Multi-timescale data assimilation for atmosphere–ocean  state estimates

Steiger, Nathan; Hakim, Gregory

doi:https://doi.org/10.5194/cp-12-1375-2016

Articles | Volume 12, issue 6

https://doi.org/10.5194/cp-12-1375-2016

© Author(s) 2016. This work is distributed under
the Creative Commons Attribution 3.0 License.

https://doi.org/10.5194/cp-12-1375-2016

© Author(s) 2016. This work is distributed under
the Creative Commons Attribution 3.0 License.

Articles | Volume 12, issue 6

Research article

|

24 Jun 2016

Research article |

| 24 Jun 2016

Multi-timescale data assimilation for atmosphere–ocean state estimates

Nathan Steiger and Gregory Hakim

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Final revised paper (published on 24 Jun 2016)
Preprint (discussion started on 19 Aug 2015)

Interactive discussion

Status: closed

AC: Author comment | RC: Referee comment | SC: Short comment | EC: Editor comment

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RC C1582: 'Review', Anonymous Referee #1, 08 Sep 2015
- AC C1745: 'Reply to Referee #1', Nathan Steiger, 21 Sep 2015
RC C1698: 'Review', James Annan, 17 Sep 2015
- AC C2371: 'RESPONSE TO JAMES ANNAN', Nathan Steiger, 19 Nov 2015
RC C1807: 'Comment on Steiger and Hakim', Anonymous Referee #3, 26 Sep 2015
- AC C2374: 'RESPONSE TO REFEREE 3', Nathan Steiger, 19 Nov 2015
RC C1822: 'Review of “Multi-time scale data assimilation for atmosphere–ocean state estimates” by Steiger and Hakim', Anonymous Referee #4, 28 Sep 2015
- AC C2376: 'RESPONSE TO REFEREE 4', Nathan Steiger, 19 Nov 2015

Peer-review completion

AR: Author's response | RR: Referee report | ED: Editor decision

ED: Reconsider after major revisions (24 Nov 2015) by Hugues Goosse

AR by Nathan Steiger on behalf of the Authors (22 Feb 2016) Author's response Manuscript

ED: Referee Nomination & Report Request started (22 Feb 2016) by Hugues Goosse

RR by James Annan (04 Mar 2016)

Suggestions for revision or reasons for rejection

Based on the authors' claim that their method is not equivalent to the augmentation method, I was prompted to reconsider their algorithm more carefully, and while I now agree with them on that point, an unfortunate but inevitable conclusion is that the algorithm is not correct. At best, perhaps one could argue that it's making a specific assumption/simplification, but no reasoning has been presented for why this is appropriate or necessary. The issue is the artificial decoupling of time scales that the algorithm performs.

I will demonstrate the flaw with a very simple example. Let us consider the model x(t)=at, which follows a simple linear trend through time t=0,...,10, where the trend a is uncertain, but we have a good observation of the initial condition x(0)=0. A natural ensemble would consist of a large number of simulations, each of which has a particular trend sampled from our prior for a. Over the simulation interval t=[0,10], the 11-year mean xbar_i for each simulation i is given by 5a_i where a_i is the value of a specified for each respective ensemble member.

If we observe the "decadal" (11y) mean xbar_obs = 15 with relatively high precision, the algorithm as described in the manuscript will shift the trend lines in order that the ensemble members take the correct mean, but the method will not affect their slopes, since the deviations from the mean are simply added back on to the mean unchanged (p4-5 in manuscript). Effectively, the algorithm is adding a constant (which varies with i but not t) on to each time series x_i(t). However, the correct posterior for this problem is actually given by a_analysis = 3, which results in a trend from x(0)=0 through the mean point (t=5,x=15) to finish at (10,30). The point of this example (and the flaw in the method) is that the authors have decoupled information provided by the decadal mean, from the sub-decadal variation around that mean. In cases where there actually is some covariation between these variables, this is incorrect.

Note that although I've couched this example such that it appears as a parameter estimation problem, my argument does not depend on this. Eg, we could consider the exponential growth of some perturbation, with known growth rate but which depends sensitivity on the initial condition. We could consider x(t+1)=2x(t), say, with the initial state x(0) ranging from 0 to 1. In this case, an observation of decadal mean should not be allowed to move x(0) more than a little, even though the decadal means of the ensemble members will range from 0 to O(1,000,000). Such an example may be loosely analogous to the birth of an el nino where its appearance a few years down the line (and consequent influence on long-term surface temperature) does not mean the surface temp at the initial time was much warmer than originally thought. Rather, there was a small perturbation which projected onto the appropriate growing mode.

The authors argue briefly against augmentation (l185-), but their criticism does not appear valid to me. Augmenting with various time averages allows observations of all of these to be assimilated at will, and this will indeed propagate information in both directions across time scales where appropriate (i.e. where there is a nonzero covariance). I suppose it might be more formally correct to refer to the system as a fixed-lag smoother rather than filter, but the principle is very straightforward. An appropriate augmented space for my example above could be the full set of x_i(t), t=0,10, together with the average xbar_i (technically even this is not required as it's observable through the x_i and thus could be generated by the observation operator applied to the individual years, but no matter). Any legitimate assimilation method applied to these augmented states that respects the covariance information will preserve the relationship between the slope and mean, and thus also preserve x(0)=0, irrespective of what value the observations take. Observations for the individual years as well as the decadal mean can be assimilated, and each one will affect the others through whatever covariances exist in the ensemble. In general, the augmentation method automatically couples the time scales and generates the correct solution. (The EnKF will add a linearisation approximation, but that's a limitation of EnKF, not augmentation per se - the PF would not have that issue). If there is in fact no covariance across time scales, this is not a problem for the augmentation method, but I see no reason for cutting any link between the time scales a priori. Indeed it seems to run contrary to the goals of the research which sets out to combine information from multiple time scales. I therefore regret I can't recommend publication of this manuscript.

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RR by Anonymous Referee #1 (14 Mar 2016)

Suggestions for revision or reasons for rejection

The authors do not give a good explanation of why the AMOC reconstruction has such a low skill.
They state a possible reason on l.392 p.12, which I doubt since GMT has also small correlations with T2m for the CCSM4 model (Fig. 4b) but the GMT reconstruction has a high skill (Fig. 6b). Possible experiment could be to use localization, since localization not only is needed for small ensemble sizes but also removes spurious correlations and might result in better skill for AMOC, especially when annual variables are assimilated. Therefore, I would like to see a better explanation of why the AMOC reconstruction has a low skill. This is my main concern. My following comment is also related to that: the authors mislead reader with the last sentence in abstract (l.17 p.1) and with the last paragraph in conclusion (l.462 p.14), since the AMOC reconstruction is poor.

The last experiment that the authors had performed (p.13) is interesting though the results are not surprising. What would be also interesting to see whether the two following experiments give better/poorer reconstructions:
1) Both the short-only and long-only reconstructions use 100 pseudoproxies randomly drawn.
For the mixed-resolution reconstructions, 100 pseudoproxies are randomly drawn for the annual time scale. And then those preudoproxies are averaged to obtain 5/20yr averaged observations.

2) Both the short-only and long-only reconstructions use 100 pseudoproxies randomly drawn.
For the mixed-resolution reconstructions, 100 pseudoproxies are randomly drawn from 274 proxies for the annual time scale, 100 pseudoproxies are randomly drawn from the left 174 proxies for long-time scales, leaving 74 pseudoproxies in total unused.

My last few suggestions concern notations:
1) namely, forward model (l.139). In DA community forward model means climatological model that is propagated forward in time. In paleoclimate community forward model is a model that estimates an observation from a proxy archive. Therefore this is confusing. I would remove forward model from l.139 leaving proxy system model only.

2) “the true value of the observations” (l.104). In DA community true value is a value of the truth from which observations are generated, thus in pseudo-observations experiments true values are the values obtained from a control model run before adding the noise in order to obtain observations. And I believe this is not what the authors mean by “the true value of the observations”. This is confusing. Please change.

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ED: Reconsider after major revisions (08 Apr 2016) by Hugues Goosse

AR by Nathan Steiger on behalf of the Authors (21 May 2016) Author's response Manuscript

ED: Publish as is (27 May 2016) by Hugues Goosse

AR by Nathan Steiger on behalf of the Authors (27 May 2016)

Short summary

We present a data assimilation algorithm that incorporates proxy data at arbitrary timescales. Within a synthetic-test framework, we find that atmosphere–ocean states are most skillfully reconstructed by incorporating proxies across multiple timescales compared to using them at short or long timescales alone. Additionally, reconstructions that incorporate long-timescale proxies improve the low-frequency components of the reconstructions relative to using only high-resolution proxies.