1
|
Rundle JB, Yazbeck J, Donnellan A, Fox G, Ludwig LG, Heflin M, Crutchfield J. Optimizing Earthquake Nowcasting With Machine Learning: The Role of Strain Hardening in the Earthquake Cycle. Earth Space Sci 2022; 9:e2022EA002343. [PMID: 36583191 PMCID: PMC9787018 DOI: 10.1029/2022ea002343] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 03/25/2022] [Revised: 07/27/2022] [Accepted: 10/02/2022] [Indexed: 06/17/2023]
Abstract
Nowcasting is a term originating from economics, finance, and meteorology. It refers to the process of determining the uncertain state of the economy, markets or the weather at the current time by indirect means. In this paper, we describe a simple two-parameter data analysis that reveals hidden order in otherwise seemingly chaotic earthquake seismicity. One of these parameters relates to a mechanism of seismic quiescence arising from the physics of strain-hardening of the crust prior to major events. We observe an earthquake cycle associated with major earthquakes in California, similar to what has long been postulated. An estimate of the earthquake hazard revealed by this state variable time series can be optimized by the use of machine learning in the form of the Receiver Operating Characteristic skill score. The ROC skill is used here as a loss function in a supervised learning mode. Our analysis is conducted in the region of 5° × 5° in latitude-longitude centered on Los Angeles, a region which we used in previous papers to build similar time series using more involved methods (Rundle & Donnellan, 2020, https://doi.org/10.1029/2020EA001097; Rundle, Donnellan et al., 2021, https://doi.org/10.1029/2021EA001757; Rundle, Stein et al., 2021, https://doi.org/10.1088/1361-6633/abf893). Here we show that not only does the state variable time series have forecast skill, the associated spatial probability densities have skill as well. In addition, use of the standard ROC and Precision (PPV) metrics allow probabilities of current earthquake hazard to be defined in a simple, straightforward, and rigorous way.
Collapse
Affiliation(s)
- John B. Rundle
- Department of PhysicsUniversity of CaliforniaDavisCAUSA
- Santa Fe InstituteSanta FeNMUSA
- Department of Earth and Planetary ScienceUniversity of CaliforniaDavisCAUSA
- Program in Public HealthUniversity of CaliforniaIrvineCAUSA
| | - Joe Yazbeck
- Department of PhysicsUniversity of CaliforniaDavisCAUSA
| | - Andrea Donnellan
- Jet Propulsion Laboratory California Institute of TechnologyPasadenaCAUSA
| | | | | | - Michael Heflin
- Jet Propulsion Laboratory California Institute of TechnologyPasadenaCAUSA
| | | |
Collapse
|
2
|
Saylor C, Rundle JB, Donnellan A. Multifractal Analysis of a Seismic Moment Distribution Obtained From InSAR Inversion. Earth Space Sci 2021; 8:e2020EA001433. [PMID: 34692923 PMCID: PMC8519160 DOI: 10.1029/2020ea001433] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 08/24/2020] [Revised: 07/05/2021] [Accepted: 08/23/2021] [Indexed: 06/13/2023]
Abstract
Interferometric synthetic aperture radar (InSAR) interferograms contain valuable information about the fault systems hidden beneath the surface of the Earth. In a new approach, we aim to fit InSAR ground deformation data using a distribution of multiple seismic point sources whose parameters are found by a genetic algorithm. The resulting source distribution could provide another useful tool in solving the difficult problem of accurately mapping earthquake faults. We apply the algorithm to an ALOS-2 InSAR interferogram and perform a multifractal analysis on the resulting distribution, finding that it exhibits multifractal properties. We report first results and discuss advantages and disadvantages of this approach.
Collapse
Affiliation(s)
- Cameron Saylor
- Department of Physics and AstronomyUniversity of California, DavisDavisCAUSA
| | - John B. Rundle
- Department of Physics and AstronomyUniversity of California, DavisDavisCAUSA
- Department of Earth and Planetary ScienceUniversity of California, DavisDavisCAUSA
- Sante Fe InstituteSante FeNMUSA
| | - Andrea Donnellan
- Jet Propulsion LaboratoryCalifornia Institute of TechnologyPasadenaCAUSA
| |
Collapse
|
3
|
Rundle JB, Stein S, Donnellan A, Turcotte DL, Klein W, Saylor C. Reports on progress in physics the complex dynamics of earthquake fault systems: new approaches to forecasting and nowcasting of earthquakes. Rep Prog Phys 2021; 84:076801. [PMID: 33857928 DOI: 10.1088/1361-6633/abf893] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/20/2020] [Accepted: 04/15/2021] [Indexed: 06/12/2023]
Abstract
Charles Richter's observation that 'only fools and charlatans predict earthquakes,' reflects the fact that despite more than 100 years of effort, seismologists remain unable to do so with reliable and accurate results. Meaningful prediction involves specifying the location, time, and size of an earthquake before it occurs to greater precision than expected purely by chance from the known statistics of earthquakes in an area. In this context, 'forecasting' implies a prediction with a specification of a probability of the time, location, and magnitude. Two general approaches have been used. In one, the rate of motion accumulating across faults and the amount of slip in past earthquakes is used to infer where and when future earthquakes will occur and the shaking that would be expected. Because the intervals between earthquakes are highly variable, these long-term forecasts are accurate to no better than a hundred years. They are thus valuable for earthquake hazard mitigation, given the long lives of structures, but have clear limitations. The second approach is to identify potentially observable changes in the Earth that precede earthquakes. Various precursors have been suggested, and may have been real in certain cases, but none have yet proved to be a general feature preceding all earthquakes or to stand out convincingly from the normal variability of the Earth's behavior. However, new types of data, models, and computational power may provide avenues for progress using machine learning that were not previously available. At present, it is unclear whether deterministic earthquake prediction is possible. The frustrations of this search have led to the observation that (echoing Yogi Berra) 'it is difficult to predict earthquakes, especially before they happen.' However, because success would be of enormous societal benefit, the search for methods of earthquake prediction and forecasting will likely continue. In this review, we note that the focus is on anticipating the earthquake rupture before it occurs, rather than characterizing it rapidly just after it occurs. The latter is the domain of earthquake early warning, which we do not treat in detail here, although we include a short discussion in the machine learning section at the end.
Collapse
Affiliation(s)
- John B Rundle
- Department of Physics and Astronomy, University of California, Davis, CA 95616, United States of America
- Department of Earth & Planetary Sciences, University of California, Davis, CA 95616, United States of America
- Santa Fe Institute, 1399 Hyde Park Rd, Santa Fe, NM 87501, United States of America
| | - Seth Stein
- Department of Earth and Planetary Sciences and Institute for Policy Research, Northwestern University, Evanston, IL 60208, United States of America
| | - Andrea Donnellan
- Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, United States of America
| | - Donald L Turcotte
- Department of Earth & Planetary Sciences, University of California, Davis, CA 95616, United States of America
| | - William Klein
- Department of Physics, Boston University, Boston, MA 02215, United States of America
| | - Cameron Saylor
- Department of Physics and Astronomy, University of California, Davis, CA 95616, United States of America
| |
Collapse
|
4
|
Ortez R, Rundle JB, Turcotte DL. Universality class for loopless invasion percolation models and a percolation avalanche burst model for hydraulic fracturing. Phys Rev E 2021; 103:012310. [PMID: 33601580 DOI: 10.1103/physreve.103.012310] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/21/2020] [Accepted: 09/07/2020] [Indexed: 11/07/2022]
Abstract
Invasion percolation is a model that was originally proposed to describe growing networks of fractures. Here we describe a loopless algorithm on random lattices, coupled with an avalanche-based model for bursts. The model reproduces the characteristic b-value seismicity and spatial distribution of bursts consistent with earthquakes resulting from hydraulic fracturing ("fracking"). We test models for both site invasion percolation and bond invasion percolation. These have differences on the scale of site and bond lengths l. But since the networks are characterized by their large-scale behavior, l≪L, we find small differences between scaling exponents. Though data may not differentiate between models, our results suggest that both models belong to different universality classes.
Collapse
Affiliation(s)
- Ronaldo Ortez
- Department of Physics, University of California, Davis, California 95616, USA
| | - John B Rundle
- Department of Physics, University of California, Davis, California 95616, USA.,Department of Earth and Planetary Science, University of California, Davis, California 95616, USA.,Santa Fe Institute, Santa Fe, New Mexico 87501, USA
| | - Donald L Turcotte
- Department of Geology, University of California, Davis, California 95616, USA
| |
Collapse
|
5
|
Rundle JB, Ortez R, Kønigslieb J, Turcotte DL. Constrained Invasion Percolation Model: Growth via Leath Bursts and the Origin of Seismic b-Value. Phys Rev Lett 2020; 124:068501. [PMID: 32109126 DOI: 10.1103/physrevlett.124.068501] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/17/2019] [Accepted: 11/27/2019] [Indexed: 06/10/2023]
Abstract
We analyze a new model for growing networks, the constrained Leath invasion percolation model. Cluster dynamics are characterized by bursts in space and time. The model quantitatively reproduces the observed frequency-magnitude scaling of earthquakes in the limit that the occupation probability approaches the critical bond percolation probability in d=2. The model may have application to other systems characterized by burst dynamics.
Collapse
Affiliation(s)
- John B Rundle
- Department of Physics, One Shields Avenue, University of California, Davis, California 95616, USA
- Department of Geology, One Shields Avenue, University of California, Davis, California 95616, USA
- Santa Fe Institute, Santa Fe, New Mexico 87501, USA
| | - Ronaldo Ortez
- Department of Physics, One Shields Avenue, University of California, Davis, California 95616, USA
| | - Joachim Kønigslieb
- Department of Physics, One Shields Avenue, University of California, Davis, California 95616, USA
| | - Donald L Turcotte
- Department of Geology, One Shields Avenue, University of California, Davis, California 95616, USA
| |
Collapse
|
6
|
Abstract
Seismic nowcasting uses counts of small earthquakes as proxy data to estimate the current dynamical state of an earthquake fault system. The result is an earthquake potential score that characterizes the current state of progress of a defined geographic region through its nominal earthquake "cycle." The count of small earthquakes since the last large earthquake is the natural time that has elapsed since the last large earthquake (Varotsos et al., 2006, https://doi.org/10.1103/PhysRevE.74.021123). In addition to natural time, earthquake sequences can also be analyzed using Shannon information entropy ("information"), an idea that was pioneered by Shannon (1948, https://doi.org/10.1002/j.1538-7305.1948.tb01338.x). As a first step to add seismic information entropy into the nowcasting method, we incorporate magnitude information into the natural time counts by using event self-information. We find in this first application of seismic information entropy that the earthquake potential score values are similar to the values using only natural time. However, other characteristics of earthquake sequences, including the interevent time intervals, or the departure of higher magnitude events from the magnitude-frequency scaling line, may contain additional information.
Collapse
Affiliation(s)
- John B Rundle
- Department of Physics University of California Davis CA USA.,Santa Fe Institute Santa Fe NM USA.,Department of Earth and Planetary Science University of California Davis CA USA.,Jet Propulsion Laboratory California Institute of Technology Pasadena CA USA.,Tohoku University Sendai Japan
| | - Alexis Giguere
- Department of Physics University of California Davis CA USA
| | - Donald L Turcotte
- Department of Earth and Planetary Science University of California Davis CA USA
| | - James P Crutchfield
- Department of Physics University of California Davis CA USA.,Santa Fe Institute Santa Fe NM USA
| | - Andrea Donnellan
- Jet Propulsion Laboratory California Institute of Technology Pasadena CA USA
| |
Collapse
|
7
|
Rundle JB, Giguere A, Turcotte DL, Crutchfield JP, Donnellan A. Global Seismic Nowcasting With Shannon Information Entropy. Earth Space Sci 2019; 6:191-197. [PMID: 30854411 PMCID: PMC6392127 DOI: 10.1029/2018ea000464] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 09/04/2018] [Revised: 10/17/2018] [Accepted: 11/20/2018] [Indexed: 06/09/2023]
Abstract
Seismic nowcasting uses counts of small earthquakes as proxy data to estimate the current dynamical state of an earthquake fault system. The result is an earthquake potential score that characterizes the current state of progress of a defined geographic region through its nominal earthquake "cycle." The count of small earthquakes since the last large earthquake is the natural time that has elapsed since the last large earthquake (Varotsos et al., 2006, https://doi.org/10.1103/PhysRevE.74.021123). In addition to natural time, earthquake sequences can also be analyzed using Shannon information entropy ("information"), an idea that was pioneered by Shannon (1948, https://doi.org/10.1002/j.1538-7305.1948.tb01338.x). As a first step to add seismic information entropy into the nowcasting method, we incorporate magnitude information into the natural time counts by using event self-information. We find in this first application of seismic information entropy that the earthquake potential score values are similar to the values using only natural time. However, other characteristics of earthquake sequences, including the interevent time intervals, or the departure of higher magnitude events from the magnitude-frequency scaling line, may contain additional information.
Collapse
Affiliation(s)
- John B. Rundle
- Department of PhysicsUniversity of CaliforniaDavisCAUSA
- Santa Fe InstituteSanta FeNMUSA
- Department of Earth and Planetary ScienceUniversity of CaliforniaDavisCAUSA
- Jet Propulsion LaboratoryCalifornia Institute of TechnologyPasadenaCAUSA
- Tohoku UniversitySendaiJapan
| | | | - Donald L. Turcotte
- Department of Earth and Planetary ScienceUniversity of CaliforniaDavisCAUSA
| | - James P. Crutchfield
- Department of PhysicsUniversity of CaliforniaDavisCAUSA
- Santa Fe InstituteSanta FeNMUSA
| | - Andrea Donnellan
- Jet Propulsion LaboratoryCalifornia Institute of TechnologyPasadenaCAUSA
| |
Collapse
|
8
|
Luginbuhl M, Rundle JB, Turcotte DL. Statistical physics models for aftershocks and induced seismicity. Philos Trans A Math Phys Eng Sci 2018; 377:rsta.2017.0397. [PMID: 30478209 PMCID: PMC6282405 DOI: 10.1098/rsta.2017.0397] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 10/07/2018] [Indexed: 06/09/2023]
Abstract
A standard approach to quantifying the seismic hazard is the relative intensity (RI) method. It is assumed that the rate of seismicity is constant in time and the rate of occurrence of small earthquakes is extrapolated to large earthquakes using Gutenberg-Richter scaling. We introduce nowcasting to extend RI forecasting to time-dependent seismicity, for example, during an aftershock sequence. Nowcasting uses 'natural time'; in seismicity natural time is the event count of small earthquakes. The event count for small earthquakes is extrapolated to larger earthquakes using Gutenberg-Richter scaling. We first review the concepts of natural time and nowcasting and then illustrate seismic nowcasting with three examples. We first consider the aftershock sequence of the 2004 Parkfield earthquake on the San Andreas fault in California. Some earthquakes have higher rates of aftershock activity than other earthquakes of the same magnitude. Our approach allows the determination of the rate in real time during the aftershock sequence. We also consider two examples of induced earthquakes. Large injections of waste water from petroleum extraction have generated high rates of induced seismicity in Oklahoma. The extraction of natural gas from the Groningen gas field in The Netherlands has also generated very damaging earthquakes. In order to reduce the seismic activity, rates of injection and withdrawal have been reduced in these two cases. We show how nowcasting can be used to assess the success of these efforts.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.
Collapse
Affiliation(s)
- Molly Luginbuhl
- Department of Physics, University of California, Davis, CA 95616, USA
| | - John B Rundle
- Department of Physics, University of California, Davis, CA 95616, USA
- Department of Earth and Planetary Sciences, University of California, Davis, CA 95616, USA
- Santa Fe Institute, Santa Fe, NM 86501, USA
| | - Donald L Turcotte
- Department of Earth and Planetary Sciences, University of California, Davis, CA 95616, USA
| |
Collapse
|
9
|
Donnellan A, Grant Ludwig L, Parker JW, Rundle JB, Wang J, Pierce M, Blewitt G, Hensley S. Potential for a large earthquake near Los Angeles inferred from the 2014 La Habra earthquake. Earth Space Sci 2015; 2:378-385. [PMID: 27981074 PMCID: PMC5125407 DOI: 10.1002/2015ea000113] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/14/2015] [Revised: 08/07/2015] [Accepted: 08/15/2015] [Indexed: 05/30/2023]
Abstract
Tectonic motion across the Los Angeles region is distributed across an intricate network of strike-slip and thrust faults that will be released in destructive earthquakes similar to or larger than the 1933 M6.4 Long Beach and 1994 M6.7 Northridge events. Here we show that Los Angeles regional thrust, strike-slip, and oblique faults are connected and move concurrently with measurable surface deformation, even in moderate magnitude earthquakes, as part of a fault system that accommodates north-south shortening and westerly tectonic escape of northern Los Angeles. The 28 March 2014 M5.1 La Habra earthquake occurred on a northeast striking, northwest dipping left-lateral oblique thrust fault northeast of Los Angeles. We present crustal deformation observation spanning the earthquake showing that concurrent deformation occurred on several structures in the shallow crust. The seismic moment of the earthquake is 82% of the total geodetic moment released. Slip within the unconsolidated upper sedimentary layer may reflect shallow release of accumulated strain on still-locked deeper structures. A future M6.1-6.3 earthquake would account for the accumulated strain. Such an event could occur on any one or several of these faults, which may not have been identified by geologic surface mapping.
Collapse
Affiliation(s)
- Andrea Donnellan
- Jet Propulsion Laboratory California Institute of Technology Pasadena California USA; Department of Earth Sciences University of Southern California Loa Angeles California USA
| | - Lisa Grant Ludwig
- Program in Public Health University of California Irvine California USA
| | - Jay W Parker
- Jet Propulsion Laboratory California Institute of Technology Pasadena California USA
| | - John B Rundle
- Departments of Physics and Geology University of California Davis California USA
| | - Jun Wang
- University Information Technology Services Indiana University Bloomington Indiana USA
| | - Marlon Pierce
- University Information Technology Services Indiana University Bloomington Indiana USA
| | - Geoffrey Blewitt
- Nevada Geodetic Laboratory Nevada Bureau of Mines and Geology, University of Nevada Reno Nevada
| | - Scott Hensley
- Jet Propulsion Laboratory California Institute of Technology Pasadena California USA
| |
Collapse
|
10
|
Norris JQ, Turcotte DL, Rundle JB. Loopless nontrapping invasion-percolation model for fracking. Phys Rev E Stat Nonlin Soft Matter Phys 2014; 89:022119. [PMID: 25353434 DOI: 10.1103/physreve.89.022119] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/22/2013] [Indexed: 06/04/2023]
Abstract
Recent developments in hydraulic fracturing (fracking) have enabled the recovery of large quantities of natural gas and oil from old, low-permeability shales. These developments include a change from low-volume, high-viscosity fluid injection to high-volume, low-viscosity injection. The injected fluid introduces distributed damage that provides fracture permeability for the extraction of the gas and oil. In order to model this process, we utilize a loopless nontrapping invasion percolation previously introduced to model optimal polymers in a strongly disordered medium and for determining minimum energy spanning trees on a lattice. We performed numerical simulations on a two-dimensional square lattice and find significant differences from other percolation models. Additionally, we find that the growing fracture network satisfies both Horton-Strahler and Tokunaga network statistics. As with other invasion percolation models, our model displays burst dynamics, in which the cluster extends rapidly into a connected region. We introduce an alternative definition of bursts to be a consecutive series of opened bonds whose strengths are all below a specified value. Using this definition of bursts, we find good agreement with a power-law frequency-area distribution. These results are generally consistent with the observed distribution of microseismicity observed during a high-volume frack.
Collapse
Affiliation(s)
- J Quinn Norris
- Department of Physics, One Shields Ave., University of California, Davis, California 95616, USA
| | - Donald L Turcotte
- Department of Geology, One Shields Ave., University of California, Davis, California 95616, USA
| | - John B Rundle
- Department of Physics, One Shields Ave., University of California, Davis, California 95616, USA and Department of Geology, One Shields Ave., University of California, Davis, California 95616, USA and Santa Fe Institute, Santa Fe, New Mexico 87501, USA
| |
Collapse
|
11
|
|
12
|
Thatcher W, Matsuda T, Kato T, Rundle JB. Lithospheric loading by the 1896 Riku-u Earthquake, northern Japan: Implications for plate flexure and asthenospheric rheology. ACTA ACUST UNITED AC 2012. [DOI: 10.1029/jb085ib11p06429] [Citation(s) in RCA: 122] [Impact Index Per Article: 10.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
|
13
|
Stein RS, King GCP, Rundle JB. The Growth of Geological Structures by Repeated Earthquakes 2. Field Examples of Continental Dip-Slip Faults. ACTA ACUST UNITED AC 2012. [DOI: 10.1029/jb093ib11p13319] [Citation(s) in RCA: 221] [Impact Index Per Article: 18.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
|
14
|
Thatcher W, Rundle JB. A viscoelastic coupling model for the cyclic deformation due to periodically repeated Earthquakes at subduction zones. ACTA ACUST UNITED AC 2012. [DOI: 10.1029/jb089ib09p07631] [Citation(s) in RCA: 159] [Impact Index Per Article: 13.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
|
15
|
|
16
|
|
17
|
|
18
|
Rundle JB, Kanamori H. Application of an inhomogeneous stress (patch) model to complex subduction zone earthquakes: A discrete interaction matrix approach. ACTA ACUST UNITED AC 2012. [DOI: 10.1029/jb092ib03p02606] [Citation(s) in RCA: 41] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
|
19
|
Rundle JB. A physical model for earthquakes: 3. Thermodynamical approach and its relation to nonclassical theories of nucleation. ACTA ACUST UNITED AC 2012. [DOI: 10.1029/jb094ib03p02839] [Citation(s) in RCA: 57] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
|
20
|
|
21
|
|
22
|
|
23
|
Rundle JB, Holliday JR, Graves WR, Turcotte DL, Tiampo KF, Klein W. Probabilities for large events in driven threshold systems. Phys Rev E Stat Nonlin Soft Matter Phys 2012; 86:021106. [PMID: 23005722 DOI: 10.1103/physreve.86.021106] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/22/2011] [Indexed: 06/01/2023]
Abstract
Many driven threshold systems display a spectrum of avalanche event sizes, often characterized by power-law scaling. An important problem is to compute probabilities of the largest events ("Black Swans"). We develop a data-driven approach to the problem by transforming to the event index frame, and relating this to Shannon information. For earthquakes, we find the 12-month probability for magnitude m>6 earthquakes in California increases from about 30% after the last event, to 40%-50% prior to the next one.
Collapse
Affiliation(s)
- John B Rundle
- Department of Physics, University of California, Davis, California 95616, USA
| | | | | | | | | | | |
Collapse
|
24
|
Yoder MR, Turcotte DL, Rundle JB. Forest-fire model with natural fire resistance. Phys Rev E Stat Nonlin Soft Matter Phys 2011; 83:046118. [PMID: 21599251 DOI: 10.1103/physreve.83.046118] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/13/2010] [Revised: 01/14/2011] [Indexed: 05/30/2023]
Abstract
Observations suggest that contemporary wildfire suppression practices in the United States have contributed to conditions that facilitate large, destructive fires. We introduce a forest-fire model with natural fire resistance that supports this theory. Fire resistance is defined with respect to the size and shape of clusters; the model yields power-law frequency-size distributions of model fires that are consistent with field observations in the United States, Canada, and Australia.
Collapse
Affiliation(s)
- Mark R Yoder
- Department of Physics, University of California, Davis, California 95616, USA.
| | | | | |
Collapse
|
25
|
Serino CA, Klein W, Rundle JB. Cellular automaton model of damage. Phys Rev E Stat Nonlin Soft Matter Phys 2010; 81:016105. [PMID: 20365430 DOI: 10.1103/physreve.81.016105] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/01/2009] [Revised: 12/10/2009] [Indexed: 05/29/2023]
Abstract
We investigate the role of equilibrium methods and stress transfer range in describing the process of damage. We find that equilibrium approaches are not applicable to the description of damage and the catastrophic failure mechanism if the stress transfer is short ranged. In the long-range limit, equilibrium methods apply only if the healing mechanism associated with ruptured elements is instantaneous. Furthermore we find that the nature of the catastrophic failure depends strongly on the stress transfer range. Long-range transfer systems have a failure mechanism that resembles nucleation. In short-range stress transfer systems, the catastrophic failure is a continuous process that, in some respects, resembles a critical point.
Collapse
Affiliation(s)
- C A Serino
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA.
| | | | | |
Collapse
|
26
|
Turcotte DL, Abaimov SG, Dobson I, Rundle JB. Implications of an inverse branching aftershock sequence model. Phys Rev E Stat Nonlin Soft Matter Phys 2009; 79:016101. [PMID: 19257101 DOI: 10.1103/physreve.79.016101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/13/2008] [Revised: 09/25/2008] [Indexed: 05/27/2023]
Abstract
The branching aftershock sequence (BASS) model is a self-similar statistical model for earthquake aftershock sequences. A prescribed parent earthquake generates a first generation of daughter aftershocks. The magnitudes and times of occurrence of the daughters are obtained from statistical distributions. The first generation daughter aftershocks then become parent earthquakes that generate second generation aftershocks. The process is then extended to higher generations. The key parameter in the BASS model is the magnitude difference Deltam* between the parent earthquake and the largest expected daughter earthquake. In the application of the BASS model to aftershocks Deltam* is positive, the largest expected daughter event is smaller than the parent, and the sequence of events (aftershocks) usually dies out, but an exponential growth in the number of events with time is also possible. In this paper we explore this behavior of the BASS model as Deltam* varies, including when Deltam* is negative and the largest expected daughter event is larger than the parent. The applications of this self-similar branching process to biology and other fields are discussed.
Collapse
Affiliation(s)
- D L Turcotte
- Department of Geology, University of California, Davis, California 95616, USA.
| | | | | | | |
Collapse
|
27
|
Xia J, Gould H, Klein W, Rundle JB. Near-mean-field behavior in the generalized Burridge-Knopoff earthquake model with variable-range stress transfer. Phys Rev E Stat Nonlin Soft Matter Phys 2008; 77:031132. [PMID: 18517354 DOI: 10.1103/physreve.77.031132] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/09/2007] [Revised: 02/20/2008] [Indexed: 05/26/2023]
Abstract
Simple models of earthquake faults are important for understanding the mechanisms for their observed behavior in nature, such as Gutenberg-Richter scaling. Because of the importance of long-range interactions in an elastic medium, we generalize the Burridge-Knopoff slider-block model to include variable range stress transfer. We find that the Burridge-Knopoff model with long-range stress transfer exhibits qualitatively different behavior than the corresponding long-range cellular automata models and the usual Burridge-Knopoff model with nearest-neighbor stress transfer, depending on how quickly the friction force weakens with increasing velocity. Extensive simulations of quasiperiodic characteristic events, mode-switching phenomena, ergodicity, and waiting-time distributions are also discussed. Our results are consistent with the existence of a mean-field critical point and have important implications for our understanding of earthquakes and other driven dissipative systems.
Collapse
Affiliation(s)
- Junchao Xia
- Department of Physics, Clark University, Worcester, Massachusetts 01610, USA
| | | | | | | |
Collapse
|
28
|
Tiampo KF, Rundle JB, Klein W, Holliday J, Sá Martins JS, Ferguson CD. Ergodicity in natural earthquake fault networks. Phys Rev E Stat Nonlin Soft Matter Phys 2007; 75:066107. [PMID: 17677325 DOI: 10.1103/physreve.75.066107] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/17/2006] [Revised: 04/08/2007] [Indexed: 05/16/2023]
Abstract
Numerical simulations have shown that certain driven nonlinear systems can be characterized by mean-field statistical properties often associated with ergodic dynamics [C. D. Ferguson, W. Klein, and J. B. Rundle, Phys. Rev. E 60, 1359 (1999); D. Egolf, Science 287, 101 (2000)]. These driven mean-field threshold systems feature long-range interactions and can be treated as equilibriumlike systems with statistically stationary dynamics over long time intervals. Recently the equilibrium property of ergodicity was identified in an earthquake fault system, a natural driven threshold system, by means of the Thirumalai-Mountain (TM) fluctuation metric developed in the study of diffusive systems [K. F. Tiampo, J. B. Rundle, W. Klein, J. S. Sá Martins, and C. D. Ferguson, Phys. Rev. Lett. 91, 238501 (2003)]. We analyze the seismicity of three naturally occurring earthquake fault networks from a variety of tectonic settings in an attempt to investigate the range of applicability of effective ergodicity, using the TM metric and other related statistics. Results suggest that, once variations in the catalog data resulting from technical and network issues are accounted for, all of these natural earthquake systems display stationary periods of metastable equilibrium and effective ergodicity that are disrupted by large events. We conclude that a constant rate of events is an important prerequisite for these periods of punctuated ergodicity and that, while the level of temporal variability in the spatial statistics is the controlling factor in the ergodic behavior of seismic networks, no single statistic is sufficient to ensure quantification of ergodicity. Ergodicity in this application not only requires that the system be stationary for these networks at the applicable spatial and temporal scales, but also implies that they are in a state of metastable equilibrium, one in which the ensemble averages can be substituted for temporal averages in studying their spatiotemporal evolution.
Collapse
Affiliation(s)
- K F Tiampo
- Department of Earth Sciences, University of Western Ontario, London, Ontario, N6A 5B7 Canada.
| | | | | | | | | | | |
Collapse
|
29
|
Klein W, Gould H, Gulbahce N, Rundle JB, Tiampo K. Structure of fluctuations near mean-field critical points and spinodals and its implication for physical processes. Phys Rev E Stat Nonlin Soft Matter Phys 2007; 75:031114. [PMID: 17500675 DOI: 10.1103/physreve.75.031114] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/03/2006] [Revised: 01/09/2007] [Indexed: 05/15/2023]
Abstract
We analyze the structure of fluctuations near critical points and spinodals in mean-field and near-mean-field systems. Unlike systems that are non-mean-field, for which a fluctuation can be represented by a single cluster in a properly chosen percolation model, a fluctuation in mean-field and near-mean-field systems consists of a large number of clusters, which we term fundamental clusters. The structure of the latter and the way that they form fluctuations has important physical consequences for phenomena as diverse as nucleation in supercooled liquids, spinodal decomposition and continuous ordering, and the statistical distribution of earthquakes. The effects due to the fundamental clusters implies that they are physical objects and not only mathematical constructs.
Collapse
Affiliation(s)
- W Klein
- Department of Physics and Center for Computational Science, Boston University, Boston, Massachusetts 02215, USA
| | | | | | | | | |
Collapse
|
30
|
Holliday JR, Rundle JB, Turcotte DL, Klein W, Tiampo KF, Donnellan A. Space-time clustering and correlations of major earthquakes. Phys Rev Lett 2006; 97:238501. [PMID: 17280253 DOI: 10.1103/physrevlett.97.238501] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/01/2006] [Indexed: 05/13/2023]
Abstract
Earthquake occurrence in nature is thought to result from correlated elastic stresses, leading to clustering in space and time. We show that the occurrence of major earthquakes in California correlates with time intervals when fluctuations in small earthquakes are suppressed relative to the long term average. We estimate a probability of less than 1% that this coincidence is due to random clustering.
Collapse
Affiliation(s)
- James R Holliday
- Center for Computational Science and Engineering, University of California, Davis, California 95616, USA.
| | | | | | | | | | | |
Collapse
|
31
|
Xia J, Gould H, Klein W, Rundle JB. Simulation of the Burridge-Knopoff model of earthquakes with variable range stress transfer. Phys Rev Lett 2005; 95:248501. [PMID: 16384429 DOI: 10.1103/physrevlett.95.248501] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/28/2005] [Indexed: 05/05/2023]
Abstract
Simple models of earthquake faults are important for understanding the mechanisms for their observed behavior, such as Gutenberg-Richter scaling and the relation between large and small events, which is the basis for various forecasting methods. Although cellular automaton models have been studied extensively in the long-range stress transfer limit, this limit has not been studied for the Burridge-Knopoff model, which includes more realistic friction forces and inertia. We find that the latter model with long-range stress transfer exhibits qualitatively different behavior than both the long-range cellular automaton models and the usual Burridge-Knopoff model with nearest-neighbor springs, depending on the nature of the velocity-weakening friction force. These results have important implications for our understanding of earthquakes and other driven dissipative systems.
Collapse
Affiliation(s)
- Junchao Xia
- Department of Physics, Clark University, Worcester, Massachusetts 01610, USA
| | | | | | | |
Collapse
|
32
|
Shcherbakov R, Yakovlev G, Turcotte DL, Rundle JB. Model for the distribution of aftershock interoccurrence times. Phys Rev Lett 2005; 95:218501. [PMID: 16384191 DOI: 10.1103/physrevlett.95.218501] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/17/2005] [Indexed: 05/05/2023]
Abstract
In this work the distribution of interoccurrence times between earthquakes in aftershock sequences is analyzed and a model based on a nonhomogeneous Poisson (NHP) process is proposed to quantify the observed scaling. In this model the generalized Omori's law for the decay of aftershocks is used as a time-dependent rate in the NHP process. The analytically derived distribution of interoccurrence times is applied to several major aftershock sequences in California to confirm the validity of the proposed hypothesis.
Collapse
Affiliation(s)
- Robert Shcherbakov
- Center for Computational Science and Engineering, University of California, Davis, California 95616, USA.
| | | | | | | |
Collapse
|
33
|
Rundle JB, Rundle PB, Donnellan A, Turcotte DL, Shcherbakov R, Li P, Malamud BD, Grant LB, Fox GC, McLeod D, Yakovlev G, Parker J, Klein W, Tiampo KF. A simulation-based approach to forecasting the next great San Francisco earthquake. Proc Natl Acad Sci U S A 2005; 102:15363-7. [PMID: 16219696 PMCID: PMC1266132 DOI: 10.1073/pnas.0507528102] [Citation(s) in RCA: 34] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
In 1906 the great San Francisco earthquake and fire destroyed much of the city. As we approach the 100-year anniversary of that event, a critical concern is the hazard posed by another such earthquake. In this article, we examine the assumptions presently used to compute the probability of occurrence of these earthquakes. We also present the results of a numerical simulation of interacting faults on the San Andreas system. Called Virtual California, this simulation can be used to compute the times, locations, and magnitudes of simulated earthquakes on the San Andreas fault in the vicinity of San Francisco. Of particular importance are results for the statistical distribution of recurrence times between great earthquakes, results that are difficult or impossible to obtain from a purely field-based approach.
Collapse
Affiliation(s)
- J B Rundle
- Center for Computational Science and Engineering, and Department of Geology, University of California-Davis, Davis, CA 95616, USA.
| | | | | | | | | | | | | | | | | | | | | | | | | | | |
Collapse
|
34
|
Abstract
Numerical simulations suggest that certain driven, dissipative mean-field threshold systems, including earthquake models, can be characterized by statistical properties often associated with ergodic dynamics, in the same sense as stochastic Brownian motion. We applied a fluctuation metric proposed by Thirumalai and Mountain [Phys. Rev. E 47, 479 (1993)]] for statistically stationary systems and find that the natural earthquake fault system in California demonstrates similar ergodic dynamics.
Collapse
Affiliation(s)
- K F Tiampo
- CIRES, University of Colorado, Boulder, Colorado 80309, USA
| | | | | | | | | |
Collapse
|
35
|
Sá Martins JS, Rundle JB, Anghel M, Klein W. Precursory dynamics in threshold systems. Phys Rev E Stat Nonlin Soft Matter Phys 2002; 65:056117. [PMID: 12059657 DOI: 10.1103/physreve.65.056117] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/22/2001] [Revised: 02/21/2002] [Indexed: 05/23/2023]
Abstract
A precursory dynamics, motivated by the analysis of recent experiments on solid-on-solid friction, is introduced in a continuous cellular automaton that mimics the physics of earthquake source processes. The resulting system of equations for the interevent cycle can be decoupled and yields an analytical solution in the mean-field limit, exhibiting a smoothing effect of the dynamics on the stress field. Simulation results show the resulting departure from scaling at the large-event end of the frequency distribution, and support claims that the field leakage may parametrize the superposition of scaling and characteristic regimes observed in real earthquake faults.
Collapse
Affiliation(s)
- J S Sá Martins
- Colorado Center for Chaos and Complexity/CIRES and Department of Physics, CB 216, University of Colorado, Boulder, Colorado 80309, USA
| | | | | | | |
Collapse
|
36
|
Rundle JB, Tiampo KF, Klein W, Sa Martins JS. Self-organization in leaky threshold systems: the influence of near-mean field dynamics and its implications for earthquakes, neurobiology, and forecasting. Proc Natl Acad Sci U S A 2002; 99 Suppl 1:2514-21. [PMID: 11875204 PMCID: PMC128570 DOI: 10.1073/pnas.012581899] [Citation(s) in RCA: 122] [Impact Index Per Article: 5.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
Threshold systems are known to be some of the most important nonlinear self-organizing systems in nature, including networks of earthquake faults, neural networks, superconductors and semiconductors, and the World Wide Web, as well as political, social, and ecological systems. All of these systems have dynamics that are strongly correlated in space and time, and all typically display a multiplicity of spatial and temporal scales. Here we discuss the physics of self-organization in earthquake threshold systems at two distinct scales: (i) The "microscopic" laboratory scale, in which consideration of results from simulations leads to dynamical equations that can be used to derive the results obtained from sliding friction experiments, and (ii) the "macroscopic" earthquake fault-system scale, in which the physics of strongly correlated earthquake fault systems can be understood by using time-dependent state vectors defined in a Hilbert space of eigenstates, similar in many respects to the mathematics of quantum mechanics. In all of these systems, long-range interactions induce the existence of locally ergodic dynamics. The existence of dissipative effects leads to the appearance of a "leaky threshold" dynamics, equivalent to a new scaling field that controls the size of nucleation events relative to the size of background fluctuations. At the macroscopic earthquake fault-system scale, these ideas show considerable promise as a means of forecasting future earthquake activity.
Collapse
Affiliation(s)
- J B Rundle
- Colorado Center for Chaos and Complexity and Cooperative Institute for Research in Environmental Sciences (CIRES) and Department of Physics, 216 UCB, University of Colorado, Boulder, CO 80309-0216, USA.
| | | | | | | |
Collapse
|
37
|
Affiliation(s)
- Donald L Turcotte
- Department of Earth and Atmospheric Sciences, Cornell University, Ithaca, NY 14853, USA.
| | | |
Collapse
|
38
|
Rundle PB, Rundle JB, Tiampo KF, Martins JS, McGinnis S, Klein W. Nonlinear network dynamics on earthquake fault systems. Phys Rev Lett 2001; 87:148501. [PMID: 11580678 DOI: 10.1103/physrevlett.87.148501] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/06/2000] [Revised: 07/16/2001] [Indexed: 05/23/2023]
Abstract
Earthquake faults occur in interacting networks having emergent space-time modes of behavior not displayed by isolated faults. Using simulations of the major faults in southern California, we find that the physics depends on the elastic interactions among the faults defined by network topology, as well as on the nonlinear physics of stress dissipation arising from friction on the faults. Our results have broad applications to other leaky threshold systems such as integrate-and-fire neural networks.
Collapse
Affiliation(s)
- P B Rundle
- Fairview High School, Bolder, Colorado 80309, USA
| | | | | | | | | | | |
Collapse
|
39
|
|
40
|
Ferguson CD, Klein W, Rundle JB. Spinodals, scaling, and ergodicity in a threshold model with long-range stress transfer. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 1999; 60:1359-73. [PMID: 11969896 DOI: 10.1103/physreve.60.1359] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/26/1998] [Indexed: 04/18/2023]
Abstract
We present both theoretical and numerical analyses of a cellular automaton version of a slider-block model or threshold model that includes long-range interactions. Theoretically we develop a coarse-grained description in the mean-field (infinite range) limit and discuss the relevance of the metastable state, limit of stability (spinodal), and nucleation to the phenomenology of the model. We also simulate the model and confirm the relevance of the theory for systems with long- but finite-range interactions. Results of particular interest include the existence of Gutenberg-Richter-like scaling consistent with that found on real earthquake fault systems, the association of large events with nucleation near the spinodal, and the result that such systems can be described, in the mean-field limit, with techniques appropriate to systems in equilibrium.
Collapse
Affiliation(s)
- C D Ferguson
- Physics Department, Center for Polymer Physics, and Center for Computational Science, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA
| | | | | |
Collapse
|
41
|
|
42
|
Bawden GW, Donnellan A, Kellogg LH, Dong D, Rundle JB. Geodetic measurements of horizontal strain near the White Wolf fault, Kern County, California, 1926-1993. ACTA ACUST UNITED AC 1997. [DOI: 10.1029/96jb03554] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
|
43
|
|
44
|
Yu TT, Rundle JB, Fernández J. Surface deformation due to a strike-slip fault in an elastic gravitational layer overlying a viscoelastic gravitational half-space. ACTA ACUST UNITED AC 1996. [DOI: 10.1029/95jb03118] [Citation(s) in RCA: 22] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
|
45
|
Rundle JB, Klein W, Gross S, Turcotte DL. Boltzmann Fluctuations in Numerical Simulations of Nonequilibrium Lattice Threshold Systems. Phys Rev Lett 1995; 75:1658-1661. [PMID: 10060353 DOI: 10.1103/physrevlett.75.1658] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
|
46
|
|
47
|
Rundle JB. Derivation of the complete Gutenberg-Richter magnitude-frequency relation using the principle of scale invariance. ACTA ACUST UNITED AC 1989. [DOI: 10.1029/jb094ib09p12337] [Citation(s) in RCA: 133] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/09/2022]
|
48
|
|
49
|
|
50
|
|