-NPI with a strength that is estimated separately for each country is treated as taking place on the same date as the last actual NPI. These country-specific pseudo-NPIs allow for variation between countries in the effectiveness of the implementation of their NPI. They are probabilistically constrained to be relatively small, making a country-specific effect large enough to cause a halving of R t exceedingly improbable.
In all 11 countries the exponential growth in infections and deaths experienced early in the epidemics slowed and then turned negative, with infections and deaths decreasing. This implies that in all 11 countries R t decreased very substantially, to below one, since the start of their epidemics.
In the Flaxman et al. model the only factor that can cause R t to decrease significantly is the effect of each NPI. Therefore, the estimated overall effect of the NPIs in reducing R t , and hence deaths resulting from COVID-19 disease, is bound to be very strong.
The only non-NPI factor that affects R t in the Flaxman et al. model is the reduction arising from the proportion of the population susceptible to infection (set at 100% initially) gradually diminishing over time due to individuals already infected by COVID-19 becoming immune to it. This reduction is very small in their model, for two reasons:
they make the very unrealistic assumption that all individuals in a country are equally susceptible to COVID-19 and, if infected, are equally likely to infect others.
the relatively high infection fatality rates they assume result in only very small proportions of countries’ populations becoming infected in their model.
Therefore, their model has to attribute almost all the overall reduction in R t to government interventions.
Factors not considered by Flaxman et al., all of which are highly likely to have caused some reduction in COVID-19 transmission, and which between them may well have caused substantial reductions in R t in all 11 countries, include:
population heterogeneity in social connectivity – which generates highly correlated heterogeneity in both susceptibility and infectivity – and in other factors determining susceptibility to COVID-19
unforced changes in the behaviour of individuals as they adjust it to reflect COVID-19 risk
seasonal factors: infections by common coronaviruses peak in the winter and diminish greatly as spring progresses.
As is well known by competent epidemiologists, the first of the above-mentioned factors causes R t to diminish faster, potentially much faster, with the number of people who have been infected than if it were proportional to the number of people remaining uninfected, as assumed by Flaxman et al. The other factors directly reduce R t .
If follows that Flaxman et al.’s counterfactual case, which predicts ~3,200,000 deaths in the absence of any NPIs (their ‘counterfactual model’), is completely unrealistic, as therefore is their estimate of 3,100,000 lives saved by interventions.
It also follows that Flaxman et al.’s claim:
Our estimates imply that the populations in Europe are not close to herd immunity (~70% if R 0 is 3.8)
may be invalid. As shown here, due to population heterogeneity in susceptibility and infectivity the herd immunity threshold it is bound to be lower – quite possibly very substantially so – than if, as required for it to be ~70% at an R 0 of 3.8, populations are homogeneous.
Flaxman et al.’s assertion that all the reduction in transmission (i.e., the reduction in R t ) was due to NPIs, other than very small reduction as more people have been infected and become immune, is unsound. Nevertheless, it seems quite likely that NPIs have had a significant, perhaps substantial, effect on R t . However, given the confounding effects of the other factors mentioned it is impossible reliably to estimate the total effect of NPIs on R t and hence on deaths.
Even when making the unrealistic assumption that almost all the reduction in R t was due to interventions, any allocation of that reduction between the NPIs is very fragile. Flaxman et al. accept this in relation to NPIs other than lockdown, writing:
Most interventions were implemented in rapid succession in many countries, and as such it is difficult to disentangle individual effect sizes of each intervention. In our analysis we find that only the effect of lockdown is identifiable, …
On their median estimates, lockdown caused an 82% reduction in R t , whereas no other NPI caused as much as a 1% reduction in R t . While it would not be particularly surprising if such a drastic intervention as lockdown had had stronger effects than other NPIs, even if lockdown had a strong effect one would expect some other NPIs to have had a significant effect. So how did Flaxman et al. find that, remarkably, almost the entire effect of interventions was due to lockdown? The answer, which turns out to be two-fold, shows that their finding is not credible.
Why Flaxman et al. found almost all reduction in COVID-19 transmission to be attributable to a single intervention
Flaxman et al. use a subjective Bayesian statistical method. I have repeatedly criticised this type of Bayesian method in the climate science field, but – probably due to its ease of use – it remains standard practice there and in many other fields.
A subjective Bayesian method requires prior probability distributions to be assigned for each unknown parameter whose value is to be inferred. These prior distributions are then modified by the likelihood function, which reflects how well the modelled deaths fit the daily deaths data at varying values of the parameters, in order to arrive at a ‘posterior’ probability distribution for the parameter values. They use a common method of achieving this that results in a large number of quasi-random draws (‘posterior draws’) from the derived posterior probability distribution.
They represent the strength of interventions by a six dimensional parameter alpha (five actual NPIs plus the synthetic first intervention NPI), with the corresponding effect of intervention i (i being 1, 2,3, 4, 5 or 6)[4] on R t being to multiply it by exp(-alpha[i]).
The combined effect of all interventions is then to multiply R t by exp[-(alpha[1] + alpha[2] + alpha[3] + alpha[4] + alpha[5] + alpha[6])][5], which depends only on the sum of the individual alpha values. Their own posterior draws show a median value of the sum of the alphas of 1.75, which corresponds to an 83% reduction in transmission (1 – e−1.75 = 0.83).
The prior distribution assigned by the authors to the strength of the reduction in R t caused by each intervention is of particular concern. Each of the six alpha values is assigned a gamma-distributed prior probability distribution; a small offset is applied, so that the gamma-distributed values inferred initially are marginally higher, but that is a cosmetic feature.[6] The authors write:
The intuition behind this prior is that it encodes our null belief that interventions could equally increase or decrease Rt, and the data should inform which.
That is not in fact true. As the left hand panel of Figure 2 shows, their prior allows each intervention to decrease R t by up to 100%, but only to increase it by less than 1%. And the combined effect on transmission of all interventions (right hand panel) can only vary between –100% and + 5%. However, since the trajectory of the deaths data is, on their assumptions, bound to result in all interventions combined being found to strongly reduce transmission, the +5% limit is of no real consequence.
Fig. 2. Reproduction of the upper panels of Flaxman et al. Supplementary Fig. 3: Cumulative distribution function F(x) of the prior for one intervention’s multiplicative effect x (= e–α) on transmission (left) or for the effect of all interventions combined (= e–Σα) (right).
On the face of it, the combined effect of the six-dimensional joint alpha prior distribution looks fairly uniform over the range in which the estimated reduction in R t could fall; it assigns a similar probability to a reduction in the range 40% to 50% and in the range 80% to 90%, for example. However, that only looks at one aspect of the six-dimensional prior distribution.
If I take the sum of the six alphas to be 1.75 (the median sum from their posterior draws) and set them to be all equal, at 1.75/6, their joint prior probability density is 0.0023. But if I set one of the alpha values to 1.70 and the remaining five to 0.01, giving the same overall reduction in transmission, the prior probability density is 64.3. That means their prior distribution assigns a 28,000 times higher prior probability assumption to this case, where one type of intervention has a completely dominating effect relative to all the others, than to a case where the same overall reduction in transmission is caused equally by all types of intervention. The reason is that the offset-gamma distribution used assigns a strongly increasing probability density as an alpha value decreases towards −0.008, its lowest permitted level, favouring cases where the effect of all but one or two NPIs is estimated to be almost zero.
So it is unsurprising that they found a single intervention to be totally dominant.
The median individual alpha values in their 2,000 archived posterior draws are −0.007, −0.007, −0.007, −0.007, 1.699 and −0.006. So all interventions except lockdown were estimated to have a completely negligible effect.
The median ratio, across their own posterior draws for alpha, of the actual prior probability to what it would have been if in each draw the total effect of the intervention had been spread evenly across them, was in fact 392 billion to one!
It is not clear that the authors realised that the prior distribution they used very strongly favoured finding that most interventions had a negligible effect, and I very much doubt that any of the peer reviewers appreciated that this was the case.
The Sweden problem
Using the code and data accompanying the Nature paper as is, except with the 8,000 draws split between 4 not 5 chains to better match my computer, I can accurately replicate Flaxman et al.’s findings, with lockdown accounting for almost the entire reduction in R t (Figure 3).
Fig. 3. Effect of interventions on R t in the base case, with all aspects of the model as per the original version (that archived for the Nature paper). The red First intervention estimate includes the effect of the synthetic first intervention NPI and so only applies for countries where the NPI concerned was the first to be implemented; it should be ignored in all other cases. Mean relative percentage reduction in R t is shown for each NPI (filled circle) together with the 95% posterior credible intervals (line). If 100% reduction is achieved, R t = 0 and there is no more transmission of COVID-19.
Sweden did not have a lockdown, but it still had a large reduction in R t , albeit one not quite as large as the average for other countries. So how did the model account for that? This is where the country specific factors, which are treated as occurring on the date of the last actual intervention and in effect are an addition to its alpha, come in.
The country specific factors are given an apparently small influence, being zero-mean normally distributed with a standard deviation that is itself zero mean normal+ distributed[7] with a standard deviation of 0.2. But for Sweden a value of 1.27, in the far tail of the resulting distribution, was inferred. The probability of such a large country factor arising by chance appears to be about 1 in 2,000. That in itself implies that their model does not adequately represent reality.
Using a less informative prior
I investigated use of a prior distribution for the six alpha parameters that was essentially flat over the alpha parameter range relevant for NPI, both for each parameter separately and for the six-dimensional joint alpha parameter. For technical reasons, rather than using a uniform distribution I chose an independent zero mean normal distribution with a standard deviation of 10 as the prior distribution for each parameter. I hereafter refer to this as the ‘flat prior distribution’, even though it is not quite flat over the parameter range of interest (approximately 0 to 2).
I then ran the model using the same assumptions, but using the flat prior distribution rather than the original offset-gamma prior distribution. Doing so should eliminate the previous strong bias towards finding that most interventions had almost no effect.
The resulting estimates of the effect of each intervention were as shown in Figure 4. The estimated effects of NPI other than lockdown all increase markedly from their near zero values when using the original prior, but the contribution of lockdown remains dominant.
Fig. 4. Effect of interventions on R t : as in Fig. 3, but with the flat prior distribution for alpha substituted for the offset-gamma prior distribution in the original model..
The country specific factor for Sweden was slightly less high than before, at 1.12. The probability of such a large country factor arising by chance appears to be about 1 in 900; still minute.
So, even when using the flat prior, the Flaxman et al. model does not adequately fit reality. The problem is that, as it still estimates lockdown to account for the vast bulk of the total reduction in R t , it cannot adequately account for the reduction in R t that occurred in Sweden, where there was no lockdown.
Why Flaxman et al. found lockdown was the intervention that dominated the reduction in COVID-19 transmission
I have explained why it to be expected, given Flaxman et al.’s choice of prior distribution for the effect of interventions on the transmission of COVID-19, that a single type of intervention (or at most two types) would account for the vast bulk of the reduction in R t . But why lockdown?
The key here seems to be that lockdown was, other than in Sweden, on average imposed at a point in time that, allowing for the assumed probabilistic delay between infection and death, would result in deaths peaking at about the time that they actually peaked. Also, the timing of lockdown, relative to the peak in recorded deaths, differed slightly less between countries that locked-down than was the case for most other interventions.
Flaxman et al. took probabilistic estimates of the delay from infection to symptoms appearing and from symptoms appearing until death, with assumed mean values of 5.1 and 17.8 days respectively, and added them to obtain the infection to death delay values. The 5.1 day delay from infection to onset of symptoms seems reasonable. But the 17.8 days mean from onset of symptoms until death looks as if it may be on the short side for European countries. Ideally, a separate onset of symptoms to death delay distribution would have been estimated for each country. However, the authors may well have been unable to find suitable European data. They actually used a value estimated by Verity et al.[8] (also members of the Imperial College COVID-19 modelling team) from just 24 cases in mainland China.
One of the peer reviewers suggested that the value Flaxman et al. were using for the delay from onset of symptoms until death of (in the originally-submitted manuscript[9] being reviewed)18.8 days, not 17.8 days, was rather short, writing:
it is smaller than preliminary estimates available from hospitalization data in Europe (about 5-6 days from onset to hospitalization, at least 2 weeks in the hospital)
I therefore increased the average delay from onset of symptoms to death slightly.
I also took the opportunity to correct the dates used in the model inputs for school/university closure in Sweden and for self-isolation in Spain to those given in Flaxman et al. Extended Data Figure 4, which agree to those in their Supplementary Table 2.
I found that adding 3 days to the infection to death delay, bringing the average onset of symptoms to death delay to ~21 days (median 19.6 days) – which is fully consistent with the peer reviewer’s comment – was adequate to reduce the problem of Sweden needing a very large country-specific factor. That factor was then estimated at ~0.4, to match the reduction in transmission in Sweden – still over twice as large as for any other country, but no longer statistically-inconsistent with their assumptions.
The resulting estimated effectiveness of the various interventions, using the authors’ original prior distribution for alpha, is shown in Figure 5.
Fig. 5. Effect of interventions on R t : as in Fig. 3 (original prior) but with the infection to death delay increased by 3 days, and one intervention date corrected for each of Spain and Sweden (see text).
School closure is now found to have a slightly stronger effect on transmission than lockdown. This may seem rather unlikely in reality, but the model has no information to go on regarding the likely relative strengths of each type of intervention – it just knows when they were implemented in each country. Other interventions are found to have almost zero mean effect, as is to be expected given the nature of the original prior distribution.
Using instead the flat prior gives slightly different estimates of the effectiveness of the various interventions (Figure 6), with school closure not having quite as strong an effect as when using the original prior. The effects of social distancing, and to a slightly lesser extent public events ban and self isolation (one of which is generally the first intervention, so the red line applies to it), all cease to be negligible.
Fig. 6. Effect of interventions on R t : as in Fig. 5, with the infection to death delay increased by 3 days, but using the flat prior distribution instead of the original prior distribution.
If the infection to death delay is increased by 5 rather than 3 days from Flaxman et al.’s assumed probabilistic magnitude – arguably still as reasonable as Flaxman et al.’s assumption – and the original prior used, the changes in the relative effectiveness of different interventions become even more marked (Figure 7). Lockdown is now estimated to have far less effect than school closure, while social distancing now has a significant effect. The country-specific factor for Sweden becomes small.
Fig. 7. Effect of interventions on R t : as in Fig. 5 (original prior) but with the infection to death delay increased by 5 days not 3 days.
When the flat prior is used instead, the estimated effect of school closure reduces while that of all other interventions increases (Figure 8).
Fig. 8. Effect of interventions on R t : as in Fig. 6 (flat prior) but with the infection to death delay increased by 5 days not 3 days.
Finally, I investigated the effects of phasing in certain of the interventions. Flaxman et al.’s assumption that all interventions immediately have their full effect on their date of implementation is questionable. It may not be too unrealistic for closing schools, banning public events and decreeing a lockdown, all of which it is feasible to enforce. However, responses to self isolation advice and social distancing encouragement (which both generally preceded a lockdown) are more within the discretion of the individuals concerned, and very arguably would take a little time to reach their final strength.
I examined phasing in over four days the effects of just those two NPIs, with their strength increasing evenly from 25% on the date of implementation to 100% three days later. The result, using the original prior distribution for alpha and making a ~3 day increase in the delay from symptoms to death, is shown in Figure 9. The strength of the reduction in transmission attributed to lockdown reduces slightly compared with the no phase-in case, while than attributed to social distancing increases.
Fig. 9. Effect of interventions on R t : as in Fig. 5 (original prior), but with the effects of self isolation and social distancing phased in over 4 days and the infection to death delay increased by 3.2 days.
Finally, I repeated this experiment using the flat prior (Figure 10). The strength of the reduction in transmission attributed to lockdown reduces noticeably compared with the no phase-in case, although it is still larger than that of school closure (the estimated effect of which reduces only marginally), while the estimated effects of banning public events and (particularly) social distancing increase markedly.
Fig.10. Effect of interventions on R t : as in Fig. 6 (flat prior), but with the effects of self isolation and social distancing phased in over 4 days and the infection to death delay increased by 3.2 days.
Conclusions
First and foremost, the failure of Flaxman et al.’s model to consider other possible causes apart from NPI of the large reductions in COVID-19 transmission that have occurred makes it conclusions as to the overall effect of NPI unscientific and unsupportable. That is because the model is bound to find that NPI together account for the entire reduction in transmission that has evidently occurred.
Secondly, their finding that almost all the large reductions in transmission that the model infers occurred were due to lockdowns, with other interventions having almost no effect, has been shown to be unsupportable, for two reasons:
the prior distribution that they used for the strength of NPI effects is hugely biased towards finding that most interventions had essentially zero effect on transmission, with almost the entire reduction being caused by just one or two NPI.
the relative strength of different interventions inferred by the model is extremely sensitive to the assumptions made regarding the average delay from infection to death, and to a lesser extent to whether self isolation and social distancing are taken to exert their full strength immediately upon implementation or are phased in over a few days.
It seems likely that the inferred relative strengths of the various NPIs are also highly sensitive to other assumptions made by Flaxman et al., and to structural features of their model. For instance, their assumption that the effect of different interventions on transmission is multiplicative rather than additive will have affected the estimated relative strengths of different types of NPI, maybe substantially so. The basic problem is that simply knowing the dates of implementation of the various NPI in each country does not provide sufficient information to enable robust estimation of their relative effects on transmission, given the many sources of uncertainty and the differences in multiple regards between the various countries.
Nicholas Lewis
[1] Flaxman, S., Mishra, S., Gandy, A. et al. Estimating the effects of non-pharmaceutical interventions on COVID-19 in Europe. Nature (2020). https://doi.org/10.1038/s41586-020-2405-7
[2] Denmark, Italy, Germany, Spain, United Kingdom, France, Norway, Belgium, Austria, Sweden and Switzerland.
[3] The seeding continues for 6 days, with the average number of seed infections per day being inferred by the model.
[4] The numbering of interventions used in their code is 1. school (and/or university) closure ordered; 2. case-based self isolation mandated; 3. public events banned; 4. first intervention; 5. lockdown ordered; and 6. social distancing encouraged.
[5] In mathematical notation, exp[-(alpha[1] + alpha[2] + alpha[3] + alpha[4] + alpha[5] + alpha[6])] is written e–Σα.
[6] The alpha distributions are defined by α i ~ Gamma( shape=1/6, scale=1) − log e (1.05)/6. Hence alpha can range between −log e (1.05)/6 (approximately −0.008) and plus infinity.
[7] “Normal+” means a normal distribution with the negative part of the distribution excluded.
[8] Verity R, Okell LC, Dorigatti I, et al. Estimates of the severity of COVID-19 disease. medRxiv 13 March 2020; https://www.medrxiv.org/content/10.1101/2020.03.09.20033357v1.
[9] The original Flaxman et al. manuscript was submitted on 30 March 2020, the same date as Imperial College published “Report 13: Estimating the number of infections and the impact of non-pharmaceutical interventions on COVID-19 in 11 European countries.”, by the same (or almost the same) authors: https://spiral.imperial.ac.uk/bitstream/10044/1/77731/9/2020-03-30-COVID19-Report-13.pdf . From the referencing of comments in the Nature peer review file, it appears that the original Flaxman et al. manuscript was almost identical to Report 13.
Originally posted here, where a pdf copy is also available
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climateetc--2020-06-20--Structural errors in global climate models

Climate model sensitivity to CO2 is heavily dependent on artificial parameterizations ( e.g .
clouds , convection ) that are implemented in global climate models that utilize the wrong atmospheric dynamical system and excessive dissipation .
The peer reviewed manuscript entitled “ The Unique , Well Posed Reduced System for Atmospheric Flows : Robustness In The Presence Of Small Scale Surface Irregularities ” is in press at the journal Dynamics of Atmospheres and Oceans ( DAO ) [ link ] and the submitted version of the manuscript is available on this site , with some slight differences from the final @ @ @ @ @ @ @ here : Manuscript Abstract : It is well known that the primitive equations ( the atmospheric equations of motion under the additional assumption of hydrostatic equilibrium for large-scale motions ) are ill posed when used in a limited area on the globe .
Yet the equations of motions for large-scale atmospheric motions are essentially a hyperbolic system , that with appropriate boundary conditions , should lead to a well-posed system in a limited area .
This apparent paradox was resolved by Kreiss through the introduction of the mathematical Bounded Derivative Theory ( BDT @ @ @ @ @ @ @ multiple time scales ( as is the case for the atmospheric equations of motion ) .
The BDT uses norm estimation techniques from the mathematical theory of symmetric hyperbolic systems to prove that if the norms of the spatial and temporal derivatives of the ensuing solution are independent of the fast time scales ( thus the concept of bounded derivatives ) , then the subsequent solution will only evolve on the advective space and time scales ( slowly evolving in time in BDT parlance ) for a period of time .
The requirement @ @ @ @ @ @ @ of the ensuing solution be independent of the fast time scales leads to a number of elliptic equations that must be satisfied by the initial conditions and ensuing solution .
In the atmospheric case this results in a 2D elliptic equation for the pressure and a 3D equation for the vertical component of the velocity .
Utilizing those constraints with an equation for the slowly evolving in time vertical component of vorticity leads to a single time scale ( reduced ) system that accurately describes the slowly evolving in time solution of the @ @ @ @ @ @ @ for a limited area domain .
The 3D elliptic equation for the vertical component of velocity is not sensitive to small scale perturbations at the lower boundary so the equation can be used all of the way to the surface in the reduced system , eliminating the discontinuity between the equations for the boundary layer and troposphere and the problem of unrealistic growth in the horizontal velocity near the surface in the hydrostatic system .
The mathematical arguments are based on the Bounded Derivative Theory ( BDT ) for symmetric hyperbolic systems introduced @ @ @ @ @ @ @ ago and on the theory of numerical approximations of partial differential equations .
What is the relevance of this research for climate modeling ?
At a minimum , climate modelers must make the following assumptions : 1 .
The numerical climate model must accurately approximate the correct dynamical system of equations Currently all global climate ( and weather ) numerical models are numerically approximating the primitive equations — the atmospheric equations of motion modified by the hydrostatic assumption .
However this is not the system of equations that satisfies the mathematical estimates required @ @ @ @ @ @ @ and subsequent solution in order to evolve as the large scale motions in the atmosphere .
The correct dynamical system is introduced in the new manuscript that goes into detail as to why the primitive equations are not the correct system .
Because the primitive equations use discontinuous columnar forcing ( parameterizations ) , excessive energy is injected into the smallest scales of the model .
This necessitates the use of unrealistically large dissipation to keep the model from blowing up .
That means the fluid is behaving more like molasses than air @ @ @ @ @ @ @ manuscript that show that this substantially reduces the accuracy of the numerical approximation .
2 .
The numerical climate model correctly approximates the transfer of energy between scales as in the actual atmosphere .
Because the dissipation in climate models is so large , the parameterizations must be tuned in order to try to artificially replicate the atmospheric spectrum .
Mathematical theory based on the turbulence equations has shown that the use of the wrong amount or type of dissipation leads to the wrong solution .
In the climate model case , this @ @ @ @ @ @ @ about climate sensitivity because the numerical solution is not behaving as the real atmosphere .
3 .
The forcing ( parameterizations ) accurately approximate the corresponding processes in the atmosphere and there is no accumulation of error over hundreds of years of simulation .
It is well known that there are serious errors in the parameterizations , especially with respect to clouds and moisture that are crucial to the simulation of the real atmospheres .
Pat Frank has addressed the accumulation of error in the climate models .
In the new manuscript , @ @ @ @ @ @ @ impacts the accuracy of the solution in a short period of time .
One might ask how can climate models apparently predict the large-scale motions of the atmosphere in the past given these issues .
I have posted a simple example on Climate Audit ( reproducible on request ) that shows that given any time dependent system ( even if it is not the correct one for the fluid being studied ) , if one is allowed to choose the forcing , one can reproduce any solution one wants .
This is essentially @ @ @ @ @ @ @ order to match the previous climate given the wrong dynamical system and excessive dissipation .
I reference a study on the accuracy of a primitive equation global forecast model by Sylvie Gravel et al .
[ link ] .
She showed that the largest source of error in the initial stages of a forecast are from the excessive growth of the horizontal velocity near the lower boundary .
Modelers have added a boundary layer drag/dissipation in an attempt to prevent this from happening .
I note in the new manuscript that this problem @ @ @ @ @ @ @ system and that in fact the correct system is not sensitive to small-scale perturbations at the lower boundary .
Biosketch : I am an independent applied mathematician trained in partial differential equations and numerical analysis concerned about the loss of integrity in science through the abuse of rigorous mathematical theory by numerical modelers .
I am not funded by any outside organization .
My previous publications can be found on google scholar by searching for Browning and Kreiss .

by Gerald Browning
Climate model sensitivity to CO2 is heavily dependent on artificial parameterizations (e.g. clouds, convection) that are implemented in global climate models that utilize the wrong atmospheric dynamical system and excessive dissipation.
The peer reviewed manuscript entitled “The Unique, Well Posed Reduced System for Atmospheric Flows: Robustness In The Presence Of Small Scale Surface Irregularities” is in press at the journal Dynamics of Atmospheres and Oceans (DAO) [link] and the submitted version of the manuscript is available on this site, with some slight differences from the final published version. Link to paper is here: Manuscript
Abstract: It is well known that the primitive equations (the atmospheric equations of motion under the additional assumption of hydrostatic equilibrium for large-scale motions) are ill posed when used in a limited area on the globe. Yet the equations of motions for large-scale atmospheric motions are essentially a hyperbolic system, that with appropriate boundary conditions, should lead to a well-posed system in a limited area. This apparent paradox was resolved by Kreiss through the introduction of the mathematical Bounded Derivative Theory (BDT) for any symmetric hyperbolic system with multiple time scales (as is the case for the atmospheric equations of motion). The BDT uses norm estimation techniques from the mathematical theory of symmetric hyperbolic systems to prove that if the norms of the spatial and temporal derivatives of the ensuing solution are independent of the fast time scales (thus the concept of bounded derivatives), then the subsequent solution will only evolve on the advective space and time scales (slowly evolving in time in BDT parlance) for a period of time. The requirement that the norm of the time derivatives of the ensuing solution be independent of the fast time scales leads to a number of elliptic equations that must be satisfied by the initial conditions and ensuing solution. In the atmospheric case this results in a 2D elliptic equation for the pressure and a 3D equation for the vertical component of the velocity.
Utilizing those constraints with an equation for the slowly evolving in time vertical component of vorticity leads to a single time scale (reduced) system that accurately describes the slowly evolving in time solution of the atmospheric equations and is automatically well posed for a limited area domain. The 3D elliptic equation for the vertical component of velocity is not sensitive to small scale perturbations at the lower boundary so the equation can be used all of the way to the surface in the reduced system, eliminating the discontinuity between the equations for the boundary layer and troposphere and the problem of unrealistic growth in the horizontal velocity near the surface in the hydrostatic system.
The mathematical arguments are based on the Bounded Derivative Theory (BDT) for symmetric hyperbolic systems introduced by Professor Heinz-Otto Kreiss over four decades ago and on the theory of numerical approximations of partial differential equations.
What is the relevance of this research for climate modeling? At a minimum, climate modelers must make the following assumptions:
1. The numerical climate model must accurately approximate the correct dynamical system of equations
Currently all global climate (and weather) numerical models are numerically approximating the primitive equations — the atmospheric equations of motion modified by the hydrostatic assumption. However this is not the system of equations that satisfies the mathematical estimates required by the BDT for the initial data and subsequent solution in order to evolve as the large scale motions in the atmosphere. The correct dynamical system is introduced in the new manuscript that goes into detail as to why the primitive equations are not the correct system.
Because the primitive equations use discontinuous columnar forcing (parameterizations), excessive energy is injected into the smallest scales of the model. This necessitates the use of unrealistically large dissipation to keep the model from blowing up. That means the fluid is behaving more like molasses than air. References are included in the new manuscript that show that this substantially reduces the accuracy of the numerical approximation.
2. The numerical climate model correctly approximates the transfer of energy between scales as in the actual atmosphere.
Because the dissipation in climate models is so large, the parameterizations must be tuned in order to try to artificially replicate the atmospheric spectrum. Mathematical theory based on the turbulence equations has shown that the use of the wrong amount or type of dissipation leads to the wrong solution. In the climate model case, this implies that no conclusions can be drawn about climate sensitivity because the numerical solution is not behaving as the real atmosphere.
3. The forcing (parameterizations) accurately approximate the corresponding processes in the atmosphere and there is no accumulation of error over hundreds of years of simulation.
It is well known that there are serious errors in the parameterizations, especially with respect to clouds and moisture that are crucial to the simulation of the real atmospheres. Pat Frank has addressed the accumulation of error in the climate models. In the new manuscript, even a small error in the system impacts the accuracy of the solution in a short period of time.
One might ask how can climate models apparently predict the large-scale motions of the atmosphere in the past given these issues. I have posted a simple example on Climate Audit (reproducible on request) that shows that given any time dependent system (even if it is not the correct one for the fluid being studied), if one is allowed to choose the forcing, one can reproduce any solution one wants. This is essentially what the climate modelers have done in order to match the previous climate given the wrong dynamical system and excessive dissipation.
I reference a study on the accuracy of a primitive equation global forecast model by Sylvie Gravel et al. [link]. She showed that the largest source of error in the initial stages of a forecast are from the excessive growth of the horizontal velocity near the lower boundary. Modelers have added a boundary layer drag/dissipation in an attempt to prevent this from happening. I note in the new manuscript that this problem does not occur with the correct dynamical system and that in fact the correct system is not sensitive to small-scale perturbations at the lower boundary.
Biosketch: I am an independent applied mathematician trained in partial differential equations and numerical analysis concerned about the loss of integrity in science through the abuse of rigorous mathematical theory by numerical modelers. I am not funded by any outside organization. My previous publications can be found on google scholar by searching for Browning and Kreiss.
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climateetc--2020-06-17--Mass spectrometry and climate science.  Part I:  Determining past climates

Mass spectrometry is essential for research in climate science .
Understanding climate requires having sufficient knowledge about past climate and about the important factors that are influencing climate today , so that reliable models can be developed to predict future climate .
Analytical chemistry enables measurement of the chemical composition of materials , from the amounts of elements and their isotopes in a sample to the identity and concentrations of substances in the most complex biological organisms .
This two-part series covers the application of a powerful analytical chemistry technology — mass spectrometry — to two important areas in climate @ @ @ @ @ @ @ of aerosols , which have a large impact on climate The examples that are included for each topic were selected out of many published papers on the study of climate using mass spectrometry , partly because they feature a very wide range of types of these instruments .
The authors were very helpful in providing me with information on their work .
[ 1 ] The technology described in this essay may at times be quite complicated !
However , I hope that the results of each study will be understandable .
Information @ @ @ @ @ @ @ The atmospheric temperature records obtained using satellites and covering nearly all of the earth start only in 1979 .
Surface temperature records cover only a small portion of the earth , perhaps 15 % going back to about 1900 , and much less before that .
The Argo buoys were deployed ~15 years ago and cover much of the oceans , which had minimal coverage before then .
Information about aspects of past climate other than temperatures is even more limited .
Analytical chemistry is providing improved information about past climate , for @ @ @ @ @ @ @ climate , and extreme weather events • Concentrations of CO2 and other atmospheric components • Extent of sea ice and glaciers over time • Impact of geological events such as volcanic eruptions and earthquakes The first step in studying a sample that was formed in the past ( such as a fossil or a layer of sediment under the ocean materials that is more than a few hundred years old and have no attached information ) is to determine its age .
The most common way to do this is using mass spectrometry @ @ @ @ @ @ @ is dependent on age .
Carbon is often used for dating once-living samples , such as plants or artifacts made of wood , as its main isotopes , carbon-12 ( 12C ) and carbon-13 ( 13C ) , are stable , while carbon-14 ( 14C ) is radioactive , with a half-life of about 5700 years .
The supply of 14C in the air is constantly replenished by cosmic rays hitting nitrogen-14 , but once an organism dies the 14C fraction will steadily go down .
[ i ] Mass spectrometry is a @ @ @ @ @ @ @ of elements , and reliably determines the fraction of 14C in the once-living sample and thus the age at the end of its life for up to about 20,000 years ago , and possibly somewhat earlier .
For older samples other elements must be used , for the 14C fraction is too small to be reliably measured .
Several other elements have naturally-occurring long-lived radioactive isotopes that could be used to determine the age of an older sample .
Potassium , for example , is a widely-distributed element with substantial concentrations throughout the @ @ @ @ @ @ @ a very long-lived isotope ( 1.25 billion years ) with ( unusually ) two forms of decay , one to stable argon-40 and the other to stable calcium-40 .
Argon in the atmosphere contains the 40Ar produced over the lifetime of the earth .
However , if the decay occurs in a solid that did not contain any air when it solidified and does not allow the argon produced by potassium decay to escape , then the amount of 40Ar reflects the date when the solidification took place .
[ ii ] The @ @ @ @ @ @ @ amount of 40Ar in air trapped in ice will reflect the age at which the ice formed .
[ iii ] This mass spectrometric technique has recently been used to determine the age of samples at different depths in an Antarctic ice core ( Figure 1 ) .
[ iv ] [ v ] The authors of these studies then measured the amounts of different atmospheric gases in those samples and plotted them as a function of sample age ( Figure 2 ) .
Figure 1 : Age of samples taken at indicated @ @ @ @ @ @ @ 2 : Results for key atmospheric gases as function of ice core age ( various techniques were used for measuring the gases ) Mass spectrometry is often used to study past temperatures .
The ratio of the trace isotope oxygen-18 ( 18O ) to the most common isotope oxygen-16 ( 16O ) in a once living organism depends on the temperature of the air at the time that the oxygen was incorporated into the organism by metabolism .
The ratio is measured using mass spectrometry .
The higher the temperature , the lower @ @ @ @ @ @ @ organism .
A representative study was carried out on mollusks found in layers of sediments near the northwest shore of Iceland , covering the period 350 B.C .
to A.D. 1600 .
The layers in the shells reflected the year-round temperatures in which the mollusks had lived ( Figure 3 ) .
From such data on many samples at different layers in the sediments the authors constructed a chart of temperatures over that time period .
Notably , they were able to correlate these temperatures with historical records in Iceland from 865 to @ @ @ @ @ @ @ authors pointed out that “ On the basis of δ18O data , reconstructed water temperatures for the Roman Warm Period in Iceland are higher than any temperatures recorded in modern times.
” [ vi ] Figure 3 : Example of temperatures derived from a shell that lived through four summers ( S ) and three winters ( W ) Figure 4 : Variation of temperatures from the Roman Warm Period to ~AD 1800.Information from historical documents is at the top on the right side .
Another study using mass spectrometry to determine temperatures @ @ @ @ @ @ @ over in fjords in Sweden , covering a 2500 year period .
One of the fjords is on the north coast of Sweden ( Atlantic Ocean ) and the other on the southwest ( North Sea near Denmark ) .
The authors state : The authors of this study include a chart relating their temperature information with others in the North Atlantic ( Figure 5 ) .
Figure 5 : Bottom water temperatures for different locations in the North Atlantic Ocean going back to about 350 BCE .
Abbreviations shown at the top @ @ @ @ @ @ @ Roman Warm Period , DA represents the Dark Ages , VA/MCA represents the Viking Age/Medieval Climate Anomaly and LIA represents the Little Ice Age .
The oxygen ratios thus allow estimating temperatures for once-living organisms .
Importantly , they allow doing so year-round .
Tree ring diameters , which are sometimes used for estimating past temperatures , primarily reflect temperatures during the growing season , and are also influenced by factors such as rainfall and location ( such as above or below the treeline at a given time ) .
A recent study @ @ @ @ @ @ @ isotope ratios for studying past climate : do volcanos have an impact on major climate factors such as the El Niño – Southern Oscillation ( ENSO ) ?
Some studies have suggested that major volcanic eruptions can impact the ENSO cycle , but only a few such events have relevant weather data .
Fossil corals in the regions impacted by ENSO dating back centuries have well-defined monthly layers .
They were dated by mass spectrometry using U/Th ratios .
Then oxygen-18 measurements for these layers allow estimation of temperatures coinciding with major volcanos @ @ @ @ @ @ @ studies to produce a temperature record covering from ~1100 to ~2000 CE .
Six major volcanos in this time period were charted against the temperatures ( Figure 6 ) .
No evidence was found that the volcanos had caused an ENSO event .
[ viii ] Figure 6 : Coral δ18O measurements of temperatures for six major volcanic eruptions in the last 900 years , with lines at the bottom showing the degree to which stratospheric aerosols reduced downwelling sunlight during the volcanos This completes the first post on applications of mass spectrometry @ @ @ @ @ @ @ will focus on studies of factors that need to be understood to be able to develop reliable models of climate , with emphasis on research on aerosols .
[ 1 ] The figures and charts and other information from the papers are the property of the authors and publishers .
[ iii ] M.L .
Bender , et al.
, “ The contemporary degassing rate of 40Ar from the solid Earth ” , PNAS ( 2008 ) 105 , 8232-8237 https : //www.pnas.org/content/105/24/8232 [ iv ] J.A .
Higgins , et al.
, @ @ @ @ @ @ @ from blue ice in the Allan Hills , Antarctica ” , PNAS ( 2015 ) 112 , 6887-6891 https : //www.pnas.org/content/112/22/6887 [ v ] Y. Yan , et al.
, “ Two-million-year-old snapshots of atmospheric gases from Antarctic ice ” , Nature ( 2019 ) 574 , 663-663 https : //www.nature.com/articles/s41586-019-1692-3 and https : //www.nature.com/articles/d41586-019-03199-8 [ vi ] W.P .
Patterson , K.A .
Dietrich , C. Holmden and J.T .
Andrews , “ Two millennia of North Atlantic seasonality and implications for Norse colonies ” PNAS ( 2010 ) 107 , 5306-5310 @ @ @ @ @ @ @ .
Asteman , H.L .
Filipsson , and K. Nordberg , “ Tracing winter temperatures over the last two millennia using a north-east Atlantic coastal record ” , Climate of the Past ( 2018 ) 14 , 1097–1118 .
https : //doi.org/10.5194/cp-14-1097-2018 [ viii ] S.G. Dee , et al.
, “ No consistent ENSO response to volcanic forcing over the last millennium ” Science ( 2020 ) , 367 , 1477-1481 https : //science.sciencemag.org/content/367/6485/1477.full Roland Hirsch has served the field of analytical chemistry in a 52-year career that spans teaching , research @ @ @ @ @ @ @ , and 33 years of government service at the National Institutes of Health and the U.S. Department of Energy .
Roland has been a leader of the ACS Division of Analytical Chemistry , as Councilor for 25 years , as Division Secretary for 4 years , Chair-Elect , Program Chair , and Chair , and as its Web Editor for 22 years .
Roland organized the 50th-anniversary celebration of the Division and 25 years later , wrote the definitive history of the first 75 years of the Division , published in Analytical Chemistry @ @ @ @ @ @ @ active in ACS Governance , including Chair of the Committee on International Activities , Secretary of the Committee on Nominations and Elections , Member of the Committee on Divisional Activities , Senior Chemists Task Force , Committee on Committees , and Liaison to the ACS Committee on Professional Training .
Based on a presentation prepared for the American Chemical Society National Meeting in Philadelphia in March 2020 .
It was to have been in the Division of Analytical Chemistry ’ s session “ Advances in Mass Spectrometry ” .
The meeting was canceled @ @ @ @ @ @ @ made available on the web site for the meeting : https : //www.morressier.com/article/mass-spectrometry-essential-research-climate-science/5e735e33cde2b641284a879e

by Roland Hirsch
Mass spectrometry is essential for research in climate science.
Understanding climate requires having sufficient knowledge about past climate and about the important factors that are influencing climate today, so that reliable models can be developed to predict future climate.
Analytical chemistry enables measurement of the chemical composition of materials, from the amounts of elements and their isotopes in a sample to the identity and concentrations of substances in the most complex biological organisms.
This two-part series covers the application of a powerful analytical chemistry technology — mass spectrometry — to two important areas in climate science:
Obtaining reliable information about past climate
Understanding composition and behavior of aerosols, which have a large impact on climate
The examples that are included for each topic were selected out of many published papers on the study of climate using mass spectrometry, partly because they feature a very wide range of types of these instruments. The authors were very helpful in providing me with information on their work.[1]
The technology described in this essay may at times be quite complicated! However, I hope that the results of each study will be understandable.
Part 1: Determining past climate
Information about past climate is quite limited. The atmospheric temperature records obtained using satellites and covering nearly all of the earth start only in 1979. Surface temperature records cover only a small portion of the earth, perhaps 15% going back to about 1900, and much less before that. The Argo buoys were deployed ~15 years ago and cover much of the oceans, which had minimal coverage before then. Information about aspects of past climate other than temperatures is even more limited.
Analytical chemistry is providing improved information about past climate, for example:
Temperatures, changes in climate, and extreme weather events
Concentrations of CO2 and other atmospheric components
Extent of sea ice and glaciers over time
Impact of geological events such as volcanic eruptions and earthquakes
The first step in studying a sample that was formed in the past (such as a fossil or a layer of sediment under the ocean materials that is more than a few hundred years old and have no attached information) is to determine its age.
The most common way to do this is using mass spectrometry to measure an elemental isotope ratio that is dependent on age. Carbon is often used for dating once-living samples, such as plants or artifacts made of wood, as its main isotopes, carbon-12 (12C) and carbon-13 (13C), are stable, while carbon-14 (14C) is radioactive, with a half-life of about 5700 years. The supply of 14C in the air is constantly replenished by cosmic rays hitting nitrogen-14, but once an organism dies the 14C fraction will steadily go down. [i]
Mass spectrometry is a preferred technique for measuring ratios of isotopes of elements, and reliably determines the fraction of 14C in the once-living sample and thus the age at the end of its life for up to about 20,000 years ago, and possibly somewhat earlier. For older samples other elements must be used, for the 14C fraction is too small to be reliably measured.
Several other elements have naturally-occurring long-lived radioactive isotopes that could be used to determine the age of an older sample. Potassium, for example, is a widely-distributed element with substantial concentrations throughout the earth. Potassium-40 (40K) is a very long-lived isotope (1.25 billion years) with (unusually) two forms of decay, one to stable argon-40 and the other to stable calcium-40.
Argon in the atmosphere contains the 40Ar produced over the lifetime of the earth. However, if the decay occurs in a solid that did not contain any air when it solidified and does not allow the argon produced by potassium decay to escape, then the amount of 40Ar reflects the date when the solidification took place.[ii]
The reverse also can be measured: the amount of 40Ar in air trapped in ice will reflect the age at which the ice formed. [iii] This mass spectrometric technique has recently been used to determine the age of samples at different depths in an Antarctic ice core (Figure 1).[iv] [v] The authors of these studies then measured the amounts of different atmospheric gases in those samples and plotted them as a function of sample age (Figure 2).
Figure 1: Age of samples taken at indicated depth below surface of ice core
Figure 2: Results for key atmospheric gases as function of ice core age (various techniques were used for measuring the gases)
Mass spectrometry is often used to study past temperatures. The ratio of the trace isotope oxygen-18 (18O) to the most common isotope oxygen-16 (16O) in a once living organism depends on the temperature of the air at the time that the oxygen was incorporated into the organism by metabolism. The ratio is measured using mass spectrometry. The higher the temperature, the lower the fraction of 18O in the once-living organism.
A representative study was carried out on mollusks found in layers of sediments near the northwest shore of Iceland, covering the period 350 B.C. to A.D. 1600. The layers in the shells reflected the year-round temperatures in which the mollusks had lived (Figure 3). From such data on many samples at different layers in the sediments the authors constructed a chart of temperatures over that time period. Notably, they were able to correlate these temperatures with historical records in Iceland from 865 to 1600 (Figure 4). The authors pointed out that “On the basis of δ18O data, reconstructed water temperatures for the Roman Warm Period in Iceland are higher than any temperatures recorded in modern times.”[vi]
Figure 3: Example of temperatures derived from a shell that lived through four summers (S) and three winters (W)
Figure 4: Variation of temperatures from the Roman Warm Period to ~AD 1800.Information from historical documents is at the top on the right side.
Another study using mass spectrometry to determine temperatures using oxygen isotope ratios was carried out over in fjords in Sweden, covering a 2500 year period. One of the fjords is on the north coast of Sweden (Atlantic Ocean) and the other on the southwest (North Sea near Denmark). The authors state:
“The record demonstrates a warming during the Roman Warm Period (~350 BCE – 450 CE), variable BWT [bottom water temperatures] during the Dark Ages (~450 – 850 CE), positive BWT anomalies during the Viking Age/Medieval Climate Anomaly (~850 – 1350 CE) and a long-term cooling with distinct multidecadal variability during the Little Ice Age (~1350 – 1850 CE). The fjord BWT record also picks up the contemporary warming of the 20th century (presented here until 1996), which does not stand out in the 2500-year perspective and is of the same magnitude as the Roman Warm Period and the Medieval Climate Anomaly.[vii]”
The authors of this study include a chart relating their temperature information with others in the North Atlantic (Figure 5).
Figure 5: Bottom water temperatures for different locations in the North Atlantic Ocean going back to about 350 BCE. Abbreviations shown at the top of the chart: RWP represents the Roman Warm Period, DA represents the Dark Ages, VA/MCA represents the Viking Age/Medieval Climate Anomaly and LIA represents the Little Ice Age.
The oxygen ratios thus allow estimating temperatures for once-living organisms. Importantly, they allow doing so year-round. Tree ring diameters, which are sometimes used for estimating past temperatures, primarily reflect temperatures during the growing season, and are also influenced by factors such as rainfall and location (such as above or below the treeline at a given time).
A recent study provides a different use of the oxygen isotope ratios for studying past climate: do volcanos have an impact on major climate factors such as the El Niño – Southern Oscillation (ENSO)? Some studies have suggested that major volcanic eruptions can impact the ENSO cycle, but only a few such events have relevant weather data. Fossil corals in the regions impacted by ENSO dating back centuries have well-defined monthly layers. They were dated by mass spectrometry using U/Th ratios. Then oxygen-18 measurements for these layers allow estimation of temperatures coinciding with major volcanos.
The results were combined with prior studies to produce a temperature record covering from ~1100 to ~2000 CE. Six major volcanos in this time period were charted against the temperatures (Figure 6). No evidence was found that the volcanos had caused an ENSO event. [viii]
Figure 6: Coral δ18O measurements of temperatures for six major volcanic eruptions in the last 900 years, with lines at the bottom showing the degree to which stratospheric aerosols reduced downwelling sunlight during the volcanos
Conclusion for Part One
This completes the first post on applications of mass spectrometry to climate science. The second post will focus on studies of factors that need to be understood to be able to develop reliable models of climate, with emphasis on research on aerosols.
Endnotes
[1] The figures and charts and other information from the papers are the property of the authors and publishers.
[i] https://en.wikipedia.org/wiki/Carbon-14
[ii] https://www.radioactivity.eu.com/site/pages/Potassium_40.htm
[iii] M.L. Bender, et al., “The contemporary degassing rate of 40Ar from the solid Earth”, PNAS (2008) 105, 8232-8237 https://www.pnas.org/content/105/24/8232
[iv] J.A. Higgins, et al., “Atmospheric composition 1 million years ago from blue ice in the Allan Hills, Antarctica”, PNAS (2015) 112, 6887-6891 https://www.pnas.org/content/112/22/6887
[v] Y. Yan , et al., “Two-million-year-old snapshots of atmospheric gases from Antarctic ice”, Nature (2019) 574, 663-663 https://www.nature.com/articles/s41586-019-1692-3 and https://www.nature.com/articles/d41586-019-03199-8
[vi] W.P. Patterson, K.A. Dietrich, C. Holmden and J.T. Andrews, “Two millennia of North Atlantic seasonality and implications for Norse colonies” PNAS (2010) 107, 5306-5310 https://www.pnas.org/content/107/12/5306
[vii] I.P. Asteman, H.L. Filipsson, and K. Nordberg, “Tracing winter temperatures over the last two millennia using a north-east Atlantic coastal record”, Climate of the Past (2018) 14, 1097–1118. https://doi.org/10.5194/cp-14-1097-2018
[viii] S.G. Dee, et al., “No consistent ENSO response to volcanic forcing over the last millennium” Science (2020), 367, 1477-1481 https://science.sciencemag.org/content/367/6485/1477.full
Biosketch
Roland Hirsch has served the field of analytical chemistry in a 52-year career that spans teaching, research, and leadership at Seton Hall University, and 33 years of government service at the National Institutes of Health and the U.S. Department of Energy. Roland has been a leader of the ACS Division of Analytical Chemistry, as Councilor for 25 years, as Division Secretary for 4 years, Chair-Elect, Program Chair, and Chair, and as its Web Editor for 22 years. Roland organized the 50th-anniversary celebration of the Division and 25 years later, wrote the definitive history of the first 75 years of the Division, published in Analytical Chemistry in 2013. Roland has also been active in ACS Governance, including Chair of the Committee on International Activities, Secretary of the Committee on Nominations and Elections, Member of the Committee on Divisional Activities, Senior Chemists Task Force, Committee on Committees, and Liaison to the ACS Committee on Professional Training.
Based on a presentation prepared for the American Chemical Society National Meeting in Philadelphia in March 2020. It was to have been in the Division of Analytical Chemistry’s session “Advances in Mass Spectrometry”. The meeting was canceled, but this presentation was revised and made available on the web site for the meeting: https://www.morressier.com/article/mass-spectrometry-essential-research-climate-science/5e735e33cde2b641284a879e
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