the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 27 Sep 2024
Multifractality of Climate Networks
Abstract. Geophysical fields are extremely variable over a wide range of space-time scales. More specifically, they are intermittent in the sense that the strongest fluctuations are increasingly concentrated on sparser and sparser fractions of the space-time domain. Multifractals have been developed to analyse and simulate across scales such multiscale intermittency, while climate networks can detect and characterise extreme event synchronisation. In contrast to multifractal analysis, climate networks are usually generated at a given observation scale, despite displaying complex structures over larger scales and likely exhibiting similar complexity at smaller scales.
In this letter, we present how to overcome this dichotomy of approaches by analysing in detail the effects of increasing the observation scale for climate networks, as allowed by empirical data, i.e. how do they upscale. This must be understood as a preliminary step to be able to downscale them, including for practical applications such as urban geosciences that require analysis and simulation of intermittent fields at very high resolution. This is one of the reasons why we are using precipitation to illustrate our multifractal climate network approach.
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RC1: 'Comment on egusphere-2024-2793', Anastasios Tsonis, 30 Sep 2024
Climate networks were introduced 20 years ago and ever since have provided a unique way to investigate dynamics, synchronization, and other properties in climate. As the authors state, climate networks are commonly generated at a given observation scale. Here the authors extend this by considering generation of networks at varying time scales akin to multifractals. The authors are leaders in the areas of climate networks and multifractals and I found their idea and approach ingenious. They show that analyzing networks using a multiscaling approach could lead to new understandings in climate dynamics.
Just a little minor point. The first time climate networks were introduced in the literature was in 2004 (“The architecture of climate networks”, Tsonis and Roebber, Physica A, 333, 497-504). The Tsonis et al., 2006 was an expansion of the above work.
I highly recommend publication of the paper as is.
Anastasios Tsonis
Citation: https://doi.org/10.5194/egusphere-2024-2793-RC1 -
AC1: 'Reply to RC1', Adarsh Jojo Thomas, 21 Oct 2024
Thank you very much for your positive and encouraging feedback on our manuscript. We greatly appreciate your kind words and the recognition of our work.
We would like to make a small clarification that while our approach involves varying scales, the scales considered in this study pertain to both space and time, rather than time alone.
Thank you also for pointing out the original reference to the introduction of climate networks in 2004. We will be sure to take this into account in the next version.
We are grateful for your recommendation to publish and your valuable comments.
Citation: https://doi.org/10.5194/egusphere-2024-2793-AC1
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AC1: 'Reply to RC1', Adarsh Jojo Thomas, 21 Oct 2024
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RC2: 'Comment on egusphere-2024-2793', Anonymous Referee #2, 28 Nov 2024
The authors describe a new formalism to study the multi-scale/multi-fractal properties of functional network representations of climatological fields for the example of precipitation data. Their reported intention to bridge the gap between two rather distinct methodological approaches is being addressed by thoroughly studying the network structures emerging at different spatial and temporal aggregation levels of the underlying observational data. I find the presented work very interesting and inspiring and generally worth being published in Nonlinear Processes in Geophysics. There are however a few aspects where I would recommend some clarification/revision of the present material prior to final publication.
Main comments:
1. I do not find Fig. 1 referenced anywhere in the text. Please ensure that this content is properly embedded and explained. With respect to the top panels, I may note a clear deviation from the suggested power-law (emphasized by the linear fits in log-log scale) for lambda<10. It would be worth mentioning and explaining the reasons for this behavior (i.e., possible reasons for the scale breaks at 16 days and 440 km, respectively).
2. I am a bit concerned about practical issues in the estimation of the marginal and joint probabilities in the mutual information for precipitation data. As I learn from the Supplementary Material (which should be referenced in the text somewhere below Eq. (9)), the authors use equi-quantile binning, which however seems incompatible with precipitation data, at least at their original (daily) temporal resolution. One should keep in mind that precipitation is, in stochastic terms, a combination of two (non-independent) stochastic processes, one occurrence process and one magnitude process. The amount of zero values in the data at a given grid point will likely not correspond well with the employed equi-quantile binning, potentially causing a bias in the TDMI estimates.
3. Page 6, l.123: Could the framework of node-splitting invariant network measures (Heitzig et al., Eur. Phys. J. B, 85, 38, 2012, also co-authored by one of the authors) provide a more natural and theoretically meaningful way to aggregate/upscale (but also to disaggregate/downscale) spatiotemporal covariability information in the network domain?
4. Page 5, l.107: It is not clear if the TDMI threshold \theta_\lambda is really considered scale-dependent or not, and if so, how exactly. This becomes potentially important along with page 6, l.128, where \theta_\lambda is used, yet in a way that I do not quite understand. When \theta_\lambda is the threshold of the similarity measure at scale \lambda (which would be what I infer from Eq. (7) applied to individual scales), how can it be at the same time the number of links in the network at that scale? To me, this does not seem to make sense. Maybe there is just something in the explanation within the text that needs to be rectified; otherwise, I would cast quite some doubts on whether the reported results of the performed case study can really be interpreted in the way they are. (For example, if the absence of commutativity of the U_\lambda and C operators is partially due to improper/inconsistent thresholding at different scales.)
Minor comments:
5. Page 1, l.14: Reference Zou et al. describes approaches for transforming individual time series into different classes of complex network representations, but not the generation of network structures from multivariate or spatio-temporal data. Citing it as a reference for climate networks thus appears inappropriate.
6. Page 1, ll.14/15: The term ”long-range dependence” is commonly used with respect to temporal dependencies, while the spatial aspect commonly addressed in climate networks (or general functional network representations of spatially distributed time series) rather refers to what is called “teleconnections” in climate science. I do not think that referring in the present context to long-range dependence is well justified.
7. Page 1, l.17: Regarding the references in the previous line 16, all cited works refer to applications of different association measures to climate network analysis. This also applies to Hlinka et al. 2013 for the case of mutual information, but not to Cellucci et al 2005. I would recommend removing this citation at this point. Alternatively, you may cite the first-time application of mutual information to climate network studies by Donges et al., EPL, 87, 48007, 2009 (also co-authored by one of the authors).
8. Page 1, l.21: If I am not mistaken, the first combination of wavelet analysis and event synchronization was Agarwal et al., Nonlin. Proc. Geophys., 24, 599-611, 2017 (again also co-authored by one of the authors). Since I generally think that the number of self-citations by Prof. Schertzer and Kurths is already rather high, I would leave it to the authors whether or not to add yet another reference from their own impressive body of past works.
9. Page 2, Section 2: If I am not mistaken, the original TRMM dataset has both a higher temporal resolution than 1 day, larger spatial domain (circumglobal sub/tropics) and complete annual coverage beyond the JJAS season (likely chosen here because it is the Indian summer monsoon (ISM) season). It might be worth clarifying that the present study uses just a part of the TRMM data for a given region, with daily aggregates of the precipitation product (likely to avoid issues with the pronounced yet space-dependent diurnal cycle) and a focus on the ISM. Besides a possible interest to tie in with previous studies by Malik et al., Stolbova et al. and others, another well acceptable reason for focusing on the Indian subcontinent and the summer monsoon is to keep the number of nodes at the highest spatial resolution in a range where computing network properties is still computationally feasible. In general, I would very much welcome some more elaborations of the motivations of the authors for choosing the described setting.
10. Page 2, l.45: Could you please clarify if the mentioned relation between the temporal and spatial scale ratios applies to 2d data, 3d data, or both?
11. Page 2, l.47: Please mention that R_\lambda refers to the rain rate at resolution lambda.
12. Page 5, ll.93-96: I do not quite get the difference between the two cases mentioned here. At least, for me “degrading” would have the same meaning as “coarse-graining” or “upscaling”. Possibly the authors might have had something different in their mind; in this case, I recommend to elaborate a bit more explicitly on these two cases and their difference.
13. Page 5, ll.100-101: “Climate Networks are typically constructed for the case of extreme rain events.” – I rather disagree with this statement, since there are various other applications not even mentioned in this paper. Even if we restrict ourselves to precipitation networks, it rather depends on the temporal scale considered if the focus is on extremes or on general co-variability. Even for daily data, there has been work on correlation based precipitation networks (e.g. Ciemer et al., Climate Dynamics, 51, 371-382, 2018; again co-authored by one of the authors, or Ekhtiari et al., Chaos, 29 063116, 2019).
14. Table 1: I am wondering about the meaning of “length scale” in this table. Is it the mean distance between neighboring grid points? (Note that the grid points are on a regular latitude-longitude grid, where the spacing depends on geographical latitude).
15. Page 6, l.115: What do you mean by “significantly change the degree structure”? Changes in the spatial pattern of node degrees?
16. Page 6, l.116: Do you really mean \tau_\lambda or rather \tau_\Lambda (with capital \Lambda)?
Technical comments:
- 2, l.49: use Latex symbol \ll instead >>
- There are a few cases of references to equations in the main text where spaces are missing (ll.63, 65,105,107,109,130,144). In l.93, the Section number should not be given in brackets.
- 3, ll.67,77: I suggest capitalization of “Delayed” in TDMI
- 3, l.72: comma in the beginning of the line
- 3, l.76: “are used”
- 2, caption, l.2: “replaced”
- References: The individual citations should not contain publisher information in the case of journal articles, and also omit the term “number” (this also applies to the Supplementary Material)
- Reference Parisi & Frisch 1985) is missing information on the pages.
- 12, l.223-224: eprint information appears redundant with the given URL/DOI.
- 12, l.225: please capitalize the journal name
Citation: https://doi.org/10.5194/egusphere-2024-2793-RC2
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