Sea surface temperature (SST) in the Pacific Ocean is a key component of many global climate models and the El Niño–Southern Oscillation (ENSO) phenomenon. We shall analyse SST for the period November 1981–December 2014. To study the temporal variability of the ENSO phenomenon, we have selected a subregion of the tropical Pacific Ocean, namely the Niño 3.4 region, as it is thought to be the area where SST anomalies indicate most clearly ENSO's influence on the global atmosphere. SST anomalies, obtained by subtracting the appropriate monthly averages from the data, are the focus of the majority of previous analyses of the Pacific and other oceans' SSTs. Preliminary data analysis showed that not only Niño 3.4 spatial means but also Niño 3.4 spatial variances varied with month of the year. In this article, we conduct an analysis of the raw SST data and introduce diagnostic plots (here, plots of variability vs. central tendency). These plots show strong negative dependence between the spatial standard deviation and the spatial mean. Outliers are present, so we consider robust regression to obtain intercept and slope estimates for the 12 individual months and for all-months-combined. Based on this mean–standard deviation relationship, we define a variance-stabilizing transformation. On the transformed scale, we describe the Niño 3.4 SST time series with a statistical model that is linear, heteroskedastic, and dynamical.

Sea surface temperature (SST) is an important component of many global
climate models. The thermal inertia of the oceanic surface layer means the
air–sea interaction that occurs at the surface of the ocean is important to
global temperature models and prediction. Monthly SST datasets are a
combination of satellite, ship, and buoy observations. Typically, these are
interpolated to produce a cohesive dataset, such as can be found on the
NOAA (National Oceanic and Atmospheric Administration) website

Comparison of the SST data to the SST anomaly data for
January 1983. The climatology base period for the anomalies is 1971–2000.
The Niño 3.4 region (

The term El Niño was originally used to refer to the upwelling of warm
water in the Pacific Ocean off the South American coast. However, the term is
now used to describe a broader range of interconnected oceanic and
atmospheric effects. ENSO describes the distribution of warmer-than-average
waters in the tropical Pacific Ocean, the associated atmospheric variations,
and the resulting weather conditions. ENSO has two canonical states or
regimes, an El Niño event (warmer eastern tropical Pacific) and a La
Niña event (cooler eastern tropical Pacific). There are also periods
where the ocean is in a transition phase between these two states, which is
referred to as the neutral phase

During an El Niño event, the trade
winds (the prevailing pattern of easterly tropical surface winds) are weaker
and the warm water in the western tropical Pacific Ocean spreads eastwards
into the central and eastern Pacific Ocean

During a La Niña event, the trade winds are stronger, the thermocline
across the Pacific Ocean gets steeper, and the western tropical Pacific Ocean
has warmer-than-normal SSTs

In the neutral phase, there is low surface pressure in the western tropical
Pacific over the warm ocean around Indonesia, high surface pressure in the
eastern tropical Pacific, and the trade winds blow across the tropical
Pacific Ocean

Although ENSO events are characterized in the tropical Pacific Ocean, the
global atmosphere and oceans are highly interconnected and, thus, climatic
effects in regions of the world outside the Pacific can be correlated with
ENSO phases. For example, the ENSO phenomenon teleconnects with precipitation
during the monsoon season in India, the rainy season in south-eastern Africa

Characterizing the ENSO phenomenon's temporal variability has, in the past,
concentrated on SST anomalies. In this article, we show that this strategy
misses a fundamental spatial mean–variability relationship that can guide us
to modelling and forecasting SST on a different scale, where the issues
associated with forecasting El Niño and La Niña events are more
transparent. Section 2 presents the dataset we analyse. In
Sect.

The dataset we shall analyse is a subset of the global monthly SST from the
Climate Modelling Branch (CMB) of NOAA using the Reynolds and Smith optimum
interpolation version 2 algorithm

The optimum interpolation analysis is produced weekly using in situ (buoys
and ships) and bias-corrected satellite datasets, combined with SST
estimations based on sea-ice cover. The buoy observations are from both
moored and drifting buoys, and they are considered to be more accurate than
ship observations

To obtain monthly SST fields, the weekly fields are linearly downscaled to
daily fields, and then the daily fields are averaged for the month. The
monthly SST data are defined on a

To study the temporal variability of
the ENSO phenomenon, we have selected a subregion of the tropical Pacific
Ocean, namely the Niño 3.4 region. The Niño 3.4 region combines part
of each of the Niño 3 region (

In what follows, the

We analyse the SST dataset for the period November 1981–December 2014. Thus,
the temporal period of interest is

The majority of previous analyses of Pacific SST focus on the SST anomalies, obtained by subtracting the appropriate monthly averages from the data. This removes seasonal effects, but some of our preliminary data analysis showed that not only spatial means but also spatial variances varied with month of the year. We conjectured that the monthly spatial variances might be related to the monthly spatial means, which led to this study. In this article, we conduct a spatio-temporal analysis of the raw SST data and introduce diagnostic plots (here, plots of spatial variability vs. spatial central tendency) for all-months-combined and for each of the 12 months separately. By going directly to anomalies, important nonlinear behaviour in the raw data may be overlooked.

Spatial standard deviation,

Define the

Define the (unbiased)

We plotted the spatial standard deviation vs. the spatial mean in the
Niño 3.4 region for all

Recall from Sect.

Spatial standard deviation,

Spatial standard deviation,

To understand this variability, we plotted the spatial standard deviation vs.
the spatial mean in the Niño 3.4 region for all

The negative slope identified in Fig.

It is generally accepted that the ENSO phenomenon is nonlinear

To obtain the slope of the linear relationships by month (Fig.

The simple-linear-regression equation based on

The OLS estimate of the intercept,

Detecting and rejecting outliers can involve subjectivity or difficult
simultaneous inferences, but robust methods provide an automatic way to deal
with them. Most robust-regression methods minimize some function of the
residuals. However, here we use the alternative expressions for the OLS
estimates of slope and intercept given by Eqs. (

Recall from Sect.

One way to robustify this estimate of the slope would be to replace the
average with the median. Hence consider

The estimate

Consider

Henceforth, we shall use

In the rest of this subsection, we summarize the statistical theory associated with WTS estimation.

From

From

In the regression setting, it is common to test
for the dependence of

A commonly used test statistic for this hypothesis test is

For a chosen significance level of

A

Results from a linear regression of the spatial standard
deviation,

In Fig.

The presence of outliers (e.g. May 1988, which was the start of a La Niña event, and November–December 1982, which were months in the middle of an El Niño event) motivated our use of the WTS robust-regression method. It should be noted that the outliers do not contradict the observed linear relationship; rather, they appear to be extreme observations caused by unusual SST conditions. Implementing robust regression provides us with some assurance that our analysis of variability will not be dominated by the outliers.

The asymptotic variance of the WTS slope estimate was given in
Sect.

The results given in Sect.

Bootstrapping in regression can be carried out in at least two ways. Draw
randomly with replacement from the

We have calculated the standard deviation and the

The resampling distributions of

Time sequences from December, January, February, …, November,
December, January, showing WTS-estimated slope coefficients with upper and
lower limits from point-wise

Consider a random variable

The delta method can be used to identify a variance-stabilizing
transformation as follows. Assuming that

Let

In the Niño 3.4 region, the spatial standard deviation of SST has an
approximately linear relationship with the spatial mean of SST. That is,

The analysis we propose in this article is to obtain the slope

Spatial standard deviation,

Because of the seasonal differences seen in Fig.

To illustrate the next stage of our analysis, we consider the month of
January. The top left-hand panel of Fig.

Results from a linear regression of the spatial standard
deviation,

The analogous plot to Fig.

Time sequences from December, January, February, …, November,
December, January, showing WTS-estimated slope coefficients with upper and
lower limits from point-wise

Time sequences from December, January, February, …, November,
December, January, showing WTS estimates of the intercept with upper and
lower
limits from point-wise

Box plots of relative skill (RS), as defined by Eq. (

Box plots of relative skill, as defined by Eq. (

In this section we fit time series models to

On the transformed scale, as defined by Eq. (

The square root absolute difference (RAD) for each predicted value was used
as a measure of the predictive skill:

For some months, specifically the austral summer months (December, January,
and February),

Our study involved fitting an autoregressive model directly to the data

The first part of this article gives an exploratory data analysis of tropical Pacific SSTs during the period November 1981–December 2014. Most analyses and models found in the literature work directly with the SST anomalies. We make a strong case here that there is structure in the raw data that these analyses miss, namely a spatial mean–variability relationship that suggests a nonlinear transformation of the data. This structure is also consistent with the generally accepted nonlinearity of the ENSO phenomenon.

Working with anomalies implies that large-scale seasonal processes and smaller-scale processes are additive. The approach based on anomalies subtracts the seasonal component, leaving behind a residual component (made up of the anomalies) that is modelled. We believe that this strategy can cause difficulties with modelling and forecasting. This is because there is a mean–standard deviation relationship that needs to be respected first, before anomalies are considered and dynamical models are built.

In this article, we give a statistical methodology that removes the
mean–standard deviation relationship by transforming the data. The
transformation is empirically driven, but it is based on 33 years of data for
which a consistent pattern is seen; outliers are apparent for less than

The transformation we derived is logarithmic, monotone, and nonlinear, and it respects the variability seen in SSTs from month-to-month during the year. At the very least, the estimated parameters of the transformation offer another characterization of the enigmatic patterns of SSTs that lead to El Niño and La Niña events.

In the latter part of this article, we fitted an autoregressive process to
the standardized anomalies on the transformed scale. Forecasting based on
autoregressive processes is quite straightforward, as is the back-transform
that yields forecasts on the original scale, in degrees Celsius. The model we
propose is fundamentally statistical, and its skill relative to a standard
forecast on the original scale is seen to be seasonal (Sect.

Sea surface temperature datasets are available from:

For further details on this algorithm, see for example

Bootstrapping regression residuals

Given observed data,

Sample, with replacement,

Use the

Repeat Steps 2 and 3,

Calculate the desired measures of uncertainty of

The authors would like to thank the referees and the guest editor, Chris Wikle, for their excellent comments. Noel Cressie's research was partially supported by a 2015–2017 Australian Research Council Discovery Grant, number DP150104576. Edited by: C. Wikle Reviewed by: J. Elsner and one anonymous referee