This review provides a summary of methods originated in (non-equilibrium) statistical mechanics and information theory, which have recently found successful applications to quantitatively studying complexity in various components of the complex system Earth. Specifically, we discuss two classes of methods: (i) entropies of different kinds (e.g., on the one hand classical Shannon and R´enyi entropies, as well as non-extensive Tsallis entropy based on symbolic dynamics techniques and, on the other hand, approximate entropy, sample entropy and fuzzy entropy); and (ii) measures of statistical interdependence and causality (e.g., mutual information and generalizations thereof, transfer entropy, momentary information transfer). We review a number of applications and case studies utilizing the above-mentioned methodological approaches for studying contemporary problems in some exemplary fields of the Earth sciences, highlighting the potentials of different techniques.

Fractal statistical analysis under the critical point (CP) hypothesis is applied to electromagnetic (EM) signals emitted before failure. A new approach to the analysis of a possible EM fractal pattern evolution toward CP is suggested. The analysis reveals characteristic signs of approaching the CP: the emergence of memory effects; the increase of the spatial correlation; the decrease of the antipersistence behavior; the appearance of persistence properties in the tail of the precursors, a loss of multifractality, and, finally, the divergence of the energy release rate. These critical features are compatible with the percolation theory of fracture process.

Electromagnetic anomalies (EMA) covering a wide range of frequencies from ULF, VLF up to VHF have been observed before recent destructive earthquakes in continental Greece. We show that the features of these signals are possibly correlated with the fault model characteristics of the associated earthquake and with the degree of geotectonic heterogeneity within the focal zone. The time evolution of these electromagnetic sequences reveals striking similarities to that observed in laboratory acoustic and electromagnetic emissions during different stages of failure preparation process in rocks. If we consider that the same dynamics governs the large-scale earthquakes and the microscopic scale sample rheological structure, the results of this analysis suggest that the recorded EMA might reflect the nucleation phase of the associated impending earthquake. We focus on the rise of the statistical view of earthquakes. We find electro-magnetic fingerprints of an underlying critical mechanism. Finally, we conclude that it is useful to combine ULF and VLF-VHF field measurements in an attempt to enhance the understanding of the physics behind these observations and thus to improve the quality of earthquake prediction. Further, the identification of an EMA as a seismogenic one supports the characterization of a sequence of shocks as foreshocks at the time they occur, further helping the earthquake prediction effort.

There is a recent thesis in the literature that an important organization of a physical system precedes a catastrophic event. In this context, one can search for signatures that imply the transition from a normal state to a main catastrophic event (e.g., earthquake). Experimental techniques are thus useful in corroborating theories from observed data. For example, recent results indicate that preseismic electromagnetic time series contain information characteristic of an ensuing earthquake event. Hereby, we attempt to demonstrate that an easily computable complexity measure, such as $T$-complexity or approximate entropy, gives evidence of state changes leading to the point of global instability. The appearance of a precatastrophic state is characterized by significant lower complexity in terms of $T$-complexity and approximate entropy. The present study confirms the conclusions of previous works based on an independent linear fractal spectral analysis. This convergence between nonlinear and linear analysis provides a more reliable detection concerning the emergence of the last phase of the earthquake preparation process. More precisely, we claim that our results suggest an important principle: significant complexity decrease and accession of persistency in electromagnetic (EM) time series can be confirmed at the tail of the preseismic EM emission, which could be used as diagnostic tools for the Earth's impending crust failure. Direct laboratory and field experimental data as well as theoretical arguments support the conclusions of the present analysis.

We suggest that the activation of a single fault by means of preseismic electromagnetic emissions (PEME) is well-described by recently introduced models for earthquake (EQ) dynamics, which have been rooted in a nonextensive framework starting from first principles. The analysis implies that the activation of a single fault is (i) a reduced self-affine image of the regional seismicity covering many geological faults, and (ii) a magnified image of the laboratory seismicity by means of acoustic and electromagnetic emissions. Finally, we study whether characteristic signatures emerged in PEME indicating the transition to the last phase of the EQ preparation process. We use the $q$-Tsallis entropy as a measure of organization. Tsallis entropy gives evidence of state changes leading to the point of global instability: it detects the pattern of alterations in the preseismic electromagnetic signals and is able to discriminate between “injury levels” of the focal area. Importantly, a significant organization increase can be confirmed at the tail of the recorded PEME by means of Tsallis entropy, which is also accompanied by the appearance of persistency. We argue that these footprints may indicate the fracture of the backbone of strong entities that hinders the relative motion of the fault planes.

Ultra low frequency, kHz and MHz electromagnetic (EM) anomalies were recorded prior to the L'Aquila catastrophic earthquake that occurred on 6 April 2009. The main aims of this paper are threefold: (i) suggest a procedure for the designation of detected EM anomalies as seismogenic ones. We do not expect to be able to provide a succinct and solid definition of a pre-seismic EM emission. Instead, we aim, through a multidisciplinary analysis, to provide the elements of a definition. (ii) Link the detected MHz and kHz EM anomalies with equivalent last stages of the earthquake preparation process. (iii) Put forward physically meaningful arguments for quantifying the time to global failure and the identification of distinguishing features beyond which the evolution towards global failure becomes irreversible. We emphasize that we try to specify not only whether a single EM anomaly is pre-seismic in itself, but also whether a combination of kHz, MHz, and ULF EM anomalies can be characterized as pre-seismic. The entire procedure unfolds in two consecutive parts. Here in Part 1 we focus on the detected kHz EM anomaly, which play a crucial role in our approach to these challenges. We try to discriminate clearly this anomaly from background noise. For this purpose, we analyze the data successively in terms of various concepts of entropy and information theory including, Shannon n-block entropy, conditional entropy, entropy of the source, Kolmogorov-Sinai entropy, T-entropy, approximate entropy, fractal spectral analysis, R/S analysis and detrended fluctuation analysis. We argue that this analysis reliably distinguishes the candidate kHz EM precursor from the noise: the launch of anomalies from the normal state is combined by a simultaneous appearance of a significantly higher level of organization, and persistency. This finding indicates that the process in which the anomalies are rooted is governed by a positive feedback mechanism. This mechanism induces a non-equilibrium process, i.e., a catastrophic event. This conclusion is supported by the fact that the two crucial signatures included in the kHz EM precursor are also hidden in other quite different, complex catastrophic events as predicted by the theory of complex systems. However, our view is that such an analysis by itself cannot establish a kHz EM anomaly as a precursor. It likely offers necessary but not sufficient criteria in order to recognize an anomaly as pre-seismic. In Part 2 we aim to provide sufficient criteria: the fracture process is characterized by fundamental universally valid scaling relationships which should be reflected in a real fracto-electromagnetic activity. Moreover, we aim to answer the following two key questions: (i) How can we link an individual EM precursor with a distinctive stage of the EQ preparation process; and (ii) How can we identify precursory symptoms in EM observations that indicate that the occurrence of the EQ is unavoidable.