The Met Office Hadley Centre provides global quality controlled ocean temperature and salinity profiles and monthly long-term objectively analyzed global fields with a one degree spatial resolution . The most recent EN4 dataset includes objectively analyzed fields formed from profile data with uncertainty estimates. The available data extend from 1900 to the present and there are separate files for each month. used a simple Analysis Correction optimal interpolation methodology to analyze global historical observations that had been methodically quality controlled. The analysis fields were constructed by combining a background field (the analysis field of the previous month) with the quality controlled profiles from the month being analyzed.

In this work, the surface salinity field (top layer of the dataset) from 1950 to 2014 is used. The 1950 cutoff was chosen to match previous trend studies and to prevent data-deficient periods.

The recently available SMOS (Soil Moisture and Ocean Salinity) satellite provides sea surface salinity data with sufficient spatial resolution (1/4∘) to characterize global and regional (e.g., Amazon: ; North Atlantic SSS maximum: ; Gulf Stream: ; northern North Atlantic: ) features. Recently, SMOS data have been used to study salinity variability in the Atlantic , the signature of La Niña in the tropical Pacific Ocean and even create surface T/S diagrams . Level 3 (global maps) data were obtained from the CP34 distribution center at the SMOS Barcelona Expert Centre (http://tarod.cmima.csic.es) for the period 2010–2014. The temporal resolution of the objectively analyzed SMOS data is one month and the spatial resolution is 1/4∘. We averaged the data to one degree resolution to match the long-term dataset and reduce noisy signals.

The SMOS satellite data exhibited deficiencies near coastal areas and in high latitudes . To prevent the introduction of bogus features, those data were removed from the analysis. The main areas affected were in high latitudes, in the proximity of continents, the Mediterranean, and in a large area of the western Pacific surrounding the Philippines and Japan.

A Gaussian Mixture Model (GMM) is a probabilistic model for which the probability density function is a combination of two or more Gaussian distributions. The Expectation–Maximization (EM) algorithm is an iterative procedure to find a Maximum Likelihood Estimate (MLE) of the parameters of a GMM. In the past, the EM algorithm was used to separate the regimes of a spatial time series and to find the best GMM describing the joint distribution of model and data in a model skill assessment scenario . In previous studies, EM was used to estimate missing values for oceanographic datasets . introduced a measurement operator (H) and error (assuming the observations were unbiased with a Gaussian measurement error ϵ(t)∼G(0,R(t)) of known covariance, R(t)), to provide a more general algorithm that could be used for missing data and interpolation problems.

In this implementation, we introduce the possibility of defining the GMM by both a mean and a linear temporal trend. After we have found the number of components (representing probability density functions), nc, component distributions (mean, trend, and covariance), G(μk,αk,Σk), and their respective likelihoods, τk, we can conduct an Empirical Orthogonal Function (EOF) analysis on the Σk.

Let d(t) denote a set of time series with a linear measurement operator, H(t), from a spatial basis on which we want to estimate the state ψ(t) at all time points, such that d(t)=H(t)ψ(t)+ϵ(t). We fit a mixture model to the data set, ψ. For an nc component Gaussian mixture model, we have in general p(ψ|μ1,…μnc,α1,…αnc,Σ1,…Σnc,τ1,…τnc)=∑k=1ncτkexp⁡(-12(ψ-μk-αkt)T[Σk]-1(ψ-μk-αkt))(2π)2nd|Σk|, where nd is the spatial dimension of ψ, τk is the probability of component distribution k, and μk, αk, and Σk are the mean, trend, and covariance of the kth component distribution. |Σk| is the determinant of the covariance.

The two steps of the EM iterative procedure are:

Expectation step: For each time point, t, in the dataset, the expected value for component k of the likelihood function, wk(t), is calculated under the current estimate of the parameters μk (mean), αk (trend), and Σk (covariance): wk(t)=exp⁡[(-12(d(t)-H(t)(μk+αkt))T[H(t)ΣkH(t)T+R(t)]-1(d(t)-H(t)(μk+αkt)))](2π)nd|H(t)ΣkH(t)T+R(t)|. Then, the likelihoods are normalized, wk(t)→wk(t)∑jwj(t).

Maximization step: The optimal parameters that maximize the current estimate given the data d(t) are calculated. Note that τk, μk, αk, and Σk may all be maximized independently of each other since they appear in separate linear terms. The frequency of the kth component distribution, nk=∑twk(t), is computed and normalized, τk=nknt=∑twk(t)nt, where nt is the length of the time series.

We enforced that the time series is an autoregressive process of order one (AR(1)) to avoid rapid switching between regimes. Thus, the salinity at time t depends linearly on the previous value, d(t)=c+φd(t-1)+εt, where c is a constant, φ is the parameter of the AR(1) model and εt is white noise.

An EOF analysis can be conducted based on the component distributions covariances, Σk. Using the eigenvectors of Σk, we obtain a new set of orthogonal basis functions that are meaningful for times when wk(t)≃1. The EOF analysis can be considered independently during different regimes in a similar approach to .

Our approach is related to the multivariate adaptive regression splines (MARS) method as both automatically determine the number and timing of the separation between regimes (breakpoints or knots) based on the data. The main difference between the two approaches is that our method includes multiple spatial location (multiple time series) and the regime extraction is across the complete dataset.

To determine the optimal number of components (regimes) in the GMM, we use the Bayesian Information Criterion (BIC, ). The BIC is an empirical approach that approximates the total probability (Bayes factor) of a probability distribution under some set data, BIC(k)=-2log⁡(p^(ψ|μ1,…,μnc,α1,…,αnc,Σ1,…,Σnc,τ1,…,τnc))-Dklog⁡(nt).

The penalty term preventing over-fitting, Dk, for a GMM with nc components and n time series is Dk=ncn(n+2)/2+nc-1, where ncn of those are for the means of each distribution, another ncn correspond to the trends, ncn(n-1)/2 are for the parameters of the covariance matrix, and nc-1 for the τk.

There are a number of potential combinations of means and trends (Fig. ) that can be fitted to any spatial time series. In the current application, the Expectation–Maximization (EM) algorithm is used to find the best GMM describing the data distribution. The method is run under several possible scenarios that separate regimes based on including only means, means and a global trend, and a combination of means and trends. The BIC approach penalizes excessive number of parameters with Dk being adjusted depending on the inclusion/exclusion of trends. BIC is used twice in the current procedure: first, to choose the optimal number of components in a particular scenario (e.g., only means, combining means and trends), and second, to choose among the different scenarios.

The Hadley Centre EN4 surface salinity interpolated fields incorporated quality controlled observations in a similar manner as the database used by to create the DW10 surface fields. DW10 fields were reported for a 50 year period (nominally 1950–2000) to simplify comparisons, even though they used data from 1950 to 2008. To establish whether the datasets were sufficiently similar, we calculated the mean and trend for the same period as DW10 (1950–2008) using the interpolated data (Fig. ). The climatological mean surface salinity (Fig. a) exhibits the same general structure and magnitude as the DW10 results.The standard deviation of the surface salinity (Fig. b) highlights the increased variability associated with river discharge, the ITCZ, and intense meandering current systems like the Gulf Stream. As in the case of the mean, the 50 year linear surface salinity trend (Fig. c) is also similar to the published fields with minimal differences. For instance, there is a slightly less negative trend in the Equatorial and North Atlantic and a larger positive trend in the Antarctic Circumpolar Current region in our results. Overall, the trend magnitude and spatial distribution for the 1950–2008 period are equivalent in both datasets. The main trend features include a predominantly positive trend in the Atlantic, southeastern Pacific, and Indian Ocean, and a predominantly negative trend in the north and western Pacific and along the Antarctic Circumpolar Current region.

When the global SSS monthly data from the Hadley Centre EN4 were analyzed, the EM method distinguished three separate regimes. These regimes are characterized by different means but also different trends (Figs.  and ). The separation (breakpoint) between the first two regimes (Regime A and B) occurs in May 1990, while the separation between the second and third regimes (Regime B and C) was found to be March 2009.

The global surface salinity means for the three regimes (Fig. ) exhibited the same general pattern with higher salinity in the subtropical gyres and lower values in the subpolar, polar regions, and under the ITCZ. The differences between regime means were on the order of 0.1 (0.099 rms difference between Regime A and B; and 0.15 between Regime B and C). The Regime B (1990–2009) average was saltier than Regime A (1950–1990) in most of the Atlantic Ocean except along the eastern US, in the central equatorial region, and south of 30∘ S. In contrast most of the Pacific (except for the southwest) was fresher in Regime B than A. The difference between the average salinities of Regime B (1990–2009) and Regime C (2009–2014) exhibited saltier values in the later period in most of the Pacific and South Atlantic, while a large part of the North Atlantic was fresher during Regime C.

The trend for Regime A (1950–1990, Fig. a) was consistent with the results obtained by DW10 ( and Fig. , note the different scale) and the references therein. The differences were a slightly larger positive trend in the Equatorial Atlantic and a lack of positive trend in the eastern Pacific. The trend for Regime B (1990–2009, Fig. b) exhibited larger positive magnitudes in the North and South Atlantic than Regime A, a negative trend (∼-0.02 yr-1) along the Equatorial Atlantic, negative trends in the areas of the Antarctic Circumpolar Current and positive trends (0.01–0.02 yr-1) in most of the Pacific between 30∘ S and 30∘ N. The trend associated with Regime B exhibited enhanced magnitudes when compared to Regime A (positive trends were more positive and negative areas were more negative during Regime B). In both regimes the trends are consistent with an intensification of the global hydrological cycle . The estimated trend during Regime C (2009–2014, Fig. c) was much larger than in any of the two early regimes. The Regime C trend was negative in the majority of the North Atlantic (up to -0.04 yr-1), eastern Indian Ocean (∼-0.05 yr-1) and western (<-0.05 yr-1) and south equatorial (∼-0.03 yr-1) Pacific. Positive trends during Regime C were estimated along the western part of the Equatorial and South Atlantic (up to 0.06 yr-1), the Pacific subtropical gyres (ranging 0.02–0.05 yr-1) and the western Indian Ocean (∼0.04 yr-1).

The salinity time evolution showed the changing temporal pattern during the full record and the potential distinction between regimes (Fig. ). For instance, the Equatorial Atlantic (Fig. a, 5∘ S–5∘ N) showed a notable difference between the trends for the three regimes, while the difference between the average salinity of the three regimes was small. The North Atlantic Subtropical Gyre (Fig. b, 15–30∘ N) in contrast exhibited noticeable differences in the trends, but especially in the means between the three regimes. Meanwhile, the most striking feature in the Mediterranean (Fig. c) and in the Equatorial Pacific (Fig. d, 5∘ S–5∘ N) was the sharp drop in salinity associated with El Niño events while the main difference between regimes was again in the magnitude of the trend. The largest trend acceleration between regimes was present in the Mediterranean (Fig. c).

As described in Sect. , the EM algorithm also allowed for the separation of the modes of variability. The 1st EOF for Regime A and B explained over 75 % of the variance (79 and 77 %, respectively) while only explaining 67 % of the Regime C variance. Meanwhile, the 2nd EOF explained 11, 12 and 14 % of the variance in each regime. The 1st EOF of Regime A was spatially consistent with Regime B, but its magnitude was smaller (Fig. ). The 1st EOF in the first two regimes suggested the Atlantic and southeastern Pacific were fluctuating in phase, while the western Pacific and most of the Southern Ocean were out of phase. The 1st EOF of Regime C exhibited a different spatial pattern and larger magnitude with the separation between basins being less apparent, with the main feature likely associated with the northern/southern migration and extent of the ITCZ. The 2nd EOF for the first two regimes was quite similar with positive values in most areas except the Antarctic and Equatorial Pacific oceans. Meanwhile, the 2nd EOF of Regime C was mostly consistent with the Pacific and Atlantic oceans being out of phase. The time series of the 1st EOF (Fig. a) showed larger short-term fluctuations for Regime B and Regime C, with Regime B exhibiting a trend in time. There was an apparent relationship between the 2nd EOF for Regime B (Fig. b) and the ENSO cycle (1998, 2003, 2005 peaks), while no clear relation appeared to be present for Regime A or C.

The recent availability of global surface salinity fields from satellites (SMOS and Aquarius) provided the possibility of using alternative data that were not included as part of the dataset. The goal was to analyze the robustness of the regime separation by replacing the recent (2010–2014) fields with satellite-derived products.

The inclusion of the available 5 years of SMOS data in the analysis in substitution of the EN4 data for 2010–2014 slightly altered the EM separation results (Figs.  and ). The EM method also distinguished three separate regimes (A', B', and C') with different means and trends. The breakpoint between Regime A' and B' was in July 1988 and between Regime B' and C' in January 2010. The difference in the separation (breakpoint) dates between the early regimes was expected as the merged dataset did not include several areas (proximity to coastal areas, Mediterranean Sea, high latitude) that were present in the original EN4 dataset.

The means and differences of the two first regimes (Fig. a, b and d) were equivalent to the results for the complete EN4 dataset (Fig. ). The average salinity for Regime C' (Fig. c) exhibited significant differences from both the Regime B' mean and also from the third regime of the original dataset (Regime C, 2009–2014). The mean salinity during Regime C' was fresher in the North (up to -0.5) and Equatorial Atlantic, the Indian and western Pacific Oceans, while being saltier in most of the Southern Ocean and in large areas of the Pacific Ocean.

As was the case with the means, the trends of the two first regimes (A' and B') of the combined time series (Fig. a, b) were equivalent to the trends extracted from the original dataset (Fig. ) for Regimes A and B. The trend for Regime C' (2010–2014, Fig. c) differed in magnitude and spatial structure from the trend for Regime C of the original EN4 dataset (2009–2014, Fig. c). The Regime C' trend was positive in the Equatorial Pacific and Atlantic while having a negative trend in most of the rest of the oceans with large negative values especially in the North Atlantic. The trend for Regime C' was likely the effect of changes in the processing methodology for global SMOS data .