Aims & Scope

First of all, factor analysis was employed to determine the number of factors (i.e., processes) required to explain the variance among a multiple of time series (Lawrence and Upchurch, 1982). Note that a factor does not necessarily correspond to a specific process, and rather that it may represent a combination of several processes. In case that multiple volcanic gas parameters (e.g., concentrations of CO₂, H₂O, SO₂, H₂S, etc.) are available, it is possible to reduce a large set of data into a smaller one based on factor analysis.

Table 1 presents the results of factor analysis for the data in Fig. 1c and 1d. As indicated by the cumulative proportions, Factors 1 and 2 in both data sets can explain nearly all of the variance among the volcanic gas parameters, with the minimal proportion of Factor 3. Accordingly, only two series are worthy for the subsequent analyses. In both sets, H₂O concentrations have positive loadings of Factors 1 and 2, whereas CO₂ concentrations have negative loadings of Factors 1 and 2. These suggest the opposite trend between H₂O and CO₂ concentrations. Indeed, the time series of H₂O concentrations are negatively correlated with those of CO₂ concentrations (see Fig. 2a and 2b). In case of Merapi volcano, the time series of CO₂ concentrations is just a mirror image of that of H₂O concentrations given that the slope of the corresponding regression line is -1.000 with R = -1.000 (R² = 1.000) in Fig. 2b.

Fig. 2. 
Correlation analysis between H₂O and CO₂ concentrations at Stromboli volcano (Aiuppa et al., 2010) (a) and at Merapi volcano (Zimmer and Erzinger, 2003) (b), and cross correlation analysis of volcanic gas data at Stromboli volcano (Aiuppa et al., 2010) (c) and at Merapi volcano (Zimmer and Erzinger, 2003) (d).

As mentioned above, two time series need to be selected in the present examples. Then, which ones are to be selected? Before answering this question, one should perform cross correlation analysis, which measures the similarity between two series as a function of lag (τ), the temporal displacement between them (Larocque et al., 1998). This analytical tool is available at a website free of charge (Wessa, 2017). By cross correlation analysis, it is possible to evaluate whether there is a stimulus-response relationship between two time series. Fig. 2c and 2d show the results of cross correlation analysis between volcanic gas data in Fig. 1c and 1d. In Fig. 2c and 2d, the highest or lowest values of cross correlation functions (CCF) occur at τ = 0, indicating that there is no stimulus-response relationship among the time series under consideration. Nonetheless, such relationships may be present in the data obtained by remote sensing techniques since the compositions of volcanic gas plumes are inevitably affected by atmospheric temperature and pressure, wind speed and direction, and precipitation (refer to section 3.3 in Lee et al. (2018)). For example, Shimoike and Notsu (2000) have found that volcanic O₂ concentration is positively correlated with the atmospheric pressure with several hour displacement. In their case, if the O₂ concentration profile is shifted to several hours earlier along the time axis, the resultant profile will match that of the atmospheric pressure. Thus, it should be noted that one cannot select any time series having stimulus-response relationships.

When choosing the parameters, a priority should be given to those that are predominantly loaded by single factors. As can be seen in Table 1, no parameters meet this criterion. Then, CO₂ and H₂S concentrations are usually good choices since both species are least affected by the groundwater scrubbing (Symonds et al., 2001). Given the dilution of volcanic gases by water vaporization and boiling during their ascents (Lee et al., 2018), it is often necessary to normalize these concentrations with respect to volcanic H₂O concentration. In the present examples, thus, the concentration ratios of CO₂/H₂O are selected. Importantly, this selection can prevent the strongly inter-correlated parameters (H₂O and CO₂ concentrations) from being considered together for the subsequent analyses.

In the present examples, one still needs to select one more parameter from both data sets. The concentration ratios of CO₂/SO₂, CO₂/HCl, CO₂/HF, SO₂/HCl, and SO₂/HF have been often used to predict volcanic eruptions (Duffell et al., 2003; Aiuppa, 2009; Notsu and Mori, 2010; Stremme et al., 2011; López et al., 2013). Notably, individual gas concentrations may fluctuate significantly in response to non-volcanological effects such as hydrothermal interactions and meteorological conditions (Shimoike and Notsu, 2000; Symonds et al., 2001). Thus, to lessen non-volcanological effects, it is good to use the concentration ratios with the expectation that such effects have an influence on individual gas concentrations in a similar way. More importantly, the ratios are more likely to reach either maxima or minima prior to volcanic eruptions than individual gas concentrations (refer to section 4.3 in Lee et al. (2018)). For the data set of Aiuppa et al. (2010), the CO₂/SO₂ ratio is the only remaining one on the aforementioned list. If more than one ratio is available, one may consider the ratio wherein two gas concentrations are characterized by the same highest loading factor but by different secondary factors, and then move to the one wherein two gas concentrations have different highest loading factors but the same secondary factor. In case of Zimmer and Erzinger (2003), volcanic gas temperature is to be chosen since no more concentration ratio is available after selecting CO₂/H₂O ratios.

Once all parameters are selected, one should make sure that they are not correlated with each other. As shown in Fig. 3a, the ratios of CO₂/H₂O and SO₂/CO₂ are not strongly correlated (R = -0.420). Also, in Fig. 3b, the ratios of CO₂/H₂O are not strongly correlated with volcanic gas temperature (R = 0.321). These results assure the validity of the selected parameters for the subsequent analyses. If selected parameters are correlated significantly, one should try other parameters and repeat correlation analysis until no strong correlation exists among them.

Fig. 3. 
Correlation analysis between CO₂/H₂O and SO₂/CO₂ at Stromboli volcano (Aiuppa et al., 2010) (a) and between CO₂/H₂O and temperature at Merapi volcano (Zimmer and Erzinger, 2003) (b).

Time series of selected parameters can be processed using a multiple of statistical tools, each of which has its own pros and cons. In this study, the selected time series were subjected to the Classical Decomposition with Moving Averages (CDMA) analysis (Wessa, 2017). This statistical tool can provide a convenient and effective way to extract trend, periodic, and random components from time series (see Fig. 4 and 5). In volcanic gas analysis, the trend component reflects the depth (pressure) of magmatic degassing and the extent of hydrothermal interactions, the key processes controlling the evolution of volcanic gas compositions (Lee et al., 2018). Note that the trend component may be long-term or short-term depending on the monitoring duration. For example, while Fig. 4a and 4b show several month long trends, Fig. 5a and 5b indicate the trends over several days. The periodic component likely represents regular rises of gas bubbles from magmas as well as diurnal and season variations in metrological conditions. The random component may result from explosive gas discharges, gas influxes from other magma sources, vent collapses due to seismic activities, and abrupt changes in metrological conditions.

Fig. 4. 
Time series analysis of CO₂/H₂O ratios (a) and SO₂/CO₂ ratios (b) by Classical Decomposition with Moving Averages (CDMA) analysis at the period (n) of 12, and the corresponding periodograms of CO₂/H₂O ratios (c) and SO₂/CO₂ ratios (d) by spectral analysis. The original data were collected at Stromboli volcano (Aiuppa et al., 2010).

Fig. 5. 
Time series analysis of CO₂/H₂O ratios (a) and temperature (b) by Classical Decomposition with Moving Averages (CDMA) analysis at the period (n) of 12, and the corresponding periodograms of CO₂/H₂O ratios (c) and temperature (d) by spectral analysis. The original data were collected at Merapi volcano (Zimmer and Erzinger, 2003).

In the CDMA analysis, one is first asked to select the period (n), the size of subset data to be averaged. In Fig. 6, it is illustrated how the period affects the apportionment among the three components. With the increasing period, the trend component becomes smoother, with the greater proportion of the other components (see Fig. 6). When choosing the period in the CDMA analysis, one should take into account the monitoring length and interval. For example, if monthly measurements are made over a few years, the seasonal periodicity (n = 12) need to be reflected. On the other hand, when the collected data are several days long with hourly intervals, the diurnal periodicity (n = 24) should be considered. Also, it is better to choose a larger period in analyzing a bigger size of data.

Fig. 6. 
Effect of the period (n) on the apportionment among the trend, periodic, and random components in the time series of CO₂/H₂O ratios by the CDMA analysis. The original data were collected at Stromboli volcano (Aiuppa et al., 2010).

Once the trend component is separated from the other components, it is necessary to evaluate whether it is significant compared to the seasonal and random components. In Fig. 4a and 4b, the ranges (i.e., spreads) of the trend components are greater than those of the other components. Thus, one may claim that the ratios of volcanic CO₂/H₂O and SO₂/CO₂ at Stromboli volcano were roughly on the upward and downward, respectively. In contrast, the trend components in Fig. 5a and 5b are largely within the ranges of the other components. Thus, despite the significant temporal variations in both CO₂/H₂O ratios and temperature, no definite trend was noted at Merapi volcano.

Despite the preceding discussion, it is not certain which feature(s) in the trend components can herald volcanic eruptions. Although this issue may not be resolved completely, it is still possible to glean an important clue. In general, abrupt changes in volcanic gas compositions precede eruptions (Lee et al., 2018), suggesting that the rates of changes in volcanic gas compositions are of potential importance in predicting eruptions. In this study, the rates of changes in CO₂/H₂O and SO₂/CO₂ ratios at Stromboli volcano were calculated for the trend components as follows:

where R_i and R_i+1 are the concentration ratios of CO₂/H₂O or SO₂/CO₂ at time t_i. and t_i+1, respectively. In Fig. 7, the calculated rates are compared to their 75th and 95th percentiles as well as the means. If the rates lie successively beyond a limit (e.g., 95th percentile), such features may be considered a sign of eruptions. In Fig. 7, it should be noted that the data located outside a cautionary limit (e.g., 95th percentile) do not necessarily foretell imminent eruptions since only a small amount of data are incorporated in these calculations. Thus, volcanic gas data monitored over a longer period are required for the better prediction of eruptions. Also, it is beneficial to have the data encompassing previous eruption events in order to determine to what extents the rates of changes in volcanic gas compositions may occur close to eruptions.

Fig. 7. 
Rates of changes in the trend components of CO₂/H₂O ratios (a) and SO₂/CO₂ ratios (b). The data sources of parts (a) and (b) are Figs. 4a and 4b, respectively. Dashed lines in red, orange, and blue represent 95th percentiles, 75th percentiles, and means, respectively.

In addition to CDMA analysis, time series can be processed and evaluated using spectral analysis, by which the data in a time domain are Fourier-transformed into those in a frequency domain (Spongberg, 2000). This statistical tool is also available at a website free of charge (Wessa, 2017). The resultant diagram obtained by this analysis is called a periodogram, in which the signals at low frequencies correspond to long-term changes whereas those at high frequencies reflect short-lived fluctuations. In Figs. 4c-4d and 5c-5d, three components as well as the standardized data are subjected to spectral analysis. It is obvious that the signals at low frequencies match the trend components and those at intermediate to high frequencies match the periodic and/or random components. In Figs. 4c-4d and 5c-5d, the signals at low frequencies are far stronger than those at high frequencies, indicating the dominance of the trend components. Also, the periodic components are shown to be weaker compared to the other components. Yet, the situations may arise that the signals at high frequencies are as strong as or stronger than those at low frequencies. Even in such cases, it is possible to selectively extract the trend components by back-transforming the signals at low frequencies in a periodogram. Although not further discussed, such back-transformation is commonly exercised to filter out noises (i.e., high frequency signals) from time series (Spongberg, 2000).