# Earthquake Forecasting with Elementary Catastrophe Theory

Earthquakes are among the most disastrous natural phenomena. Because they are impossible to control, earthquake forecasting draws a great deal of attention from both specialists and the general community. An unquestionable advancement in the area is the implementation of earthquake early warning systems (EEWS) [3, 7]. Yet because the process of forecasting is so complex, much work remains to be done. An EEWS triangulates the epicenter from where the large tremor could feasibly come (in terms of probability) and issues a maximum advance warning of one and a half minutes, based on information that is available the day of the event.

We aim to amend the EEWS and provide a new prediction method to increase warning time by one order of magnitude. Our idea involves either obtaining certain advantageous information from historical earthquake records or accruing new data, assuming the existence of implicit regularities (which were formerly omitted). Researchers could then weigh these records with ongoing seismogram behavior to search for certain analogies and ultimately forecast earthquakes before they occur. Our latest results indicate that such information is hidden inside the seismograms’ increasing segments, which we wrongly believed were useless until now. Figure 1 depicts a “traditional” seismogram presentation. The outlines that describe the propagation of the seismic waves are certainly close to vertical, though the presence of measurable time intervals from crest to valley and vice versa are evident.

**Figure 1.**Corresponding seismograms from three perpendicular detecting sensors for the high-magnitude earthquake (with a moment magnitude of 8.1) that occurred in the Sandwich Islands on August 12, 2021. Records are courtesy of support from the University of Jaén’s Seismic Risk and Active Tectonics Research Group.

Let us focus our attention on our method for extracting the additional information via *elementary catastrophe theory*. This theory is part of differential topology and is thus quite sophisticated, though it is not frequently used in the field of applied mathematics. While we cannot provide many details here, our results are simple to understand and we are confident that our proposed method is completely viable.

Elementary catastrophe theory examines the stability of the real functions \(f: \mathbb{R}^m \longmapsto \mathbb{R}\) under infinitesimal variations \(f \longmapsto f + \delta f\) [5, 6]. We address the mathematical existence of elementary catastrophes in scenarios with functions whose infinitesimal perturbations can change their “quality” (for example, a monotonic function after the variation becomes oscillating). The simplest catastrophe—and the only one that we utilize here—is called *fold catastrophe* and is presented by the function \(f(t) = t^3 + \chi t\). When \(\chi \ge 0\), this function is strictly monotone increasing; for any negative values, it has two extremums. This function usually manifests as a potential (energy). However, there are exceptions; for instance, use of the fold catastrophe (corresponding to the soil liquefaction) to determine the resistance at low speeds [1]—which we believe would be impossible to do otherwise—is one such example.

At least one drawback exists when the variable \(t\) has a *temporal dimension*: the process is not repeatable, meaning that one cannot speak with certainty about the establishment of a physical law. Repetitive and separate application of the fold catastrophe to the processes that occur in each quasi-period of the seismograms is therefore necessary. The question is thus as follows: “In which domain of the seismograms would an analysis based on elementary catastrophe theory be relevant?”

**Figure 2.**The low activity period corresponds to a partial-small rupture of a plate. If this activity continues, an earthquake will occur; if not, the situation calms down. Figure courtesy of the authors, with records courtesy of support from the University of Jaén’s Seismic Risk and Active Tectonics Research Group.

Within certain limits, epicenters are characterized by both large earthquakes and more numerous series of small tremors (fixed by the Gutenberg-Richter formula). The latter correspond to local plates that begin to break, but do not continue further (see Figure 2).

A comparison between Figures 1 and 2 shows that a major earthquake starts in the same way as a partial-small rupture but does not stop after the rupture — one can observe the complete characteristic form of the seismogram at this point. These “beginnings” are hence the pertinent segments that must be realized in our analysis.

Let \(t_{-n}\) denote the relevant instant that corresponds to our selected first period’s time lapse (here, \(n\) is a natural number). The final instant—which is associated with the start of the real earthquake—will fix the zero time \((t_0 = 0)\). Figure 3 depicts the fold catastrophes from \(t_{-n}\) up to the “change in quality” \((k = n,n - 1,...,0)\).

**Figure 3.**The best method for the approximation of two different curves on the seismogram with catastrophes at a given point involves matching their derivatives at this point. Figure courtesy of the authors.

*two extremums*(maximum and minimum); the folds change the function’s quality. In any case, \(\chi_{-k}t - \chi_{-k}t_{-k}\) traces the tangent line’s equation. Of course, the tangent of the polar angle expresses the parameter \(\chi_{-k}\). In this sense, the jump from a “regular” situation to the beginning of an earthquake is sufficiently characterized by the transition of the tangent polar angle through \(\frac{\pi}{2}\). According to all of these factors that contribute to earthquake forecasting, the discrimination (including triangulation) of the seismic velocities/amplitudes must account for the low level that is evident in Figure 2. And due to the \(p\) waves’ frequencies, an accurate graph reproduction requires a sampling of 1,000 Hertz instead of the typical of 100 Hertz.

The initial instant \(t_{-n}\) is positioned at the beginning of any small activity in Figures 1 or 2. As in Figure 3, let us draw the respective tangents at the temporal axis crossings in the seismogram when the readings begin to rise. As was aforementioned, we must couple the fold catastrophe within each seismogram’s (quasi) period by “merging” respective tangents. The coefficients \(\chi_{-l}\)—which determine the slopes of the respective tangents to the seismogram \(\varphi_{-l} = \tanh^{-1} (\chi_{-l})\), where \(l = n,n - 1,...,0\)—serve as the characteristic presentations of the corresponding fold catastrophes. In sufficiently small time lapses around these points, the concept of analyzing a current tremor’s evolution with the help of the fold catastrophe is therefore promising. In this way, the angles that are formed by the abscissa and aforementioned tangents can illustrate some of the behavior of the processes that precede an earthquake — and hopefully forecast them as a result [2]. The idea involves the application of heuristic algorithms (in the artificial intelligence sense) and subsequent elaboration with effective software to discover regularities in the sequence:

\[\chi_{-n},\chi_{-n + 1},...\chi_0.\tag1\]

**Figure 4.**A diagram that provides a physical sense of the \(\chi\) parameters. Figure courtesy of the authors, with records courtesy of support from the University of Jaén’s Seismic Risk and Active Tectonics Research Group.

Of course, this task could become increasingly complex and require much more statistical treatment. Nevertheless, the periodicity of earthquakes brings hope that some regularities in the behavior of \(\chi\) must exist. Moreover, a recent study has theoretically proven the existence of cyclic consequences that encompass all epicenters of a local plate [4]. Unfortunately, we have found no historical records to compare with said theory. But all of this insight allows us to suppose the existence of currently hidden regularities. We are confident that with appropriate financing, EEWS in the near future will become increasingly effective.

*Ricardo Ceballos presented this research during a contributed presentation at the 2022 SIAM Conference on Mathematics of Planet Earth (MPE22), which took place concurrently with the 2022 SIAM Annual Meeting in Pittsburgh, Pa., in July 2022. He received funding to attend MPE22 through a SIAM Student Travel Award. To learn more about Student Travel Awards and submit an application, visit the online page.*

**References**

[1] Ceballos G., R., & Netchev, P.N. (2020). On the Burridge and Knopoff model and the theoretical seismicity. Preprint, *Earth and Space Science Open Archive*.

[2] Dick, S. (2019). Artificial intelligence. *Harvard Data Science Review*, *1*(1).

[3] Kagan, Y.Y. (1997). Are earthquakes predictable? *Geophys. J. Int.*, *131*(3), 505-525.

[4] Netchev, P. (2020). *Tectonics, earthquakes and paleogeography*. Chisinau, Moldova: Lambert Academic Publishing.

[5] Thom, R. (1989). *Structural stability and morphogenesis: An outline of a general theory of models *(D.H. Fowler, Trans.). Boca Raton, FL: CRC press. (Original work published 1975).

[6] Zeeman, E.C. (1976, April 1). Catastrophe theory. *Scientific American*, 65-83.

[7] Zollo, A., Iannaccone, G., Convertito, V., Elia, L., Iervolino, I., Lancieri, M., … Gasparini, P. Earthquake early warning system in Southern Italy. In R. Meyers (Ed.), *Extreme environmental events *(pp. 175-201). New York, NY: Springer.