We apply tipping point analysis to a large record of ocean acoustic data to identify the main components of the acoustic dynamical system and study possible bifurcations and transitions of the system. The analysis is based on a statistical physics framework with stochastic modelling, where we represent the observed data as a composition of deterministic and stochastic components estimated from the data using time-series techniques. We analyse long-term and seasonal trends, system states and acoustic fluctuations to reconstruct a one-dimensional stochastic equation to approximate the acoustic dynamical system. We apply potential analysis to acoustic fluctuations and detect several changes in the system states in the past 14 years. These are most likely caused by climatic phenomena. We analyse trends in sound pressure level within different frequency bands and hypothesize a possible anthropogenic impact on the acoustic environment. The tipping point analysis framework provides insight into the structure of the acoustic data and helps identify its dynamic phenomena, correctly reproducing the probability distribution and scaling properties (power-law correlations) of the time series.

The Preparatory Commission of the Comprehensive Nuclear-Test-Ban
Treaty Organization (CTBTO) has established a global network of
underwater hydrophones as a part of its hydroacoustic observations
(others being seismic, infrasound, and radionuclide), with the goal
of continuous monitoring for possible nuclear explosions

Tipping points in climatic subsystems have become a widely publicized topic
of high societal interest related to climate change; see, for example,

A stochastic model combining deterministic and stochastic components
is a powerful yet simple tool for modelling time series of real-world
dynamical systems. Given a one-dimensional trajectory of a dynamical
system (the recorded time series), the system dynamics can be modelled
by the stochastic equation with state variable

The probability distribution of the trajectory (time series) of
such a system, however complex it may be, can in the majority
of cases be approximated using a so-called system potential in
the form of a polynomial of even order. Tipping points can be
identified in terms of the variability of the underlying system
potential

The tipping point analysis consists of the following three stages: (1) anticipating (pre-tipping, or analysis of early-warning signals), (2) detecting (tipping), and (3) forecasting (post-tipping).

Lag-1 autocorrelation is estimated by fitting an autoregressive process of order 1 (AR1):

For the purposes of potential analysis, the dynamics of the
system is approximated by a potential stochastic model with a
polynomial

The potential can be reconstructed from time series data of the
system using the following relation to the probability density function:

The structural changes of the potential are often not visible
in the time series, yet they may lead to a dramatic evolution
of the system. Detecting such changes gives an advantage in
understanding of the dynamical system. The potential coloured
map

This stochastic approximation of the system structural dynamics
has remarkable accuracy for data subsets of length as short as
400 to 500 data points, demonstrating above 90 %
rate of successful detection, as was shown in an experiment
with double-well-potential artificial data

The technique of potential forecasting is based on dynamical
propagation of the probability density function of the time series.
We employ the coefficients of the Chebyshev polynomial approximation
of the empirical probability distribution and extrapolate them in
order to forecast the future probability distribution of the data.
After reconstruction of the system kernel distribution,
a time series is generated using rejection sampling technique,
and then the obtained dataset is sorted according to the initial
data in order to reconstruct the temporal correlations in the
time series. The detailed mathematical description of the potential
forecasting technique is given in

We study the large CTBTO record (2003–2016) of the Cape
Leeuwin hydrophone, series H01W1, which is a 250 Hz sampled
time series of ocean sound pressure. The raw data represent
3 TB of binary waveforms, which after extraction constitute
95 billion points in the time series. We analyse 1 min
averages of sound pressure level (SPL) in five frequency bands
(5–115 Hz broadband as well as 10–30, 40–60, 56–70, and 85–105 Hz),
of about 7 million points per time series. These data have
pronounced seasonality and some small gaps, and therefore we
perform interpolation and deseasonalization of all five
time series, the result of which can be seen in Fig.

Initial

The data samples were scaled using their calibration factors (provided by
CTBTO), and an inverse filter of the recording system's frequency response
was applied to eliminate the effect of the acquisition chain on the frequency
response of the recordings. The fast Fourier transform (FFT) of the signal was
computed using rectangular windows of 15 000 samples (i.e. 1 min
intervals at 250 Hz sampling rate) and the broadband signal was then
filtered in five frequency bands (5–115, 10–30, 40–60, 56–70, 85–105 Hz)
via selection of the corresponding FFT bins within each frequency band. Then
the resulting sound pressure level (SPL) in dB re 1

Because of the data gaps, we interpolate the SPL data to achieve
equidistant 1 min temporal resolution. We then remove the
seasonal periodicity by subtracting the averaged seasonal cycle
over the 14 years of observation to obtain the fluctuations

We analyse the global trends of these five datasets, assuming
the simplest linear model in a least-squares regression.
To estimate the uncertainty in the trends, we apply
the “jackknife”
technique (see

The resulting trends show a small annual decline in SPL for
all five datasets, as shown in Fig.

Trend estimation in SPL bands (deseasonalized data)
using “delete-

The above trend analysis was applied to deseasonalized
fluctuations (SPL broadband). It is interesting that the
average annual cycle of the initial broadband data, too,
has a declining trend, which is illustrated in Fig.

Average annual cycle of the SPL broadband, its linear regression line (red) and horizontal line (cyan) for comparison.

The origin of the seasonality in the acoustic data from a hydrophone installed at depth is a subject of discussion, because the seasonal ocean temperature fluctuations at the surface would barely influence the sound propagation towards hydrophones. There are various possible mechanisms through which seasonal variability may manifest in the hydroacoustic data. For example, seasonal variations in shipping frequency, recreational vehicle use, iceberg breakup. Seasonally, there may be a slight warming/cooling in the top few tens of metres of water surface layer, but at the depth of the SOFAR (sound fixing and ranging) channel, where the hydrophone is located, temperature is stable on a seasonal timescale. Some seasonal effects in the sound record may be originating from iceberg formation as the edges of the Antarctic, as ice breaks up at a higher rate in the Southern Hemisphere summertime. Furthermore, seasonal variations in whale song are plausible, as well as in fauna migration due to seasonal fluctuations in food supply.

We next apply the pre-tipping analysis (early-warning signals)
to analyse lag-1 autocorrelations and variance of the broadband
SPL record, with estimation of uncertainty. We vary the length
of the sliding windows for calculating these indicators between
one-fourth and three-fourths of the record length to obtain the averaged curves
and standard uncertainties and display the indicator values at
the end of each window, as shown in Fig.

Early-warning indicators of the broadband
SPL dataset: lag-1 autocorrelation (upper panel) and variance
(lower panel), calculated with variable window lengths, from

The noticeable change at the end of these early-warning indicators may be
related to the unusually large El Niño event of 2015–2016. One can see
that the variance decline slows down and autocorrelation sharply rises, which
means that the increase in memory is not accompanied by increasing amplitude
of acoustic fluctuations. Such effects may happen when a dynamical system
experiences critical slowing down prior to a bifurcational tipping. As we
hypothesize that the El Niño signature may be related to changes in both
oceanic dynamics and fauna, the increasing memory in the acoustic data may
reflect, for instance, the observation that during the El Niño the Cape
Leeuwin current slows down

Similar to what is seen in the CTBTO data, the effect of increasing autocorrelation and
decreasing variance was earlier observed in bifurcating artificial data
changing from white noise to random walk, in

Further, we apply potential analysis to identify smaller-scale
variability, varying the length of the sliding window from 3 days to 1 year. The resulting potential plot is shown in Fig.

El Niño–Southern Oscillation (ENSO) can be monitored using
several indices, which are obtained by averaging climatic variables
to make the presence of El Niño more visible in the series.
We show in Fig.

We calculate, for easier
comparison of El Niño indices and potential analysis,
two binary indices derived
from the ONI and from the single level
of the potential plot at the scale 0.5 years. The bars in the bottom panel
of Fig.

The vertical span of the features of the potential plot (the specks of different colours) corresponds to the timescale of the change, i.e. the size of the time window, within which the change has been detected. As El Niño is a seasonal phenomenon (except the unusually long event of 2015–2016), most of such specks are located within the window of size 1 year. The large event of 2015–2016, indeed, extends higher than that. To address this timescale, we derived the binary potential index using the detection data at fixed timescale of 0.5 year, at which most El Niño events should be present in the detection statistics.

We do not claim that the potential colour plot could be used for early-warning signals (such as prognosis of El Niño), in this system or in others. Moreover, there may be other factors causing structural changes in the acoustic data, rather than El Niño or La Niña. On the other hand, detection of such changes can indeed be useful useful for other studies that could investigate attribution of structural variability, and here the technique of potential analysis might be very useful.

To understand better what dynamical changes occur in the
acoustic fluctuations, it is useful to plot the histograms
of the corresponding subsets of data. Figure

The difference between two-well-potential
(first half of the year 2016, black curve) and three-well-potential
(second half of 2016, red curve) subsets of the broadband SPL
record in the two upper panels. In the lower panel, the build-up
of the new potential well can be seen in the red histogram,
where the main mode becomes broader and starts building two
new modes around SPL values

The variability of the potential can be understood as appearance and disappearance of the SPL fluctuations, which are present in the three-well-potential subsets and disappear in sub-periods of two-well-potential dynamics. These periods of change seem to coincide with some of the recent El Niño events, in particular the strong oscillation in 2015–2016. Since in these short periods data become two-well potential during El Niño, one can hypothesize that the El Niño event reduces acoustic fluctuations events in the tails of the probability distribution (higher and lower values) and intensifies the events in the middle range of values.

It is known

The hydrophone, located 100 km off-coast from Cape Leeuwin at the south-western corner of Australia, has unobstructed reception of acoustic signals arriving via the SOFAR channel at angles between 110 and 355

The detailed analysis of directional acoustic propagation is beyond the scope of the current paper and may be analysed later elsewhere.

Finally, we analyse the scaling properties of the deseasonalized
fluctuations of the broadband SPL to identify the type of noise
present in this dynamical system. When the noise is white, the
DFA scaling exponent has value
0.5, whereas red noise has values of the exponent higher than
0.5

When we apply the
scaling analysis to the deseasonalized broadband SPL,
in both short and long temporal range it has a high exponent
(about 0.9), which means that the acoustic fluctuations are
stationary red noise, and this is how they should be modelled
to represent accurately the stochastic term in Eq. (

Detrended fluctuation analysis scaling curve of the broadband SPL, with estimated scaling exponent values. The straight line denotes the slope with scaling exponent 1. The scaling exponents of the curve (short term and long term) are much higher than 0.5, hence the noise is not white but red (presence of correlations); the exponent is slightly smaller than 1, which means that the fluctuations are stationary (unlike a nonstationary random walk).

It is important to model the climatic variables with colour noise rather than
with basic white noise, especially when a system like this exhibits highly
correlated long-term persistence (as estimated by DFA in Fig. 7 with

Acoustic noise at the depth of 1 km can be influenced by multiple factors of natural and anthropogenic origin, and we investigate some of the possible components that could represent the dynamics of the acoustic system. While there may be various equivalent models reproducing the observed time series, we choose the simple stochastic dynamic system which generates simulated time series with closely matching statistical properties.

Based on the above analysis, we can formulate a stochastic
model for the acoustic oceanic noise. We adopt an additive
model with the following terms:

Five samples of SPL data (real, black; modelled, red).

The model (

We have applied tipping point analysis and identified deterministic and stochastic components of the ocean acoustic data. We have discovered a possible signature of El Niño in the deep-ocean acoustic data, which is an interesting observation confirmed by both potential analysis and direct estimation of the probability density function of the broadband SPL. Given that the hydrophones are located at depth, and the number of factors influencing the hydroacoustic system in conjunction with the global climate system is large, the investigation of the transitional mechanisms between the surface multiannual phenomena and deep-water acoustic processes may be a subject of a separate paper. The current dynamics of the acoustic fluctuations, which demonstrate slow but steady changes in early-warning indicators, suggests of an upcoming tipping point in this hydroacoustic system, with possible appearance/disappearance of system states, which in this context denote higher/lower SPL fluctuations. Because Cape Leeuwin is a busy shipping junction in world trade, and as trading processes intensify (at the same time requiring more modern ships, with more efficient and less noisy engines), we hypothesize that frequency ranges of the oceanic acoustic noise will be affected unequally, due to multiple factors related to anthropogenic activities.

In

Some frequency bands may decrease in level because of the technological
changes: new developments in quieter engine technology, establishment of
noise mitigation standards, and renewal of the fleets. In particular, there
is global large-scale replacement of heavy-tonnage ships, where in some
categories, like “cargo”, “containers” and “bulk carriers”, ships of age
above 25 years are no longer present; see the report of the European Maritime
Safety Agency

Other potential causes of trends in ocean noise levels include changes in the frequency of other anthropogenic sources such as geophysical surveying, changes in the number and distributions of biological sources such as large cetaceans, changes in natural sources of sound such ice breakup and ice formation, and changes in the ocean environment which may affect the propagation of sound (for example, sea temperature).

By taking into account all the components of the proposed model (

The hypothesis of the possible influence of El Niño appeared in the course of our research and was unexpected. Therefore, our modelling approach, in principle, is capable of discovering such interesting signatures in the data for further investigation. This demonstrates the capability of the proposed data analysis, on its part, to stimulate geophysical research.

CTBTO data are not openly accessible, but access can be requested by submitting a
research proposal via

The authors declare that they have no conflict of interest.

The views expressed herein are not the opinion of the CTBTO. Valerie N. Livina, Peter Harris, Kostas Sotirakopoulos, Lian Wang, and Stephen Robinson are funded by the UK Department for Business Energy and Industrial Strategy. The authors are grateful to the specialists of the virtual Data Exploitation Centre (vDEC) for providing the CTBTO data in their servers.Edited by: Stéphane Vannitsem Reviewed by: two anonymous referees