Here we focus on a network-based methodology, δ-MAPS, that we developed to robustly compare spatial consistent gridded fields. Our goal is to exemplify how data mining methods can assist with discovering important linkages, or their absence, in climate data.

δ-MAPS identifies the spatially contiguous components of a system, or domains, that contribute in a homogenous way to the system’ dynamics, and then infers their connections accounting for autocorrelations. It refines a previously proposed methodology41,42 and allows for overlapping domains and weighted links at a temporal lag, both relevant to climate fields. After the domains are identified, δ-MAPS infers a functional network between them by examining the statistical significance of each lagged cross-correlation between any two domains, calculating a range of potential lag values for each edge, and assigning a weight that is based on the covariance of the signal of the corresponding two domains. While a temporally ordered correlation does not imply causation, it provides information on the plausible directionality of interactions. Finally each domain has a ‘strength’ calculated as the sum of the absolute weights of all links ignoring their directionality. The greater the strength, the larger is the domain influence on the system at the temporal scales considered.

Details about the methodology are provided as a Supplementary file (Supplementary Methods) and illustrations of advantages of δ-MAPS compared to standard techniques such as principal component analysis, clustering and community detection are presented in Fountalis et al.49

We present a sample of networks from two global monthly sea surface temperature (SST) reanalysis datasets, the HadISST50 and COBE-SST2,51 from the fractional ice content within clouds from the MERRA-2 project52 available from 1980 onward and corresponding variables from a representative member of the Community Earth System Model (CESM) large ensemble.53 The resolution is 1.25°x1° and the focus on the latitudinal range [60°S-60°N] for SST and [55°S-55°N] for clouds to avoid regions where the correlation across reanalyzes is widely low51 or data are not continuously available. All networks are built using detrended monthly anomalies.

Figure 1 presents strength maps over the period 1971–2015. Domains are similar in the reanalyzes, but generally weaker in COBE. The strongest domain covers the El Niño Southern Oscillation (ENSO) region extending to 60°N with a pattern reminiscent of the Pacific Decadal Oscillation (PDO) footprint. Strong domains include the horseshoe areas north and south of the equator, the eastern portion of the South Pacific, the tropical Indian Ocean, the north Tropical Atlantic, and in the reanalyzes the south Tropical Atlantic. A domain occupies the Warm Pool only in HadISST. We verified that also the ERSSTv454 reanalysis network and the MERRA-2 cloud fields presented later do not include it. In the randomly chosen CESM member no domain occupies the Warm Pool region and the south Tropical Atlantic area is extremely weak. Both features are common to all other CESM runs analyzed.

Fig. 1 SST domains identified by δ-MAPS and their strength in a HadISST, b COBE, and c one member of the CESM ensemble over the 1971–2015 period. The strength of the domain occupying the ENSO region (E) is off-scale and indicated atop of each panel Full size image

The connections between the strongest domains including the Warm Pool for HadISST, and their lags are shown in Fig. 2. In the reanalyzes the ENSO/PDO area is linked to all others at zero or positive lags except for the south Tropical Atlantic, which is anticorrelated and leads by 8 to 10 months. Positive (negative) spring SST anomalies in the Equatorial Tropical Atlantic and in the Gulf of Guinea indeed strengthen (weaken) the Walker circulation, modifying the equatorial winds and the eastern Pacific upwelling and favoring La Niña (El Niño) conditions the following winter55,56 through a Gill-Matsuno-type response.57 Such connection is only partially counteracted by the thermodynamic link from the ENSO area into the Tropical Atlantic through the warming of the entire tropical troposphere following El Niños58,59 and by the dynamical response of the tropical Atlantic trades to the Pacific warming.59,60,61 In CESM links from the Pacific to the Indian Ocean and north Tropical Atlantic are stronger than observed, while the connection from the south Tropical Atlantic is missing. The relation between ENSO and south Atlantic domains is indeed weak and opposite in sign.

Fig. 2 SST network across the a seven of the strongest domains in HadISST (the Warm Pool domain is excluded), b the seven strongest domains in COBE, and c six strongest domains in CESM, where TA S has no links. The color of each link represents the corresponding cross-correlation. Arrows indicate signed definitive (positive or negative) lags. The absence of arrow indicates that connections are significant also at zero lags. Some (not all for clarity) lags are indicated Full size image

The network analysis of cloud fields can contribute to diagnose this common model bias.62 Despite the higher level of noise and intermittency of cloud fields compared to SST, the δ-MAPS outcome is insightful. Figure 3 presents maps of strength for all domains and links from the ENSO area for the ice cloud fraction. Focusing on the Equatorial and south Tropical Atlantic, two domains are identified in MERRA-2, with the first negatively connected to the Equatorial Pacific, and the southern one positively correlated as expected in the thermodynamic response to ENSO; in SST these domains are merged due to the oceanic circulation. In the CESM ice cloud fraction network there is only one domain, positively, but statistically insignificantly, linked to ENSO; a weakly anticorrelated one is found entirely shifted into the northern hemisphere. The domains in MERRA-2 are used to define boxes to evaluate correlograms of SST anomalies with respect to those from the E domain (Fig. 3e–f). In HadISST (or COBE) both the thermodynamic feedback, lead by ENSO and mostly effective into the southern box, and the dynamical Gill-Matsuno teleconnection, lead by the Equatorial Atlantic, are identified. The second dominates the total domain signal. In CESM the dynamical connection is mostly absent, the Equatorial Atlantic evolves independently of ENSO and the thermodynamic link is stronger than observed63 but not sufficient to achieve statistical significance. All other 29 members of the large-ensemble confirm that CESM overestimates the thermodynamic feedback and underestimates the dynamic teleconnection, which prevails only in one run. In several integrations the thermodynamic feedback is so strong that a significant link from ENSO to the south Tropical Atlantic domain characterizes the SST network.