Spatiotemporal complexity and time-dependent networks in sea surface temperature from mid- to late Holocene

Abstract

In climate science regime transitions include abrupt changes in modes of climate variability and shifts in the connectivity of the whole system. While important, their identification remains challenging. This paper proposes a new framework to investigate regime transitions and connectivity patterns in spatiotemporal climate fields. Firstly, local regime shifts are quantified by means of information entropy; secondly, their spatial heterogeneity is examined by identifying the underlying spatial domains of the entropy field; finally, a weighted, direct and time-dependent network is inferred to capture the linkages between these domains. The spatiotemporal variability in sea surface temperature (SST) in two simulations of the last 6000 years is investigated with the proposed approach. The largest regional regime shifts emerge as abrupt transitions from low to high-frequency SST oscillations, or vice versa, in both simulations. Furthermore, the variability in time of the two climate networks is studied in terms of their network density. Generally, rapid and sudden transitions in the degree of connectivity of the system are observed in both simulations but, in most cases, at different times, with few exceptions. This suggests that our ability to predict the climate system may be hampered by its inherent complexity resulting from internal variability.

Appendices

Appendix 1: δ-MAPS

1.1 Cores identification

The domain identification algorithm starts from epicenters or cores. Given a K-neighborhood $\Gamma _{K} \left( i \right)$ of a cell $i$, we compute its local homogeneity as:

$$ r_{k} \left( i \right) = \frac{{\mathop \sum \nolimits_{{m \ne n \in \;\Gamma _{K} \left( i \right) }} r_{m,n} }}{{K\left( {K + 1} \right)}} . $$

(5)

A similar notation is used for the homogeneity of a set of grid cells:

$$ r\left( A \right) = \frac{{\mathop \sum \nolimits_{m \ne n \in A } r_{m,n} }}{{\left| A \right|\left( {\left| A \right| - 1} \right)}} , $$

(6)

$\left| A \right|$ being the cardinality of the set and $r_{m,n}$ the correlation (at lag $\tau = 0$) between the time series embedded in grid cells m and n. A grid cell is marked as a core if its local homogeneity is a local maximum and greater than a threshold $\delta$. More formally, a grid cell $i$ is a core if $r_{k} \left( i \right) > \delta$ and $r_{k} \left( i \right) > r_{k} \left( j \right),\quad i \ne j,\quad \forall j \in\Gamma _{K} \left( i \right)$. Cores are then iteratively expanded and merged to identify domains [22].

1.2 Domain identification

Given one or more identified cores $c$, a spatial grid G and a threshold $\delta$, a domain $A$ is the maximal set of grid cells satisfying three constraint: (a) $c \in A$, (b) $I_{G} \left( A \right) = 1$ and (c) $r\left( A \right) > \delta$. $I_{G} \left( A \right) = 1$ denotes that the set $A$ is spatially contiguous. Fountalis et al. [22] proved that this problem is NP-hard and relied on a heuristic to solve it. The heuristic algorithm iteratively expands and merges cores to find the full extent of domains. The process starts from the domain with largest homogeneity, let it be $A$. The expansion algorithm considers all adjacent grid cells of $A$ and then adds the grid cell $i$ which maximizes the homogeneity $r\left( {A \cup i} \right)$ and for which $r\left( {A \cup i} \right) > \delta$. If two domains $A$ and $B$ are adjacent, the merging algorithm determines that the two domains should be merged whenever $I_{G} \left( {A \cup B} \right) = 1$ and $r\left( {A \cup B} \right) > \delta$. This process stops when no more merging and expansions are possible. Two heuristics to infer the $\delta$ threshold and the K parameter have been proposed in [22] and [38]. Given a significance level $\alpha$, the threshold $\delta$ is computed as a sample average of the statistically significant cross-correlations between randomly chosen grid cells in the dataset considered.

Appendix 2: Heuristic for the vicinity threshold $\varepsilon$

A crucial step in the computation of a recurrence plot (RP) is the definition of the vicinity threshold $\varepsilon$. The $\varepsilon$ parameter defines the neighborhood of each state $\varvec{x}_{i}$ in a trajectory of a dynamical system. It must be carefully chosen since a value that is too small may include noisy fluctuations in the RP, and one that is too large could hide the recurrence structure of the time series [30]. Several “rules of thumb” for the choice of the $\varepsilon$ threshold have been proposed: ~ 5% of the maximal state space diameter [54] no more than 10% of the mean (or maximum) state space diameter [55, 56] or a value ensuring a recurrence point density of ~ 1% [57]. Moreover, choosing $\varepsilon$ strongly depends on the system under study [30].

Here, we present a new simple heuristic tailored for the selection of a different $\varepsilon_{i}$ for each (not-embedded) time series $x_{i} \left( t \right)$ of a spatiotemporal field $\varvec{X}\left( t \right)$.

The proposed strategy relies on the following steps:

• (a) We consider a monthly spatiotemporal climate field $\varvec{X}\left( t \right)$ embedded in a two-dimensional grid at each time step t. For every grid cell $i$ we have a linearly detrended, anomaly time series $x_{i} \left( t \right)$ with $T$ data points. We define an $\varepsilon_{i}$ as a percentage $\rho$ of the standard deviation $\sigma_{i}$ of $x_{i} \left( t \right)$. This step defines a new spatial field ${\mathbf{E}}\left( \rho \right)$, referred to as $\varepsilon$-map.

• (b) For a given $\varepsilon$-map ${\mathbf{E}}\left( \rho \right)$, we compute the RP and its information entropy for every $x_{i} \left( t \right)$ in $\varvec{X}\left( t \right)$ to get a spatial entropy field $\varvec{S} = \varvec{S}\left( \rho \right)$. Since the entropy field depends on $\rho$, we assess its sensitivity to $\rho$ as follows: first, we compute $\varvec{S}\left( \rho \right)$ for $\rho$ ranging from $\rho_{ \hbox{min} } = 0.01$ to $\rho_{ {\rm max} } = 1.5$ every $\Delta \rho = 0.01$; second, we define a pairwise distance matrix between all $\varvec{S}\left( \rho \right)$. Given two fields, $\varvec{S}\left( {\rho_{1} } \right)$ and $\varvec{S}\left( {\rho_{2} } \right)$, we compute their distance as $D\left( {\varvec{S}\left( {\rho_{1} } \right), \varvec{S}\left( {\rho_{2} } \right)} \right) = \mathop \sum \limits_{i} | s_{i} \left( {\rho_{1} } \right) - s_{i} \left( {\rho_{2} } \right) |$, where $s_{i} \left( \rho \right)$ denotes the entropy at grid cell $i$.

• (c) Finally, we select an optimal $\rho = \rho^{*}$ such that the computed entropy field does not depend on perturbations around $\rho^{*}$. This assures that, for every $s_{i} \left( {\varepsilon_{i} } \right)$ at grid cell i, $s_{i} \left( {\varepsilon_{i} } \right) \sim s_{i} \left( {\varepsilon_{i} \pm \delta_{i} } \right),\quad \delta_{i} \ll \varepsilon_{i}$. The proposed heuristic for the $\varepsilon$-selection is then based on the rationale that the computed entropy field should not depend on perturbations around the chosen $\varepsilon_{i}$, for every grid cell i.

In case of results presented in Sect. 5, we obtain a value of $\rho^{*} = 75\%$ both for COBEv2 and for the Sr02 simulation. We then assign to each grid cell i a value of $\varepsilon_{i} = \rho^{*} \cdot \sigma_{i}$, with $\rho^{*} = 0.75$.

For results in Sect. 4, we face an additional problem: we have to compute a spatiotemporal entropy field and not just one time instant. Therefore, for this step we compute ${\mathbf{E}}\left( \rho \right)$ using the complete simulation length. We consider a time series $\hat{x}_{i} \left( t \right)$ of monthly SST defined at grid cell i (with length $T = 72{,}000$  months) and compute a new time series $x_{i} \left( t \right)$ representing the evolution of detrended anomalies at all time $t$. SST anomalies $x_{i} \left( t \right)$ are computed by removing both the seasonal cycle and a linear trend. This is done for each 100-year non-overlapping window instead of the 6000-year time series to account, at first order, for changes in SST seasonality and (small) trends induced by orbital changes from mid- to late Holocene [29]. The parameter $\varepsilon_{i}$ is thus constant in time. Given a $x_{i} \left( t \right)$, this implies a notion of neighborhood as a function of all possible states explored in the 6000-year-long simulation (i.e., the closeness between two states is conditioned on distances between all states in the 6000-year simulation).

The heuristic is then tested for the first $W = 100$ years of the simulation. This analysis is found to be independent of resolution. A global minimum in the pairwise distance matrix is identified around the values of $\rho^{*} = 75\%$ (Figure A1). We then assign to each grid cell i a value of $\varepsilon_{i} = \rho^{*} \cdot \sigma_{i}$, with $\rho^{*} = 0.75$.

It should be noted that in the analyzed simulations the standard deviations of time series in each grid cell i do not undergo profound changes: the $\varepsilon$-map ${\mathbf{E}}$ obtained using the 6000 years of the simulations is qualitative similar to the one obtained using a randomly chosen period of 100 years. Therefore, the optimization problem should not depend on the century analyzed. We verified this assumption by recomputing the heuristic for three, randomly picked centuries. Results are indistinguishable from those shown in Fig. 8.

Fig. 8

Pairwise distance matrix between the spatial complexity fields computed for the first 100 years as a function of the percentage of standard deviation of the 6000 years detrended anomalies time series defined at each grid cell. Similar results are obtained for the COBEv2 dataset and for other centuries in the simulations

Appendix 3

Here, we present a domain in the Sr02 simulation similar to the one shown in Fig. 3 for the Vlr01 simulation. The two complexity signals shown in Figs. 3 and 9 have same mean (0.68), but shifts are found at different times.

Fig. 9

a Domain in the low resolution (Sr02) simulation, b mean complexity signals for the domain in a over time (x-axis: time in years; 6000 = year 1950; the year reported indicates the first year of the period considered, i.e., year = 1 for the period going from year 1 to 100), c, d associated monthly SST anomalies corresponding to centuries of minimum and maximum complexity, respectively; x-axis: time in months. The black lines indicate a 10-year moving average