In general, I am supportive of the efforts of the authors to evaluate (from the abstract) “the dynamic upwelling of the unfiltered error from outside of the span of the anomalies into the filtered subspace.” In particular, the work begins to formalize a concept of ‘dynamic upwelling’ or mixing that occurs between the unfiltered and filtered subspaces that has been mentioned in previous literature in passing, but never deeply explored.
I like the work beginning to extend the mathematical framework for the AUS to the scenario with the presence of additive model errors (though, I think it should be made very clear at the start that this is a simple subset of the full range of possible model error types, e.g. around page 2, line 12). I find this portion of the work very intriguing.
However, I have a bit of difficulty seeing the connection with multiplicative inflation. I agree with the previous reviewer in that there is a bit of a jump from section 3 into the results. I was expecting a bit more mathematical development to show exactly why multiplicative inflation is the choice made by the authors to address the problem of dynamic upwelling. As a reader, the motivation is not clear. I would appreciate at least one more subsection at the end of section 3 walking the reader through the motivation and justification for turning to multiplicative inflation to resolve the issue of dynamic upwelling.
From the second referee:
“My main concern is about the connection between unfiltered errors and covariance inflation.
I agree with the authors that “multiplicative inflation (in the ensemble span) neglects the fundamental issue that the unfiltered error lying outside of the ensemble span can be the major driver of the uncertainty in a reduced rank filter with model error”. This could be seen from eq. (38) and eq. (39). However, it should be studied analytically rather than numerically how these missing terms are linked to the inflation, meaning that inflation should be derived analytically rather than empirically, since the authors did develop all the mathematical tools.
Moreover, the empirical inflation parameter is chosen to be between 1 and 4, which means the covariance was inflated 4 times, while typically it is inflated by 20% at most.
Apart from that I am surprised that EKF-AUS performs equivalently for inflation between 1.5 and 4 (only slightly worse for high inflation). I would expect that the RMSE substantially increases for higher inflation.”
I am generally of the opinion that multiplicative inflation is a simple but inappropriate tool for use in general ensemble-based data assimilation. I am aware of the original motivation for its origins, but my experience has shown in many circumstances that in trying to correct one problem (e.g. an under-dispersive ensemble spread due to under-sampling or model error), it has the potential to lead to catastrophic filter divergence. I am open to being convinced to the contrary with a rigorous justification, but what I have seen so far has not quite succeeded.
For this reason, I agree with the second referee that if any statement about the validity of multiplicative inflation method is to be made, then it should have a rigorous analytical justification. However, I believe the authors may be very close to that goal for a specific subset of problem relating to the case of additive model errors. As long as the authors make this limitation clear from the beginning, then I believe the work should be published.
Response, page 2:
The terminology is conflated and should be decomposed.
“The interval between observations δ controls the nonlinearity of the map”
This is not correct. It is the interval between random impulsive forcing that controls the nonlinearity of the map. You happen to have equal matching intervals between the observing time and the random forcing, but these need not be identical. The interval between observations is only relevant to the dynamics if you are referring to the dynamics of the combined forecast/analysis data assimilation system.
Response, page 5:
“as demonstrated in Fig. 1 of our manuscript, for a reduced rank filter in the presence of model error, there is additional structure which gives a refinement to this set of matrices.”
This may be true, but it appears that the additional structure must be dependent on the choice of the presumed model error covariance Q. Is this correct?
Response, page 7:
“major difference in the results with reduced observations lies only in the minimum rank of the filtered subspace to prevent filter divergence”
Again, I believe that making the distinction between the analysis update interval, observing interval, and impulsive model forcing interval should help to clarify some of the issues. In this case, reducing the observations can change the effective forecast window in a ‘staggered’ sense, so that any point in the domain has a different time since last having an analysis update.
Response, page 8:
I agree with the reviewer, the use of italics for emphasis is unnecessary. My general opinion regarding a scientific or technical paper is that if something is unnecessary then it should not be included.
Definition 1 should be reworded so that the description of the indices k and i are not intertwined, e.g.
“Define the matrix Ek to be the orthogonal matrix at time k whose i-th column is the i-th backward Lyapunov vector (BLV) , corresponding to the Lyapunov exponent λi.”
Page 5, lines 1-8, section 2.1:
I’m not sure that ‘projection’ is the correct term to use here, since E^T itself is not a projection operator. Perhaps you can alter the terminology slightly to be more precise.
Equation (11) is missing a superscript “b” on the first Hx term, otherwise it appears incorrect. And then based on equation (7) the vk term should be positive, and the He term negative, which I think then corresponds correctly with equation (14) below.
The term vk should be known if it is the difference between the observation and the forecast mean.
I follow equation (17) as long as E is orthonormal. Is that the case?
Page 11, line 4:
Should this be K^?
I agree with the previous reviewer that the italics for emphasis are unnecessary and can be removed.
“Nevertheless, this design is artificial and would lead to poor filter performance”
This is an absolute statement that would require significantly more evidence in order to state it here. I suggest removing the assertion.
Page 14, line 6:
Again, the italics are unnecessary.
Page 15 (line 32) -16 (line 9)
I like some of the discussion here that was removed describing how previous authors have viewed the use of inflation. It would be nice to put this last paragraph back into the manuscript.
P. 27, Line 10:
“(i) sufficiently increasing the ensemble size to include asymptotically stable modes that produce transient instabilities”
Do you have a proof that the stable modes with larger Lyapunov exponents are more likely to produce the ‘dynamic upwelling’ than the stable modes with smaller LEs? If not, than modestly increasing the ensemble size may not be a guaranteed solution.