The deterministic behaviour of earthquake rupture beginning

The deterministic behaviour of earthquake rupture beginning | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article The deterministic behaviour of earthquake rupture beginning Simona Colombelli, Valeria Longobardi, Aldo Zollo This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3967674/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Earthquakes are among the most destructive natural hazards. The energy released by an earthquake can be quantified by its magnitude. However, predicting how much energy the earthquake will release before the end of its rupture process represents a challenging question in geohazards. The way earthquake ruptures grow and arrest determines the final event size: small-to-moderate ruptures evolve in few seconds with typical lengths of few kilometers, while large-to-huge events develop in tens of seconds or more, involving lengths of several hundred kilometers. If the rupture process starts in the same way for small and large earthquakes, no deterministic prediction of the final size is feasible, until the process has finished. On the contrary, if the source mechanism starts differently from its early beginning, real-time proxies can be measured on seismic waves to discriminate the final event size. Here we show that the initial ground displacement growth is differently for small and large earthquakes, based on the analysis of an unprecedented catalog of seismic waveforms from worldwide earthquakes. The result supports the hypothesis of early predictable event magnitude for a wide range of different size earthquakes in diverse geological settings. This study confirms that the measure of the initial growth of displacement can be used as a parameter for a fast magnitude estimation, making it feasible for future implementation in early warning systems. Earth and environmental sciences/Solid Earth sciences/Seismology Earth and environmental sciences/Natural hazards Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Introduction The nucleation of earthquakes is a long-standing issue in seismology, being one of the open questions for which a unique and unquestionable answer has not yet been provided. It is known, indeed, that seismic ruptures begin with a process of quasi-static slip accumulation over a limited region of the fault. Here, the slip slowly accumulates until reaching a critical threshold, beyond which the fracture becomes unstable and triggers the dynamic propagation 1 2 3 4 . While the quasi-static phase (referred to as the preparatory phase ) has no definite beginning and duration, the time during which the rupture accelerates to the dynamic propagation (referred to as the nucleation phase ), in contrast, is thought to be well defined and relatively short 3 .The controversial point, however, is whether the nucleation phase of seismic ruptures is a similar process for all earthquakes, or if a different mechanism is responsible for the generation of small and large events 5 6 7 8 . Two distinct models of earthquake nucleation have indeed been proposed and discussed among the seismological community 7 9 . In the “ cascade model ” 7 , all earthquakes start in the same way and there is no difference in their nucleation. Local friction conditions and geometrical discontinuities of the fault interface determine the final rupture extent 10 . The nucleation here is a stochastic process, which makes it impossible to estimate the final size of an earthquake until the rupture has completely stopped. In the “ pre-slip model ” 7 instead, the rupture beginning for small and large events is different, likely resulting from a different nucleation phase. In this view, the nucleation phase of an earthquake is informative of the later evolution of the rupture 11 12 13 14 . This model supports the capacity of predicting the earthquake magnitude since the very beginning of the process. Observations of the P-wave onset of real earthquakes provided so far are contradictory, due to the use of different types of data, different approaches and frequency scales of the observations to infer the features of the nucleation phase from earthquake recordings. Several authors 6 15 16 17 8 18 19 did not observe any scaling of the P-wave onset with the final earthquake size, or they interpreted it in terms of biases due to data processing and/or wave propagation effects. These authors supported the idea of a universal behaviour of the seismic rupture onset, for which there is not a natural predisposition for a rupture to grow to a large earthquake since the beginning of the rupture process. Other authors 11 20 12 13 14 provided evidence for a deterministic nature of the seismic rupture showing straightforward relationships between the early signals radiated by the source and the final earthquake size. Among them, at the time of scale of seconds-to-tens of seconds, Melgar and Hayes ( 2019 ) 21 provided evidence that the initial growth-rate amplitude is higher for larger events and lower for smaller ones. At a much shorter time scale, (few seconds or shorter), observations of the early recorded amplitude on limited catalog of past Japanese earthquakes 20 22 have shown that the initial growth-rate amplitude is lower for larger events and higher for smaller ones. While these recent results support the idea of a deterministic process of earthquake nucleation, the related observations are grounded on limited datasets and show a large variability among data, so that the this idea of determinism is still considered weak and not consolidated. Here we show that the initial ground displacement growth behaves differently for small and large earthquakes, by analyzing an unprecedented catalog of seismic waveforms from worldwide earthquakes. Our results support the hypothesis of early predictable event magnitude for a wide range of different size earthquakes in diverse geological settings, regardless of the distance from event location. The initial ground displacement rate is interpreted here in terms of a characteristic time of the rupture process, during which changes of the critical slip weakening distance and/or rupture /slip velocity can occur and increase with the final earthquake size. This study confirms that the measurement of the initial growth of displacement can be used as a proxy for a fast magnitude estimation, making it feasible for future implementation in early warning systems. Results We analyzed 200 earthquakes with magnitude between 4 and 9 occurred worldwide. For each event, we looked at the closest five stations, using both velocimetric or accelerometric waveforms, depending on the availability of data (Fig. 1 ) and performing a single or a double integration of the signal to get displacement, respectively. We computed the logarithm of the peak of the absolute displacement starting from the P-wave arrival time and then averaging the station curves, to obtain a single event curve, hereinafter referred to as the LPDT curve (Fig. 2 ). For each curve, we identified the plateau time and then measured the initial slope on the curves from the line that crosses the LPDT curve in its early part (see Methods section). The main results of this study are illustrated by Figs. 3 and 4 . Figure 3 shows the evolution of the LPDT curves with time, for different magnitude classes and different distance ranges. The curves follow a monotonic increase with time, starting from small amplitude values and reaching a plateau level that is generally higher for larger magnitudes (red-purple curves) and smaller for small events (cyan-green curves). The time needed to reach the plateau level also increases with magnitude, being less than 1 sec for small magnitudes and of the order of tens of second for the largest ones, as it can be seen from panels a,b,c. The typical pattern of LPDT curves is observed at all distances ranges, although for the two extreme ranges (40–60 km and 200-250km) not all the magnitude classes are represented, due to the limited availability of records within that distance range. In addition to the plateau time and level, the most relevant difference among the curves is in the way the curves increase at their beginning. The zoom on the first 1.5 seconds of curves (Fig. 3 , panels d,e,f) provides the proper representation to appreciate this difference. When observed at this time scale, the curves of small events clearly begin with a higher amplitude growth rate, while for the largest events the initial amplitude increase is slower. This difference with magnitude is well evident for the distance range of 100–150 km, for which all magnitude classes are represented by a sufficient number of available records. Other examples of curves for different distance ranges are provided in Figure S1 of the Supplemental Material. The average curves of Fig. 3 are well separated for different magnitude classes. However, the uncertainties on amplitude values at each time along the curves (computed as the standard deviation of the mean value (see Supplemental Material, Figure S2) are relatively large, such that, in some cases, the error bars overlap. The width of the error bar depends on the natural variability of recorded amplitudes, but it may also reflect the intrinsic variability of the magnitude range considered for each class (0.5 magnitude units). A more quantitative analysis of the difference in the initial rise of LPDT curves is provided by the slope measurements as a function of magnitude (Fig. 4 a). When evaluated over a large spectrum of magnitudes and for different tectonic areas worldwide, the initial rise of the log-displacement vs. time curves measured along the P-wave portion of near-source stations (distances smaller than 50–100 km) show a clear decreasing linear trend with the earthquake magnitude. The time window in which this estimate is done is also depending on the final earthquake magnitude, with larger time windows being necessary to measure the initial slope for larger magnitude events (Fig. 4 b). Given the variability and uncertainties of measurements, the observed standard error of the linear regressions, suggests an uncertainty of about one magnitude unit associated with as the slope measurement obtained by averaging the values at the five stations closest to the source epicenter. Overall, a P-wave time window of about 1 sec is sufficient to discriminate between small-moderate (M 6.5) earthquakes and to determine its magnitude with the specified uncertainty. Discussion We show here that the P-wave signals associated to large earthquakes typically begin with a slow initial amplitude growth in the first few seconds, while the P-wave signals radiated by small events are mainly characterized by a rapid amplitude increase, in a shorter time. The results are consistent with observations from previous works 20 22 , that focused on the analysis of a limited number of Japanese earthquakes, recorded at regional distances (R < 100-200km), with most of the events having magnitude below 6 and very few of them having larger magnitudes ( 7 – 9 ). We used a massive catalogue of worldwide earthquakes, from different tectonic areas and geological settings, including normal fault zones, strike-slip environments, as well as major earthquakes from subduction zones. We included here more than 7 thousands records, covering a broad distance range (0-500 km), that provided us with a robust catalogue, in which all magnitude classes are represented by a consistent number of data. We manually picked the P-wave arrival time at all the available waveforms, resulting into an unprecedent dataset, in terms of both number and quality of recorded data. In terms of data processing and analysis, here we tried to keep the methodology as much as possible free from artificial contaminations that could produce biased results. We adopted a simple scheme for the early P-wave slope measurement, that does not require complex signal processing or manipulations. We identified the plateau time on the LPDT curves and then directly measured the initial slope on the curves, at a fixed time, corresponding to half of the estimated plateau time. Slope measurements are directly obtained from the observed log-displacement evolution, with no use of interpolated models or fitted curves. Our results show that the initial slope of LPDT curves decreases with magnitude, with an average value of about 20 \({s}^{-1}\) for M = 4 earthquakes and about 2 \({s}^{-1}\) for M = 9 events (Fig. 4 a). The time at which the slope is measured increases with magnitude, being approximately 0.2 s for M = 4 earthquakes and about 1 s for M = 9 events. (Fig. 4 b). The times measured here are related to the plateau time of LPDT curves which are, in turn, a proxy for the peak of the MRF 23 20 . The choice of the time to measure the slope is arbitrary (half of the plateau time of the curves), thus both the times and the slope values do not have an obvious physical meaning. However, their scaling over the entire magnitude range (as shown in Fig. 4 ) provides a frame to interpret our observations. During the preparatory phase of earthquake ruptures, the crack size increases slowly at few percentages of the shear wave speed 34 until it reaches a critical size related to the frictional parameters (e.g., the slip-weakening critical distance, D c ). At this point, the unstable fracture expands at an increasing velocity, with an acceleration stage that triggers the dynamic propagation 1234 . The time during which rupture accelerates to the dynamic propagation is well defined and related to the final slip (and therefore to the earthquake magnitude) 3 . In this view, the observed scaling of both times and slopes could be the footprint of this unstable acceleration phase and corroborates the idea that earthquakes are different already at the beginning of the rupture process. The slip-weakening critical distance, D c , increases with magnitude 3 . Intuitively, longer times are needed to reach this condition on a limited portion of the fault for larger events, thus implying a smaller rate of amplitude growth. Therefore, a long duration of the rupture acceleration (i.e. large D c , large magnitude) is associated to a low rate of amplitude increase (i.e., a small slope value). Conversely, for small events, the smaller D c value is reached in a shorter time on a limited portion of the fault, with consequent smaller rates of amplitude growth. A short duration of the acceleration phase is therefore associated to high values of the amplitude growth rate (large slopes). Thus, the nucleation phase is different for small and large events and this difference can be seen within a “characteristic time” of the process itself. The observed slope decrease with magnitude can be related to the effect of variable dynamic stress drop and/or rupture velocity in the initial stage of the rupture of small and large earthquakes, possibly triggered by the acceleration phase during the quasi-static rupture nucleation. Indeed, based on the dynamic-consistent model for an expanding shear circular crack of Sato & Hirasawa (1973) 24 , the relation between the early P radiated displacement pulse \({{\Omega }}_{P}\left(t\right)\) , the dynamic stress drop ( \({\tau }_{e}\) ) and the rupture velocity ( \({v}_{R}\) ) during the rupture growth phase can be written as: 25 : $$\begin{array}{c}{{\Omega }}_{P}\left(t\right)=\frac{2\pi {\Delta }{v}_{o}{v}_{R}^{2}}{{\left(1-{\zeta }^{2}\right)}^{2}}{t}^{2}\approx \frac{2\pi {v}_{R}^{3}{\tau }_{e}}{\mu {\left(1-{\zeta }^{2}\right)}^{2}}{t}^{2}\#\left(1\right)\end{array}$$ where \({\Delta }{v}_{o}\) is peak slip velocity, \(\mu\) the rigidity at the source region and \(\zeta =\frac{{v}_{R}}{{v}_{P}}\text{sin}\theta\) is the P-wave apparent Mach number 25 , with \(\theta\) being the angle between the ray take-off direction and the normal to the circular fault. The term including the apparent Mach number accounts for rupture directivity, depending on the receiver view angle and the rupture to wave velocity ratio. In ( 1 ) the relation between the peak slip velocity and dynamic stress drop is inferred from the dynamic models of a propagating shear crack by Kostrov(1964) 26 and Dahlen(1974) 27 . Averaging over \(\theta\) , Eq. 1 changes to: $$\begin{array}{c}{{\Omega }}_{c}\left(t\right)\approx \frac{2\pi C\left({v}_{R}\right){\tau }_{e}}{\mu }{t}^{2}\#\left(2\right)\end{array}$$ where the rupture velocity factor $$\begin{array}{c}C\left({v}_{R}\right)=\frac{\pi }{2}\frac{{v}_{R}^{3}\left(2-{\left(\frac{{v}_{R}}{{v}_{P}}\right)}^{2}\right)}{{\left(1-{\left(\frac{{v}_{R}}{{v}_{P}}\right)}^{2}\right)}^{\frac{3}{2}}}\#\left(3\right)\end{array}$$ shows an exponential increase with \({v}_{R}\) . According to Eq. 2, both the rupture velocity and/or dynamic stress drop control the rise of the radiated P pulse displacement and therefore its initial slope. An example of the effect of a varying rupture velocity with magnitude on LPDT curves and slope by assuming a Sato and Hirasawa (1973) 24 kinematic source model of a circular shear crack is presented in the Supplemental Material (Text S1 and Figure S5). Synthetic tests support the idea that a decreasing value of rupture velocity with magnitude could be responsible for the observed decrease of the initial slope of P-wave displacement. Additional constraints provided by more complex and realistic numerical simulations are necessary to understand in which physical conditions and and how these two parameters may play a role during the initial stage of the rupture propagation. The results obtained here contrast with the findings of previous studies 16,17,19 showing that the initial rate of the P-wave amplitude is almost independent of the earthquake magnitude. While we do not question the validity of these previous studies, significant differences, related to the data selection and processing may justify the different observations we have obtained here. An extensive discussion about the possible effects of data processing is provided in Colombelli et al 22 and similar considerations can be applied to the results of the current analysis. Our results, instead, do not exclude the evolution of the source process on a longer time scale, as seen by other authors 17 21 . At the time scale of seconds to tens-of-seconds, Meier at al 17 suggested that the MRFs of different events with increasing magnitude have all the same initial, growth-rate amplitude (or initial amplitude rate slope). At the same time scale, Melgar and Hayes 21 provided evidence that the initial growth-rate amplitude is higher for larger events and lower for smaller ones. In Fig. 5 we suggest a hybrid model for the MRFs that could bring together observations at short 20 22 and long time-scales 17 21 . In this model, at short time scales, the growth-rate amplitude would follow an inverse scaling with magnitude, with large events starting with a slower amplitude increase and small events beginning with a higher amplitude growth-rate. Figure 5 a shows the early evolution of the earthquake rupture, as resulting from our observations. The figure shows the initial slopes of LPDT curves a s function of time for all earthquakes used in this work. Each curve is plotted up to the time t half as resulting from our analysis. Circle markers are single event data; squares are averaged value in the magnitude bin. Panel (b) suggests a schematic representation of the later evolution of the earthquake source time function. At longer time scales, the growth-rate amplitude would then change either in the sense of increasing with magnitude, or maintaining the same growth rate with magnitude, leading to the generation of absolute peak values of the MRFs, which are consistent (in time and amplitude) with those predicted from the scaling laws 28 . With current data and methods, we do not have adequate elements to discriminate the growing behavior of MRFs at longer time scales, approaching to the peak values. Relevant implications of the proposed model are also related to the rapid assessment of the earthquake size for Earthquake Early Warning Systems (EEWS), for which the possibility of discriminating a large event from a small one is crucial to provide reliable prediction of the incoming ground shaking level, for the prompt activation of emergency procedures and real-time risk mitigation actions. This could be achieved either using direct amplitude measurement or passing through the computation and modelling of amplitude vs. time curves. A simple threshold-based warning system, based on the initial slope measurement, would allow for the discrimination of small/large events within very short time windows (< 1 s), opening to new perspectives for the practical applications of EEWS. Declarations Acknowledgements. The research was funded by the University of Naples Federico II and by the Italian Ministry of University and Research (within the framework of the National Operative Programme PON-AIM AIM1834927 – 3). Author contributions . V.L., S.C. and A.Z. equally contributed to the concept of the work and to the methodological developments. V.L. analysed data, prepared the figures and wrote the original version of the manuscript. S.C. and A.Z. contributed to the interpretation and discussion of results and to the revision of the manuscript. Competing interests. The authors declare no competing interests. Materials & Correspondence. Correspondence and material requests should be addressed to Dr. Simona Colombelli ( [email protected] ). Data availability. The waveform data used for the analysis in the study are availableat the IRIS web data service (http://service.iris.edu, last access 2024/02/18), at the Northern California Earthquake Data Center (http://service.ncedc.org, last access 2024/02/18) and at the European Mediterranean Seismological Centre (http://www.seismicportal.eu, last access 2024/02/18). All analysis and figures are performed using Phyton. References Ampuero, J. ‐P., Vilotte, J. ‐P. & Sánchez‐Sesma, F. J. Nucleation of rupture under slip dependent friction law: Simple models of fault zone. J Geophys Res Solid Earth 107 , (2002). Dascalu, C., Ionescu, I. R. & Campillo, M. Fault finiteness and initiation of dynamic shear instability. Earth Planet Sci Lett 177 , 163–176 (2000). Latour, S., Schubnel, A., Nielsen, S., Madariaga, R. & Vinciguerra, S. Characterization of nucleation during laboratory earthquakes. Geophys Res Lett 40 , 5064–5069 (2013). Nielsen, S., Taddeucci, J. & Vinciguerra, S. Experimental observation of stick-slip instability fronts. Geophys J Int 180 , 697–702 (2010). Ishihara, Y., Fukao, Y., Yamada, I. & Aoki, H. Rising slope of moment rate functions: The 1989 earthquakes off east coast of Honshu. Geophys Res Lett 19 , 873–876 (1992). Abercrombie, R. & Mori, J. Local Observations of the Onset of a Large Earthquake: 28 June 1992 Landers, California . Bulletin of the Seismological Society of America vol. 84 (1994). Ellsworth, W. L. & Beroza, G. C. Seismic Evidence for an Earthquake Nucleation Phase. Science (1979) 268 , 851–855 (1995). Mori, J. & Kanamori, H. Initial rupture of earthquakes in the 1995 Ridgecrest, California Sequence. Geophys Res Lett 23 , 2437–2440 (1996). Gomberg, J. Unsettled earthquake nucleation. Nat Geosci 11 , 463–464 (2018). Wesnousky, S. G. Predicting the endpoints of earthquake ruptures. Nature 444 , 358–360 (2006). Beroza, G. C. & Ellsworth, W. L. Properties of the seismic nucleation phase. Tectonophysics 261 , 209–227 (1996). Iio, Y. Observations of the slow initial phase generated by microearthquakes: Implications for earthquake nucleation and propagation. J Geophys Res Solid Earth 100 , 15333–15349 (1995). Olson, E. L. & Allen, R. M. The deterministic nature of earthquake rupture. Nature 438 , 212–215 (2005). Rice, J. R. Elastic wave emission from damage processes. J Nondestr Eval 1 , 215–224 (1980). Aagaard, B. T. & Heaton, T. H. Constraining fault constitutive behavior with slip and stress heterogeneity. J Geophys Res Solid Earth 113 , (2008). Meier, M., Heaton, T. & Clinton, J. Evidence for universal earthquake rupture initiation behavior. Geophys Res Lett 43 , 7991–7996 (2016). Meier, M.-A., Ampuero, J. P. & Heaton, T. H. The hidden simplicity of subduction megathrust earthquakes. Science (1979) 357 , 1277–1281 (2017). Scherbaum, F. & Bouin, M.-P. FIR filter effects and nucleation phases. Geophys J Int 130 , 661–668 (1997). Trugman, D. T., Page, M. T., Minson, S. E. & Cochran, E. S. Peak Ground Displacement Saturates Exactly When Expected: Implications for Earthquake Early Warning. J Geophys Res Solid Earth 124 , 4642–4653 (2019). Colombelli, S., Zollo, A., Festa, G. & Picozzi, M. Evidence for a difference in rupture initiation between small and large earthquakes. Nat Commun 5 , (2014). Melgar, D. & Hayes, G. P. Characterizing large earthquakes before rupture is complete. Sci Adv 5 , (2019). Colombelli, S., Festa, G. & Zollo, A. Early rupture signals predict the final earthquake size. Geophys J Int 223 , 692–706 (2020). Nazeri, S., Colombelli, S. & Zollo, A. Fast and accurate determination of earthquake moment, rupture length and stress release for the 2016-2017 Central Italy seismic sequence. Geophys J Int 217 , 1425–1432 (2019). SATO, T. & HIRASAWA, T. Body wave spectra from propagating shear cracks. Journal of Physics of the Earth 21 , 415–431 (1973). Boatwright, J. A spectral theory for circular seismic sources; simple estimates of source dimension, dynamic stress drop, and radiated seismic energy. Bulletin of the Seismological Society of America (1980). Kostrov, B. V. Selfsimilar problems of propagation of shear cracks. Journal of Applied Mathematics and Mechanics 28 , 1077–1087 (1964). Dahlen, F. A. On the ratio of P -wave to S -wave corner frequencies for shallow earthquake sources. Bulletin of the Seismological Society of America 64 , 1159–1180 (1974). Scholz, C. H. & Cowie, P. A. Determination of total strain from faulting using slip measurements. Nature 346 , 837–839 (1990). Method The initial dataset consists of 200 earthquakes from worldwide locations (see Fig. 1). The events span a time ranging between 2003 and 2023 and moment magnitude ranging between 4 and 9 (see Supplemental Material, Table S1 ). The waveforms are recorded either from velocimeter networks or accelerometer networks, depending on data availability. For each earthquake, we select the closest five stations to event epicenter. Depending on the available recording sensor (velocimeter or accelerometer) (see Fig. 1) we perform a single or double integration of signal, respectively, to get displacement and we finally apply a high-pass Butterworth filter with cut-off frequency of 0.075Hz to remove possible baseline effects. We compute the logarithm of the peak of the absolute displacement starting from the P-arrival time at the station. We average the obtained LPDT curves by stations to get the final LPDT curve on which we evaluate the intial slope, as explained below. We perform an interpolation of the real LPDT curve with an exponential function of the form: $${\varvec{L}\varvec{P}\varvec{D}\varvec{T}}_{\varvec{t}\varvec{e}\varvec{o}}={\varvec{L}\varvec{P}\varvec{D}\varvec{T}}_{\varvec{e}\varvec{n}\varvec{d}}\left(1-{\varvec{e}}^{-\raisebox{1ex}{$\varvec{t}$}\!\left/ \!\raisebox{-1ex}{${\varvec{t}}_{1}$}\right.}\right)-{\varvec{L}\varvec{P}\varvec{D}\varvec{T}}_{0}$$ 4 where \({LPDT}_{0}\) and \({LPDT}_{end}\) are the first and the last point of the curve respectively, \({t}_{1}\) is a fit parameter that simply allows the function to bend towards the plateau level. The interpolation is used to avoid numerical noise caused by the discontinuity of real curves. For each curve, we compute the curvature as \(curvature= \frac{\left|{y}^{{\prime }{\prime }}\right|}{{\left(1+{\left({y}^{{\prime }}\right)}^{2}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\) where \({y}^{{\prime }}\) and \({y}^{{\prime }{\prime }}\) are the numerical first and second derivative of the interpolated LPDT curve. The curvature allows to quantify how much a function is deviated from a linear behaviour. Figure S3 and S4 of the Supplemental Material show examples of curvature computation, for different curves. We then measure the time \({t}_{HALF}\) , corresponding to half of the time where the maximum of the curvature is reached. Finally, we evaluate the slope of the real LPDT curve between \({t}_{MIN}=0.05 s\) and \({t}_{HALF}\) . The choice of the starting point for the slope evaluation is done to account for errors in affect the manual picking of waveforms. The slope is obtained from the line that crosses the LPDT between t MIN and t HALF , as it follows: \(Slope= \frac{LPDT\left({t}_{HALF}\right)-LPDT\left({t}_{MIN}\right)}{{t}_{HALF}-{t}_{MIN}}\) ( 5 ). Additional Declarations There is NO Competing Interest. Supplementary Files SupplementalMaterial.docx Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3967674","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":274506424,"identity":"1f66bdbc-77e0-40f9-870d-2f13b85a2593","order_by":0,"name":"Simona Colombelli","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/ElEQVRIiWNgGAWjYDACZhiDHcy2kGFjb2BgSAAKsBHUwgxGEjxsPAegWnDqQdfCIJEAFcGhRb6dO/FxQQ2DPD8z89HNBRUSPHySb8wePNxhx8An34BVi8Fh3s3GM44xGM5sZku7PeMM0GHSOeYGiWeScTrMgJl3mzQPG0OCwWEes9u8bWAtZhKJbcw4tcg3g7T8g2n5B9QieQakpR639w8DtfC2wbQ0ALVI8IC0HMbtMJBfePskoH45BgrktDKgluM8bGwJWLXI95/d+Jjnm408P3vzsdsFNTZy8u2Ht0n+bKuWk28+gMNlYCCBKcSDT/0oGAWjYBSMAvwAAEcERa33/yQVAAAAAElFTkSuQmCC","orcid":"https://orcid.org/0000-0002-5946-0795","institution":"University of Naples Federico II","correspondingAuthor":true,"prefix":"","firstName":"Simona","middleName":"","lastName":"Colombelli","suffix":""},{"id":274506425,"identity":"64a10efe-c08c-4282-bbca-20cadff494a2","order_by":1,"name":"Valeria Longobardi","email":"","orcid":"","institution":"University of Naples Federico II","correspondingAuthor":false,"prefix":"","firstName":"Valeria","middleName":"","lastName":"Longobardi","suffix":""},{"id":274506426,"identity":"6fe4477b-b4d2-49b8-8972-c7521c79738b","order_by":2,"name":"Aldo Zollo","email":"","orcid":"https://orcid.org/0000-0002-8191-9566","institution":"University of Naples Federico II","correspondingAuthor":false,"prefix":"","firstName":"Aldo","middleName":"","lastName":"Zollo","suffix":""}],"badges":[],"createdAt":"2024-02-18 18:10:51","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-3967674/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3967674/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":51621662,"identity":"be364ebe-d6f2-42d6-94a0-f10c7ccaf29c","added_by":"auto","created_at":"2024-02-26 06:00:23","extension":"jpeg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":2189617,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eMap of the events\u003c/strong\u003e. The figure shows the epicentral position of the events used in this study (colored circles). The size of the circles is proportional to the earthquake magnitude and the color shows the event depth. Black triangles are the velocimeter sensors whilemagenta squares represent accelerometer sensors. The histogram in the middle shows the distribution of records in each magnitude bin.\u003c/p\u003e","description":"","filename":"floatimage1.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-3967674/v1/55c86fd2e25d82dba4550856.jpeg"},{"id":51621270,"identity":"e933c8bc-c4ad-49cb-a770-bab773814da8","added_by":"auto","created_at":"2024-02-26 05:52:23","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":365793,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eComputation of LPDT curves and slope measurement. \u003c/strong\u003ePanel (a) shows the vertical acceleration records at the five closest stations for a single event. Panel (b) represents the absolute displacement of signals. Panel (c) shows the LPDT curves at each station (thin colored lines) and the averaged curve (thick black line). The red circle (corresponding to t\u003csub\u003eMIN\u003c/sub\u003e=0.05s) is the starting point for the slope evaluation. The magenta diamond\u0026nbsp; (t\u003csub\u003eHALFf\u003c/sub\u003e) is the ending point for slope evaluation. In all panels, the colors of the lines represents the epicentral distance of each station.\u0026nbsp;\u003c/p\u003e","description":"","filename":"floatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-3967674/v1/acfb4a57ade2910c66813363.png"},{"id":51621268,"identity":"193363b2-e229-45ba-81ea-5f1fe0d92b79","added_by":"auto","created_at":"2024-02-26 05:52:23","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":440676,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eExample of LPDT curves.\u003c/strong\u003e The figure shows the evolution of LPDT curves for different magnitude classes and distance ranges.. Panels (a) (b) (c) show the entire duration of LPDT curves, while panels (d) (e) (f) show a zoom on the first 1.5 seconds of the curves (represented by the gray area in top panels). In all panels, errorbars are shown for some reference times along the curves and the number of records used to obtain each average curve is also reported in the box, in top panels. In bottom panels the LPDT are initialized to their first point, for a matter of representation, to appreciate the slope variation with magnitude.\u003c/p\u003e","description":"","filename":"floatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-3967674/v1/5a158b4f39778b2e87d12845.png"},{"id":51620850,"identity":"0ca07e23-0836-4c56-bddb-a34853d8c4e3","added_by":"auto","created_at":"2024-02-26 05:44:23","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":1257483,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eInitial Slope pf LPDT cureves and related time. \u003c/strong\u003e\u0026nbsp;a) initial slope of LPDT curves as a function of magnitude. b) the time at which LPDT curves reach half of the maximum of their curvature (t\u003csub\u003eHALF\u003c/sub\u003e) as a function of magnitude. In both panels grey dots are single slope and time measurements, while red squares are the average values for each magnitude bin (bin width = 1). The\u0026nbsp; solid, black line represents the best fit line, along with ± one-SE thin lines. The fit parameters and the SE values are shown at top left of the panel.\u003c/p\u003e","description":"","filename":"floatimage4.png","url":"https://assets-eu.researchsquare.com/files/rs-3967674/v1/60340450e19bf0a24d1762be.png"},{"id":51620854,"identity":"3f922c9b-47d2-4a69-9d13-22b6cadee36d","added_by":"auto","created_at":"2024-02-26 05:44:24","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":617138,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eEarly evolution of rupture\u003c/strong\u003e. Panel (a) shows a plot of logarithm of initial LPDT slope versus t\u003csub\u003ehalf\u003c/sub\u003e of earthquakes used in this work. Circle markers are single event data; squares are averaged value in the magnitude bin. The average slope value ranges from 20 for magnitude=4 events to 2 for magnitude=9 events; the average time required to evaluate the initial slope ranges from 0.2 second for magnitude = 4 events to 1 second for magnitude = 9 events. Panel (b) is a schematic representation of earthquake source time function. The behaviour of source functions in grey shaded area is still under debate in literature.\u003c/p\u003e","description":"","filename":"floatimage5.png","url":"https://assets-eu.researchsquare.com/files/rs-3967674/v1/e48cbe84ec48651e76d7d6dc.png"},{"id":54903077,"identity":"8ea9cf30-e3a1-44cc-b5f4-d65c2ff363ce","added_by":"auto","created_at":"2024-04-18 10:49:29","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1493211,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3967674/v1/3996e9d6-6f07-44f3-9c78-815f7ee99252.pdf"},{"id":51620855,"identity":"721922fc-6ef9-47b6-83f1-2d641da1c2b1","added_by":"auto","created_at":"2024-02-26 05:44:24","extension":"docx","order_by":7,"title":"","display":"","copyAsset":false,"role":"supplement","size":1520899,"visible":true,"origin":"","legend":"","description":"","filename":"SupplementalMaterial.docx","url":"https://assets-eu.researchsquare.com/files/rs-3967674/v1/fd14fd2a60ff573818e57737.docx"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"The deterministic behaviour of earthquake rupture beginning","fulltext":[{"header":"Introduction","content":"\u003cp\u003eThe nucleation of earthquakes is a long-standing issue in seismology, being one of the open questions for which a unique and unquestionable answer has not yet been provided. It is known, indeed, that seismic ruptures begin with a process of quasi-static slip accumulation over a limited region of the fault. Here, the slip slowly accumulates until reaching a critical threshold, beyond which the fracture becomes unstable and triggers the dynamic propagation\u003csup\u003e1 2 3 4\u003c/sup\u003e. While the quasi-static phase (referred to as \u003cem\u003ethe preparatory phase\u003c/em\u003e) has no definite beginning and duration, the time during which the rupture accelerates to the dynamic propagation (referred to as the \u003cem\u003enucleation phase\u003c/em\u003e), in contrast, is thought to be well defined and relatively short\u003csup\u003e3\u003c/sup\u003e .The controversial point, however, is whether the nucleation phase of seismic ruptures is a similar process for all earthquakes, or if a different mechanism is responsible for the generation of small and large events\u003csup\u003e5 6 7 8\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eTwo distinct models of earthquake nucleation have indeed been proposed and discussed among the seismological community \u003csup\u003e7 9\u003c/sup\u003e. In the \u0026ldquo;\u003cem\u003ecascade model\u003c/em\u003e\u0026rdquo; \u003csup\u003e7\u003c/sup\u003e, all earthquakes start in the same way and there is no difference in their nucleation. Local friction conditions and geometrical discontinuities of the fault interface determine the final rupture extent \u003csup\u003e10\u003c/sup\u003e. The nucleation here is a stochastic process, which makes it impossible to estimate the final size of an earthquake until the rupture has completely stopped. In the \u0026ldquo;\u003cem\u003epre-slip model\u003c/em\u003e\u0026rdquo; \u003csup\u003e7\u003c/sup\u003e instead, the rupture beginning for small and large events is different, likely resulting from a different nucleation phase. In this view, the nucleation phase of an earthquake is informative of the later evolution of the rupture \u003csup\u003e11 12 13 14\u003c/sup\u003e. This model supports the capacity of predicting the earthquake magnitude since the very beginning of the process.\u003c/p\u003e \u003cp\u003eObservations of the P-wave onset of real earthquakes provided so far are contradictory, due to the use of different types of data, different approaches and frequency scales of the observations to infer the features of the nucleation phase from earthquake recordings. Several authors \u003csup\u003e6 15 16 17 8 18 19\u003c/sup\u003e did not observe any scaling of the P-wave onset with the final earthquake size, or they interpreted it in terms of biases due to data processing and/or wave propagation effects. These authors supported the idea of a universal behaviour of the seismic rupture onset, for which there is not a natural predisposition for a rupture to grow to a large earthquake since the beginning of the rupture process. Other authors \u003csup\u003e11 20 12 13 14\u003c/sup\u003e provided evidence for a deterministic nature of the seismic rupture showing straightforward relationships between the early signals radiated by the source and the final earthquake size. Among them, at the time of scale of seconds-to-tens of seconds, Melgar and Hayes (\u003cem\u003e2019\u003c/em\u003e) \u003csup\u003e21\u003c/sup\u003e provided evidence that the initial growth-rate amplitude is higher for larger events and lower for smaller ones. At a much shorter time scale, (few seconds or shorter), observations of the early recorded amplitude on limited catalog of past Japanese earthquakes \u003csup\u003e20 22\u003c/sup\u003e have shown that the initial growth-rate amplitude is lower for larger events and higher for smaller ones. While these recent results support the idea of a deterministic process of earthquake nucleation, the related observations are grounded on limited datasets and show a large variability among data, so that the this idea of determinism is still considered weak and not consolidated.\u003c/p\u003e \u003cp\u003eHere we show that the initial ground displacement growth behaves differently for small and large earthquakes, by analyzing an unprecedented catalog of seismic waveforms from worldwide earthquakes. Our results support the hypothesis of early predictable event magnitude for a wide range of different size earthquakes in diverse geological settings, regardless of the distance from event location. The initial ground displacement rate is interpreted here in terms of a characteristic time of the rupture process, during which changes of the critical slip weakening distance and/or rupture /slip velocity can occur and increase with the final earthquake size. This study confirms that the measurement of the initial growth of displacement can be used as a proxy for a fast magnitude estimation, making it feasible for future implementation in early warning systems.\u003c/p\u003e"},{"header":"Results","content":"\u003cp\u003eWe analyzed 200 earthquakes with magnitude between 4 and 9 occurred worldwide. For each event, we looked at the closest five stations, using both velocimetric or accelerometric waveforms, depending on the availability of data (Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e) and performing a single or a double integration of the signal to get displacement, respectively. We computed the logarithm of the peak of the absolute displacement starting from the P-wave arrival time and then averaging the station curves, to obtain a single event curve, hereinafter referred to as the LPDT curve (Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e). For each curve, we identified the plateau time and then measured the initial slope on the curves from the line that crosses the LPDT curve in its early part (see Methods section).\u003c/p\u003e\n\u003cp\u003eThe main results of this study are illustrated by Figs.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e and \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e. Figure\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e shows the evolution of the LPDT curves with time, for different magnitude classes and different distance ranges. The curves follow a monotonic increase with time, starting from small amplitude values and reaching a plateau level that is generally higher for larger magnitudes (red-purple curves) and smaller for small events (cyan-green curves). The time needed to reach the plateau level also increases with magnitude, being less than 1 sec for small magnitudes and of the order of tens of second for the largest ones, as it can be seen from panels a,b,c. The typical pattern of LPDT curves is observed at all distances ranges, although for the two extreme ranges (40\u0026ndash;60 km and 200-250km) not all the magnitude classes are represented, due to the limited availability of records within that distance range.\u003c/p\u003e\n\u003cp\u003eIn addition to the plateau time and level, the most relevant difference among the curves is in the way the curves increase at their beginning. The zoom on the first 1.5 seconds of curves (Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e, panels d,e,f) provides the proper representation to appreciate this difference. When observed at this time scale, the curves of small events clearly begin with a higher amplitude growth rate, while for the largest events the initial amplitude increase is slower. This difference with magnitude is well evident for the distance range of 100\u0026ndash;150 km, for which all magnitude classes are represented by a sufficient number of available records. Other examples of curves for different distance ranges are provided in Figure \u003cspan class=\"InternalRef\"\u003eS1\u003c/span\u003e of the Supplemental Material.\u003c/p\u003e\n\u003cp\u003eThe average curves of Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e are well separated for different magnitude classes. However, the uncertainties on amplitude values at each time along the curves (computed as the standard deviation of the mean value (see Supplemental Material, Figure S2) are relatively large, such that, in some cases, the error bars overlap. The width of the error bar depends on the natural variability of recorded amplitudes, but it may also reflect the intrinsic variability of the magnitude range considered for each class (0.5 magnitude units).\u003c/p\u003e\n\u003cp\u003eA more quantitative analysis of the difference in the initial rise of LPDT curves is provided by the slope measurements as a function of magnitude (Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003ea). When evaluated over a large spectrum of magnitudes and for different tectonic areas worldwide, the initial rise of the log-displacement vs. time curves measured along the P-wave portion of near-source stations (distances smaller than 50\u0026ndash;100 km) show a clear decreasing linear trend with the earthquake magnitude. The time window in which this estimate is done is also depending on the final earthquake magnitude, with larger time windows being necessary to measure the initial slope for larger magnitude events (Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003eb). Given the variability and uncertainties of measurements, the observed standard error of the linear regressions, suggests an uncertainty of about one magnitude unit associated with as the slope measurement obtained by averaging the values at the five stations closest to the source epicenter. Overall, a P-wave time window of about 1 sec is sufficient to discriminate between small-moderate (M\u0026thinsp;\u0026lt;\u0026thinsp;5.5) and large (M\u0026thinsp;\u0026gt;\u0026thinsp;6.5) earthquakes and to determine its magnitude with the specified uncertainty.\u003c/p\u003e"},{"header":"Discussion","content":"\u003cp\u003eWe show here that the P-wave signals associated to large earthquakes typically begin with a slow initial amplitude growth in the first few seconds, while the P-wave signals radiated by small events are mainly characterized by a rapid amplitude increase, in a shorter time. The results are consistent with observations from previous works\u003csup\u003e20 22\u003c/sup\u003e, that focused on the analysis of a limited number of Japanese earthquakes, recorded at regional distances (R\u0026thinsp;\u0026lt;\u0026thinsp;100-200km), with most of the events having magnitude below 6 and very few of them having larger magnitudes (\u003cspan additionalcitationids=\"CR8\" citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eWe used a massive catalogue of worldwide earthquakes, from different tectonic areas and geological settings, including normal fault zones, strike-slip environments, as well as major earthquakes from subduction zones. We included here more than 7 thousands records, covering a broad distance range (0-500 km), that provided us with a robust catalogue, in which all magnitude classes are represented by a consistent number of data. We manually picked the P-wave arrival time at all the available waveforms, resulting into an unprecedent dataset, in terms of both number and quality of recorded data. In terms of data processing and analysis, here we tried to keep the methodology as much as possible free from artificial contaminations that could produce biased results. We adopted a simple scheme for the early P-wave slope measurement, that does not require complex signal processing or manipulations. We identified the plateau time on the LPDT curves and then directly measured the initial slope on the curves, at a fixed time, corresponding to half of the estimated plateau time. Slope measurements are directly obtained from the observed log-displacement evolution, with no use of interpolated models or fitted curves.\u003c/p\u003e \u003cp\u003eOur results show that the initial slope of LPDT curves decreases with magnitude, with an average value of about 20 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({s}^{-1}\\)\u003c/span\u003e\u003c/span\u003e for M\u0026thinsp;=\u0026thinsp;4 earthquakes and about 2 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({s}^{-1}\\)\u003c/span\u003e\u003c/span\u003e for M\u0026thinsp;=\u0026thinsp;9 events (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003ea). The time at which the slope is measured increases with magnitude, being approximately 0.2 s for M\u0026thinsp;=\u0026thinsp;4 earthquakes and about 1 s for M\u0026thinsp;=\u0026thinsp;9 events. (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003eb). The times measured here are related to the plateau time of LPDT curves which are, in turn, a proxy for the peak of the MRF\u003csup\u003e23 20\u003c/sup\u003e. The choice of the time to measure the slope is arbitrary (half of the plateau time of the curves), thus both the times and the slope values do not have an obvious physical meaning. However, their scaling over the entire magnitude range (as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e) provides a frame to interpret our observations.\u003c/p\u003e \u003cp\u003eDuring the preparatory phase of earthquake ruptures, the crack size increases slowly at few percentages of the shear wave speed\u003csup\u003e34\u003c/sup\u003e until it reaches a critical size related to the frictional parameters (e.g., the slip-weakening critical distance, D\u003csub\u003ec\u003c/sub\u003e). At this point, the unstable fracture expands at an increasing velocity, with an acceleration stage that triggers the dynamic propagation\u003csup\u003e1234\u003c/sup\u003e. The time during which rupture accelerates to the dynamic propagation is well defined and related to the final slip (and therefore to the earthquake magnitude)\u003csup\u003e3\u003c/sup\u003e. In this view, the observed scaling of both times and slopes could be the footprint of this unstable acceleration phase and corroborates the idea that earthquakes are different already at the beginning of the rupture process. The slip-weakening critical distance, D\u003csub\u003ec\u003c/sub\u003e, increases with magnitude \u003csup\u003e3\u003c/sup\u003e. Intuitively, longer times are needed to reach this condition on a limited portion of the fault for larger events, thus implying a smaller rate of amplitude growth. Therefore, a long duration of the rupture acceleration (i.e. large D\u003csub\u003ec\u003c/sub\u003e, large magnitude) is associated to a low rate of amplitude increase (i.e., a small slope value). Conversely, for small events, the smaller D\u003csub\u003ec\u003c/sub\u003e value is reached in a shorter time on a limited portion of the fault, with consequent smaller rates of amplitude growth. A short duration of the acceleration phase is therefore associated to high values of the amplitude growth rate (large slopes). Thus, the nucleation phase is different for small and large events and this difference can be seen within a \u0026ldquo;characteristic time\u0026rdquo; of the process itself.\u003c/p\u003e \u003cp\u003eThe observed slope decrease with magnitude can be related to the effect of variable dynamic stress drop and/or rupture velocity in the initial stage of the rupture of small and large earthquakes, possibly triggered by the acceleration phase during the quasi-static rupture nucleation. Indeed, based on the dynamic-consistent model for an expanding shear circular crack of Sato \u0026amp; Hirasawa (1973) \u003csup\u003e24\u003c/sup\u003e, the relation between the early P radiated displacement pulse \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({{\\Omega }}_{P}\\left(t\\right)\\)\u003c/span\u003e\u003c/span\u003e, the dynamic stress drop (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\tau }_{e}\\)\u003c/span\u003e\u003c/span\u003e) and the rupture velocity (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({v}_{R}\\)\u003c/span\u003e\u003c/span\u003e) during the rupture growth phase can be written as: \u003csup\u003e25\u003c/sup\u003e:\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$\\begin{array}{c}{{\\Omega }}_{P}\\left(t\\right)=\\frac{2\\pi {\\Delta }{v}_{o}{v}_{R}^{2}}{{\\left(1-{\\zeta }^{2}\\right)}^{2}}{t}^{2}\\approx \\frac{2\\pi {v}_{R}^{3}{\\tau }_{e}}{\\mu {\\left(1-{\\zeta }^{2}\\right)}^{2}}{t}^{2}\\#\\left(1\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\Delta }{v}_{o}\\)\u003c/span\u003e\u003c/span\u003eis peak slip velocity, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\mu\\)\u003c/span\u003e\u003c/span\u003e the rigidity at the source region and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\zeta =\\frac{{v}_{R}}{{v}_{P}}\\text{sin}\\theta\\)\u003c/span\u003e\u003c/span\u003e is the P-wave \u003cem\u003eapparent\u003c/em\u003e Mach number \u003csup\u003e25\u003c/sup\u003e, with \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\theta\\)\u003c/span\u003e\u003c/span\u003e being the angle between the ray take-off direction and the normal to the circular fault. The term including the apparent Mach number accounts for rupture directivity, depending on the receiver view angle and the rupture to wave velocity ratio. In (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e) the relation between the peak slip velocity and dynamic stress drop is inferred from the dynamic models of a propagating shear crack by Kostrov(1964)\u003csup\u003e26\u003c/sup\u003e and Dahlen(1974) \u003csup\u003e27\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eAveraging over \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\theta\\)\u003c/span\u003e\u003c/span\u003e, Eq.\u0026nbsp;1 changes to:\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$$\\begin{array}{c}{{\\Omega }}_{c}\\left(t\\right)\\approx \\frac{2\\pi C\\left({v}_{R}\\right){\\tau }_{e}}{\\mu }{t}^{2}\\#\\left(2\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere the rupture velocity factor\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equc\" name=\"EquationSource\"\u003e\n$$\\begin{array}{c}C\\left({v}_{R}\\right)=\\frac{\\pi }{2}\\frac{{v}_{R}^{3}\\left(2-{\\left(\\frac{{v}_{R}}{{v}_{P}}\\right)}^{2}\\right)}{{\\left(1-{\\left(\\frac{{v}_{R}}{{v}_{P}}\\right)}^{2}\\right)}^{\\frac{3}{2}}}\\#\\left(3\\right)\\end{array}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eshows an exponential increase with \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({v}_{R}\\)\u003c/span\u003e\u003c/span\u003e. According to Eq.\u0026nbsp;2, both the rupture velocity and/or dynamic stress drop control the rise of the radiated P pulse displacement and therefore its initial slope.\u003c/p\u003e \u003cp\u003eAn example of the effect of a varying rupture velocity with magnitude on LPDT curves and slope by assuming a Sato and Hirasawa (1973)\u003csup\u003e24\u003c/sup\u003e kinematic source model of a circular shear crack is presented in the Supplemental Material (Text S1 and Figure S5). Synthetic tests support the idea that a decreasing value of rupture velocity with magnitude could be responsible for the observed decrease of the initial slope of P-wave displacement. Additional constraints provided by more complex and realistic numerical simulations are necessary to understand in which physical conditions and and how these two parameters may play a role during the initial stage of the rupture propagation.\u003c/p\u003e \u003cp\u003eThe results obtained here contrast with the findings of previous studies \u003csup\u003e16,17,19\u003c/sup\u003e showing that the initial rate of the P-wave amplitude is almost independent of the earthquake magnitude. While we do not question the validity of these previous studies, significant differences, related to the data selection and processing may justify the different observations we have obtained here. An extensive discussion about the possible effects of data processing is provided in Colombelli et al \u003csup\u003e22\u003c/sup\u003e and similar considerations can be applied to the results of the current analysis. Our results, instead, do not exclude the evolution of the source process on a longer time scale, as seen by other authors\u003csup\u003e17 21\u003c/sup\u003e. At the time scale of seconds to tens-of-seconds, Meier at al \u003csup\u003e17\u003c/sup\u003e suggested that the MRFs of different events with increasing magnitude have all the same initial, growth-rate amplitude (or initial amplitude rate slope). At the same time scale, Melgar and Hayes \u003csup\u003e21\u003c/sup\u003eprovided evidence that the initial growth-rate amplitude is higher for larger events and lower for smaller ones.\u003c/p\u003e \u003cp\u003eIn Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e we suggest a hybrid model for the MRFs that could bring together observations at short \u003csup\u003e20 22\u003c/sup\u003eand long time-scales \u003csup\u003e17 21\u003c/sup\u003e. In this model, at short time scales, the growth-rate amplitude would follow an inverse scaling with magnitude, with large events starting with a slower amplitude increase and small events beginning with a higher amplitude growth-rate. Figure\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003ea shows the early evolution of the earthquake rupture, as resulting from our observations. The figure shows the initial slopes of LPDT curves a s function of time for all earthquakes used in this work. Each curve is plotted up to the time t\u003csub\u003ehalf\u003c/sub\u003e as resulting from our analysis. Circle markers are single event data; squares are averaged value in the magnitude bin. Panel (b) suggests a schematic representation of the later evolution of the earthquake source time function. At longer time scales, the growth-rate amplitude would then change either in the sense of increasing with magnitude, or maintaining the same growth rate with magnitude, leading to the generation of absolute peak values of the MRFs, which are consistent (in time and amplitude) with those predicted from the scaling laws \u003csup\u003e28\u003c/sup\u003e. With current data and methods, we do not have adequate elements to discriminate the growing behavior of MRFs at longer time scales, approaching to the peak values.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eRelevant implications of the proposed model are also related to the rapid assessment of the earthquake size for Earthquake Early Warning Systems (EEWS), for which the possibility of discriminating a large event from a small one is crucial to provide reliable prediction of the incoming ground shaking level, for the prompt activation of emergency procedures and real-time risk mitigation actions. This could be achieved either using direct amplitude measurement or passing through the computation and modelling of amplitude vs. time curves. A simple threshold-based warning system, based on the initial slope measurement, would allow for the discrimination of small/large events within very short time windows (\u0026lt;\u0026thinsp;1 s), opening to new perspectives for the practical applications of EEWS.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAcknowledgements.\u003c/strong\u003e The research was funded by the University of Naples Federico II and by the Italian Ministry of University and Research (within the framework of the National Operative Programme PON-AIM AIM1834927 \u0026ndash; 3).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor contributions\u003c/strong\u003e. V.L., S.C. and A.Z. equally contributed to the concept of the work and to the methodological developments. V.L. analysed data, prepared the figures and wrote the original version of the manuscript. S.C. and A.Z. contributed to the interpretation and discussion of results and to the revision of the manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting interests.\u003c/strong\u003e The authors declare no competing interests.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMaterials \u0026amp; Correspondence.\u003c/strong\u003e Correspondence and material requests should be addressed to Dr. Simona Colombelli ([email protected]).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData availability.\u0026nbsp;\u003c/strong\u003eThe waveform data used for the analysis in the study are availableat the IRIS web data service (http://service.iris.edu, last access 2024/02/18), at the Northern California Earthquake Data Center (http://service.ncedc.org, last access 2024/02/18) and at the European Mediterranean Seismological Centre (http://www.seismicportal.eu, last access 2024/02/18). All analysis and figures are performed using Phyton.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eAmpuero, J. ‐P., Vilotte, J. ‐P. \u0026amp; S\u0026aacute;nchez‐Sesma, F. J. Nucleation of rupture under slip dependent friction law: Simple models of fault zone. \u003cem\u003eJ Geophys Res Solid Earth\u003c/em\u003e \u003cstrong\u003e107\u003c/strong\u003e, (2002).\u003c/li\u003e\n\u003cli\u003eDascalu, C., Ionescu, I. R. \u0026amp; Campillo, M. Fault finiteness and initiation of dynamic shear instability. \u003cem\u003eEarth Planet Sci Lett\u003c/em\u003e \u003cstrong\u003e177\u003c/strong\u003e, 163\u0026ndash;176 (2000).\u003c/li\u003e\n\u003cli\u003eLatour, S., Schubnel, A., Nielsen, S., Madariaga, R. \u0026amp; Vinciguerra, S. Characterization of nucleation during laboratory earthquakes. \u003cem\u003eGeophys Res Lett\u003c/em\u003e \u003cstrong\u003e40\u003c/strong\u003e, 5064\u0026ndash;5069 (2013).\u003c/li\u003e\n\u003cli\u003eNielsen, S., Taddeucci, J. \u0026amp; Vinciguerra, S. Experimental observation of stick-slip instability fronts. \u003cem\u003eGeophys J Int\u003c/em\u003e \u003cstrong\u003e180\u003c/strong\u003e, 697\u0026ndash;702 (2010).\u003c/li\u003e\n\u003cli\u003eIshihara, Y., Fukao, Y., Yamada, I. \u0026amp; Aoki, H. Rising slope of moment rate functions: The 1989 earthquakes off east coast of Honshu. \u003cem\u003eGeophys Res Lett\u003c/em\u003e \u003cstrong\u003e19\u003c/strong\u003e, 873\u0026ndash;876 (1992).\u003c/li\u003e\n\u003cli\u003eAbercrombie, R. \u0026amp; Mori, J. \u003cem\u003eLocal Observations of the Onset of a Large Earthquake: 28 June 1992 Landers, California\u003c/em\u003e. \u003cem\u003eBulletin of the Seismological Society of America\u003c/em\u003e vol. 84 (1994).\u003c/li\u003e\n\u003cli\u003eEllsworth, W. L. \u0026amp; Beroza, G. C. Seismic Evidence for an Earthquake Nucleation Phase. \u003cem\u003eScience (1979)\u003c/em\u003e \u003cstrong\u003e268\u003c/strong\u003e, 851\u0026ndash;855 (1995).\u003c/li\u003e\n\u003cli\u003eMori, J. \u0026amp; Kanamori, H. Initial rupture of earthquakes in the 1995 Ridgecrest, California Sequence. \u003cem\u003eGeophys Res Lett\u003c/em\u003e \u003cstrong\u003e23\u003c/strong\u003e, 2437\u0026ndash;2440 (1996).\u003c/li\u003e\n\u003cli\u003eGomberg, J. Unsettled earthquake nucleation. \u003cem\u003eNat Geosci\u003c/em\u003e \u003cstrong\u003e11\u003c/strong\u003e, 463\u0026ndash;464 (2018).\u003c/li\u003e\n\u003cli\u003eWesnousky, S. G. Predicting the endpoints of earthquake ruptures. \u003cem\u003eNature\u003c/em\u003e \u003cstrong\u003e444\u003c/strong\u003e, 358\u0026ndash;360 (2006).\u003c/li\u003e\n\u003cli\u003eBeroza, G. C. \u0026amp; Ellsworth, W. L. Properties of the seismic nucleation phase. \u003cem\u003eTectonophysics\u003c/em\u003e \u003cstrong\u003e261\u003c/strong\u003e, 209\u0026ndash;227 (1996).\u003c/li\u003e\n\u003cli\u003eIio, Y. Observations of the slow initial phase generated by microearthquakes: Implications for earthquake nucleation and propagation. \u003cem\u003eJ Geophys Res Solid Earth\u003c/em\u003e \u003cstrong\u003e100\u003c/strong\u003e, 15333\u0026ndash;15349 (1995).\u003c/li\u003e\n\u003cli\u003eOlson, E. L. \u0026amp; Allen, R. M. The deterministic nature of earthquake rupture. \u003cem\u003eNature\u003c/em\u003e \u003cstrong\u003e438\u003c/strong\u003e, 212\u0026ndash;215 (2005).\u003c/li\u003e\n\u003cli\u003eRice, J. R. Elastic wave emission from damage processes. \u003cem\u003eJ Nondestr Eval\u003c/em\u003e \u003cstrong\u003e1\u003c/strong\u003e, 215\u0026ndash;224 (1980).\u003c/li\u003e\n\u003cli\u003eAagaard, B. T. \u0026amp; Heaton, T. H. Constraining fault constitutive behavior with slip and stress heterogeneity. \u003cem\u003eJ Geophys Res Solid Earth\u003c/em\u003e \u003cstrong\u003e113\u003c/strong\u003e, (2008).\u003c/li\u003e\n\u003cli\u003eMeier, M., Heaton, T. \u0026amp; Clinton, J. Evidence for universal earthquake rupture initiation behavior. \u003cem\u003eGeophys Res Lett\u003c/em\u003e \u003cstrong\u003e43\u003c/strong\u003e, 7991\u0026ndash;7996 (2016).\u003c/li\u003e\n\u003cli\u003eMeier, M.-A., Ampuero, J. P. \u0026amp; Heaton, T. H. The hidden simplicity of subduction megathrust earthquakes. \u003cem\u003eScience (1979)\u003c/em\u003e \u003cstrong\u003e357\u003c/strong\u003e, 1277\u0026ndash;1281 (2017).\u003c/li\u003e\n\u003cli\u003eScherbaum, F. \u0026amp; Bouin, M.-P. FIR filter effects and nucleation phases. \u003cem\u003eGeophys J Int\u003c/em\u003e \u003cstrong\u003e130\u003c/strong\u003e, 661\u0026ndash;668 (1997).\u003c/li\u003e\n\u003cli\u003eTrugman, D. T., Page, M. T., Minson, S. E. \u0026amp; Cochran, E. S. Peak Ground Displacement Saturates Exactly When Expected: Implications for Earthquake Early Warning. \u003cem\u003eJ Geophys Res Solid Earth\u003c/em\u003e \u003cstrong\u003e124\u003c/strong\u003e, 4642\u0026ndash;4653 (2019).\u003c/li\u003e\n\u003cli\u003eColombelli, S., Zollo, A., Festa, G. \u0026amp; Picozzi, M. Evidence for a difference in rupture initiation between small and large earthquakes. \u003cem\u003eNat Commun\u003c/em\u003e \u003cstrong\u003e5\u003c/strong\u003e, (2014).\u003c/li\u003e\n\u003cli\u003eMelgar, D. \u0026amp; Hayes, G. P. Characterizing large earthquakes before rupture is complete. \u003cem\u003eSci Adv\u003c/em\u003e \u003cstrong\u003e5\u003c/strong\u003e, (2019).\u003c/li\u003e\n\u003cli\u003eColombelli, S., Festa, G. \u0026amp; Zollo, A. Early rupture signals predict the final earthquake size. \u003cem\u003eGeophys J Int\u003c/em\u003e \u003cstrong\u003e223\u003c/strong\u003e, 692\u0026ndash;706 (2020).\u003c/li\u003e\n\u003cli\u003eNazeri, S., Colombelli, S. \u0026amp; Zollo, A. Fast and accurate determination of earthquake moment, rupture length and stress release for the 2016-2017 Central Italy seismic sequence. \u003cem\u003eGeophys J Int\u003c/em\u003e \u003cstrong\u003e217\u003c/strong\u003e, 1425\u0026ndash;1432 (2019).\u003c/li\u003e\n\u003cli\u003eSATO, T. \u0026amp; HIRASAWA, T. Body wave spectra from propagating shear cracks. \u003cem\u003eJournal of Physics of the Earth\u003c/em\u003e \u003cstrong\u003e21\u003c/strong\u003e, 415\u0026ndash;431 (1973).\u003c/li\u003e\n\u003cli\u003eBoatwright, J. A spectral theory for circular seismic sources; simple estimates of source dimension, dynamic stress drop, and radiated seismic energy. \u003cem\u003eBulletin of the Seismological Society of America\u003c/em\u003e (1980).\u003c/li\u003e\n\u003cli\u003eKostrov, B. V. Selfsimilar problems of propagation of shear cracks. \u003cem\u003eJournal of Applied Mathematics and Mechanics\u003c/em\u003e \u003cstrong\u003e28\u003c/strong\u003e, 1077\u0026ndash;1087 (1964).\u003c/li\u003e\n\u003cli\u003eDahlen, F. A. On the ratio of \u003cem\u003eP\u003c/em\u003e -wave to \u003cem\u003eS\u003c/em\u003e -wave corner frequencies for shallow earthquake sources. \u003cem\u003eBulletin of the Seismological Society of America\u003c/em\u003e \u003cstrong\u003e64\u003c/strong\u003e, 1159\u0026ndash;1180 (1974).\u003c/li\u003e\n\u003cli\u003eScholz, C. H. \u0026amp; Cowie, P. A. Determination of total strain from faulting using slip measurements. \u003cem\u003eNature\u003c/em\u003e \u003cstrong\u003e346\u003c/strong\u003e, 837\u0026ndash;839 (1990).\u003c/li\u003e\n\u003c/ol\u003e"},{"header":"Method","content":"\u003cp\u003eThe initial dataset consists of 200 earthquakes from worldwide locations (see Fig.\u0026nbsp;1). The events span a time ranging between 2003 and 2023 and moment magnitude ranging between 4 and 9 (see Supplemental Material, Table \u003cspan class=\"InternalRef\"\u003eS1\u003c/span\u003e). The waveforms are recorded either from velocimeter networks or accelerometer networks, depending on data availability. For each earthquake, we select the closest five stations to event epicenter. Depending on the available recording sensor (velocimeter or accelerometer) (see Fig.\u0026nbsp;1) we perform a single or double integration of signal, respectively, to get displacement and we finally apply a high-pass Butterworth filter with cut-off frequency of 0.075Hz to remove possible baseline effects. We compute the logarithm of the peak of the absolute displacement starting from the P-arrival time at the station. We average the obtained LPDT curves by stations to get the final LPDT curve on which we evaluate the intial slope, as explained below. We perform an interpolation of the real LPDT curve with an exponential function of the form:\u003c/p\u003e\n\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ1\" class=\"mathdisplay\"\u003e$${\\varvec{L}\\varvec{P}\\varvec{D}\\varvec{T}}_{\\varvec{t}\\varvec{e}\\varvec{o}}={\\varvec{L}\\varvec{P}\\varvec{D}\\varvec{T}}_{\\varvec{e}\\varvec{n}\\varvec{d}}\\left(1-{\\varvec{e}}^{-\\raisebox{1ex}{$\\varvec{t}$}\\!\\left/ \\!\\raisebox{-1ex}{${\\varvec{t}}_{1}$}\\right.}\\right)-{\\varvec{L}\\varvec{P}\\varvec{D}\\varvec{T}}_{0}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({LPDT}_{0}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({LPDT}_{end}\\)\u003c/span\u003e\u003c/span\u003e are the first and the last point of the curve respectively, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({t}_{1}\\)\u003c/span\u003e\u003c/span\u003e is a fit parameter that simply allows the function to bend towards the plateau level. The interpolation is used to avoid numerical noise caused by the discontinuity of real curves.\u003c/p\u003e\n\u003cp\u003eFor each curve, we compute the curvature as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(curvature= \\frac{\\left|{y}^{{\\prime }{\\prime }}\\right|}{{\\left(1+{\\left({y}^{{\\prime }}\\right)}^{2}\\right)}^{\\raisebox{1ex}{$3$}\\!\\left/ \\!\\raisebox{-1ex}{$2$}\\right.}}\\)\u003c/span\u003e\u003c/span\u003e where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({y}^{{\\prime }}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({y}^{{\\prime }{\\prime }}\\)\u003c/span\u003e\u003c/span\u003e are the numerical first and second derivative of the interpolated LPDT curve. The curvature allows to quantify how much a function is deviated from a linear behaviour. Figure S3 and S4 of the Supplemental Material show examples of curvature computation, for different curves.\u003c/p\u003e\n\u003cp\u003eWe then measure the time \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({t}_{HALF}\\)\u003c/span\u003e\u003c/span\u003e, corresponding to half of the time where the maximum of the curvature is reached. Finally, we evaluate the slope of the real LPDT curve between \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({t}_{MIN}=0.05 s\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({t}_{HALF}\\)\u003c/span\u003e\u003c/span\u003e. The choice of the starting point for the slope evaluation is done to account for errors in affect the manual picking of waveforms. The slope is obtained from the line that crosses the LPDT between \u003cem\u003et\u003c/em\u003e\u003csub\u003e\u003cem\u003eMIN\u003c/em\u003e\u003c/sub\u003e and t\u003csub\u003eHALF\u003c/sub\u003e, as it follows:\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(Slope= \\frac{LPDT\\left({t}_{HALF}\\right)-LPDT\\left({t}_{MIN}\\right)}{{t}_{HALF}-{t}_{MIN}}\\)\u003c/span\u003e \u003c/span\u003e (\u003cspan class=\"CitationRef\"\u003e5\u003c/span\u003e).\u003c/p\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-3967674/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3967674/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Earthquakes are among the most destructive natural hazards. The energy released by an earthquake can be quantified by its magnitude. However, predicting how much energy the earthquake will release before the end of its rupture process represents a challenging question in geohazards. The way earthquake ruptures grow and arrest determines the final event size: small-to-moderate ruptures evolve in few seconds with typical lengths of few kilometers, while large-to-huge events develop in tens of seconds or more, involving lengths of several hundred kilometers. If the rupture process starts in the same way for small and large earthquakes, no deterministic prediction of the final size is feasible, until the process has finished. On the contrary, if the source mechanism starts differently from its early beginning, real-time proxies can be measured on seismic waves to discriminate the final event size. Here we show that the initial ground displacement growth is differently for small and large earthquakes, based on the analysis of an unprecedented catalog of seismic waveforms from worldwide earthquakes. The result supports the hypothesis of early predictable event magnitude for a wide range of different size earthquakes in diverse geological settings. This study confirms that the measure of the initial growth of displacement can be used as a parameter for a fast magnitude estimation, making it feasible for future implementation in early warning systems.","manuscriptTitle":"The deterministic behaviour of earthquake rupture beginning","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-02-26 05:44:19","doi":"10.21203/rs.3.rs-3967674/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"17cd7c39-0944-4001-abab-4585f589de3e","owner":[],"postedDate":"February 26th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":28937807,"name":"Earth and environmental sciences/Solid Earth sciences/Seismology"},{"id":28937808,"name":"Earth and environmental sciences/Natural hazards"}],"tags":[],"updatedAt":"2024-11-20T09:01:35+00:00","versionOfRecord":[],"versionCreatedAt":"2024-02-26 05:44:19","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3967674","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3967674","identity":"rs-3967674","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}