**1. Introduction**

Remote sensing allows one to acquire information from a distance from cameras, sensors, microphones, and other external devices. The data may originate from satellites, aeroplanes, and sonar systems, among other sources. Satellite-based instruments are commonly used to monitor various parameters of the Earth's surface, such as temperatures in various infrared frequencies, and to take images. Global coverage is offered with frequency being as low as once per day. Stationary satellites are even capable of providing continuous monitoring of specific locations. The main problems associated with satellite data include their heterogeneity and large volumes. The data need to be analyzed on a daily basis, sometimes in real time, if timely response is needed.

In general, multisensor data are voluminous, heterogeneous, and sometimes incomplete. Prior to being analyzed, they need to be processed and represented in a specific form. They may be represented as discrete data by temporal series, i.e., by sequences of data usually taken in equal time intervals, or as continuous time-dependent functions and stochastic processes. Numerous algorithms have been developed, for example, concerning volcano monitoring (cf., e.g., [1]).

In the literature, heterogeneous means are used, such as means diagrams, functions, tables, and textual descriptions, to specify those heterogeneous types of phenomena. Dependencies between factors of different kind, such as similarities, regularities, and

**Citation:** Kosiuczenko, P. An Interval Temporal Logic for Time Series Specification and Data Integration. *Remote Sens.* **2021**, *13*, 2236. https://doi.org/10.3390/ rs13122236

Academic Editors: Stefano Mattoccia, Piotr Kaniewski and Mateusz Pasternak

Received: 15 April 2021 Accepted: 2 June 2021 Published: 8 June 2021

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**Copyright:** © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

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periodicity, are quite often described with the use of text only, meaning that various problems stemming from such an approach, such as imprecision and the lack of a unifying framework for specification, validation, and reasoning need to be dealt with. Textual specifications are inherently imprecise, as this is the feature of natural languages. They do not rely on formal semantics in the form of comprehensive mathematical models. Consequently, they do not allow for precise evaluation or a formal reasoning, and thus the reasoning lacks due formality.

Logics provide uniform formal languages, models, and reasoning methods, and thus provide a solution enabling to address above mentioned problems. Logics are associated with well-defined classes of models constituting their semantics. They provide precise formal languages for the description of the models and capabilities for correct reasoning abut properties of the models. In general, a logic is a system consisting of a formal language for specification, a class of models corresponding to the language, and a set of sound rules for a correct reasoning about the about the models.

Temporal logics are used to express change over time, properties of behaviors, and sequences of actions. Their languages provide temporal modalities for specifying future or past events. They are used also to define and synthesize system controllers. On the other hand, they provide rules for correct reasoning about temporal properties. Various kinds of temporal logics exist (see [2] for an overview).

Interval temporal logics (cf., e.g., [2,3]) are used for specifying time-dependent processes relative to time intervals. Duration calculus (DC) is an interval logic that is widely used for the specifying, modeling, and reasoning about discrete and continuous processes. It allows to specify propositional functions with Boolean values changing over time. There is an operator corresponding to the integral (cf., e.g., [4]) that measures how long such a propositional function remains true. It may be used to study periodicity of system states.

In this paper, we present duration calculus for functions (DC4F), a natural extension of the duration calculus for dealing with general integrable functions, not only Boolean-valued ones, as is the case of DC. The idea is simple; we take the integral operator on Riemann integrable functions and use it within a frame of an interval logic such as DC. Thus, the integral operator is used within a well-suited logical framework. DC4F, in addition to the expressive capabilities of DC, allows one to characterize the behavior of functions over time intervals in terms of their integrals. Consequently, we can characterize not only the duration of a certain property, as in the case of DC, but characterize the behavior of functions over time intervals in terms of integrals. The proposed extension is conservative in the logical sense: the DC part is unchanged meaning that its valid formulas remain valid and all its invalid formulas remain invalid. Even though the extension of DC is natural, we are, to our best knowledge, the first ones who propose it.

To evaluate the proposed logic, we investigated various phenomena, multisensor data, and facts concerning volcano monitoring, as this is a popular research topic and the degree of complexity of data is significant. Periodic degassing and temperature increases are common characteristic of active volcanoes. Distinct periodicity patterns concerning measurable parameters of volcanoes' activity have been widely identified. The timescales are ranging from seconds to weeks and months. The development of temperature and gas measurement techniques is aimed at enabling a robust quantification of high-frequency processes. Paper [5] presents an overview of the current state of knowledge regarding periodic volcanic degassing and evaluates the methods aiming at detecting periodicity. It summarizes and statistically analyzes published studies. Periodicity analysis of volcano activity (cf., e.g., [5]) is one of the challenges.

It turns out that such phenomena and their dependencies may be conveniently specified using DC4F, as it provides a expressive and uniform language for expressing various phenomena in a precise way, models for various data, and evaluation and reasoning capabilities. It provides a convenient specification language to express hypothesis concerning expected temporal properties. If data (in particular, multisensor data) are provided, then DC4F formulas may be validated for them and, thus, their truthfulness may be checked.

On the other hand, the reasoning rules provide for convenient reasoning possibilities. Thus, DC4F may be perceived as a unifying logical framework for data integration and for formulating hypotheses, their evaluation, and reasoning.

However, high expressivity of languages always comes at the cost of complexity of reasoning rules. The higher the expressivity, the more difficult the reasoning. More precisely, the question if a formula follows from a set of other formulas has high computational complexity. DC4F, like DC, is an expressive language and is, therefore, undecidable, i.e., there is no general algorithm for deciding the question mentioned above. Nonetheless, this does not hinder its use for specification, nor does it hinder the validation of formulas for concrete data.

The paper is organized as follows. In Section 2, we discuss related works. Section 3 contains a brief presentation of duration calculus and the way it is used. In Section 4, we present the main idea of DC4F and some examples of its application. In Section 5, we define its formal syntax and outline its semantics in an informal and exemplary way; we also list some of its properties and show how it applies to the multidimensional case. Section 6 is devoted to applications of DC4F in the area of volcano monitoring, in particular to the validation of the proposed concepts. In particular, we present an exemplary reasoning in DC4F. We conclude the paper with Section 7. The paper also contains an Appendix in which we define the formal semantics of DC4F.
