**2. Related Works**

Data fusion means the integration of data and knowledge from different sources. It is a widely studied topic (cf. the overview papers [6,7] and the references there). Data fusion approaches may be classified in various ways—for example, data association, state estimation, and decision fusion. Data fusion has been studied in the context of temporal series as well. In paper [8], the authors study the fusion of long-term data in the form of dense time series from the Moderate Resolution Imaging Spectroradiometer (MODIS) and Landsat imagery. They investigate a spatiotemporal adaptive fusion algorithm in a regionalization study in which MODIS was used. They show the correlation of the time series achieved by different observation methods. In paper [8], the authors propose a data fusion method for producing high spatiotemporal resolution values for the normalized difference vegetation index in the case of time series.

We approach this topic from the perspective of temporal logic, rather than from the practical angle, as in the papers mentioned above and various others. Temporal logic (TL) describes behavior over time (cf., e.g., [2]). Their languages offer temporal modalities for specifying past or future events, as well as their temporal relations, such as: event A will happen or event A may happen in the future, property A has to be always present, property A holds until property B will be satisfied, and so on. They are also used to define and synthesize system controllers; the controllers are then guaranteed to monitor and control the underlying systems according to the specific requirements. On the other hand, they are used to ensure that conclusions drawn based on the assumptions made are correct, that the implementation satisfies the specifications, and so on.

There are different types of temporal logics (see [2] for the overview). The linear temporal logic describes behaviors by referring to linear sequences of events. The computation tree logic (CTL) describes the possibilities and, more precisely, the branching time structures. The interval temporal logic (ITL) (cf., e.g., [2,3]) describes behaviors relative to time intervals, stating, for example, that a property is valid during the entire time period or that it occurs within a certain subperiod. It is also possible to express the fact that a property holds within a given time period, which is then followed by another time period over which another property holds.

Duration calculus (DC) is a popular and widely studied kind of interval logic (cf., e.g., [4]). Duration calculus may be considered to be a form of ITL. In addition to the above-mentioned potential of ITL, DC contains an integral operator that allows one to express durations of properties in a quantitative manner.

In [9], automata-based semantics of DC have been defined covering data, real-time, and communication-related aspects. A model-checking algorithm has been presented for a subset of DC that may be model-checked. The algorithm has been implemented and its use demonstrated.

DC and ITL are expressive but not decidable, i.e., no procedure exists for figuring out whether a given formula is always valid or not. Thus, as usual, expensivity is at the cost of complexity—in this case, decidability. A restricted form of DC, known as RDC, has decidable inference relation [10]. In fact, formulas of this type may be reduced to the regular expressions. The problem of DC model checking is a topic of ongoing research (cf. e.g., [11] and the references there). In [12], the authors propose a method for solving minimal and maximal reachability problems for the multipriced timed automata. The automata are an extension of timed automata with multiple cost variables that may evolve according to specified rates.

The range of applications varies from the above-mentioned real-time systems (cf. e.g., [13] and the references there) to air traffic control (cf. e.g., [14]) and hybrid systems [15].

Duration calculus is often used for the synthesis of real-time system controllers (see, for example, [16]) and the references there). It is capable of specifying time constraints of dynamic systems. Various methods are used—for example, integer linear problem-solving and optimization problem-solving methods (cf. [16]). DC is used also for traffic system specification (cf., e.g., [13]). There is also a wide range of tools supporting DC (see, for example, [10] and the references there).

In [17], a variant of duration calculus was presented for system discounting, i.e., the idea that something happening earlier is more important than similar events happening later. The idea was introduced earlier into temporal logics such as LTL and CTL. The authors demonstrated decidability of the model-checking for timed automata and a dedicated fragment of discounted duration calculus.

We illustrate our ideas with data concerning volcano monitoring. In fact, volcano monitoring is one of the prime examples for the application of multisensor systems placed in satellites, aeroplanes, balloons, and on the ground. They provide huge amounts of data that have to be preprocessed, analyzed, evaluated, and reasoned about. Thus, it is a proper area to test ideas such as ours. Of particular interest are the Strombolian effects, i.e., periodic volcanic activity phenomena such as temperature picks, gas bursts, and lava eruptions (cf., e.g., [18–20]).

In [1], Koeppen et al. propose a fully automated interactive algorithm called MOD-VOLC for the analysis of thermal satellite time-series data. The algorithm is aimed at detecting and quantifying the excess energy radiated from the thermal anomalies such as active volcanoes. The algorithm enhances the previously developed approaches (see the references in [1]). It is characterized by the law rate of false positives (see [1,21]). It flags thermal anomalies—in particular, volcanic eruptions. The algorithm was tested for different localizations such as the Anatahan volcano, the K¯ılauea volcano, and the Cantarell oil field in the Gulf of Mexico.

In [22], Laiolo et al. compared thermal satellite images of the Etna eruption that took place in 2018 with ground-based geophysical data of summit craters. The Moderate Resolution Imaging Spectroradiometer (MODIS) provided infrared images and helped to identify pixels including the possible hot spots.

Temporal series corresponding to Strombolian effects may be compared using similarity measures. In fact, various similarity measures are used to compare temporal series and functions in general (cf., e.g., [23] and the references there). One can use the dynamic time warping algorithm (cf., e.g., [24] and the references there). This algorithm is used in temporal series analysis for measuring similarity between temporal series that may vary in terms of speed. Algorithms of this type were applied to compare graphical data representing temporal series, but also in comparisons of audio and video materials (cf., e.g., [25,26]).

Dyea et al. [19] proposed a method of detecting Strombolian eruptions based on training a convolutional neural network. The method automatically categorizes eruptions based on infrared images taken at the rim of a crater atop Mount Erebus. The authors show that machine learning may be effectively used to classify the characteristics of Strombolian eruptions, to facilitate the process of studying their origins, and to assess the hazards posed by volcanic eruptions.
