The effusive–explosive energy emission process in a volcano is a dynamic and complex physical phenomenon. The importance of quantifying this complexity in terms of the physical and mathematical mechanisms that govern these emissions should be a requirement for deciding to apply a possible forecasting strategy with a sufficient degree of certainty. The complexity of this process is determined in this research by means of the reconstruction theorem and statistical procedures applied to the effusive–explosive volcanic energy emissions corresponding to the activity in the Volcán de Colima (western segment of the Trans-Mexican Volcanic Belt) along the years 2013–2015. The analysis is focused on measuring the degree of persistence or randomness of the series, the degree of predictability of energy emissions, and the quantification of the degree of complexity and “memory loss” of the physical mechanism throughout an episode of volcanic emissions. The results indicate that the analysed time series depict a high degree of persistence and low memory loss, making the mentioned effusive–explosive volcanic emission structure a candidate for successfully applying a forecasting strategy.

Right forecasting of dangerous long drought episodes, high-magnitude earthquakes or great volcanic emissions should be one of the main objectives of the scientific fields of climatology, seismology or volcanology to prevent disasters which could affect the environment and human life. Several examples of forecasting algorithms could be cited, among them the nowcasting strategy (Rundle et al., 2016, 2017), the multi-fractal analysis in seismology (Monterrubio-Velasco et al., 2020), the ARIMA (auto-regressive integrated moving average based on a reconstruction theorem) process in climatic research (Lana et al., 2021) and neural algorithms (Lipton et al., 2015; Lei, 2021), which are also useful for predicting monthly rainfall. These cited algorithms systematically forecast the next episode, taking into account a certain number of previously recorded data, this number being strongly associated with the characteristics of the physical mechanism. These forecasting results should be validated by previously analysing the degree of complexity and the “loss of memory” of the physical mechanisms along the evolution of the physical process. In other words, how many previous data would be necessary for right use of a forecasting algorithm, and what could the range of uncertainties in the predictions be?

The database, analysed in this research in Sect. 2, is the time series of explosive volcanic events (Vulcanian explosions) emitted by the Volcán de Colima (western segment of the Trans-Mexican Volcanic Belt) during the years 2013–2015. A Vulcanian explosion is an eruption where fragmented material is expelled into the atmosphere as a result of overpressure into the conduit or lava dome (Arámbula-Mendoza et al., 2018). The event releases energy in several ways based on elastic, seismic, acoustic and thermal processes. The next explosion will occur when the overpressure breaks the impermeable cap again. The Volcán de Colima has emitted many Vulcanian explosions, some of them with generation of pyroclastic density currents (PDCs) until 5 km of runout (Arámbula-Mendoza et al., 2019). For these reasons, strategies for right forecasting of the mentioned Vulcanian explosions are important.

The reconstruction theorem (Sect. 3) is a mathematical strategy that allows quantification of the degree of complexity in a time series and its loss of memory, which are both required to validate a possible forecasting strategy (Diks, 1999). Additionally, the nowcasting algorithm (Sect. 5), a statistical process developed by Rundle et al. (2016, 2017) to detect the risk of imminent high-magnitude earthquakes, could also be applied to quantify the probability of an imminent high-magnitude volcanic emission.

The main objective of this work is to detect the degree of difficulty for a forecasting of volcanic emissions associated with energies close to or exceeding 10

The most relevant results of the reconstruction theorem and their effects on forecasting algorithms are discussed in Sect. 6. Finally, Sect. 7 (“Conclusions”) summarises the most relevant results with respect to the expected success in preventing volcanic energy emissions based on forecasting and nowcasting processes.

A time series of volcanic explosions, named Vulcanian explosions (Clarke et al., 2015), emitted by the Volcán de Colima (western segment of the Trans-Mexican Volcanic Belt, years 2013–2015) (Arámbula-Mendoza et al., 2018) is analysed. Figure 1a depicts the histogram of the logarithm of the emitted energy. The dataset contains 6182 observations of the emissions equalling or exceeding approximately

Figure 2a describes the six segments to be analysed with 1000 samples and a seventh segment that is excluded from the analysis due to a lack of data. The highest explosions can be observed at the beginning of the first segment [log

The statistical distribution of these emissions is analysed by means of the L-skewness or L-kurtosis formulation (Hosking and Wallis, 1997). The statistical analysis of these emissions shows that the complete series of emissions, including those not fulfilling the Gutenberg–Richter law, are well fitted to the generalised logistic, GL, function (Fig. 3a and b). Additionally, three different empirical distributions of extreme emissions, equalling or exceeding 90 %, 95 % and 99 % respectively of the data (Fig. 3b), can be associated with the generalised extreme value, GEV, function. Figure 4 shows the evolution of these three expected extreme emissions with the increasing return periods (given as the number of events equalling or exceeding 90 %, 95 % and 99 % respectively). For instance, the expected values of emissions for the three percentage levels and return periods with up to 200 extreme emissions fit the theoretical evolution quite well, with emissions close to

Return period curves (90 %, 95 % and 99 %) of extreme emissions.

Prior to the reconstruction theorem (Diks, 1999) based on monofractal theory, the degree of randomness, anti-persistence or persistence of the analysed data is established by taking into account the concept of the Hurst exponent (Turcotte, 1997), which is defined as the exponent

The analysis of the monofractal structure of a series, by means of the reconstruction theorem (Diks, 1999), permits quantification of its complex forecasting by means of the following parameters.

The necessary minimum number of non-linear equations governing the physical mechanism, usually referenced as a correlation dimension

The embedding dimension,

The Kolmogorov entropy,

Some examples of the Hurst exponent for

The results of the Hurst exponent for the whole series and the six data segments are described in Fig. 5a and b, obtaining a clear sign of persistence for the complete series of Vulcanian explosions, with

Embedding dimension and Kolmogorov coefficient for the 21 moving windows.

Embedding dimension and Kolmogorov coefficient for the six segments of volcanic emissions.

An example of the evolution of embedding dimensions (first segment of 1000 elements) for reconstruction dimensions

With respect to the embedding dimension, Fig. 7 illustrates five examples of the first segment of 1000 recorded emissions, where the slope,

The obtained values of the Kolmogorov entropy exponent, based on Eq. (6) and summarised in Tables 1 and 2, are also illustrated with some examples (Fig. 8). In these four examples, the loss of memory of the physical mechanism is quite similar for the 6th segment and the 10th moving window, with values of

Mean and standard deviation for the first 10 Lyapunov exponents after 975 iterations.

Fifteen Lyapunov exponents for the third segment of the effusive–explosive volcanic emissions.

Right computation of the Lyapunov exponents needs an iterative process, with the aim of minimising the final uncertainty on every exponent. In the present computations, 975 iterations have been good enough to obtain the first 15 exponents with very small oscillations at the end of the iterative process. An example of this process is shown in Fig. 9, which describes the evolution of the exponents for the third segment of emissions up to

The results are summarised in Table 3, which shows the mean and standard deviation for every one of the first 10 Lyapunov exponents obtained for the 21 moving windows and the six data segments after 975 iterations of the corresponding computational algorithm to obtain accurate and confident values. First, in agreement with the results shown in the mentioned table, every one of the

While for the six trams,

Basic characteristics of the energy emissions (logarithm of energy) for the whole database, the six segments of 1000 elements and the last segment of 182 elements. The empirical results of the Kolmogorov–Smirnov, K–S, test are compared with the significance levels of 95 % and 99 %, KS_0.05 and KS_0.01, corresponding to the Gaussian distribution.

A comparison of nowcasting results for

The nowcasting process (Rundle et al., 2016, 2017) is based on the computation of the “natural time” or, in other words, the number of consecutive earthquakes (seismic cycle length) with magnitudes within a determined interval. In this way, the empirical cumulative distribution function, CDF, of these natural times is established by the high-magnitude earthquakes interrupting these seismic cycle lengths. Consequently, the nowcasting process does not exactly predict a forthcoming high magnitude but quantifies the probability of an imminent high earthquake magnitude based on the empirical CDF curves.

A first illustrative example of the nowcasting algorithm, from the point of view of the seismic activity, is depicted in Fig. 10a. It corresponds to the recorded seismic activity in Waitaha / Canterbury (National Earthquake Information Database,

The probability of forthcoming extreme magnitudes (7.2 and 7.8) interrupting a cycle length exceeds 80 %. Consequently, the probability of an earthquake of a similar extreme magnitude should be more or less imminent if the real cycling length ranges between approximately 100 and 1000 natural times, depending on the chosen maximum magnitude

Two examples of nowcasting corresponding to volcanic energy explosions are shown in Fig. 10b. The first one corresponds to the volcanic activity of the third segment (Fig. 2a) and the second one includes the whole series of volcanic emissions. In both cases, the cycle lengths are obtained by considering the minimum and maximum levels of volcanic emissions of 10

The results obtained by the reconstruction theorem and the possible relationships between the fractal reconstruction exponents (

The Hurst exponent (Fig. 11a) is characterised by a continuous increase, finally achieving oscillations close to 0.7 with an evident structure of persistence from the 7th to 21th moving windows, all of them including two high emissions, log

With respect to the results of the nowcasting, the return period curves (90 %, 95 % and 99 % of extreme emissions) could be, as cited before, a relatively similar strategy. Nevertheless, the nowcasting process permits us to decide on the minimum and maximum emissions of energy levels to define the best empirical distributions of cycle lengths of natural waiting times, detecting in this way the probability, in percentage, of a probable imminent volcanic emission of high energy. In spite of the nowcasting method not determining a concrete next volcanic emission, given that it is not a forecasting process, it takes into account that future high emissions will be expected with similar natural waiting times. Although some emissions (Fig. 10b) exceeding log

The fractal parameters, obtained by means of the reconstruction theorem of six segments and 21 moving windows of the analysed volcanic explosions in the Volcán de Colima (Mexico) as well as the nowcasting strategy, are the first steps in the application of different forecasting processes.

The results suggest that different strategies for future forecasting emissions can be applied, especially for high-energy emissions. These forecasting strategies can be based on different algorithms (Box and Jenkins, 1976; Lipton et al., 2015; Rundel et al., 2017; Lei, 2021) and multi-fractal analysis of moving window data (Monterrubio-Velasco et al., 2020). In spite of the uncertainties with respect to the waiting time of an emission and its corresponding energy estimated by means of forecasting, which are expected to be non-negligible, these algorithms should depict reasonably good approaches to real energy emissions, bearing in mind the obtained reconstruction theory results. Additionally, the analysis of the multi-fractal structure is expected to be a warning factor for volcanic activities associated with high emissions of energy, quite similar to the analysis of consecutive seismic magnitudes (Monterrubio-Velasco et al., 2020) and usefully applied to analyse climatic data as well as thermometric and pluviometric data (Burgueño et al., 2014; Lana et al., 2023) and also bearing in mind (Shimizu et al., 2002) the concept of a multi-fractal complexity index, which could also contribute to detecting imminent extreme volcanic emissions of energy. In short, the reconstruction theory applied in this research, together with nowcasting and forecasting algorithms and multi-fractal theory, could be a very important process for preventing extreme emissions of volcanic energy.

Data sharing is not applicable to this article as no new data were created or analysed in this study.

XL contributed in the conceptualization of the work, formal analysis, visualization, writing the original draft preparation, and the review and editing. MMV contributed in the conceptualization of the work, funding acquisition, visualization, and review and editing. RA participated in the data curation, investigation, supervision, and review and editing.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The authors thank the Center of Excellence for Exascale in Solid Earth for supporting this research.

This research has been supported by the European High-Performance Computing Joint Undertaking (JU) as well as Spain, Italy, Iceland, Germany, Norway, France, Finland and Croatia under grant agreement no. 101093038, (ChEESE-CoE).

This paper was edited by Luciano Telesca and reviewed by two anonymous referees.