 Review
 Open Access
Nonlinear dynamical analysis of GNSS data: quantification, precursors and synchronisation
 Bruce Hobbs^{1, 3}Email authorView ORCID ID profile and
 Alison Ord^{1, 2}
https://doi.org/10.1186/s4064501801936
© The Author(s). 2018
 Received: 13 December 2017
 Accepted: 25 June 2018
 Published: 23 July 2018
Abstract
The goal of any nonlinear dynamical analysis of a data series is to extract features of the dynamics of the underlying physical and chemical processes that produce that spatial pattern or time series; a byproduct is to characterise the signal in terms of quantitative measures. In this paper, we briefly review the methodology involved in nonlinear analysis and explore time series for GNSS crustal displacements with a view to constraining the dynamics of the underlying tectonic processes responsible for the kinematics. We use recurrence plots and their quantification to extract the invariant measures of the tectonic system including the embedding dimension, the maximum Lyapunov exponent and the entropy and characterise the system using recurrence quantification analysis (RQA). These measures are used to develop a data model for some GNSS data sets in New Zealand. The resulting dynamical model is tested using nonlinear prediction algorithms. The behaviours of some RQA measures are shown to act as precursors to major jumps in crustal displacement rate. We explore synchronisation using cross and jointrecurrence analyses between stations and show that generalised synchronisation occurs between GNSS time series separated by up to 600 km. Synchronisation between stations begins up to 250 to 400 days before a large displacement event and decreases immediately before the event. The results are used to speculate on the coupled processes that may be responsible for the tectonics of the observed crustal deformations and that are compatible with the results of nonlinear analysis. The overall aim is to place constraints on the nature of the global attractor that describes plate motions on the Earth.
Keywords
 GNSS time series
 Nonlinear analysis
 Dynamical systems
 Recurrence plots
 Recurrence quantification analysis (RQA)
 Cross and joint recurrence plots
 Crustal deformation
 Precursors
 Synchronisation
Introduction
The general nature of the dynamics of the mantle of the Earth along with the interaction of the mantle with the lithosphere is thought to be well known; broadly, convective motion in the mantle with coupled thermal and mass transport results in tractions on the bases of the lithospheric plates. These tractions together with other tractions generated by instabilities, such as subducting slabs along with forces generated by spreading from midocean ridges, lead to plate motions expressed as plate deformations observed at the surface of the Earth in the form of GNSS (Global Navigation Satellite System) measurements. Fundamental questions are do the displacements we observe synchronise in some way from one place to another? And if so, on what spatial and time scales does synchronisation occur? Can the pattern of synchronisation be used to define precursors to major and commonly destructive displacement events? The global array of GNSS measurements and their time series should, on principle, give enough information to construct the dynamics of the underlying processes and answer such questions. However, in order to be more specific, one needs to better express the partial differential equations that describe the processes responsible for the dynamics and the ways in which these processes are coupled and evolve with time. With the present uncertainties regarding constitutive relations and properties and the temperature distribution within the Earth, it is difficult to constrain possible geometric and kinematic models for plate development and evolution with computer models based on current knowledge of these issues and with the available computing power. The results of any such models should at least be compatible with the results of detailed analysis of observed crustal displacement data which is the purpose of this paper.
The main data sets we have at present that are useful in developing such constraints are geophysical data sets (gravity, magnetics and seismic), heat flow measurements, the distribution of topography on the surface of the Earth and GNSS data on crustal displacements. These latter data sets are now well distributed over the Earth, and in some instances, continuous time series go back at least a decade. We concentrate on GNSS data in this paper with the aim of establishing how much of the dynamics of the plate tectonic processes is reflected in such data. Future papers attempt to integrate these data sets. Just as the global weather system is an expression of the NavierStokes equations for a viscous fluid with coupled heat and mass transport and which result in highly nonlinear behaviour, we expect the dynamics underlying plate tectonics to be highly nonlinear. The aim is to characterise and quantify this behaviour and, as far as is possible, move towards identifying the mathematical expression of the coupled processes that operate to produce crustal deformation driven by plate motions.
Nonlinear time series analysis and dynamical systems
The nature of nonlinear time series analysis
The outcome of any time or spatial series analysis is a data model which enables one to characterise the statistical measures (mean, standard deviation, autocorrelation function, power spectrum and so on) of the data and if possible undertake forecasts, interpolations and extrapolations of the data. We distinguish two classes of data models; one is a parametric stochastic data model that assumes an underlying statistical distribution and has no relation to the underlying processes that produced the data. The other is a nonparametric deterministic data model that makes no assumptions about the underlying statistics and directly reflects the dynamics of the system. The linear, stochastic procedures of kriging, cokriging, autoregressive and moving average methods work well for linear systems where the law of superposition holds and Fourier methods clearly delineate discrete periodicities in the data. These are methods of constructing a stochastic, parametric data model. However, in nonlinear systems, especially those that are chaotic, these methods fail; the assumptions of Gaussian or lognormal distributions with no longrange correlations break down. Nonlinear signal processing methods (Small 2005) become not only essential but are capable of delineating the nature of the processes that operated or of testing models of processes that might be proposed (Judd and Stemler 2009; Small 2005). We paraphrase Judd and Stemler (2010): Understanding: it is not about the statistics, it is about the dynamics.
Part of the reason why linear parametric procedures fail for nonlinear systems that arise from a number of coupled processes is that in nonlinear systems the data for a particular quantity are a projection of processes from a higher dimensional state space on to that single quantity. Thus, for a sliding frictional surface where the only processes might be velocitydependent frictional softening, accompanied by heat production and chemical healing of damage, the behaviour is described in a fourdimensional state space with coordinates comprising the state variables, velocity, temperature, friction coefficient and degree of chemical healing. A time series for temperature is a projection from the fourdimensional state space on to a onedimensional time series. Quantities that appear close together in the time series may in fact be widely separated in state space. With respect to the GNSS time series from New Zealand that we examine in this paper, the deforming crustal system operates in a state space where at least the state variables velocity, stress, strainrate, temperature, damagerate, healingrate and fluid pressure are needed to define the system; there probably are others involving the ways in which one part of the system is coupled to other parts. The GNSS displacement signal we observe is the projection from a space defined by these state variables on to a single displacement record that we observe at a particular station as a one dimensional time series.
As opposed to stochastic data models based on Gaussian statistics, lack of longrange correlations and the principle of superposition, the nonlinear systems we are interested in studying in the geosciences result from clearly defined physical and chemical processes. Although we may have considerable trouble in discovering and characterising these processes, the system is deterministic rather than stochastic. Hence, in principle, we should be able to define for a system of interest the invariant measures that characterise the system. An invariant measure remains the same independently of the way in which the system is observed and so remains the same independently of the dimensions of the state space in which we observe the system. Such measures include the Rényi generalised dimensions (including the fractal support dimension and the correlation dimension for the system) that characterise the geometry and are defined from a multifractal spectrum for the system (Beck and Schlögl 1995; Arneodo et al. 1995; Ord et al., 2016), the Lyapunov exponents that are related to the dynamics of the system and define the stability of the system and how far prediction is possible (Small 2005) and the KolmogorovSinai entropy, related to information theory, that tells us how much information exists in the signal and is also related to predictability (Beck and Schlögl 1995; Small 2005). We will estimate these invariant measures for GNSS time series together with a number of other quantitative measures but not dwell too heavily on the mathematics behind the theory. Readers who require indepth treatments should consult (Abarbanel 1996; Beck and Schlögl 1995; Sprott 2003; Kantz and Schreiber 2004; Small 2005; Judd and Stemler 2010). This paper is a brief review of nonlinear analysis with an emphasis on recurrence methods (Marwan et al. 2007a). The principles are illustrated using specific examples from the Lorentz system (Sprott 2003, p. 205) and several GNSS time series from New Zealand.
The invariant measures
In this process, every point in the signal is compared with a point distant τ away. These vectors define the attractor for the system; this is the manifold that all possible states of the system can occupy independently of the initial conditions. If in this construction, the delay, τ, is small, the coordinates comprising M are strongly correlated and so the reconstructed attractor lies close to the diagonal of the reconstruction space. It is something of an art form to select τ such that the dynamics unfold off that diagonal. The attractor describing the complete dynamics of the system is embedded in a space which has a dimension that reflects the number of state variables in the system dynamics. The state space in which the attractor “lives” has dimensions, \( \mathbb{D} \). If \( \mathbb{M} \) exceeds \( \mathbb{D} \), the attractor does not change and \( \mathbb{D} \) is called the embedding dimension. For very large dimension systems, it may prove very difficult to construct an attractor that looks interesting or meaningful. This is simply because we are projecting a \( \mathbb{D} \)dimensional object into two or three dimensions. If we explore the system in a space that has dimensions less than \( \mathbb{D} \) then evolutionary trajectories of the system appear to cross one another because of the problems in projecting the trajectories from a higher dimensional space. Points on trajectories that appear close in the observational space but in fact are far apart in \( \mathbb{D} \)—space are called false neighbours. Algorithms for calculating the number of false neighbours in a given data set are given by Sprott (2003) and Small (2005). If one can identify a dimension where the number of false neighbours is zero then one has a good estimate of the embedding dimension. For white noise, the percentage of false neighbours remains at 50% independent of the dimension of the space in which the signal is observed.
One can also identify another dimension that we call the dynamical or state dimension, D. This is the dimension, it may or may not be an integer, that is the true dimension of the attractor. Generally, D is difficult to measure because of noise and \( \mathbb{D} \) is attenuated because of nonstationary behaviour or local variability in attractor density of states so that \( \mathbb{D} \) ≥ D. \( \mathbb{D} \) can be estimated directly from the time series whereas D can only be measured if we have access to a welldefined attractor (Packard et al. 1980; Ord 1994).
Recurrence plots and recurrence quantification
Recurrence plots
Although nonlinear signal processing is at least 30 years old (Abarbanel 1996; Beck and Schlögl 1995; Sprott 2003; Kantz and Schreiber 2004; Small 2005; Judd and Stemler 2010), most approaches are fairly opaque to potential users. Hence, particularly in the geosciences, the inertia involved in using such developments is very large. However, a step in overcoming the inertia was made by Eckmann et al. (1987) who introduced the concept of recurrence plots which are generalised autocorrelation functions based on Takens’ theorem and derived from the conclusion reached by Poincaré (1890) for nonlinear systems that ….., neglecting some exceptional trajectories, the occurrence of which is infinitely improbable, it can be shown, that the system recurs infinitely many times as close as one wishes to its initial state. Since Eckmann’s classical paper, the subject has expanded dramatically with important contributions from Casdagli (1997), Webber and Zbilut (2005) and Marwan et al. (2007a). The literature is now very large especially in climate studies, biology and medicine; applications to seismic studies are Chelidze and Matcharashvili (2015) and Garcia et al. (2013) but other applications in the geosciences are rare. Generalised recurrence plots for ndimensional spatial data sets are discussed in Marwan et al. (2007b). If the dimensions of the system are n then the generalised recurrence plot is in 2nspace. Thus, a recurrence plot for threedimensional data exists in 6space. We only consider onedimensional data sets in this paper.
Recurrence quantification
Summary of quantities used in recurrence quantification analysis. Modified after Webber and Zbilut (2005): https://www.nsf.gov/pubs/2005/nsf05057/nmbs/nmbs.pdf
%recurrence, %REC  Percentage of recurrent points falling within the specified radius, ε.  \( \%\mathrm{REC}=100\frac{number\ of\ points\ in\ triangle}{\varepsilon \left(\varepsilon 1\right)/2} \) 
%determinism, %DET  Percentage of recurrent points forming diagonal line structures. This is a measure of determinism in the signal.  \( \%\mathrm{DET}=100\frac{number\ of\ points\ in\ diagonal\ lines}{number\ of\ recurrent\ points} \) 
Linemax, DMAX  The length of the longest diagonal line in the plot (except main diagonal).  DMAX = length of longest diagonal line in the recurrence plot 
Entropy, ENT  The Shannon information entropy of all diagonal line lengths over integer bins in a histogram. This is a measure of signal complexity with units bits/bin.  ENT = − ∑ (P_{bin})log_{2}(P_{bin}) 
Trend. TND  A measure of system stationarity.  \( \mathrm{TND}=1000\left(\begin{array}{l} slope\ of\% local\ recurrence\ \\ {}\kern2.25em vs. displacement\end{array}\right) \) 
%laminarity, %LAM  The percentage of recurrent points forming vertical line structures.  \( \%\mathrm{LAM}=100\frac{number\ of\ points\ in\ vertical\ lines}{number\ of\ recurrent\ points} \) 
VMAX  The length of the longest vertical line in the plot.  VMAX = length of longest vertical line in the recurrence plot 
Trapping time, TT  The average length of vertical line structures.  TT = Average length of vertical lines ≥ parameter line 
Note that two invariant measures may be derived from the recurrence plot. The entropy is given by ENT: for periodic signals, ENT = 0 bits/bin, and for the Hénon attractor (Sprott 2003, p. 421), ENT = 2.557 bits/bin (Webber and Zbilut 2005). The first positive Lyapunov exponent is proportional to (1/DMAX). The smaller DMAX, the more chaotic is the signal. For the Hénon attractor, DMAX = 12 points (Webber and Zbilut 2005).
Significance of patterns in recurrence plots (after Marwan et al. 2007a)
Pattern  Significance 

Homogeneous  The process is stationary 
Fading pattern to upper right or lower left  Nonstationary data; the process contains a trend or drift 
Disruptions (horizontal or vertical)  Nonstationary data; some states are far from the normal; transitions may have occurred 
Periodic or quasiperiodic patterns  The process is cyclic. The vertical (or horizontal) distance between periodic lines corresponds to the period. Variations in the distance mean quasiperiodicity in the process. 
Single isolated points  Strong fluctuations in the process. The process may be uncorrelated or anticorrelated. 
Diagonal lines (parallel to the LOI)  The evolution of the system is similar over the length of the line. If lines appear next to single isolated points the process may be chaotic. 
Diagonal lines (orthogonal to the LOI)  The evolution of states at different times is similar but with reverse timing. 
Vertical and horizontal lines or clusters  States do not change with time or change slowly 
Lines not parallel to the LOIsometimes curved.  The evolution of states is similar at different times but the rate of evolution changes with time. The dynamics of the system is changing with time. 
Recurrence plots and their quantification in this paper have been prepared using the software VRA: http://visualrecurrenceanalysis.software.informer.com/4.9/ and RQA software: http://cwebber.sites.luc.edu/. Other software is available at http://tocsy.pikpotsdam.de/, http://tocsy.pikpotsdam.de/CRPtoolbox/, http://tocsy.pikpotsdam.de/pyunicorn.php and https://www.pks.mpg.de/~tisean/.
Prediction and noise reduction
Most signals, especially those from natural systems, contain some form of noise which consists of the addition of a stochastic (originates from uncorrelated processes) signal. It is generally considered as an adulteration to the signal and needs to be removed or reduced as far as possible. This notion arises from a linear view of the world where the solutions to linear differential equations are smoothly varying functions and any irregularity must be the result of externally imposed random input. However, irregular behaviour including nonperiodicity and intermittency can arise from nonlinear systems with no externally imposed noise. The problem that arises in nonlinear systems is to understand if some of the noise results from deterministic processes of interest and hence should be retained.
Dynamical noise might be generated in a system where different processes dominate at different time and/or length scales so that some frequencies and/or parts of the system evolve in different ways and rates to others. This results in probability distributions for some time/length scales diffusing (broadening) and drifting (shifting the mean) with different diffusivities as described by FokkerPlanck equations (Moss and McClintock 1989). Such processes add a stochastic but dynamic noise to the system behaviour but such noise is a fundamental part of the processes operating in the system and should be preserved in any noise reduction algorithm. This kind of noise is generally, but not always, coloured noise (Moss and McClintock 1989).
To a large extent, the process of nonlinear noise reduction is the inverse of the nonlinear prediction (or forecasting) problem. For nonlinear noise reduction, one can determine the dynamics of the system given the whole signal up to its current state and then search for parts of the signal in the past that are not part of the dynamics. These parts are removed as noise. This means we take the whole signal and work backwards. For prediction, we take one part of the signal, determine the dynamics and then see if we can find a part of the dynamics that fits the way in which the signal is evolving into the future and use that to make a forward prediction or forecast. Nonlinear prediction is particularly useful if one needs to “fill in” short gaps in data sets in a manner that honours the deterministic dynamics of the system.
An approach to prediction in chaotic systems is spelt out here based on Casdagli and Eubank (1992), Weigend and Gershenfeld (1994), Fan and Gijbels (1996) and Abarbanel (1996). To predict a point x_{n + 1}, we determine the last known state of the system as represented by the vector \( \mathbf{X}=\left[{x}_n,{x}_{n\tau },{x}_{n2\tau },.\dots \dots, {x}_{n\left(\mathbb{D}1\right)\tau}\right] \), where \( \mathbb{D} \) is the embedding dimension and τ is the delay. We then search the series to find k similar states that have occurred in the past, where “similarity” is determined by evaluating the distance between the vector X and its neighbour vector X’ in the Ddimensional state space. The concept is that if the observable signal was generated by some deterministic map, \( \mathrm{M}:\left(.\dots \left(\left({x}_n,{x}_{n\tau}\right),{x}_{n2\tau}\right),\dots, {x}_{n\left(\mathbb{D}1\right)\tau}\right)={x}_{n+1} \); that map can be reconstructed from the data by looking at the signal behaviour in the neighbourhood of X. We find the approximation of M by fitting a loworder polynomial (Fan and Gijbels 1996) which maps k nearest neighbours (similar states) of X onto their next immediate values. Now, we can use this map to predict x_{n + 1}. In other words, we make an assumption that M is fairly smooth around X, and so if a state \( {\mathbf{X}}^{\prime }=\left[{x}_n^{\prime },{x}_{n\tau}^{\prime },{x}_{n2\tau}^{\prime },\dots, {x}_{n\left(\mathbb{D}1\right)\tau}^{\prime}\right] \) in the neighbourhood of X resulted in the observation, x’_{n + 1}, in the past, then the point x_{n + 1} which we want to predict must be somewhere near x’_{n + 1}. In any chaotic system, we expect the error in prediction to increase exponentially (as measured by the Lyapunov exponent) as we move away from known data.
The above approach is based on intensive work on prediction in chaotic systems largely carried out in the 1990s and relies on finding local states in the past that resemble current states of the system. A relatively recent approach to nonlinear filtering is the shadowing filter (Stemler and Judd 2009). A shadowing filter (Davies 1993; Bröcker et al. 2002; Judd 2003; Judd 2008a, 2008b) searches in state space for a trajectory (defined by a sequence of z_{t} for the system), rather than local states, that remains close to (that is, the trajectory shadows) a sequence of observations, s_{t}, on the system. The algorithm is discussed by Judd and Stemler (2009). We do not use a shadowing filter in this paper, but its use in future work promises to give better results than reported here.
Synchronisation

Phase synchronisation: the two signals are phase locked but amplitudes are not identical.

Frequency synchronisation: the two signals are frequency locked.

Lag synchronisation: there is a time or space lag between similar or identical states.

Generalised synchronisation: the synchronisation comprises nonlinear locking between similar or identical states.

Chaotic transition synchronisation: similar behaviour in the signal is locked into chaotic transitions in the respective recurrence plots that occur at similar times in two or more time series.
In many systems, synchronisation switches from one of these five types to another as the system evolves and the coupling between parts of the system changes strength (Romano Blasco 2004). We will see that cross recurrence plots and particularly joint recurrence plots are powerful ways of investigating such synchronisation (Marwan et al. 2007a). Just as a recurrence plot identifies recurrences at different parts of the same signal, cross recurrence plots identify recurrences at identical times on two different signals. In other words, a cross recurrence plot identifies those times when a state in one system recurs in the other. Joint recurrence plots identify recurrences in the recurrence histograms of two signals; they are somewhat similar to identifying simultaneously occurring maxima in power spectra from two different signals in linear systems. Clearly, the plots only reflect something of the dynamics if both signals originate from similar processes and belong to state spaces with similar or identical attractors.
In Fig. 4, we show several different cross and joint recurrence plots so that the reader might obtain some insight into how to read such plots. In Fig. 4a recurrences between the signals y = sin(x) and y = cos(x) are plotted. The recurrences plot on straight diagonal lines and the vertical distance between these lines is the (identical) period of both signals. The straight diagonal lines are referred to (Marwan et al. 2007a) as lines of identity (LOI). In a more general recurrence plot for a dynamical system, the LOIs represent segments of the trajectories of both systems that run parallel for some time. The frequency and lengths of these lines are measures of the similarity and nonlinear interactions between the two systems.
In Fig. 4c, recurrences between y = sin(x) and y = sin(x) + sin(5x^{2}) are plotted. The straight LOIs of Fig. 4a, b are now curved and are referred to as lines of synchronisation (LOSs). Thus, the details of the cross recurrence plot can give information on whether the signals that are compared are linear or nonlinear and also give an indication of both the absolute and the relative time scales associated with the two systems. Figure 4d is a cross recurrence plot between the two quasiperiodic signals: \( y=\sin (x)+\sin \left(\sqrt{2}x\right) \) and y = sin(x) + sin(πx). Figure 4e is a cross recurrence plot between two logistic signals given by x_{n + 1} = αx_{n}(1 + x_{n}) with α = 3.7 and 3.8 and Fig. 4f is a joint recurrence plot between the signals: y = sin(x) and y = sin(20x).
Plotting the changes in slopes of LOSs is a powerful way of tracking the evolution of two synchronised systems and of observing the ways in which time scales that characterise each system change with time.
Examples of joint recurrence plots are given in Fig. 4g, h, i for the same signals in the cross recurrence plots of Fig. 4a, b, c. In contrast to the cross recurrence plots (a to c) which express the ways in which two signals occupy similar states synchronously, a joint recurrence plot expresses (in the form of blue lines or dots in g to i) the ways in which recurrences on two different signals occur synchronously.
An example: the Lorentz attractor—quantification and prediction
Significance of a linear trend in the data
Analysis of GNSS data
Nature of the data
The raw data used here and reported at 1 day intervals contains relatively small gaps (about 6 days at most in the signals we investigated) that presumably arise from station downtime. We have retained these gaps for most analyses but have explored the effect of removing them. Such a process seems to make little difference to the details of both recurrence and cross recurrence plots but clearly is important if one wants to match events in cross and joint recurrence plots. Future work should explore nonlinear prediction methods in filling these gaps.
The emphasis in the use of GNSS time series for geotectonic purposes in most published literature is to establish the velocity imposed on the crust by plate tectonic processes. As such the data are processed (Beavan and Haines 2001; Wallace et al. 2004) in order to arrive at a velocity field that is smooth and continuous over substantial parts of the surface of the Earth. From such studies, important constraints can be placed on that part of the deformation of the crust that is commonly referred to as the rigid body motions (Wallace et al. 2004, 2010). Many studies propose that the crust is made of microplates that may have slightly different rigid body motions (Thatcher 1995, 2007; Chen et al. 2004; Wallace et al. 2004, 2010) and although some may offer more continuous models (Zhang et al. 2004) the case for such microplates existing in New Zealand seems to be well established (Wallace et al. 2004, 2010). The deformation within such microplates is commonly thought of as elastic (McCaffrey 2002; Wallace et al. 2010) and such an assumption is reasonable if one is seeking a smooth, continuous distribution of velocities on the scale of the microplate. However, in this paper, we seek to understand something of the system dynamics of crustal deformation processes by examining the history of deformation, continuous and discontinuous, within these microplates together with the coupling between these microplates over time. As such, the rigid plate tectonic motions are, in a sense, noise as far as the signal is concerned whereas for geotectonic purposes the details of the signal, which are our interest, are noise that is commonly removed by intensive processing (Wallace et al. 2010).
The rigid body motions of the crust arising from plate tectonic motions constitute a vector field on the surface of the Earth whereas the history of displacements within a microplate can be represented as an attractor that describes the dynamics in phase space. In principle, the characteristics of the attractor should not be altered by the subtraction of rigid body velocities but there is an issue in defining how much of an observed trend in a GNSS signal arises from a rigid body motion is a contribution from regional plate tectonic motions and how much arises from elastic deformation or even from other internal permanent plastic/viscous deformation of the microplate. This is particularly the case if the overall trend is not linear.
For the recurrence plots presented here, we have elected not to remove the overall trend since the RQA measures for such plots are influenced by the trend although the trend, if pronounced, is clear in the plot. This is particularly true for KAHU_e and PAWA_e (Fig. 11e, f, but for other plots, the influence of the trend is minimal. We have removed the trend from the KAIK_e plot when we examine synchronisation between stations but recognise that such removal may have an influence on the apparent dynamics of the system; we examine the influence of such trend removal later in the paper.
It would seem from a cursory examination of many of the GNSS records from the North and South Islands of New Zealand that New Zealand is composed of an interlocking mosaic of blocks and within each block the history of GNSS displacements have a similar history. It appears that each block moves whilst maintaining deformation compatibility at the boundaries of these blocks by combinations of boundary slip, block rotation and internal elastic, brittle and plastic deformation. There is evidence (Wallace et al. 2017 and this paper) that the motions of individual blocks are synchronised with others over quite large distances. The question arises therefore: How much of the average trend is to be attributed to the overall plate tectonic motions and how much is to be attributed to the nonlinear dynamics of the microplate? Although such a question is fundamental and is in need of detailed examination we elect to sidestep the issue and unless indicated otherwise treat the raw data as an input to analyses.
Recurrence analysis of GNSS data
We begin by analysing the data from one station (CNST) in some detail to illustrate the procedures spelt out in Fig. 2 and then proceed to examine the other four other stations shown as yellow triangles in Fig. 11a in less detail. We then proceed to examine nonlinear synchronisation of displacement histories between stations CNST and PAWA, a distance of ≈ 220 km, and between stations CNST, PAWA and KAIK, a distance of ≈ 440 to 650 km.
In all the recurrence/crossrecurrence/jointrecurrence plots for GNSS data, the embedding dimension is 10 and the time delay is 5. The scaling is maximum distance, and the radius is 20% (Webber and Zbilut 2005). The parameter c is 5 so that four levels of contours appear in each plot. The signal for the raw data together with the time scale is shown at the base of each recurrence plot at the same linear scale as the plot. Cross reference to Fig. 11 gives finer detail of the absolute time scale for each plot.
The recurrence plot for CNST_u (Fig. 12b) is much more highly populated with recurrences. Regions of no recurrence (black) tend to occur immediately prior to changes in the patterns in the raw data but these gaps are ≈ 50 days wide as opposed to up to a year in the CNST_e recurrence plot.
RQA measures for selected GNSS data sets on the North Island of New Zealand
Station  Data set  %REC  %DET  DMAX  ENT  TREND  %LAM  VMAX  TTIME 

CNST  e  39.49  97.88  3588  4.4  − 6  98.5  837  51.56 
CNST*  e_trunc  38.44  97.48  2687  4.28  − 7.05  98.21  411  40.1 
CNST**  e_trunc_n  36.44  97.48  2688  4.28  − 7.02  98.21  411  40.11 
CNST***  e_reg_n  36.48  97.31  2676  3.8  − 10.02  97.99  446  31.85 
CNST  u  6.9  29.13  45  0.89  − 1.92  45.88  42  2.59 
CNST****  u_n  27.77  65.81  114  1.6  − 6.95  76.54  154  3.58 
PARI  e  39.26  98.73  3380  4.5  − 13.77  99.07  898  70.36 
MAHI  e  42.38  98.26  3685  4.38  − 9.77  98.79  841  60.7 
KAHU  e  37.47  99.47  4183  4.51  − 27.64  99.62  1315  181.64 
PAWA  e  38.16  99.50  4526  5.14  − 25.24  99.65  1326  200.18 
PAWA  u  56.32  95.54  1796  2.92  − 15.05  96.76  1027  18.05 
These values of RQA measures for CNST_e are to be contrasted with those for CNST_u which reflects the more diffuse nature of the latter signal. In particular, the first Lyapunov exponent indicates that predictability may be difficult.
Similar observations to the above hold for the other signals examined: large gaps (black areas) in recurrence tend to occur prior to large discontinuities in displacement, determinism is high and the first Lyapunov exponent is small. Obvious differences in *_e recurrence plots exist for data sets that show significant nonstationarity: recurrence tends to be restricted to a relatively narrow zone either side of the main diagonal LOI but again discontinuities in the signal are preceded on the recurrence plots by gaps in recurrence.
As a final way of analysing recurrence plots, we show in Fig. 12h a series of windows along the main identity diagonal. Within each window, a different pattern of recurrence exists that reflects the details of the signal. One can undertake an RQA within each window and map the way in which the RQA measures evolve with time. This is done in Figs. 16 and 19. The procedure is known as a sliding window analysis (Webber and Zbilut 2005).
Embedding dimension and the nature of the attractor
Noise reduction
Comparison of RQA measures for data set CNST_e, as raw data (column 1), with nonlinear noise reduction (column 2) and with noise reduction using the regional filter method (column 3; Beavan et al. 2004)
Data set  1. CNST_e Raw oneday data  2. CNST_e Nonlinear noise removal  3. CNST_e Regional filter  % change of # 2 with respect to 1  % change of # 3 with respect to 1 

%REC  38.44  38.44  36.48  0  5.10 
%DET  97.48  97.48  97.31  0  0.17 
LMAX  2687  2688  2676  − 0.04  0.41 
ENT  4.28  4.28  3.8  0  11.21 
TREND  − 7.05  − 7.02  − 10.02  − 0.43  − 42.13 
%LAM  98.21  98.21  97.99  0  0.22 
VMAX  411  411  446  0  − 8.52 
TTIME  40.1  40.1  31.85  0  20.57 
Sliding window analysis of CNST_e data
In this analysis, the window size is 100 days and there is a 50 day overlap between windows so that each window can “see” 100 days ahead; this represents two black dots on the RQA signals shown in Fig. 16. Each dot is plotted at the beginning of the 100 day window so each dot represents the RQA measure of the signal 100 days ahead. Here, we concentrate on the main displacement discontinuity that begins about day 3307 and continues until about day 3320. One can see that the mean and standard deviation track the signal precisely and so are of little use as precursors. However, the %LAM and DMAX measures both behave anomalously 100 days before the displacement discontinuity at ~ 3307 days and so are candidates as precursors for this event although it is appreciated that this event is not sharp and extends in a compound manner starting at ≈ 3200 days; others are possible but higher resolution (say 0.1 day binning) is necessary before one can be definitive.
Nonlinear prediction of CNST_e data
Synchronisation of data sets
It is clear that some of the large displacement events in New Zealand occur at near to the same time. Thus the same large displacement events can be observed synchronously at several stations (Wallace et al. 2017, Fig. 2). This represents synchronisation of large displacements over distances of at least 220 km. Recently, Wallace et al. (2017) have proposed that the magnitude 7.8 seismic Kaikōura event triggered large displacement events 250 to 600 km away on the North Island for 1 to 2 weeks after the South Island event. Whilst such synchronisation is clear, it is of interest to see if more subtle forms of synchronisation exist and, if so, over what length and time scales does synchronisation occur? Also an understanding of where such synchronisation sits in the classification of synchronisation types described earlier in the paper and details of the frequencies at which synchronisation occurs would shed light on the dynamics of crustal deformation. In what follows, we employ crossrecurrence and joint recurrence plots to detect synchronisation between stations, to clarify details of the synchronisation and to classify the mode of synchronisation.
Synchronisation between stations on the North Island
The joint recurrence plot is shown in Fig. 18b and shows a high degree of joint recurrence along a single LOS for the early part of the history and widens out to have a higher proportion of joint recurrences as the major event is approached. After the major event, the joint recurrences are still strongly synchronised but the pattern of joint recurrences has broadened even further.
normalised recurrence probability is proportional to (normalised frequency)^{11.6}
indicating that all of the power in the signal is partitioned into the highest frequencies. Since almost all (99%) of the joint recurrences occur in the four or five highest frequencies in Fig. 18e, the partitioning of power into the highest frequencies is far more pronounced (as indicated by the dotted red line) than indicated by the above scaling relation. These observations indicate that the synchronisation between CNST and PAWA is a form of generalised synchronisation.
Synchronisation between stations on the North Island and KAIK on the South Island
The spectacular observations of Wallace et al. (2017) that the magnitude 7.8 Kaikōura seismic event in the South Island of New Zealand is followed for a few weeks by slow displacement events 250–600 km distant on the North Island indicates clear lag synchronisation over large distances. The questions we want to address are the following: do other forms of synchronisation exist over these large distances and, if so, what form do they take and is there information in the form of precursors in the synchronisation patterns? We first characterise the KAIK displacement history using a sliding window with RQA and then investigate cross recurrence plots between CNST and KAIK and between PAWA and KAIK.
We see that the following RQA measures are not suitable as precursors to the Kaikōura event: mean, standard deviation, TTIME and TREND. The other RQA measures %REC, %DET, DMAX, ENT, %LAM and VMAX seem to be useful precursors and increase to well over the mean of the measure 100 days before the Kaikōura event. These measures are connected to the determinism and the organisation of recurrence states and indicate that the processes operating in the system are becoming more organised for about 3 months at least before the Kaikōura event. The question then arises: do stations in the North Island “know” about this organisation process?
We conclude that there is strong nonlinear generalised synchronisation between stations CNST and PAWA on the North Island with station KAIK on the South Island in a punctuated manner for a period of at least 9 months before the magnitude 7.8 Kaikōura event. For ≈ 270 days before the Kaikōura event, the degree of synchronisation between CNST and KAIK intensifies dramatically and ceases at the Kaikōura event. However, similar patterns of synchronisation occur associated with every displacement event at CNST for the previous 3 years. These synchronisation events act as powerful precursors to both minor and major displacement events.
We attribute changes in β in cross recurrence plots between two signals to changes in the frequency content of one or both signals. For instance, consider a situation where β changes for 45° to 90° as occurs in Fig. 18a, c. This can result from a change where the frequency contents of both signals are equal to a situation where the frequency content of one signal does not change but all the power in the other signal is partitioned into the highest frequency as occurs in Fig. 18e.
Discussion
Linear time series analysis concerns the manipulation of time series in order to characterise the statistics of the signal (mean, standard deviation, Fourier components of power spectrum) and/or remove noise (make the signal smoother, remove inliers) and/or make predictions or forecasts. Usually, techniques such as the Kalman filter or other forms of sequential Bayesian filters are employed (Jazwinski 1970; Young 2011). These methods do not always assume Gaussian distributions for the original data and associated noise but commonly work best when the underlying statistics are Gaussian and the system is linear.
Nonlinear time series analysis, in contrast, is concerned with defining the dynamics of the processes that produced the signal rather than emphasising the statistics of the data. The dynamics are embodied in the attractor for the system which is the manifold that defines all possible states the system can occupy as it evolves no matter what the initial conditions are. The precise form of the attractor may be difficult to discern especially if the embedding dimension is large or has many “outliers” but some indications and/or constraints on its nature can be investigated using the ways in which the correlation dimension and false nearest neighbours scale with the embedding dimension. The output from a nonlinear signal analysis comprises estimates of the dimensions of the space in which the attractor exists (the embedding dimension) and other invariants such as the Rényi generalised dimensions, the entropy and Lyapunov exponent. If the embedding dimensions are small and the attractor is not too complicated, in the sense that state space is not too heterogeneous with respect to the density of states, then existing methods of nonlinear analysis work quite well. For high dimensional systems and complicated attractors it may be difficult to reach precise conclusions unless the data set is large enough to have completely sampled the attractor. Many of the pitfalls and problems are discussed by Small (2005) and McSharry (2011).
In this paper, we have reviewed, in a condensed manner, many aspects of nonlinear time series analysis with a view to focussing on a specific application, namely, GNSS data from New Zealand. The motivation for such studies is to characterise the dynamics of the processes that underlie the GNSS time series so that we learn more about the mechanisms that drive plate tectonics. GNSS data from five stations on the North Island of New Zealand have been analysed using recurrence plots and recurrence quantification analysis (RQA). The results of this analysis are shown in Table 3. We have also compared signals from two North Island stations with a station on the South Island that recorded displacements associated with the November 2016, magnitude 7.8 Kaikōura earthquake (Wallace et al. 2017) using cross and joint recurrence analysis.
For the five stations on the North Island, the embedding dimension (estimated as the dimension where the correlation dimension reaches a plateau when plotted against the embedding dimension) is approximately 10. This value is comparable to that of many biological systems (Webber and Zbilut 2005). Although the embedding dimension is commonly inflated over the attractor dimension because of noise and nonstationary effects, this constrains the number of variables involved in the processes responsible for the underlying dynamics to ≤ 10. We consider such processes later in the “Discussion” section. The high dimensions of the attractor and its complexity are indicated by a delay construction of the attractor in three dimensions (Fig. 14). The attractor seems to have a number of knotlike outliers that may be visited only rarely so that it is necessary to have a very long time series to ensure all parts of the attractor have been sampled. The high dimensions of the attractor are confirmed by nonlinear prediction that gives reasonable results only if an embedding dimension of about 15 is used.
A second invariant of importance is the first positive Lyapunov exponent which is a measure of how fast adjacent trajectories diverge as the system evolves; the larger this (positive) Lyapunov exponent the more chaotic the system (that is, the faster adjacent trajectories diverge). The RQA measure, DMAX, is inversely proportional to the largest positive Lyapunov exponent (Eckmann et al. 1987; Trulla et al. 1996) and so Table 2 shows that the signals from all five stations are chaotic from this point of view.
The entropy is the third invariant of importance and is a measure of the amount of information in the signal or of signal complexity or of how rapidly the information encoded in the current state of the system becomes irrelevant. The range for ENT in Table 2 is 4–5 bits/bin. By comparison, the entropy for the classical Hénon attractor (Sprott 2003) is 2.557 bits/bin. These invariants enable us to define a chaotic system (or at least the chaotic systems we are interested in as geoscientists) as a bounded (does not grow without limits) system with deterministic dynamics (defined by an underlying set of physical and chemical processes) with a positive first Lyapunov exponent (neighbouring trajectories diverge exponentially with time).
The RQA measures for the five stations confirm that these data sets arise from chaotic dynamics. Generally the determinism is high and the systems are characterised by a large number of chaotic transitions (characterised by high percentage laminarity) which correspond to jumps in displacement at all scales.
Synchronisation
In most nonlinear systems, where subsystems exist that are coupled together by the transfer of energy or mass, some form of synchronisation develops if the coupling strength is large enough (Romano Blasco 2004; Marwan et al. 2007a). Synchronisation in seismic systems has been discussed by Perez et al. (1996), Sammis and Smith (2013), Scholz (2010) and Bendick and Bilham (2017). Synchronisation develops because the coupling inhibits some frequencies and enhances others so that a particular range of frequencies survives in the system as a whole. Phase and/or frequency synchronisation are the commonly envisaged forms of synchronisation but these are actually relatively rare and the more common form is generalised synchronisation where the behaviour of one subsystem is a (generally nonlinear) function of the behaviour of other subsystems. Generalised synchronisation is difficult to detect but cross and joint recurrence plots offer fast and efficient means of detection and investigation. In cross recurrent plots lines of synchronisation (LOSs) develop and the slope of these lines is tan^{−1}(T_{1}/T_{2}) where T_{1} and T_{2} are the characteristic times of processes operating in two synchronised subsystems.
Analysis (Fig. 18) shows that generalised synchronisation exists between CNST_e and PAWA_e signals in the North Island of New Zealand. The value of (T_{CNST}/T_{PAWA}) varies with time for these two subsystems but averages about 3:5 (Fig. 18a). The synchronisation is well developed throughout the recorded displacement histories of the two stations. There are large gaps in synchronisation before PAWA events. Similar less defined gaps exist before major events for CNST_e. These gaps act as precursors to the main displacement events.
Generalised synchronisation also occurs between CNST and PAWA stations on the North Island and KAIK station on the South Island. For PAWA_e and KAIK_e synchronisation, a single LOS is developed with the ratio, T_{PAWA}/T_{KAIK}, varying from 1:1 to > 20:1 over the time records that are available. For CNST_e/KAIK_e synchronisation, the situation is a little more complicated. The ratio, T_{CNST}/T_{KAIK}, remains large (≥ 20) throughout the recorded history of displacements but synchronisation occurs in bursts with each burst beginning about 100 days before a major displacement event at CNST and ending as the event begins. The Kaikōura event is preceded by 9 months of intense synchronisation and the synchronisation ceases as the Kaikōura event occurs.
The synchronisation between CNST and KAIK is made even more evident by a joint recurrence plot (Fig. 21f) and joint RQA (Fig. 22) where the system shows clear evidence of synchronisation beginning at least 100 days before the Kaikōura event.
Precursors, predictions and forecasting
Nonlinear forecasting or prediction, particularly with respect to process control, is now widely used in industrial applications ranging from control of lathe tool chatter (Abarbanel 1996), brake squeal (Oberst and Lai 2015) and chemical reactions (Petrov and Showalter 1997) to weather forecasting (Yoden 2007). With the development of new forecasting procedures such as shadowing algorithms (Stemler and Judd 2009), one can only expect forecasting to improve. The accuracy of such methods depends on how well the observed time series for the system has sampled the underlying attractor that describes the dynamics of that particular system. For attractors that are characterised by large embedding dimensions or that are topologically complicated, forecasting may be difficult. Moreover, we do not yet know if the attractors for GNSS systems evolve with time. There is a fundamental restriction that the trajectory of any chaotic system will diverge exponentially with time from any observed trajectory. The rate of divergence is measured by the magnitudes of the (positive) Lyapunov exponents. Thus, there is always an “horizon of predictability” for any chaotic system. At present, we do not know how far this “horizon” extends for GNSS time series but the single result reported in this paper shows promise; much more work is needed.
Despite the limitations mentioned above, the time histories of RQA measures for individual stations show that some act as precursors for main displacement events up to 100 days before an event. These results need to be studied and confirmed using time series with greater time resolution. Since data is collected at 1 s intervals a reasonable refinement of the data would to be to bin results at 0.1 day intervals rather than the present 1 day binning procedure. In doing so, it is important to pay attention to the binning procedure and ensure that dynamical noise is retained during the aggregation procedure. The definition of RQA for single stations that precede a major displacement event by considerable time periods is clearly an important goal for nonlinear time series analysis.
With respect to the overall strategy for forecasting of GNSS time series, the application of nonlinear analysis to data from one station can only aim to forecast future events for that particular station. A far more promising approach, motivated by the cross and joint recurrence results for CNST_e and KAIK_e and for PAWa_e and KAIK_e is to monitor cross and joint recurrence histories for a network of stations. This regional network approach has the promise to track the spatial evolution of synchronicity across the network and detect sites where abnormal synchronicity is evolving. The results of this paper (Figs. 21 and 22) show the intensity of synchronicity growing dramatically (such that the intensity of previous synchronous recurrences in the network are relatively small) for approximately 9 months before the Kaikōura event. The precise time of the Kaikōura event cannot be defined by the analysis but the risk could be quantified well ahead of the event. An extension of this cross/joint recurrence study to the whole of the GNSS network in New Zealand, so that patterns of synchronisation developed in historical time series over the past decade could be understood, would be a fundamental next step.
The advantage of a networked cross and joint recurrence system of monitoring is that the site of an imminent event can be pinpointed whereas the use of only two stations results in ambiguity regarding which of the two stations will host the event. Ambiguity can be further reduced if reliable RQA precursors for events at individual stations by sliding window analysis are integrated with results from synchronisation in the GNSS network.
Underlying processes
The processes that have been proposed to produce the spectrum of behaviours observed in the New Zealand tectonic system ranging from slow continuous displacement histories with small but distinct variability (PAWA prior to the 14 November, 2016, M7.8 Kaikōura earthquake), to repeated slow discontinuous events (CNST over a 10 year period), to major seismic events (the M7.8 Kaikōura earthquake) have been discussed by many authors (see papers referred to in Wallace et al. 2017). We break these processes into the following high level categories.
Deformation processes involving elasticbrittleviscous behaviour of rocks in the upper crust with fluids present. The brittle behaviour seems to be associated with rate dependent weakening of the constitutive parameters (commonly expressed as velocity dependent frictional state variables) and some form of chemical healing of damage that has resulted from the brittle behaviour. The deformation process is exothermic (in the extreme case producing pseudotachylite melts) and involves the generation of dilatancy. The heat generated during deformation together with dilatancy influences the fluid pore pressure in the deforming rock mass.
Chemical processes involving the devolatilisation of hydrous and carbonate minerals including calcite, clays, micas and, in particular, serpentinite. These reactions are commonly endothermic and involve a positive ΔV of reaction. Another chemical process is melting to produce pseudotachylites, also an endothermic process. Such processes compete in the heat budget with deformation. Another fundamental chemical process is the healing of damage which may involve hydrolysis and hence is exothermic and competes with the devolatilisation and melting processes for heat.
Hydraulic processes involving fluid flow within the deforming rock mass and especially away from chemical devolatilisation sites.
Thermal processes involving heat transport by both conduction and advection of heat in moving fluids. Heat is generated during deformation and hydrolysing reactions and removed from a deforming site by simultaneous endothermic reactions and heat transport.

The Damköhler number, Da, that is a measure of the importance of the time scale of fluid flow relative to that for chemical reaction. In many systems with coupled chemical reactions and fluid/heat transport changes in Da result in episodic behaviour of the chemical reaction and temperature (Gray and Scott 1990; Aris 1999).

The Newtonian cooling time that measures the importance of the thermal cooling time scale relative to the chemical time scale. This is also important in controlling chemical reaction rates (Aris 1999). It is influenced strongly by the shape of the system and by the thermal conductivity of the wall rocks and so is important in studies of specific tectonic systems; we expect systems with different geometries to behave in different manners.

The Arrhenius number, Ar, that measures the importance of the thermal energy relative to the activation energy for chemical reactions and for rate sensitive deformation processes. Ar defines the spectrum of chemical reactions that occur in a particular system (Law 2006, pp. 60–62) and hence is fundamental in defining the order in which chemical reactions occur in a particular system. Ar is also fundamental in defining the deformation behaviour of the system (Veveakis et al. 2010).

The Gruntfest number, Gr, that measures the importance of the heat generated by deformation to that absorbed or generated by chemical reactions. Gr is important in defining the magnitude of the temperature increase possible for coupled deformation/chemical reactions and hence influences the deformation rate in systems where rate dependent constitutive behaviour is included (Veveakis et al. 2010, 2014; Alevizos et al. 2014; Poulet et al. 2014).

The Lewis number, Le, that is a measure of the importance of heat transport by diffusion to mass transport. Le is important in defining the magnitude of the pore pressure increase possible for a particular temperature increase as fluid diffuses from the site of the temperature increase (Veveakis et al. 2014, 2017; Alevizos et al. 2014; Poulet et al. 2014). In tectonic systems, the permeability of the wall rocks for the system controls whether pore pressure leaks away as devolatilisation proceeds and the temperature increases. If the wall rocks are relatively impermeable, the Lewis number is small and the pore pressure increases would be large thus promoting brittle failure.

In addition, other evolutionary processes need to be parameterised including rates of damage generation and of chemical healing, the temperature dependence of deformation and chemical reaction rates, and the influence of temperature changes and dilatancy on fluid pressure.
We propose that these dimensionless groups and associated evolution equations define the state space for crustal systems such as that examined for New Zealand so that the dimensions of the state space are relatively large as indicated by the recurrence analysis. Any computer models need to be constrained by the magnitude of the dimensions of state space indicated by the displacement histories. Some beginnings of coupled modelling are presented by Veveakis et al. (2010, 2014); Alevizos et al. (2014); Poulet et al. (2014) where close to periodic behaviour is modelled and such studies need to be extended to include the complete parameter space.
Future directions

Better definition of the attractor for the system.

Better definition of precursors to major discontinuities in the displacement history.

Better definition and understanding of the synchronisation system and dynamics of synchronisation, particularly the distribution of coupling strength between sites.

Multidimensional recurrence network analysis aimed at establishing the underlying dynamics for the regional system.
These objectives can be addressed by a combination of longer time series (in order to ensure all parts of the attractor are sampled by the data) and better resolution within existing data sets in order to better resolve precursors and synchronisation frequencies. The former clearly will occur with continued data collection and the second requires binning the data in sizes smaller than 1 day (say 0.1 day intervals). Binning to small sizes requires attention to the process of aggregating data from the 1 s collection frequency to the 0.1 day time scale so that dynamical noise is preserved whilst observational noise is discarded.
Understanding the synchronisation dynamics requires nonlinear analysis of the complete networked New Zealand GeoNet GNSS measurement array. The synthesis of such a regional analysis might best be undertaken with some form of fusion between the concepts of recurrence networks (Small et al. 2009; Donner et al. 2010, 2011; Donges et al. 2012, 2015) and spatial network theory (Barthélemy 2011). This means that the regional GNSS array is to be regarded as a nonlinear dynamical network in which each node is coupled to every other node with variable coupling strengths between nodes. At each node, there is a nonlinear time series of displacements. Such a synthesis would better define the attractor for the integrated system as well as providing an understanding of the detailed dynamics of the system and of how the dynamics leads to synchronisation. An undertaking such as this is a major software development project and parts are being addressed at present. Some important developments for seismic spatial patterns are in Banish and Conrad (2014).
We conclude by speculating on a tectonic model that duplicates the synchronisation patterns revealed by the recurrence analysis in this paper. We propose, following Ben Zion (2008), that the crust of the Earth in the New Zealand region is close to a phase transition but is dominantly subcritical so that there are no longrange correlations with respect to distortion of the plate. The plate as a whole continuously undergoes rigid body translations driven by plate tectonics and revealed by the calculations of Wallace et al. (2004). Internally, the plate undergoes a heterogeneous deformation with local shear strain rates varying from zero to ≈ 10^{−14} s^{−1} in places along the Great Alpine Fault (Beavan and Haines 2001). The plate is composed of microplates that undergo a spatially coordinated pattern of rotation, translation and internal deformation that maintains compatibility of the deformation gradient tensor.
The heterogeneous velocity field imposed on the crust by plate tectonic motions drives parts of the plate towards critical behaviour and this is expressed as increasing length scales for the correlation of distortions together with increasing length scales of synchronisation across the plate until criticality is reached at one or more places in the plate. The synchronisation stops and the plate becomes subcritical; the process then repeats itself. A test of such a model would be to show that some RQA measures scale as a power law with time before a large event as is the case for many systems as criticality is approached (Sethna 2011). Unfortunately, the data used here is not sufficiently closely spaced as to give the resolution for such a test. Perhaps, sampling at 0.1 day intervals would be sufficient to carry out such a study.
Finally, prediction within the system may be significantly enhanced by the use of new forecasting procedures such as the use of shadowing algorithms (Stemler and Judd 2009) coupled with ensemble modelling (Yoden 2007). In addition, other data sets, in particular, seismic data sets, need to be integrated with the GNSS data sets using cross and jointrecurrence procedures.
Conclusions

The data are chaotic in the sense that they express the behaviour of a large bounded system, with a positive first Lyapunov exponent and result from deterministic dynamics.

The embedding dimension for the attractor for this system is relatively large (≈ 10) which is comparable with many biological systems and is to be expected for a system where coupled deformationchemicalhydraulicthermal processes are responsible for the dynamics.

The entropy of these systems is moderate and comparable with that of many classical, relatively simple systems such as the Lorentz attractor. This means that predictive procedures have considerable potential.

Precursors to strong discontinuities in the signal exist and in some cases precede the event by 100 days. Finer time resolution in the data sets (say binning at 0.1 days) would refine the precursors.

Synchronisation between stations is common even over distances of 600 km. The synchronisation is generalised synchronisation and the analysis reveals details of how the absolute and relative frequencies of recurrence vary both spatially and in time during synchronisation.

Synchronisation between CNST and KAIK begins approximately 9 months before the Kaikōura magnitude 7.8 event. Such synchronisation (as revealed by crossrecurrence plots) provides a powerful means of forecasting this major seismic event and should be explored for other events.

The displacement signals from the regional array are the response of a networked dynamical system where each node interacts with every other node with variable and evolving relations between the strength of coupling between nodes. Thus the behaviour of a specific site can only be understood and forecasted if the evolution of the whole system is monitored.
This paper shows that there is enough information in this regional nonlinear synchronised network to extract details of the underlying dynamics of the crustal deformation system and to provide precursors for major displacement events many months prior to an event. Investigations involving higher timeresolution and spatial recurrence network analysis should be pursued with vigour.
Declarations
Acknowledgements
We thank Margaret Boettcher, Virginia Toy, Mark Munro and Klaus Gessner for the discussions that lead to this paper. Sebastian Orbest and Robert Niven are thanked for introducing us to recurrence analysis. We thank two anonymous reviewers for constructive comments that greatly improved the paper.
Funding
No funding from an external source was involved in this study.
Availability of data and materials
All raw GNSS data used in this paper can be downloaded from http://fits.geonet.org.nz/apidocs/
Authors’ contributions
The topic was conceived jointly between BEH and AO. The first draft was written by BEH and subsequently added to and edited by AO who contributed substantially to the data processing. Both authors read and approved the final manuscript.
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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