*6.1. Volcano Monitoring*

To demonstrate the applicability of DC4F, we looked at several examples of signal processing and the corresponding temporal series. It turned out that volcano monitoring is an interesting and challenging task due to the complexity of the process, various monitoring methods used, and the heterogeneity of the acquired data. In this subsection, we present a brief presentation of volcano monitoring processes, as they are instrumental in illustrating our ideas.

As pointed out by Corradini et al. in [18,28], volcano monitoring processes require the correlation of satellite data, provided that satellites are present over the specific target. Satellite data are heterogeneous in nature and relate to temperatures measured in different wavelengths, among other measurements. Several phenomena, such as temperature picks, gas explosions, and magma eruptions, need to be considered (cf., e.g., [19,20]).

Strombolian volcano eruptions are a widely studied topic. They are characterized by specific cyclical patterns. Therefore, their specification is a challenging task. A Strombolian volcano eruption is characterized by regular, relatively mild blasts. It consists of

the ejection of fragments of solidified lava, material deposited by previous volcanic eruptions, and masses of molten rock to altitudes ranging from tens to hundreds of meters (cf., e.g., [18–20]). Eruptions of this type are named after the Stromboli volcano. They are commonly observed in volcanoes fed by low to moderate viscosity magma. The explosions are caused by the bursting of gas slugs, which rise to the surface faster than the surrounding magma. Quakes associated with such explosions occur at low depths (cf. e.g., [20]). Spectroscopic measurements performed on the Stromboli volcano were used to demonstrate that gas slugs originate from the volcano–crust interface at the depth of approximately 3 kilometers, which may promote slug coalescence. In the case of the Stromboli volcano, quantitative constraints for the depth of conglomerates of high-pressure gas bubbles inducing Strombolian activity can be identified on the basis of spectroscopic measurements of the magmatic gas phase driving the explosions.

#### *6.2. Dealing with Estimates*

Various parameters are considered when a volcano eruption is monitored. These parameters need to be measured and compared. In this subsection, we show that functional limits may be dealt with in CD4F.

Thermal satellite images of the Etna eruption that took place in 2018 were compared with ground-based geophysical data of summit craters using the Moderate Resolution Imaging Spectroradiometer (MODIS) (see [22]). The technology provides infrared images and makes the identification of hot spots including pixels possible. The thermal anomalies may be quantified in terms of volcanic radiative power (VRP), expressed in watts (see [22] and the references there). The VRP value is calculated as follows:

$$VRP\_{pIX} = Ap\_{IX} \times 18.7 \times (L\_{4alert} - L\_{4bk}) \tag{9}$$

where *ApIX* is the pixel area of approximately 10<sup>6</sup> m2, *L*4*alert* is the recorded the middle infrared radiance (W m<sup>−</sup>2sr−1m−1), and *L*4*bk* are the background pixels. These values allow to estimate the average lava erupted volume, i.e., the time average discharge rate (TADR), through a unique parameter called radiant density *crad* (J m3):

$$TADR = \frac{VRP}{c\_{rad}}\tag{10}$$

This parameter represents capacity of the lava body to radiate heat (see [22] and the references there). It allows to estimate the erupted lava volume by assuming the appropriate parameter *crad* for the lava flow in question.

The fact that the actual lava flow volume *TADR* is between limits *LApp* and *UpApp* (see Figure 5) can be specified by the following formula, which is analogous to Formula (5) from Section 4.3. This fact cannot be expressed in DC, since DC only allows to consider the duration properties on intervals but not the values of integrals:

$$
\Box \int L App \not\sim \int TADR \not\sim \int \mathcal{U}p App \tag{11}
$$

The inequalities express the desired property in terms of intervals. The formula means that the inequalities hold for every subinterval of the given interval (see Section 3.2).

**Figure 5.** TADR with its lower and upper approximation.

#### *6.3. Dealing with Two-Dimensional Spaces*

There exist algorithms for detecting thermal anomalies in two-dimensional pictures related to volcano eruptions. In this subsection, we discuss the application of DC4F to such cases. We specify the units of measurement, but it should be noted that for the presentation, the units of measurement and exact numeric values obtained do not play a significant role in the presentation data, since in our framework we can deal with values of type R independently of their interpretation.

MODVOLC is a fully automated algorithm for the analysis of thermal satellite timeseries data [1]. The algorithm is based on spectral wavelengths centred at 4 and 11–12 μm emitted by high-temperature volcanic sources and the surrounding Earth surface. In the case of a volcanic eruption, the radiance is measured from the corresponding pixel. One pixel corresponds to approximately 4 km. Images are collected every 15 min. In pixels that show a thermal anomaly, the radiance measured in the mid-infrared (4 μm) is significantly higher than in surrounding pixels where no thermal anomalies are depicted [1,21].

Figure 6 shows an excerpt of Figure 6 from [1]. It contains results obtained by the MODVOLC algorithm detecting thermal anomalies for different thresholds 2.0, 2.6, and 3.0, respectively, in case of the Anatahan volcano monitoring data. The following formula presents the Local Index of Change of the Environment <sup>⊗</sup>*v*(*<sup>x</sup>*, *y*,*<sup>t</sup>*) representing the degree to which each pixel deviates from its normal behavior normalized by its variability (cf. [1,29]).

$$\log\_v(\mathbf{x}, y, t) = \frac{V(\mathbf{x}, y, t) - V\_{REF}(\mathbf{x}, y)}{\sigma\_v(\mathbf{x}, y)}\tag{12}$$

An individual observation is modeled by the function *V*(*<sup>x</sup>*, *y*,*<sup>t</sup>*) specifying the state of a pixel (*<sup>x</sup>*, *y*) at time *t*. Its normal behavior is expressed by function *VREF*(*<sup>x</sup>*, *y*) and its variability by *<sup>σ</sup>v*(*<sup>x</sup>*, *y*). One can then identify pixels (*<sup>x</sup>*, *y*) for which the function exceeds a given threshold *Tr* ⩽ <sup>⊗</sup>*v*(*<sup>x</sup>*, *y*,*<sup>t</sup>*). The number of such pixels can be then computed for the entire area of interest *A* using its characteristic function 1*A*, i.e., (*<sup>x</sup>*, *y*) ∈ *A* ⇔ <sup>1</sup>*A*(*<sup>x</sup>*, *y*) = 1 in the discrete case when single pixels are considered and each pixel has measure 1. A thermal anomaly is a pixel with a temperature measurement value above a defined threshold. The total number of thermal anomalies occurring at time *t* with values above threshold *Tr* is modeled by function *anomalies*(*Tr*,*<sup>t</sup>*) defined by the following integral:

$$
tau \text{momalies}(Tr, t) = \int\_A \mathbf{1}\_{\{\{\mathbf{x}, y\} \mid Tr \leqslant v\_{\mathbf{c}}\{\mathbf{x}, y, t\}\}} \, d\mathbf{x} dy \\
= \int\_{\mathbb{R} \times \mathbb{R}} \mathbf{1}\_A \, \mathbf{1}\_{\{\{\mathbf{x}, y\} \mid Tr \leqslant v\_{\mathbf{c}}\{\mathbf{x}, y, t\}\}} \, d\mathbf{x} dy \tag{13}$$

Thus, the number of anomalies is the integral of the product of the characteristic function of set *A*, corresponding to the area of interest, and of the characteristic function identifying points with temperature above threshold *Tr*. In the continuous case, the value of the integral is equal to the measure of the set of thermal anomalies in the area of interest.

**Figure 6.** Number of thermal anomalies detected by the hybrid algorithm over MODVOLC using thresholds of 2.0, 2.6, and 3.0, respectively.

#### *6.4. Specification of Strombolian Effects*

Strombolian effects are characterized by cyclic periods of repetitive behaviors. Repetitive gas explosions and temperature peaks occurring in a periodic manner are one of

such aspects. These explosions are caused by deep slag-driven explosive activity (cf., e.g., [20]). Those effects include also consecutive magma eruptions. In this subsection, we demonstrate that these kinds of effects can be specified in DC4F. We utilize the temporal properties formulated in the preceding section. The data that we use were obtained on 9 April 2002 during hours of passive and explosive degassing on the Stromboli volcano that was most active at the time [20]. The spectrometer was placed at a distance of 240 m.

**Figure 7.** Periodic peaks in radiation corresponding to source temperature.

Repetitive temperature increases are shown in Figure 7. It shows the diagram presented in [20] (see Figure 1 and Figure S1 there) representing peaks in radiation corresponding to source temperature in Celsius. Figure 7 shows a function *g*(*t*) with cyclic periods of consecutive value peaks and drops representing the level of radiation. The peaks can reach different maximal values, the drops can reach different minimal values, and their numbers are not fixed in a cycle. The time scale of Strombolian effects is usually hours and days (cf., e.g., [18,20,28]). In Figure 7 and below, we use only numeric time values, as the exact timescale does not play any role in the demonstration and may be changed in an arbitrary manner.

Each peak formation is characterized by a monotone increase of *g* followed by its monotone decrease. In consequence, a peak can be specified in DC4F by formula *MIncreasing*(*g*); *MDecreasing*(*g*) (see Section 4.2). For example, Figure 7 shows a peak corresponding to interval [*<sup>a</sup>*1, *<sup>a</sup>*2].

We can also require that the peak values be above a certain threshold *tr* and the amplitude be above the value of *d*. These requirements can be formulated as *PeakAbove*(*g*,*tr*) and *Amplitude*(*g*, *d*), respectively. The first formula states that function *g* reaches a value above threshold *tr*, and the second one states that its amplitude is at least equal to *d* (see Section 4.3). In Figure 7, the first peak is above 400 and the amplitude is above 100; of course, these parameters can be adjusted arbitrarily. The fact of reaching such a peak with the constraints concerning values of threshold and amplitude can be expressed by the formula *MIncreasing*(*g*)⌢ *MDecreasing*(*g*) ∧ *PeakAbove*(*g*,*tr*) ∧ *Amplitude*(*g*, *d*). This formula is the logical conjunction of the previous formulas; we abbreviate it as *Peak*(*g*,*tr*, *d*). We use capital letters to indicate that functions have functional arguments, not only arguments of type R.

Now, the peaks and the corresponding drops form a repetitive behavior. The left-hand side of Figure 7 shows three peaks, modeling COS bursts, which correspond to the intervals [*ai*, *ai*+<sup>1</sup>], for *i* = 1, ... , 3. We can write the corresponding formula as *Peak*(*g*,*tr*, *d*)⌢ *Peak*(*g*,*tr*, *d*)⌢ *Peak*(*g*,*tr*, *d*). It means that the interval may be split into three subintervals such that each satisfies the formula *Peak*(*g*,*tr*, *d*). We abbreviate the formula as *Peak*(*g*,*tr*, *d*)3.

In general, for a formula *F*, *n* repetitions of the form *F* ⌢ ... ⌢ *F* are abbreviated as *Fn*. This formula means that the interval may be split into *n* subintervals in which *F* is satisfied. If the number of repetitions is arbitrary, then we write *F*<sup>∗</sup>. If it equals at least *n*, then we write *Fn*<sup>+</sup>, which is equivalent to *Fn* ⌢ *F*<sup>∗</sup>.

The peaks occur in cyclic series separated by periods of low volcanic activity. Let formula *Below*(*g*, *ltr*, *lb*, *ub*) be the abbreviation of (◻∫ *g* ⩽ *ltr*) ∧ *lb* ⩽ - ⩽ *ub*. This means that *g* is always below threshold *ltr* in a given time interval and that the length of the interval is between time bounds *lb* and *ub* (cf. Section 4.3). We can compose the formulas again to specify the behavior shown in Figure 7; for *m* = 3 and *n* = 3, the formula has the form:

$$\{Peak(\lg, tr, d)^{m+} \stackrel{\sim}{\smile} Below(\lg, ltr, lb, ub)\}^n\tag{14}$$

This Strombolian activity consists of three cycles; each cycle is composed of at least three bursts followed by a period of a low volcanic activity. The three cycles of repetitive bursts are indicated in Figure 7 by brackets below the time axis. The periods of limited volcanic activity are indicated in a similar way.

Strombolian activities may be also detected using satellites equipped with thermal imaging cameras, which sometimes are the source of more reliable indicators than sensors on the ground (cf., e.g., [18,28]). However, if the satellites are not stationary or if visibility is reduced by clouds, then data from thermal imaging cameras cannot be used. We can specify the conditional visibility of thermal peaks by formula ∫ *visibility* = - ⇒ *Peak*( *f* ,*tr*′ , *d*′ ). In this formula, *visibility* is a Boolean-valued time-dependent function modeling visibility; it is true if there is visibility and false otherwise. Function *f* models temperature measurements at a certain frequency. Parameter *tr*′ is the corresponding temperature threshold and parameter *d*′ is the corresponding amplitude. The formula is conditional and reads as follows: if there is visibility, then a thermal peak is observed. We abbreviate this formula by *CondPeak*( *f* ,*tr*′ , *d*′ ). Analogously, we can define *CondBelow*(*b*, *ltr*′ , *lb*′ , *ub*′ ) for the case in which the thermal radiance measurement is below a certain threshold. If the peaks in thermal anomalies coincide with the peaks in COS bursts, then we can transform formula (14), taking the thermal anomalies into account as well:

$$\begin{aligned} \{ \text{\textquotedblleft Comlet} (\text{g}, \text{tr}, \text{d})^{\text{m}+} \} & \quad \{ \text{\textquotedblleft Conl-lebilew} (\text{g}, \text{ltr}, \text{lb}, \text{ub}) \} \}^{n} \land \\ & \quad \{ \text{\textquotedblleft Conl-lebilex} (\text{f}, \text{tr}', \text{d}')^{\text{m}+} \} \}^{n} \land \end{aligned} \tag{15}$$

The sometime-modality is used here to specify that we do not require anything for the time periods before and after the characteristic behavior. We may also specify here the minimum and maximum lengths of the periods of low activity, if needed.

Of course, the formulas become more and more complicated as we specify more kinds of behaviors and data, e.g., multisensor data, and constraints. However, the point is that the formulas may be specified in DC4F. Thus, we may precisely formulate fine hypotheses concerning complex behaviors. Furthermore, if implemented, the hypotheses may be verified in respect to existing data. They may also be used to automatically mine the available data, historic or incoming.

#### *6.5. Reasoning in DC4F*

In the previous subsections, we presented the capabilities of DC4F for the specification of periodic behaviors and the hypothesis framing. In this subsection, we present a relatively simple example of reasoning in DC4F utilizing the properties of DC4F defined in Section 5.3. The example is somewhat artificial, as we did not find proper examples in the literature, and we do not claim that the formulas truly express actual volcanic behavior. We simplify the proof, as its full version would be rather long. As in the previous section, we skip units of measurement as they do not play any role for the illustration.

Suppose that we know that a Strombolian effect is characterized by two properties occurring within 24 h: a series of COS bursts and an observation of thermal anomalies. The data is on COS burst is collected by sensors on the ground and the data on thermal anomalies are collected by a sensor placed on a satellite. The behavior is specified by functions *fCOS*(*t*) and *fanom* representing the emission level of COS and the number of observed anomalies, respectively. We use abbreviation *COSB* for the formula obtained from formula (14) by substituting *fCOS* for *g*, 1 for *n*, and by setting other parameters somehow. We assume that *fanom*(*t*) = *anomalies*(*<sup>t</sup>*, *Tr*) for a certain threshold *tr* (cf. Section 6.3).

Let the Strombolian effect be characterized by at least 10 COS bursts and thermal anomalies of value 5 occurring within 24 h. Moreover, let it be an established fact that 12 COS bursts are always accompanied by thermal anomalies of value 7. The following formulas specify these two properties:

$$\ell \lhd \mathbf{24} \land \diamondsuit \text{ COSB}^{10} \land \dots \nwarrow \int f\_{\text{anom}} \tag{16}$$

$$2.6 \times 24 \land \diamondsuit \text{COSB}^{12} \implies 7 \prec \int f\_{\text{anom}} \tag{17}$$

Now, let the observation made by sensor on the ground be that within 24 h, there were 14 COS bursts, as specified by the following formula:

$$\ell \lhd \text{24} \land \diamondsuit \text{COSB}^{14} \tag{18}$$

Suppose that the data on thermal anomalies are not available due to bad weather and, thus, the integral ∫ *fanom* cannot be computed. We will show that, despite that obstacle, the Strombolian behavior can be proved.

The formula ◊ *COSB*<sup>14</sup> can be presented in the form *true*⌢ *COSB*<sup>12</sup> ⌢ *COSB*<sup>2</sup> ⌢ *true* (we apply here the definition of ◊ and unfold the exponent). The formulas *COSB*<sup>2</sup> ⇒ *true* and *true*⌢ *true* ⇒ *true* are tautologies of the predicate calculus (see point (6) in Section 5.3). Due to these two tautologies and the monotonicity of the chop operator (see point (7) in Section 5.3), we deduce the implication

$$\text{true} \stackrel{\frown}{\phantom{\frown}} \text{COSB}^{12} \stackrel{\frown}{\phantom{\frown}} \text{COSB}^{2} \stackrel{\frown}{\phantom{\frown}} \text{true} \stackrel{\frown}{\phantom{\frown}} \text{COSB}^{12} \stackrel{\frown}{\phantom{\frown}} \text{true}$$

Further, applying the monotonicity property of conjunction (see point (6) two times in Section 5.3) to the above implications, we derive the following implications:

$$\ell \lhd 24 \land \lozenge \text{COSB}^{14} \Rightarrow \ell \lhd 24 \land \lozenge \text{COSB}^{12}, \; \ell \lhd 24 \land \lozenge \text{COSB}^{12} \Rightarrow 7 \not\subset \int f\_{2\text{non-}1}$$

Moreover, 7 ⩽ ∫ *fanom* ⇒ 5 ⩽ ∫ *fanom* and ◊*COSB*<sup>12</sup> ⇒ ◊*COSB*10. From the above implications, we derived the following implication:

$$\ell \lhd 24 \land \Diamond \text{COSB}^{14} \Rightarrow \text{ } L\mathfrak{c} \lhd 24 \land \Diamond \text{COSB}^{10} \land \dots \lhd \int f\_{20000}$$

Antecedent of this implication is the observation (18). The consequent of this implication is the formula to be proved. The consequent follows from the antecedent and the implication by the application of modus ponens rule (see point (8) in Section 5.3). Thus, observation (18) and fact (17) imply that the Strombolian property holds.

#### *6.6. Detecting Similar Behaviors*

In this subsection, we consider the methods used in different areas, volcano monitoring included, for detecting similarities between functions and, in particular, between temporal series. Various similarity measures can be used in this case (cf., e.g., [23–26]). Different types of volcanic activity may be characterized not only in absolute terms, i.e., by general characteristics satisfied by all behaviors, but also relative to one another, or in relation to known examples of behaviors. We consider neither the application of specific similarity measures to specific cases, nor their adequacy, as we are not aiming to identify similarities in volcanic behavior. We are not in the position to judge which similarity measures would be most suitable for the Strombolian eruption patterns. We merely intend to stress that such measures and the corresponding algorithms can be used within the framework of DC4F.

Similarity measures may be formalized by functionals **FlSym**, which are interpreted as real-valued functions defined on intervals (see Section 5.1 and also the Appendix A). Similarity measures *m* can be seen as functionals that, for a given interval *I* and two functions *f* and *g* to be compared, return a real value being the measure of similarity. Thus, we can consider the application of a measure functional *m* to the two measured functions *f* and *g* and an interval of interest *I*: *m*( *f* , *g*, *<sup>I</sup>*). Consequently, the result can be presented as a real-valued function on intervals *Xf* ,*g* ∈ **FlSym**, which returns a real value *r* for the interval *I* (see Section 5.1 and also Appendix A). The measured level of similarity can be then used in framing descriptions and hypotheses concerning the time series—in particular, concerning volcanic eruptions.

Similarly, the standard Pearson correlation coefficient used in statistics may be integrated into CD4F as well. The correlation coefficient *ρf* ,*g* of two functions *f* , *g* is specified by integral ∫( *f* − ∫ *f* )(*g* − ∫ *g*)
( ∫ ( *f* − ∫ *f* )2 ∫ (*g* − ∫ *<sup>g</sup>*)<sup>2</sup>). The integral returns for an interval *I* a value of type real, provided that the denominator is different from zero. Thus, coefficient *ρf* ,*g* depends on the interval in question, and as was the case in the example presented above, it may be represented as a real-valued function on intervals.

In general, some of the similarity functions can be defined directly in DC4F. Others, such as algorithms computing the similarity measures, can be defined simply by the interval mapping operators corresponding to functionals **FlSym**. Thus, if there is an algorithm computing a similarity measure for intervals, then we can associate interval operator *X* ∈ **FlSym** with the algorithm: *<sup>X</sup>*([*<sup>a</sup>*, *b*]) = *x* if *x* is the value computed by the algorithm for interval [*a*, *b*]. The algorithm proposed in [19] can be treated along these lines.
