Formulation of scale transformation in a stochastic data assimilation framework

Liu, Feng; Li, Xin

doi:https://doi.org/10.5194/npg-24-279-2017

Articles | Volume 24, issue 2

https://doi.org/10.5194/npg-24-279-2017

© Author(s) 2017. This work is distributed under
the Creative Commons Attribution 3.0 License.

https://doi.org/10.5194/npg-24-279-2017

© Author(s) 2017. This work is distributed under
the Creative Commons Attribution 3.0 License.

Articles | Volume 24, issue 2

Research article

|

15 Jun 2017

Research article |

| 15 Jun 2017

Formulation of scale transformation in a stochastic data assimilation framework

Feng Liu and Xin Li

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Final revised paper (published on 15 Jun 2017)
Preprint (discussion started on 30 Jun 2016)

Interactive discussion

Status: closed

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

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- Supplement

RC1: 'The Stochastic Calculus Reformulation of Data Assimilation: On Scale', Anonymous Referee #1, 12 Jul 2016
- AC1: 'Author's comments for reviewer 1', Feng LIU, 28 Jul 2016
- AC2: 'Author's new responses to Referee 1', Feng LIU, 30 Oct 2016
RC2: 'Comment on The Stochastic Calculus Reformulation of Data Assimilation: on Scale', Peter Jan van Leeuwen, 31 Aug 2016
- AC3: 'Author's responses to Referee 2', Feng LIU, 30 Oct 2016
EC1: 'Editor's comment', Olivier Talagrand, 04 Sep 2016
- AC4: 'Author's responses to the editor's comments', Feng LIU, 30 Oct 2016

Peer-review completion

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

AR by Feng LIU on behalf of the Authors (30 Oct 2016) Manuscript

ED: Referee Nomination & Report Request started (01 Nov 2016) by Olivier Talagrand

RR by Peter Jan van Leeuwen (20 Nov 2016)

Suggestions for revision or reasons for rejection

This manuscript describes a new way to incorporate scale into data assimilation by defining it via the Lebesgue measure on the state and observation spaces. Although the mathematical derivations make perfect sense to me now, and the connection to the literature on representation errors is made, I must confess I still miss the point. My main issue is that I don’t understand the stochastic equation in scale space, neither where it comes form nor how it helps solve the representation error problem. Detailed comments are given below. Each page is denotes with a capital P (e.g. P1, P2, etc), followed by line numbers on that page.

Abstract:
Define the spatial scale issue, Do you mean that models and observations define scale differently? Or do you mean that scale is not well defined in general, which then hampers model-observation comparisons and data assimilation?

P1
25 ‘increases with the difference’
28 parameters are chosen constants in time, variables are varying in time. So I guess the authors mean ‘geophysical variables’, or perhaps both. This runs through the whole manuscript.

P2
14 Parameters cannot be collected. Values for variables can be collected. And they are not collected by Earth Observation techniques but by Earth observations.
15 The ‘therefore’ is not a logical consequence. What does ‘them’ refer to?
16 The fact that observation operators are nonlinear and complex has in principle nothing to do with mismatches between model units and observation footprints. The logical connection is not clear.
16 Model units should be defined a bit better (I know this is difficult because it is unclear what scales a model with a certain grid box size represents.)

P3
4 Van Leeuwen also discussed the spatial and temporal resolution differences as giving rise to representation error, and Lorenc 1986 was the first to discuss the observation operator as source of an extra error on top of the measurement error in data assimilation.
7 ‘According to the above…’ The land surface dynamical processes have not been discussed, so the logical link is missing.
25 Van Leeuwen and I think also Hodyss and Nichols state the nonlinear problem and Van Leeuwen gives general equations that are exact in the nonlinear case (the convolutions). However, indeed, as the authors mention, the theory is only illustrated in the linear case. This could be mentioned.

P4
7 Data assimilation does not necessarily result in first and second moments, the solution is the full pdf. For instance, data assimilation can describe multimodal pdfs.
16 ‘the definition of scale uses measure theory’

P6
19 typo in first equation: ‘the first s1’ should be ‘s2’.

P7
I assume phi(t) is deterministic in eq (1), it is the drift term, so ‘transition probability’ is perhaps misleading.

P8
11 scale is defined as an area, so has physical dimension m^2. Typically scale is defined in terms of distance. That might be mentioned.
14 The standard scale depends on the units used, it is a different thing using meters of millimetres. Is this a useful definition? Or do you assume that all physical scales have been normalised? Please clarify.

P10
16 ‘discovered’ should perhaps be ‘described the origin of’

P11
7 the definition of ‘variable’ is unconventional and perhaps misleading. An element of state vector X is typically called a variable in the data-assimilation literature. Another name would be preferable.
7 What is the exact relation between state vector X and variable V?
23 The authors introduce a stochastic process in scale space that operates at a time instance, so time is a constant, and the variable changes due to a process in scale space. Is this interpretation correct, and if so, what is this process physically? This is a crucial point for me, and the point where I get lost.
15 ‘the variable varies with scale because of the scale issue’ is unnecessary vague. Perhaps remove this part of the sentence?
31 What are the exact relations between M and p, and eta and q?

P12
I understand this as a formal derivation of the stochastic process in scale space up to line 23 (but, as mentioned, I don’t understand the physical process behind this).
I don’t understand what assumption 1 means, ‘in data assimilation’ seems rather vague to me.
Assumption 2 assumes that the model has a constant grid in space and time. That should perhaps me mentioned explicitly. So the grid is not finite element or finite volume, or adaptive in time.
Assumption 3: What does ‘scale dependent’ mean here? That the scale changes when applying Bayes Theorem?

P13
4 What does this equation mean? That X(s) changes due to changes in scale in the analysis step? If so, phi should also depend on Y, or at least on the scales in Y. Or does this equation describe a scale relation in X? So how X depends on scale? If so, what is the stochastic forcing?
Also, how to choose or estimate phi and sigma in an application?
9 The `authors state that ‘Assumption 3 implies that the scales of the state and observation are invariant when observational information is added in the analysis step’. I don’t see that, please clarify.
13 Typo in equation (15): X(s_X) = X_0 + …
22 What does this equation mean? Is this the prior marginal pdf of X in scale space?
I am lost as this page. Where is the measurement uncertainty? That should also appear somewhere in p(y|x). I see only the scale part of the error. Is the assumption that the scale part is dominant?
And again, what is this process in scale space that happens at observation time prior to calculating p(y|x). Note that assumption 2 mentions explicitly that the scale of state and observation do not change before the observation time.
28 I’m not sure what happens here. Why is there a stochastic equation for H? Ehy not use the pdf of X directly to find the uncertainty in H?
Formally I can follow the derivations, they a very clear, but I don’t understand what the equations mean.

Concluding, I seem to miss something fundamental related to the stochastic equation in scale space and hope the authors can clarify that. What I would understand is a transformation from state to observation space, which might be modelled by a stochastic process. The rationale for that is that the state and/or observation subgrid processes are unknown and treated randomly. For that, only equations (19) and (20) are needed, although I still don’t see why a stochastic differential equation is used to model this transformation, why not define it directly as a nonlinear function from state to observation space? Then the likelihood can be formulated.

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RR by Anonymous Referee #1 (24 Nov 2016)

ED: Reconsider after major revisions (further review by Editor and Referees) (28 Nov 2016) by Olivier Talagrand

AR by Feng LIU on behalf of the Authors (27 Jan 2017) Author's response

ED: Referee Nomination & Report Request started (01 Feb 2017) by Olivier Talagrand

RR by Anonymous Referee #1 (14 Feb 2017)

RR by Peter Jan van Leeuwen (19 Feb 2017)

Suggestions for revision or reasons for rejection

The authors have done a fantastic job in clarifying all mathematics used, and I find the manuscript very easy to read as the argument s are now presented in a logical order (for me). Furthermore, the results are very interesting. However, I suggest a modifications along the following lines.

1. I still expect the readers get lost. Let me explain what I think their reasoning could be:

They understand that a geophysical variable is a random variable as function of scale. The reason is two-fold, at a certain scale the variable can have different values in different parts of the domain, and our prior knowledge of the random variable value at a certain scale is limited. This, then leads to a pdf p(V,s) at each time instant. This joint pdf gives the likely range of V given likely ranges of s. It’s marginal p_s(V) gives the pdf of V at each scale value s. This is as far as they understand the reasoning until page 10 line 17.

However, at 18 on that page the authors introduce an SDE for V(s). This description has a different interpretation from the one above. Now the uncertainty, so the width of the pdf at a certain scale depends on the scale at which the SDE is started. This, then, means that the pdf p(V,s) depends also on a starting scale s_0, and a different s_0 leads to a different pdf. But, as far as they can see, s_0 is arbitrary, so this cannot work.

Even if one defines s_0 as the model unit scale or the observational footprint, there are still problems. Why would the uncertainty grow the further we are away from that scale? My guess would be that the pdf p_s(V) as defined above will be narrower at large scales, but this is not what the SDE will give us. Furthermore, the manuscript assumes s_0 to be the same for observation and model, and it is unclear why that is a good choice.

So their question will be: why is modelling the random character of the geophysical variable as function of scale via an SDE the right thing to do? Why can’t one work directly with p(V,s) as defined above?

If one does use p(V,s) as defined above for both state and observation they won't see the extra terms in 20-27 arising, apart from those terms already present in the representation error literature.

It is important that the authors make sure that this interpretation is avoided.

2. A completely different interpretation, and the one that the authors have in mind I think, is that one does know the variable at one scale and needs to transform that to a value for the variable at another scale. This is indeed what is needed to compare model results with observations. The interpretation of equation (10) is now that it described how to transform the variable value from the model unit, or the observational footprint, to another scale where the comparison will be made. That will in general be a deterministic part to this (e.g. due to statistical knowledge about the averaging process), and a stochastic part due to the heterogeneity of the variable. This should be made more clear.

3. The assumption is that the scale at which the comparison is made is larger than both the scale at which we know the model and the observation. This makes this problem very different from the main issue in dealing with representation errors in data assimilation, in which we do know the model only at a much courser scale then the observations, and the observations only at the fine scale. Then a direct comparison is not possible and extra information is needed to solve this problem.
This assumption is crucial to mention, and, although a limitation, it still makes this an interesting problem to study.

4. The authors need to add a motivation why an SDE is a good model for changing scales. What does the drift term represent? Does it represent that we can use more model units or more observations from different positions to describe the scale change via averaging? And if so, the SDE leads to an increasing stochastic variance for an increasing scale separation. Is that realistic if one can do spatial averaging with increasing scale?

5. The authors implicitly assume that both model unit and observational footprint are at scale s_0. Otherwise the Y_0 in equation (21) is unknown and the integrations in (21) from s_0 cannot be performed, so the equations cannot be used in practise. I think this assumption is not necessary, as long as s_0 for Y and s_0 for X are smaller than the scale at which the comparison is made they can be different.

6. Further on point 5, I actually think the authors should reformulate page 13. Assume the model is available at scale s_x and the observation at scale s_y. One cannot refer to scale s_0 as it is smaller then both of them, so x_0 and Y_0 are not known, so cannot be used.
Let us first look at the case s_x < s_y. We first have to transform the model values from scale s_x to s_y before we can use H. This is done via the SDE. Equation (19) needs to be rewritten as (assuming sigma_x = sigma_y = 1, and phi_x=phi_y=0 as in the paper):

H(s_y,X(s_y)) = equation (19) with s_0 replaced by s_x, and s_x by s_y.

Equation (20) becomes

Y(s_y)- H(s_y,X(s_x)) = Y(s_y)- H(s_x,X(s_x)) + ½ int_s_x^s_y H_xx du +
int sx^s_y dW – int_s_x^s_y H_x dW

This leads to a new drift and variance term easily extracted from the above.
In this way all reference to Y_0 and X_0 is disappeared, and all results become intuitive.

Now consider s_x> s_y. In that case one uses the SDE on Y, and finds:

Y(s_y)- H(s_y,X(s_x)) = Y(s_x) + int_s_y^s_x dW )- H(s_y,X(s_x))

And again drift and variance can easily be extracted.

(In fact, I would suggest the authors keep the phi’s and sigma’s as it is good to have the full set of equations.)

7. It would be good if the authors would discuss methods to find the sigma’s and the phi’s.

8. Example 3.4 would benefit from specifying the phi’s and sigma’s and pushing the example to the end, perhaps showing pictures of the pdf’s involved.

9. I don’t see how 3.5 is relevant and/or needed. Why all this trouble, why the one-dimensional rule? Different variables will have different scaling rules, but as far as I can see one can follow the scaling rule ideas for each independent observation separately. What is the problem?

10. As far as I can see, one can just define scale as the Lebesque integral, why do we need section 2.1?

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ED: Publish subject to minor revisions (further review by Editor) (22 Feb 2017) by Olivier Talagrand

AR by Feng LIU on behalf of the Authors (14 Mar 2017) Author's response Manuscript

ED: Publish subject to minor revisions (further review by Editor) (10 Apr 2017) by Olivier Talagrand

AR by Feng LIU on behalf of the Authors (20 Apr 2017) Author's response Manuscript

ED: Publish subject to technical corrections (03 May 2017) by Olivier Talagrand

AR by Feng LIU on behalf of the Authors (11 May 2017) Author's response Manuscript

Short summary

This is the first mathematical definitions of the spatial scale and its transformation based on Lebesgue measure. An Ito process-formed geophysical variable with respect to scale was also provided. The stochastic calculus for data assimilation discovered the new expressions of error caused by spatial scale transformation. The results improve the ability to understand the spatial scale transformation and related uncertainties in Earth observation, modelling and data assimilation.