A New Model of Campi Flegrei Inflation and Deflation Episodes Based on Brownian Motion Driven by the Telegraph Process

Abstract

A stochastic model to describe the vertical motions in the Campi Flegrei volcanic region is proposed herein, consisting of a Brownian motion process driven by a generalized telegraph process. Knowledge on the probability law of this process enables quantitative investigation of some basic parameters regulating the inflation/deflation processes, such as velocities and time constants. Statistical analysis was carried out based on linear regression with constraints. Predictions of ground displacements and their changing tendency at future time instants were also made. Finally, a statistical test on the Brownian component of the process confirmed the goodness of the model.

1 Introduction

The Campi Flegrei active caldera is located near the City of Naples in Italy. This region, famous worldwide for its slow vertical motion recorded since Roman times, is characterized by the longest ground deformation time series near a volcanic region (Guidoboni and Ciuccarelli 2011; Orsi et al. 1999). After various centuries of subsidence following the last eruption (Monte Nuovo 1538), the Campi Flegrei caldera has shown irregular episodes of activity since at least 1950 (Del Gaudio et al. 2010). The first recent uplift episode dates back to the period 1950–1953, amounting to 73 cm, without any reports or records of seismic activity. In the periods of 1970–1972 and 1982–1984, two strong inflation episodes occurred. The first was accompanied by moderately low seismicity (Corrado et al. 1977), with only a few events being felt by residents, whereas the second was characterized by relatively intense swarms of volcano tectonic (VT) earthquakes, reaching magnitude 4 (Barberi et al. 1986). This seismic activity caused alarm among the population and a spontaneous nocturnal evacuation of part of the City of Pozzuoli (44,000 residents). Since this last episode, subsidence has been recorded for several years, interrupted by some small mini-uplift episodes, with duration of several weeks, all accompanied by seismic swarms of low-magnitude VT events. In recent years, some high-sensitivity instruments have been installed to detect slow earthquake transients and other low-intensity mechanical/temperature precursory signals (Scarpa et al. 2007; Amoruso et al. 2015). Since late 2004, another moderate uplift episode has been occurring at a very low rate, amounting to about $1-2$ cm/year, revealing the presence of clear long-period (LP) events (Amoruso et al. 2007; Saccorotti et al. 2007; D’Auria et al. 2011). The aim of the work presented herein is to provide a quantitative model for this phenomenon based on the recently formulated Brownian motion theory driven by a generalized telegraph process. This approach is very important for deriving a quantitative formulation of some basic parameters regulating the inflation/deflation processes, such as their velocities and relative time constants. This is of fundamental relevance for understanding the source mechanism of this activity.

There is a singular ground deformation time series in the Campi Flegrei region. The archeological ruins of the Serapeo Roman temple clearly demonstrate that three columns, built above ground level around 200 BC, were below sea level for centuries until the Middle Ages. The dominant deflation was interrupted around 88 AC, and in early 1500, when a rapid uplift amounting to approximately 8 m preceded the last eruption several decades later in 1538. The subsidence is clearly documented from 1800 to 1950, when this trend was interrupted by three main episodes of uplift and several minor inflation/deflation trends. Detailed reports of these processes have been made by Dvorak and Gasparini (1991), Dvorak and Berrino (1991) and more recently by Del Gaudio et al. (2010). Figure 1 illustrates the ground deformation reported in the last 2400 years (after Woo and Kilburn 2010). A comparable rate of uplift can be observed before the 1538 eruption and in the present.

The quality of the data on these vertical movements has improved dramatically since 1970, due to alarm among the population caused by the effects of water drainage in the local harbor and the occurrence of some felt earthquakes. These effects were particularly evident during the uplift episode in the period 1982–1984. Leveling measurements improved in these years, together with the installation of other geophysical monitoring networks, managed by Vesuvius Observatory. Continuous Global Positioning System (GPS) measurements have been performed since 2000, and these data have been integrated using interferometric synthetic aperture radar (InSAR) techniques since 1992, thus providing a more complete and homogeneous picture of ground deformations. The dataset under investigation here relates to weekly averaged GPS data recorded at the RITE station, located near Pozzuoli Harbor. The data reported in this paper refer to the period from May 2000 to March 2017. These values are average GPS data recorded by the Campi Flegrei network, composed of ten stations. The precision of these data is close to 1–2 mm. Data were retrieved from the technical reports of Vesuvius Observatory.

Fig. 1

Secular behavior of uplift and subsidence observed in Campi Flegrei region at the Serapeo Roman temple, Pozzuoli in the period from 200 BC to the present

More time series of ground deformation are available, but they are not homogeneous and can be basically divided into three parts. A first series, with precision of around 1 cm, referred to vertical leveling in the period 1970–1995, was obtained from sea tide gauge data recorded at Pozzuoli Harbor and compared with another tide gauge instrument located in Napoli Harbor. A second dataset consisting of the measurements at a benchmark located near Pozzuoli Harbor, deduced from high-precision altimetric leveling in the period from April 1950 to July 2010. The precision of these data is around 1 cm, and the data are referred to a benchmark located away from the center of the caldera, near Napoli Harbor. A third dataset is deduced from leveling data, but for yearly measurements performed in the period from March 1970 to December 1994, corresponding to the most intense phase of unrest. However, the work presented herein was based only on the dataset for the period from 2000 to March 2017, whose measurements are homogeneous and more reliable. Future investigations will be conducted also on the other time series, taking into account their different precision and reliability.

The rest of this manuscript is structured as follows: Section 2 presents the stochastic model of the ground displacements, based on the superposition of an asymmetric telegraph process and a Brownian motion process. Analysis of the considered dataset using the proposed stochastic model is performed in Sect. 3. A linear regression method with constraints is adopted to estimate the relevant parameters, such as the velocities and time constants of the inflation/deflation processes. The investigation also includes a prediction of ground movements and the change of their tendency in future time instants, both based on the proposed model and the analysis conducted so far. Section 4 is devoted to a statistical test on the Brownian component of the ground motion dynamics, whose result confirms the goodness of the model. Below, times are expressed in days and lengths in centimeters.

2 Stochastic Model

The Brownian motion (process) is well known as a stochastic process that is useful to describe a large variety of natural phenomena that undergo regular diffusion driven by random noise. Herein, the term “Brownian motion” refers to the so-called mathematical Brownian motion or Wiener process, which is exempt from inertial processes. Derived stochastic processes are suitable to model more complex situations, such as diffusion in the presence of jumps or time-varying behavior; For instance, Markov-modulated diffusion processes (and more generally, regime-switching Lévy processes) are widely employed in financial modeling (Asmussen et al. 2004; Jobert and Rogers 2006; Ratanov 2010), insurance (Bäuerle 2007), and the theory of queueing networks (Ren and Kobayashi 1998). Moreover, the telegraph process is often employed as a model for alternating random trends in several application fields, such as finance (Kolesnik and Ratanov 2013). Analysis of the telegraph process with drift was performed in Beghin et al. (2001) by means of the Lorentz transformation. Recent advances in this field deal with the telegraph process subject to randomly occurring jumps (Di Crescenzo et al. 2013a; Di Crescenzo and Martinucci 2013b; Ratanov 2015).

To describe the alternating random trend exhibited by the measurements performed in the Campi Flegrei area, the aim of this work is to develop a suitable stochastic model, defined as the superposition of a pure alternating trend process (telegraph process) and a diffusive noise component (Brownian motion). The available data do not exhibit drastic displacements, so the considered model is characterized by continuous sample paths. Specifically, the model is based on a stochastic process $\{X(t), t\ge 0\}$ consisting in a Brownian motion whose drift alternates randomly between a positive and negative value (c and $-v$), according to a generalized telegraph process. Formally, this assumes that the ground position at time t, with respect to sea level, is described by

$$\begin{aligned} X(t)=y_0+Y(t)+b_0+\sigma B(t), \qquad t\ge 0, \end{aligned}$$

(1)

where

• (i) $\{Y(t), t\ge 0\}$ is a generalized (integrated) telegraph process, with $Y(0)=0$;

• (ii) $\{B(t), t\ge 0\}$ is a standard Brownian motion;

• (iii) $\{Y(t)\}$ and $\{B(t)\}$ are independent processes;

• (iv) $y_0, b_0\in \mathbb {R}$, $\sigma >0$ and $X(0)=x_0:=y_0+b_0$, since $Y(0)=B(0)=0$.

In Eq. (1), $y_0+Y(t)$ describes the randomly alternating trend of the observed displacements, whereas the term $b_0+\sigma B(t)$ is a diffusive noise component that represents the total sum of all other small independent perturbation factors that are involved in the considered phenomenon. Physically, these factors may include cumulative effects of small pressure perturbations in a shallow magma chamber, as suggested by Amoruso et al. (2007). Clearly, the Gaussian distribution of $b_0+\sigma B(t)$ might be motivated by central-limit arguments. Similar arguments justify the role of the Brownian motion in other seismological contexts, as pointed out in Kagan and Knopoff (1987) and Matthews et al. (2002).

The telegraph process Y(t) is characterized by velocities $c>0$ and $-v<0$, which alternate according to an independent alternating Poisson process $\{N(t), t\ge 0\}$. The latter process is governed by the sequences of positive independent random times $\{U_1,U_2,\ldots \}$ and $\{D_1,D_2,\ldots \}$, which in turn are assumed to be independent. For all $i=1,2,\ldots$, the random variable $U_i$ ($D_i$) is exponentially distributed with parameter $\lambda >0$ ($\mu >0$), and describes the duration of the i-th random period during which Y(t) has positive (negative) velocity, and thus X(t) has positive (negative) drift. Furthermore, V(t) denotes the velocity of Y(t) at time $t\ge 0$. As a general model, if the initial velocity of Y(t) is randomly chosen by an independent Bernoulli trial, with

$$\begin{aligned} \mathbb {P}\{V(0)=c\}=\theta , \quad \mathbb {P}\{V(0)=-v\}=1-\theta , \qquad 0\le \theta \le 1, \end{aligned}$$

(2)

then, the following stochastic equations hold

$$\begin{aligned} Y(t)= & {} \int _0^t V(s)\,\mathrm{d}s, \qquad t>0, \end{aligned}$$

(3)

$$\begin{aligned} V(t)= & {} \frac{c-v}{2}+\mathrm{sgn}[V(0)]\,\frac{c+v}{2}\,(-1)^{N(t)}, \qquad t>0. \end{aligned}$$

(4)

The alternating Poisson process $\{N(t)\}$ involved in the right-hand side of (4) counts the number of velocity changes of Y(t) in [0, t]; see, for instance, Fig. 2 for simulated paths of Y(t) and X(t) when the initial velocity of Y(t) is positive. Denoting by

$$\begin{aligned} p(x,t):=\frac{\partial }{\partial x}\mathbb {P}\{X(t)\le x\} \end{aligned}$$

the transition density of X(t), for all $(x,t)\in \mathbb {R}\times (0,+\infty )$, one has (see, for instance, Di Crescenzo and Zacks 2015, and Zacks 2017)

$$\begin{aligned} \begin{aligned} p(x,t)&=\frac{1}{\sigma \sqrt{t}}\biggl \{ \theta \,\mathrm {e}^{-\lambda t}\phi \biggl (\frac{x-x_0-ct}{\sigma \sqrt{t}}\biggr ) +(1-\theta )\,\mathrm {e}^{-\mu t}\phi \biggl (\frac{x-x_0+vt}{\sigma \sqrt{t}}\biggr ) \biggr \} \\&\qquad +\frac{1}{\sigma \sqrt{t}}\int _{0}^{t}\psi (w,t) \phi \biggl (\frac{x-x_0+vt-(c+v)w}{\sigma \sqrt{t}}\biggr )\hbox {d}w, \end{aligned} \end{aligned}$$

(5)

Fig. 2

Simulated sample paths of Y(t) and X(t) for $c=2$, $v=1$, $\lambda =1$, $\mu =0.5$, $\sigma =0.5$, and $y_0=x_0=0$

where $\phi (\cdot )$ is the standard normal density, and for $0<x<t$,

(6)

In Eq. (6), $I_0(z)$ and $I_1(z)$ denote the modified Bessel functions given by

$$\begin{aligned} I_0(z)=\sum _{k=0}^{+\infty }\frac{(z/2)^{2k}}{(k!)^2}, \qquad I_1(z)=\sum _{k=0}^{+\infty }\frac{(z/2)^{2k+1}}{k!\,(k+1)!}=I_0'(z). \end{aligned}$$

(7)

The function (6) is the probability density of the sojourn time of V(t) in the state c in the time interval [0, t]. Other issues on the stochastic process (1) have been treated by Di Crescenzo et al. (2005).

3 Data Analysis

Various statistical analyses on the parameters of the (discretely observed) telegraph process have been performed in the past by means of a least-squares approach, pseudo maximum-likelihood, and moment (or approximate moment) estimations (De Gregorio and Iacus 2008, 2011; Iacus and Yoshida 2009). Since the model (1) also involves a Brownian motion component, a different approach is considered here for estimation of the relevant parameters. An alternative estimation procedure based on likelihood inference has been developed by Pozdnyakov et al. (2017) for a different alternating Brownian motion, whose infinitesimal variance—rather than the drift—alternates according to a telegraph process.

3.1 Model with Constant Velocities

Recalling that the proposed position process (1) is the sum of a telegraph process $y_0+Y(t)$ and an independent Brownian motion $b_0+\sigma B(t)$, the available data (shown in Fig. 3) can be interpreted as a trajectory of such an alternating Brownian process X(t). The initial and final values are − 0.49 and 35.8 (cm). The alternating behavior suggests identification of five time periods, consisting in three inflation episodes and two deflation episodes. In a preliminary approach, the turning times of the alternating periods were estimated empirically by the local minima and maxima of the observed sample path; the corresponding estimates are

$$\begin{aligned} t_1=2000.09.26, \;\; t_2=2002.10.22, \;\; t_3=2007.01.23, \;\; t_4=2007.10.23. \end{aligned}$$

(8)

Fig. 3

Observed ground position, with alternating inflation and deflation episodes, where the indication of the turning times is given in (8). (Year ticks are the closest to May 30)

A linear regression approach with constraints was adopted to estimate the trajectory of the underlying telegraph process. The coefficients of the alternating trend were detected by imposing that the linear regression curves related to each time period intersect at the turning points. Hence, denoting by $\alpha _i+t \,\beta _i$ the regression line of the i-th time period, the resulting system of constraints is

$$\begin{aligned} \begin{array}{l} \alpha _1+t_1 \beta _1=\alpha _2+t_1\beta _2, \\ \alpha _2+t_2 \beta _2=\alpha _3+t_2\beta _3, \\ \alpha _3+t_3 \beta _3=\alpha _4+t_3\beta _4, \\ \alpha _4+t_4 \beta _4=\alpha _5+t_4\beta _5, \end{array} \end{aligned}$$

so that

$$\begin{aligned} \begin{array}{l} \alpha _2=\alpha _1+t_1 \beta _1-t_1\beta _2, \\ \alpha _3=\alpha _1+t_1\beta _1+(t_2-t_1)\beta _2-t_2\beta _3, \\ \alpha _4=\alpha _1+t_1\beta _1+(t_2-t_1)\beta _2+(t_3-t_2)\beta _3-t_3\beta _4, \\ \alpha _5=\alpha _1+t_1\beta _1+(t_2-t_1)\beta _2+(t_3-t_2)\beta _3+(t_4-t_3)\beta _4-t_4\beta _5. \end{array} \end{aligned}$$

The least-squares estimates of the relevant parameters $(\alpha _1,\beta _1,\beta _2,\beta _3,\beta _4,\beta _5)$ can be obtained using (8).

To estimate the velocity c of the telegraph process Y(t) in the inflation episodes, the adopted procedure requires computation of the weighted average of the coefficients $\beta _1$, $\beta _3$, and $\beta _5$, weighted on the number of observed data in the relative time intervals. Similarly, the velocity $-v$ in the deflation episodes is obtained as the weighted average of the coefficients $\beta _2$ and $\beta _4$. Furthermore, the rate $\lambda$ ($\mu$) is estimated by computing the reciprocal average of the duration of the inflation (deflation) episodes. Figure 4a shows the trajectory of the telegraph process thus obtained, compared with the observed data. Estimates of the four parameters are listed in Table 1.

Fig. 4

Trajectories of a the telegraph process $y_0+Y(t)$, with the observed data, and b the resulting Brownian motion $b_0+\sigma B(t)$

As a preliminary index of the goodness of the model, it is interesting to determine the coefficient of determination, given by $R^2:=1- S_\mathrm{res}/S_\mathrm{tot}$, where $S_\mathrm{res}$ is the residual deviance and $S_\mathrm{tot}$ is the total deviance of data. Note that the value of $R^2$ is concerning the approximation of the observed data by the telegraph process Y(t) alone.

Due to Eq. (1), the trajectory of the Brownian component $\sigma B(t)$ is obtained simply as a difference, as shown in Fig. 4b. The maximum-likelihood estimate (MLE) of $\sigma$ can be obtained from the increments of the Brownian motion over the available intervals, viewed as a Gaussian sample. Such an estimate is presented in Table 1, together with $R^2$ and the values of the other parameters. For the considered model, it is now possible to perform some predictions based on the results obtained so far.

Table 1 Estimates of the velocities of the telegraph process in the inflation and deflation episodes, of turning rates, and of the infinitesimal standard deviation of the Brownian component, with the coefficient of determination for the telegraph process

The probability that the motion exhibits no changes of tendency in time is evaluated first. To this purpose, $P_0(t)$ denotes the probability of having no changes of tendency of the motion in the time interval $(t_\mathrm{f},t]$, where $t_\mathrm{f}=2017.03.14$ is the last available observation time. According to the given assumptions, since at time $t_\mathrm{f}$ the trend is increasing (Fig. 3), $P_0(t)$ is related to an exponential distribution with parameter $\lambda$. Hence, at time t, this probability is given by

$$\begin{aligned} P_0(t) = \mathrm {e}^{- \lambda (t-t_\mathrm{f})}, \qquad t>t_\mathrm{f}, \end{aligned}$$

where the estimate $\lambda =5.88\cdot 10^{-4}$ is provided in Table 1. Some values of $P_0(t)$ are listed in Table 2. Due to the given results, during the year 2021, this probability becomes smaller than 1 / 2, so that a change of tendency becomes more likely.

Table 2 Probability that the current motion exhibits no changes of tendency up to time t

Table 3 Estimated intervals for $\mathbb {P}[x_1<X(t)<x_2]$ conditional on the observed data

Some predictive intervals where the estimated location of X(t) is more likely are now determined for suitable choices of t. Recalling (5), for $x_1<x_2$ one has

$$\begin{aligned} \begin{aligned}&\mathbb {P}[x_1<X(t)<x_2] = \int _{x_1}^{x_2}p(x,t)\hbox {d}x \\&\quad = \theta \,\mathrm {e}^{-\lambda t} \left[ \Phi \biggl (\frac{x_2-x_0-ct}{\sigma \sqrt{t}}\biggr )-\Phi \biggl (\frac{x_1-x_0-ct}{\sigma \sqrt{t}}\biggr )\right] \\&\qquad + (1-\theta ) \,\mathrm {e}^{-\mu t} \left[ \Phi \biggl (\frac{x_2-x_0+vt}{\sigma \sqrt{t}}\biggr )-\Phi \biggl (\frac{x_1-x_0+vt}{\sigma \sqrt{t}}\biggr )\right] \\&\qquad + \int _{0}^{t}\psi (w,t) \left[ \Phi \biggl (\frac{x_2-x_0+vt-(c+v)w}{\sigma \sqrt{t}}\biggr )\right. \\&\qquad \left. -\Phi \biggl (\frac{x_1-x_0+vt-(c+v)w}{\sigma \sqrt{t}}\biggr )\right] \hbox {d}w, \end{aligned} \end{aligned}$$

(9)

where $\psi (w,t)$ is defined in (6), and $\Phi (\cdot )$ is the standard normal cumulative distribution function. Probability (9) conditional on the observed data can then be evaluated for various choices of t and $(x_1,x_2)$. This is performed by making use of the Markov property of the Brownian process, taking into account that at time $t_\mathrm{f}= 2017.03.14$ the position is $X(t_\mathrm{f})=x_\mathrm{f}=35.8$, and that the tendency is increasing. Moreover, the other involved parameters are chosen as in Table 1.

Finally, Table 3 presents the estimated intervals $(x_1,x_2)$ for various values of the probability (9) conditional on the observed data. The extremes $x_1$ and $x_2$ are taken as $m\pm h$, where m is the mode of the density p(x, t). By analysis of the density, one has

$$\begin{aligned} m= \left\{ \begin{array}{ll} 41.8 &{} \hbox { for }t = 2018.12.31, \\ 45.1 &{} \hbox { for }t = 2019.12.31. \end{array} \right. \end{aligned}$$

The density p(x, t) is plotted in Fig. 5 for the above specified choices of the parameters.

Fig. 5

Density p(x, t) for $t =$ 2018.12.31 and $t =$ 2019.12.31

The adopted prediction procedure consists in a statistical method based on the hypothesis that no catastrophic event occurs in the reference time interval. Indeed, it is typically confirmed that catastrophic events modify the ground dynamics substantially.

3.2 Model with Variable Velocities

This section deals with a more advanced model in which the telegraph process Y(t) can assume different velocities $c_i$ ($-v_i$) in each inflation (deflation) episode. The aim of this model is to improve the correspondence of the observed data with a refined version of the telegraph process.

Fig. 6

Trajectories of (a) the telegraph process $y_0+Y(t)$ with variable velocities, together with the observed data, and (b) the resulting Brownian motion $b_0+\sigma B(t)$

The procedure for the construction of the trajectories is the same as provided in Sect. 3.1, but it does not include the weighted average adopted for the estimates of the slopes $\beta _i$. The results of this analysis are shown in Fig. 6, whereas Table 4 presents the estimates of the parameter values and the coefficient of determination. In this case, an explicit form for the transition density of the process X(t) is not available, so investigation on the predictive intervals was not performed.

Table 4 Estimates of the variable velocities of the telegraph process in the inflation and deflation episodes and the infinitesimal standard deviation of the Brownian component, with the coefficient of determination for the telegraph process

4 Testing the Brownian Component

For the stochastic model considered so far, denote the difference between the position process X(t) and the telegraph (trend) process $y_0+Y(t)$ by $\{D(t), t\ge 0\}$. In this section, for both cases analyzed in Sect. 3, a statistical test is performed based on observed data to verify whether D(t) is a Brownian motion. Specifically, the following test is considered

$$\begin{aligned} H_0: \; D(t)=b_0+\sigma B(t) \qquad \hbox {vs} \qquad H_1: \; D(t) \ \hbox {is a } \ \left\{ \begin{array}{l} \hbox {confined} \\ \hbox {directed} \end{array} \hbox {diffusion}. \right. \end{aligned}$$

In a sense, this aims to test whether the noise component is a Brownian motion or a confined or directed diffusion. This method was recently proposed by Briane et al. (2016). It is based on the dataset for D(t), denoted by $(D(\tau _0),D(\tau _1),\ldots ,D(\tau _n))$, where the $\tau _i$ are the observation time instants, with $n+1=823$. To this purpose, consider the following standardized statistic

$$\begin{aligned} T^n_D(\tau _n)=\frac{S^n_D(\tau _n)}{\hat{\sigma }\sqrt{\tau _n}}, \end{aligned}$$

(10)

where

$$\begin{aligned} S^n_D(\tau _n)=\max _{i=1,2,\ldots ,n} |D(\tau _i)-D(\tau _0)| \end{aligned}$$

and

$$\begin{aligned} \hat{\sigma }=\left\{ \frac{1}{n} \sum _{i=1}^n \frac{[D(\tau _i) - D(\tau _{i-1})]^2}{\tau _i- \tau _{i-1}}\right\} ^{1/2}. \end{aligned}$$

(11)

Equation (11) is the maximum-likelihood estimator (MLE) of $\sigma$, whose values are presented in Tables 1 and 4 for the two considered cases.

As the sample size is large, according to Briane et al. (2016), the asymptotic acceptance region of amplitude $1-\alpha$ of the null hypothesis $H_0$ is defined as

$$\begin{aligned} \left\{ q\left( \frac{\alpha }{2}\right) \le T^n_D(\tau _n)\le q\left( 1-\frac{\alpha }{2}\right) \right\} , \end{aligned}$$

(12)

where $q(\alpha )$ is the lower quantile of level $\alpha$ of $\sup _{0\le s\le 1}|B(s)-B(0)|$. Since the estimate of (10) is

$$\begin{aligned} t^n_D(\tau _n)=\left\{ \begin{array}{ll} 1.749 &{} \hbox {(case with constant velocities)},\\ 1.092 &{} \hbox {(case with variable velocities)}, \end{array} \right. \end{aligned}$$

and since, for $\alpha =0.05$, one has (cf. Table 1 of Briane et al. (2016))

$$\begin{aligned} q(0.025)=0.834, \qquad q(0.975)=2.940, \end{aligned}$$

due to (12) the null hypothesis $H_0$ can be accepted with level 0.95 in both cases. Therefore, this confirms that the observed trajectory of D(t) can be viewed as a realization of a Brownian motion.

5 Conclusions

Data on the vertical ground motion of Campi Flegrei, analyzed using different methodologies, provide a unique example of episodes of unrest in an active caldera due to the length of the available time series. It is very difficult to determine whether the current episodes of unrest are a long-term precursor of a new eruption, and if so, when such an event might occur. The main results obtained in this work provide a more precise quantification of uplift and subsidence rates, in good agreement with previous estimates made by different authors. The beginning of unrest dates back to 1950, and it has not been marked by significant seismicity, whereas later episodes (in particular the 1982–1984 event) were characterized by seismicity with progressively higher magnitude and more significant seismic energy release. However, these values are somewhat low than those observed in other calderas which experienced eruptions, such as that in Rabaul in 1994. The evolution of the deformation and seismicity seems to indicate a slow approach to more unstable conditions of the volcano. The time duration of the current period of unrest, its size, and its trend are similar to those that preceded the eruption of 1538 in the period 1400–1536 (i.e. between 2.9 and 9.1 cm/year, as estimated by Di Vito et al. 2016) and may have a similar conclusion, but it is quite difficult, in the absence of additional data, to provide a more precise determination of the required time. It is only roughly expected that, before the onset of an eruption, an accelerated rate of uplift and seismicity will be observed. Another relevant observation made by Amoruso et al. (2015) is that seismicity occurs several minutes after inflation and deflation episodes. The quantitative model presented here offers a relevant possibility to precisely and quantitatively estimate the alternation of the uplift and subsidence rates characterizing this volcanic region.