Ordinal analysis is a symbolic data analysis technique proposed by that has been extensively applied in a wide variety of different scientific fields . The application of ordinal analysis to a climatological spatio-temporal dataset is schematically illustrated in Fig. . Ordinal analysis takes an ordered series of N data values, x={x1,x2,…,xi,…,xN}, and translates it into a sequence of symbols, s={s1,s2,…,si,…,…sn}. For example, considering a grid point in the geographical region shown in Fig. a, x can be the time series of SST anomalies at this grid point, as schematically shown in Fig. b. On the other hand, the ordered series x can be the sequence of values of SST anomalies at time t, along a row (or a line), from right to left (from top to bottom), in the region shown in Fig. a, as schematically shown in Fig. e and f.

The ordinal symbolic transformation requires defining only two parameters: the symbol length, L, and the lag, δ. These parameters are used to associate, to a vector vi whose components are L data points lagged by δ, a symbol si that is known as ordinal pattern (OP), vi(L,δ)=[xi,xi+δ,xi+2δ,…,xi+(L-1)δ]→si(L,δ)=[π(i),π(i+δ),…,π(i+(L-1)δ)]. Here π(⋅) is the permutation index that sorts the components of vi in ascending order: xπ(i)≤xπ(i+δ)≤…≤xπ(i+(L-1)δ). The number of possible permutations, that is, of possible patterns, grows as L!. The six possible patterns of length L=3 are shown in Fig. c and examples are shown in Fig. 2f: the vector [1.1,-1.4,4.1] is represented by pattern “102” while [3.2,4.4,1.3] is represented by “120”, etc. We remark that the ordinal transformation takes into account relative values, not absolute ones.

Applying this transformation to every vi(L,δ) with i=1,…,n=N-(L-1)δ, gives a sequence of n ordinal patterns, s(L,δ)={s1,s2,…,si,…,…sn}. Labeling the possible patterns (symbols) from k=1 to k=L! (for L=3, k=1 corresponds to pattern “012”, k=2 to pattern “021”, etc., as illustrated in Fig. c), we can count the number of times each pattern appears in s. Let nk be the number of times the kth pattern appears. Then, the probability of this pattern is pk(L,δ)=nk/n. Figure d displays an example of six probabilities of patterns of length L=3. The permutation entropy (PE) is then defined as Shannon's entropy of the probabilities : 1H(L,δ)=-1log⁡L!∑k=1L!pk(L,δ)log⁡pk(L,δ). The coefficient 1/log⁡(L!) normalizes H(L,δ) between 0–1 and enables the comparison between values obtained from ordinal pattens of different lengths. A small entropy value is obtained where one pattern predominates (this occurs when the sequence x={x1,x2,…,xi,…,xN} is periodic, or when it has a strong trend), whereas a high entropy value is obtained when all symbols are almost equally probable, which normally occurs when the sequence x is fully stochastic.

To estimate the patterns' probabilities with good statistics, the number of symbols, n=N-(L-1)δ, needs to be much larger than the number of possible patterns, L!. In practical terms, this limits the values of L to the range from 3 to 6. Here, unless explicitly stated, we use L=4 and use different values of δ to tune the spatial scale of the analysis.

The SST anomalies in the two regions of interest are represented as the time evolution of 2-dimensional N×M gridded datasets, Xi,j(t) with i∈{1,…,N},j∈{1,…,M},t∈{1,…,T}, where the index i corresponds to different latitudes, the index j to different longitudes, and T is the number of time steps in the series. Given L and δ, at each time step, t, we apply the ordinal transformation using two spatial orientations: in the North–South (NS) direction, the ordered series x is defined from the values, at time t, of the columns of the grid, and in the West–East (WE) direction, from the rows of the grid, see Fig. e and f. In each column we can define N-(L-1)δ OPs, while in each row, we can define M-(L-1)δ OPs. Defining OPs over all the columns the total number of OPs defined along the NS direction is MN-(L-1)δ, and defining OPs over all the rows, the total number of OPs defined along the WE direction is NM-(L-1)δ. An important advantage of this methodology for the analysis of climatological data is its flexibility, because the OP orientation is not limited to NS and WE directions, but other orientations can be selected for the analysis.

We also remark that L and δ allow tuning the spatial scale of the analysis. In the original application of permutation entropy to time series analysis , we can interpret L and δ as parameters that allow a nonlinear embedding of a time series in a L! dimensional space, because at each time, an ordinal pattern can be defined, and the number of possible patterns is L!. In the spatial approach, the parameters L and δ have a similar role: they allow to embed a set of L gridded data points in a L! dimensional space.

From the probabilities of the OPs defined at time t along the NS and WE directions, pWEk(t) and pNSk(t) respectively, we compute two spatial permutation entropies at time t: 2HWEL,δ(t)=-1log⁡L!∑k=1L!pWEk(t)log⁡pWEk(t),3HNSL,δ(t)=-1log⁡L!∑k=1L!pNSk(t)log⁡pNSk(t). Since SST anomalies vary over time, the ordinal probabilities vary over time and thus, the entropies vary over time. HNSL,δ(t) and HWEL,δ(t) will be close to 1 when there is no spatial order in the data (all OPs are equally probable), and will be <1 when there are spatial structures, such as gradients in the NS or in the WE direction, that make some OPs more or less probable.

In this work, we consider symbols defined by L=4 grid points and a spatial lag up to δ=8, as these are the largest values that allow us, given the size of the two regions studied, to estimate with good statistics the probabilities of the L!=24 possible patterns. Table  shows the number of symbols (defined by L=4 grid points in the geographical region analyzed) when a spatial lag of δ=1, 2, 4, and 8 is used. We see that the lowest number (in the Gulf stream with δ=8) is ≫24. We tested the robustness of our results with respect to L, and found similar results with L=3 and L=5 (see Figs. – of Appendix ).

In this work we also compare the permutation entropy with the conventional way of calculating Shannon entropy of a spatio-temporal field, Xi,j(t), from the distribution of its data values, 4Hhist(t)=-1log⁡m∑k=1mpk(t)log⁡pk(t). Here pk(t) is the probability that Xi,j(t) is in the k state and m is the number of possible states. pk(t) is estimated from histograms of m bins of equal size, that represent the possible states. Hhist(t)=1 when pk(t)=1 and pj(t)=0∀j≠k (only one bin is occupied), and Hhist(t)=0 if pk(t)=1/m∀k (the distribution of values is uniform). To compare with SPE values calculated with ordinal patterns of length L, we select the number of bins equal to the number of possible ordinal patterns, m=L!. To calculate Hhist(t) all the data points of Xi,j(t) in the regions of interest are used. As explained in Sect. 2, ERA5 and NOAA have 40×200 grid points in the Niño region and 40×90 in the Gulf Stream region.

In this work we demonstrate that spatial permutation entropy can detect differences in ERA5 and NOAA SST products, and an important question is whether such differences can also be detected using standard correlation or distance measures. Therefore, we also compare SST anomaly values using the Average Absolute Difference (AAD), the Pearson's spatial cross-correlation coefficient (r), and the Spatial Mutual Information (SMI), which is a non-linear cross-correlation measure .

The Average Absolute Difference is defined as: 5AAD(t)=Xi,j(t)-Yi,j(t)i,j, where Xi,j(t) and Yi,j(t) represent the two gridded datasets, ERA5 and NOAA, and ⋅i,j represents the spatial average in the analyzed region, at time t.

Pearson's spatial cross-correlation coefficient is defined as: 6r(t)=σX,Y(t)σX(t)σY(t), where σX(t), σY(t) and σX,Y(t) are the spatial standard deviations and the spatial covariance of Xi,j(t) and Yi,j(t) at time t. To calculate ADD and r, all the data points of Xi,j(t) and Yi,j(t) in the regions of interest are used. As explained in Sect. 2, ERA5 and NOAA both have 40×200 grid points in the Niño region and 40×90 in the Gulf Stream region.

The spatial mutual information (SMI) of Xi,j(t) and Yi,j(t) at time t is defined as: 7SMI(t)=HX(t)+HY(t)-HX,Y(t), where HX(t)=-∑k=1mXpXklog⁡pXk and HY(t)=-∑l=1mYpYllog⁡pYl are the Shannon entropies of Xi,j(t) and Yi,j(t), and HX,Y(t) denote their joint entropy, 8HX,Y(t)=-∑k=1mX∑l=1mYpk,l(t)log⁡pk,l(t), where mX and mY are the numbers of possible states of X and Y respectively, and pk,l(t) denotes the joint probability that Xi,j(t) is in state k and Yi,j(t) in state l.

We calculate SMI using two approaches: the conventional one, in which the distributions of values of Xi,j(t) and Yi,j(t) are divided in an equal number of bins, mX=mY=m, of equal size, that represent the possible states, and the ordinal approach, in which the possible states are the possible ordinal patterns, defined over Lδ-lagged gridded points, aligned along the NS or WE directions. Then, in the ordinal approach, mX=mY=L!. We refer to the mutual information calculated in these ways as SMIhist(t), SMINSL,δ(t) and SMIWEL,δ(t).

Since for L=4 OPs there are m=L!=24 possible patterns, which are the possible “states”, to calculate SMIhist the probabilities pXk(t) and pYl(t) were estimated from histograms with m=24 bins of equal size, and the joint probability, pk,l(t), was estimated from 2D histograms with 24×24 bins of equal size. In this way, the SMI values were obtained from probabilities defined over the same number of possible states. To calculate all the histograms, all the data points of in the regions of interest were used. To test the robustness of the results, we also calculated SMINSL,δ(t) and SMIWEL,δ(t) using ordinal patterns of length L=3 and compared with SMIhist(t) using histograms with 6 bins and joint histograms with 6×6 bins, and obtained very similar results (shown in Fig.  of Appendix ).

Figure  displays the temporal evolution of the spatial permutation entropy, calculated from the probabilities of L=4 OPs defined from the values of SST anomalies in neighboring grid point (i.e. the spatial lag is δ=1). Panels (a) and (d) display HNSL=4,δ=1 for the two datasets analyzed, ERA5 in blue and NOAA OI v2 in red, in the two regions analyzed: panel (a) corresponds to the Niño 3.4 region, and panel (c), to the Gulf Stream region. Results are presented from the beginning of the datasets (1940 for ERA5 and 1981 for NOAA OI v2). Panels (b) and (e), instead, display HWEL=4,δ=1, also in the two regions and for the two datasets under analysis. For comparison, panels (c) and (f) show the entropy calculated from the distribution of data values, Hhist. As explained in Sect. , to estimate the distribution we use histograms with 24 equal bins.

In the Niño 3.4 region, panel (a) shows that the evolution of HNSL=4,δ=1 for ERA5 and NOAA OI v2 is quite similar. However, panel (b) shows differences in the evolution of HWEL=4,δ=1 which persist until 2007. After 2007 the differences reduce until 2022, when the two values of HWEL=4,δ=1 diverge again. The years when these transitions occur coincide with the switch of the sea-surface boundary condition of ERA5, from HadISST2 to OSTIA, in 2007 , and with the inclusion of MeteOp-C satellite data in NOAA's dataset in November 2021 .

Although many transitions can be identified by visual inspection, to objectively identify change points in the temporal evolution of the entropies (and in all the quantifiers used), we employed a well-known unsupervised change point detection (CPD) algorithm, the Pruned Exact Linear Time (PELT) . We tested the significance of the detected change points by PELT using a surrogate analysis, and their robustness against variations of the penalty parameter (see Appendix  for details).

The arrows in Fig.  indicate the change points detected by the PELT algorithm. We note that no change point is detected in panel (a), but a change point is detected in panel (b), in 2007. In the Gulf Stream region, panel (d) shows differences between the values of HNSL=4,δ=1 of ERA5 and NOAA OI v2, which become relatively small after 2013, a fact that could be due to the update of the background error covariances in OSTIA in January 2013 . Panel (e) also shows considerable differences between the values of HWE, from 2021 onward, and which could be due to the inclusion of MetOp-C AVHRR data in NOAA OI v2 . While change points in 2007 and 2013 are returned by the PELT algorithm, the one in 2021, although significant in the NOAA signal, does not pass the robustness test (See Table.  in Appendix ). For the same level of statistical significance with respect to surrogates and robustness with respect to variations of the penalty parameter, no change points are detected for Hhist (panels c and f).

Summarizing, most changes that can be identified by visual inspection are consistent with changes returned by the PELT algorithm when used to analyze HNSL=4,δ=1(ERA5), HNSL=4,δ=1(NOAA OI v2), HWEL=4,δ=1(ERA5) and HWEL=4,δ=1(NOAA OI v2) in Niño 3.4 and Gulf Stream regions. The PELT change points are:

A first transition occurs in ERA5 in 2007, which we interpreted as due to the change of the sea-surface boundary conditions: the inclusion of OSTIA in 2007. This transition is associated to change points that are identified in HNSL=4,δ=1 and HWEL=4,δ=1 in the Gulf Stream region, and in HWEL=4,δ=1 in the Niño 3.4 region, but not in HNSL=4,δ=1 in the Niño 3.4 region. We remark that ENSO is a large-scale phenomenon characterized by large zonal temperature changes, and in the Niño 3.4 region, the spatial permutation entropy best captures differences along the direction of the largest gradients.

Figure  displays the entropies as in Fig. , but now the ordinal patterns are defined with non-neighboring grid points. Specifically, the grid points are spaced by a lag δ=8 that corresponds to 2°. Now we see that the temporal evolution of the entropies of ERA5 and NOAA OI v2 is consistent, in the two regions and for the two orientations. Discrepancies are small: not only the fluctuations are highly correlated, but also, the trends are similar, but no change points are detected at this scale. For the Nino 3.4 region, although HNSL=4,δ=8 (panel a) does not present a significant trend, in panel (b) we see that HWEL=4,δ=8 from the two datasets display a negative trend. This trend is interpreted as due to the inhomogeneous long-term SST variations over the equatorial Pacific, since over the years, SST has warmed in the west and cooled in the east . This represents an intensifying westward large-scale gradient over the Niño 3.4 region that can make the patterns that encode monotonic increasing or decreasing, such as “0123” and “3210”, more prevalent than those that encode an oscillation (See Fig.  of Appendix ), thus decreasing the entropy. However, low SPE values do not always imply that the “0123” and “3210” patterns dominate, and a careful inspection of the patterns' probabilities is needed in order to be able to confidently draw conclusions.

In the Gulf Stream region, which is also heating due to global warming , HWE and HNS in ERA5 and in NOAA present a positive trend, which reveals a decrease of spatial structure, as it means that the different symbols become equally probable. In this case, since the northwest part is warming faster than the rest of this area , it decreases the climatological gradient of the SST across the Gulf stream, thus homogenizing the entire region, and the resulting symbols resemble more closely those of random fluctuations (See Fig.  of Appendix ).

To check the robustness of our results, we also performed the analysis after removing the linear temporal trend in each grid point, and obtained similar results, shown in Figs. – of Appendix . In these figures we also include the analysis of the raw SST data – including the seasonal cycle.

Figure  shows the same entropy signals as Figs.  and  for the Niño 3.4 region, but in the period 1981–2025, and also shows the mean SST anomaly of this region (black line, right vertical axis) computed from the NOAA dataset.Table  displays the Pearson cross-correlation coefficients between the entropies computed from NOAA and ERA5, and the corresponding SST anomaly. In this Table we observe a good agreement between NOAA and ERA5 in Niño 3.4, in terms of the sign and magnitude of the cross-correlation coefficient, for all the entropies. In panel (b) of Fig. , we can see that HWEL=4,δ=1 from the NOAA dataset is positively correlated with the SST anomaly (the Pearson cross-correlation coefficient is r=0.42), most clearly between 2007–2022 (r=0.70). The HWEL=4,δ=1 values are the largest during El Niño years, capturing the decrease in SST gradients along the longitudinal direction. For ERA5, this correlation is strongest from 2007 onward (r=0.57). In contrast, HNSL=4,δ=1 (panel a) is negatively correlated with the SST anomaly (r=-0.23 for NOAA, and r=-0.20 for ERA5), thus during some El Niño events (such as 1997–1998 and 2015–2016 years) the entropy decreases, reflecting the north-south gradients, one in the Northern Hemisphere and one in the Southern Hemisphere, that occur as the equatorial zone is warmer than the north-south edges of the region. This is also observed for some La Niña events, such as the one in 1988–1989 or in 1998–1999.

For the same analysis, but with δ=8, sudden drops of HWEL=4,δ=8 (Fig. d) can be observed, and some correlate with El Niño events, such as the ones in 1982–1983 or 1997–1998, but also at the end of a strong La Niña in 2008. In these cases (see Fig. ), a WE gradient is formed due to the strong warming of the eastern Pacific, which dominates at this long spatial scale, and in consequence, pattern “0123” prevails (thus, the entropy drops). In contrast, for δ=1, the distribution of ordinal probabilities presents larger contributions of two patterns, “0123” and “3210”, so the entropy is higher. Some drops are also seen in the temporal evolution of HNSL=4,δ=8 (Fig. c), although at this scale HNSL=4,δ=8 cannot capture the NS gradients appropriately. In fact, for δ=8 the ordinal patterns span 6° (more than half of the region length in this direction), because the data points are separated by 0.25°×8=2° and thus, L=4 data points that define a pattern cover a distance of 3×2°=6°). As El Niño develops in a waveguide between 5° S–5° N, the SST anomalies evolve particularly in that area, and the NS gradients are highly concentrated near the equator. Therefore, they can be seen with δ=1, but if δ=8 is used, the OPs can include the SST variations north and south of the equator (SST decreases north and south of the equator), mixing both gradients. This does not occur in the longitudinal WE direction, as the WE gradients have a larger spatial scale. To gain further insight, we inspected the temporal evolution of the ordinal probabilities, and found that their variation is consistent with the interpretation of the gradients that are captured when using using different δs. Videos showing how the probabilities of NS and WE OPs change over time can be found in the Video Supplement.

We also performed this analysis in the Gulf Stream region. The results, displayed in Fig.  and Table  reveal that there is no consistent correlation between the entropy signals and the SST anomalies in the two datasets (i.e. the cross-correlation coefficients obtained from NOAA and ERA5 differ both in magnitude and sign). We speculate that this is due to the fact that SST anomalies, even when aggregated at the basin scale, do not present a smooth behavior (as in the Niño 3.4 region).

To analyze similarities and differences between ERA5 and NOAA, in this section we only consider the period when both datasets are available (1981–2024). To visualize the effect of the spatial lag between the grid points, Fig.  displays the entropies (as done in Figs.  and ), for grid points that are consecutive (i.e. they are separated by 0.25°, that is, δ=1), that are separated by 0.5° (δ=2), by 1° (δ=4), and by 2° (δ=8). Therefore, in this figure, the left and right columns display the same entropies as in Figs.  and .

We observe that as δ increases the behavior of HNSL,δ and HWEL,δ for the two datasets converge, which indicates that the differences found between ERA5 and NOAA occur mainly at short spatial scales, while at long scales, linear trends in the temporal evolution of the entropies are consistently identified in both, ERA5 and NOAA. We also notice that, as δ increases, so do the entropies, suggesting that the gradients become less pronounced as the spatial scale increases.

To perform a quantitative comparison between the datasets, we begin by analyzing their SMI, reported in Fig. . In panels (a)–(h) we report SMIL,δ estimated from ordinal probabilities, in particular δ=1 for panels (a)–(d) and δ=8 for panels (e)–(h), while panels (i) and (j) display SMIhist. estimated from the probabilities of values of SST anomalies.

When SMI is calculated from the ordinal probabilities, in the two regions and for the two orientations, we can see that it increases with time, which reveals that the amount of information shared by the datasets. In particular, the probability that the same symbols occur in the same locations and at the same time in both datasets increases with time (see Fig.  of Appendix ). PELT detects a transition in 2007 in all the panels.

For the Niño 3.4 region, we observe that the PELT analysis now also detects a transition at the small scale (δ=1) in SMINSL=4,δ=1 (Fig. c), which was not detected in the analysis of the HNSL=4,δ=1 signal (Fig. ). We note that El Niño events have an impact on SMIWEL,δ, as we observe that the coherence between the datasets decreases during such events, as we can observe, for example, during the El Niño events of 1997, 2010, and 2015. We highlight that this effect is observed at all scales and in the current version of the products since it is observed during the 2023 warm ENSO event. Regarding SMIhist. (Fig. i), although we observe some increase in the average value in 2007 and 2015, no robust and significant change points are detected in this signal.

For the Gulf Stream region (Fig. b, d, f, h, and j), all the SMI values reported present oscillations around a relatively constant value before 2007, which appears to increase consistently from this year on. Additionally, on the large scale, we detected two other transitions: one at end of 2015/beginning 2016 (the same observed in SMINS for δ=1 in the Niño 3.4 region), and another one in 2021. Both transitions correlate with changes in the satellites experienced by NOAA SST those years . We also note that 2016 corresponds to the start date of version 2.1 of the NOAA OI product, which also includes the Argo observational data , and several changes in the conventional and radiance observations assimilated by ERA5 also occurred that year .

Regarding SMIhist., we observe that the annual cycle has some impact on this signal, but not on SMINS, nor SMIWE.

Finally, to demonstrate the added value of using symbolic ordinal analysis, in Fig.  we present the results obtained with two well-known measures of linear relationship between two datasets: the average absolute difference, AAD, and the spatial Pearson's correlation coefficient, r. They provide complementary information because when AAD and r are both high, the data sets differ in values but their spatial distributions are consistent, whereas when both are low, there is agreement between the values, but not in the spatial distributions. In Fig.  we see that both measures show continuous improvement of the agreement between the two datasets in the two regions; however, there are oscillations and no clear transitions are observed. In the Gulf Stream region, AAD and r show a strong coupling with the annual cycle, with disagreement (high AAD and low r) peaking during northern winters. A possible explanation could be due to the high cloud coverage in this region during winters, which could difficult infrared measurements, leading to larger differences between ERA5 and NOAA datasets. On the other hand, in the Niño 3.4 region ENSO events affect the agreement between the datasets, but their effect is captured differently by the two measures. For example, AAD peaks during El Niño in 1988, 1997 and during La Niña of 2011, but these events do not affect r, and vice-versa, La Niña in 1996 and 1999, and El Niño in 2004, affect r but not AAD.

For the same levels of statistical significance (with respect to surrogate data) and robustness (with respect to variations of the penalty parameter) used for the analysis of SMI signals, no robust and significant change points are detected in the analysis of AAD and r signals.