A volcanic eruption is usually preceded by seismic precursors, but their interpretation and use for forecasting the eruption onset time remain a challenge. A part of the eruptive processes in open conduits of volcanoes may be similar to those encountered in geysers. Since geysers erupt more often, they are useful sites for testing new forecasting methods. We tested the application of Permutation Entropy (PE) as a robust method to assess the complexity in seismic recordings of the Strokkur geyser, Iceland. Strokkur features several minute‐long eruptive cycles, enabling us to verify in 63 recorded cycles whether PE behaves consistently from one eruption to the next one. We performed synthetic tests to understand the effect of different parameter settings in the PE calculation. Our application to Strokkur shows a distinct, repeating PE pattern consistent with previously identified phases in the eruptive cycle. We find a systematic increase in PE within the last 15 s before the eruption, indicating that an eruption will occur. We quantified the predictive power of PE, showing that PE performs better than seismic signal strength or quiescence when it comes to forecasting eruptions.

<p>The complexity of time series can be quantitatively measured using Permutation Entropy (PE). PE has recently been introduced as a potential tool in eruption forecasting by applying it to the seismic time series. Examples of successful applications are the eruptions at Strokkur Geyser, Iceland, the 1996 eruption of Gjálp, Iceland, and the eruptions of Shinmoedake volcano, Japan, in 2011, 2017, and 2018. While PE is able to show temporal changes prior to an eruption, these features are not always prominent. To improve this method, we calculate PE not only for the amplitudes of the seismic signals but also for the seismic instantaneous phases, called Phase Permutation Entropy (PPE). To understand the difference between PE and PPE, we performed synthetic tests by creating several synthetic waveforms using different numbers of sin wave superposition. We used more wave superposition with different frequencies to create complex waves containing broader frequency spectrum, while less superposition is used to create simpler waves containing narrower frequency spectrum. PE and PPE values are both low for simple waves and high for complex waves, but their absolute values differ, which might contain valuable information. The gap, dP = PE-PPE, is found to be smaller for complex waves compared to the more simple waves.  We then calculated PE and PPE for seismic data recorded from January 2014 to December 2015, which covered the eruption period of Holuhraun in Iceland. During the time of quiescence, both PE and PPE exhibit a long period variation which seems to be seasonal. Calculating dP weakens the long period noise and generates a more stable baseline.  We observe that the temporal variation of dP started to decrease below the baseline after 24 May 2014, indicating that the ground motion got more complex. An abrupt drop of dP to its lowest level was observed on 16 August 2014, when the dyke started to propagate from Bardarbunga to Holuhraun. While dP increases after the onset of eruption on 29 and 31<sup> </sup>August 2014, there is no prominent feature between the dyke propagation and the onset of the eruptions.  During the eruption period, dP stays lower than the background dP, indicating a higher ground motion complexity compared to the quiescence time. After January 2015, the gradual increase of dP back to the baseline level is clearly observed, showing the method’s potential to foresee the end of the eruption.</p>

When a volcano becomes restless, it is challenging to assess whether it will lead to an actual eruption and determine the timing of the eruption onset. A magmatic intrusion starting at depth can (a) remain at depth, (b) stall just before reaching the surface, (c) erupt in sluggish and viscous extrusion, or (d) erupt rapidly or explosively (Moran et al., 2011). The process of magma migration involves interactions with the surrounding country rock, cooling magma bodies from previous eruptions, and (or) hydrothermal system (Moran et al., 2008). These interactions generate natural phenomena such as earthquakes, deformation, temperature changes, and gas emissions. These phenomena can be observed by geophysical and geochemical measurements (Moran et al., 2008) and integrated with the history of past eruptions in a framework of eruption forecasting (Whitehead & Bebbington, 2021).From a seismic point of view, eruptions can show precursors such as accelerating or decelerating earthquake rates. To assess this, monitoring institutes conventionally use methods to tabulate daily event counts (McNutt, 1996) and calculate the average amplitude for a certain window length (Endo & Murray, 1991). The Failure Forecast Method estimates the onset time of eruption by using the rate and the acceleration of seismic precursors associated with the rock failure caused by magma propagation (Boué et al., 2015). However, this method cannot deal with complex precursory signals, for example, that exhibits fluctuations or deceleration (Boué et al., 2015). Furthermore, due to the uncertainty of the eruption forecast and numerous false alarms (Bell et al., 2013), this method is not recommended to be stand-alone (Whitehead & Bebbington, 2021). Dempsey et al. (2020) tested a real-time Machine Learning framework to detect eruption precursors of five major eruptions at Whakaari

When a volcano becomes restless, it is challenging to assess whether it will lead to an actual eruption and determine the timing of the eruption onset. A magmatic intrusion starting at depth can (a) remain at depth, (b) stall just before reaching the surface, (c) erupt in sluggish and viscous extrusion, or (d) erupt rapidly or explosively (Moran et al., 2011). The process of magma migration involves interactions with the surrounding country rock, cooling magma bodies from previous eruptions, and (or) hydrothermal system (Moran et al., 2008). These interactions generate natural phenomena such as earthquakes, deformation, temperature changes, and gas emissions. These phenomena can be observed by geophysical and geochemical measurements (Moran et al., 2008) and integrated with the history of past eruptions in a framework of eruption forecasting (Whitehead & Bebbington, 2021).From a seismic point of view, eruptions can show precursors such as accelerating or decelerating earthquake rates. To assess this, monitoring institutes conventionally use methods to tabulate daily event counts (McNutt, 1996) and calculate the average amplitude for a certain window length (Endo & Murray, 1991). The Failure Forecast Method estimates the onset time of eruption by using the rate and the acceleration of seismic precursors associated with the rock failure caused by magma propagation (Boué et al., 2015). However, this method cannot deal with complex precursory signals, for example, that exhibits fluctuations or deceleration (Boué et al., 2015). Furthermore, due to the uncertainty of the eruption forecast and numerous false alarms (Bell et al., 2013), this method is not recommended to be stand-alone (Whitehead & Bebbington, 2021). Dempsey et al. (2020) tested a real-time Machine Learning framework to detect eruption precursors of five major eruptions at Whakaari

<p>A volcanic eruption is usually preceded by increased seismic activity resulting from magma propagation. Although these precursors can be detected by a modern seismometer, it is still a challenge to answer whether they will be followed by an actual eruption and when the eruption will occur after precursors are detected. The time between the start of volcanic unrest and the actual eruption is crucial. Therefore, there is a need for an assessment tool that is applicable in real-time. Permutation Entropy (PE) has been recently suggested to be a promising tool for the prediction of volcanic eruptions. It is a robust yet simple tool to quantify the complexity of time series. We aim to find out whether there is a distinct feature in the temporal variation of PE that is useful for eruption forecasting. We performed several synthetic tests to understand how PE works and how to choose the optimum input parameters for a signal with certain properties. We then applied this knowledge to calculate PE of seismic data that recorded eruptions of Strokkur geyser, Iceland on the 10<sup>th</sup> of June 2018. 78 eruptions occurred within five hours of observation. We used this fast-repeating process to check if the eruptions cause a repetitive pattern of PE. The input parameters used for PE calculation are a window length of 1 second, an embedding dimension of 5, and a delay time of 0.067 seconds. Our results show a distinct, repeating pattern of the PE that is consistent with the phases in the eruptive cycle of Strokkur as described by Eibl et al. (2021). The PE drops in the stage of bubble accumulation at depth, then undergoes repeated increasing and decreasing patterns during regular bubble collapses at depth in the conduit, and finally continuously increases as a precursor towards the time of eruption on the surface. The average duration of this precursor to the eruption is about 10 seconds.</p>