Mechanics of Earthquakes and Faulting
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1 Mechanics of Earthquakes and Faulting Lecture 20, 30 Nov Seismic Spectra & Earthquake Scaling laws.
2 Seismic Spectra & Earthquake Scaling laws. Aki, Scaling law of seismic spectrum, JGR, 72, , Hanks, b Values and ω -γ seismic source models: implications for tectonic stress variations along active crustal fault zones and the estimation of high-frequency strong ground motion, JGR, 84, , Scaling and Self-Similarity of Earthquake Rupture: Implications for Rupture Dynamics and the Mode of Rupture Propagation 0 Self-similar: Are small earthquakes the same as large ones? Do small ones become large ones or are large eq s different from the start? 1 Geometric self-similarity: aspect ratio of rupture area 2 Physical self-similarity: stress drop, seismic strain, scaling of slip with rupture dimension 3 Observation of constant b-value over a wide range of inferred source dimension. 4 Same physical processes operate during shear rupture of very small (lab scale, mining induced seismicity) and very large earthquakes? 5 Expectation of scaling break if rupture physics/dynamics change in at a critical size (or slip velocity, etc.). Shimazaki result. (Fig. 4.12). Length-Moment scaling and transition at L 60km (Romanowicz, 1992; Scholz, 1994). 6 Gutenberg-Richter frequency-magnitude scaling, b-values. 7 G-R scaling, b-value data. Single-fault versus fault population. G-R versus characteristic earthquake model. 8 Crack vs. slip-pulse models
3 Earthquake Source Properties, Spectra, Scaling, Self-similarity Displacement and acceleration source spectra. Spectra: zero-frequency intercept (M o ), corner frequency (ω o or f c ), high frequency decay (ω γ ), maximum (observed, emitted) frequency f max ω -n ω-square model, ω -2 ω-cube model, ω -3 log u at R ω ο Far-field body-wave spectra and relation to source slip function log freq. (ω) Displacement waveform for P & S waves: In general, very complex. Ω(x, t) and Ω(ω) depend on slip function, azimuth to observer and relative importance of nucleation and stopping phases
4 Scaling and Self-Similarity Are small earthquakes the same as large ones? 1 Geometric self-similarity: rupture aspect ratio 2 Physical self-similarity: stress drop, seismic strain, scaling of slip with rupture dimension Circular ruptures (small) Hanks, 1977 Abercrombie & Leary, 1993
5 Earthquake Source Properties, Spectra, Scaling, Self-similarity Displacement and acceleration source spectra. Spectra: zero-frequency intercept (M o ), corner frequency (ω o or f c ), high frequency decay (ω γ ), maximum (observed, emitted) frequency f max f max log u at R ω ο ω -n log a at R ω ο log freq. (ω) log freq. (ω) Aki, Scaling law of seismic spectrum, JGR, 72, , Hanks, b Values and ω -γ seismic source models: implications for tectonic stress variations along active crustal fault zones and the estimation of high-frequency strong ground motion, JGR, 84, , 1979.
6 Seismic Source Spectra. Saturation occurs for large events, particularly saturation of M s (T=20 s) Corner frequency, Brune Stress drop. Aki, 1967
7 N L W Rupture area, A Slip contours, u Magnitude and Seismic Moment. Moment is a most robust measure of earthquake size because magnitude is a measure of size at only one frequency. M o = µ A u, where µ is shear modulus, A is fault Area and u is mean slip. Moment and Moment Magnitude (Hanks and Kanamori, JGR, 1979): M w = 2/3 log M o 6 or M o = 3/2 M w +9 (for M o in N-m)
8 Earthquakes represent failure on geologic faults. The rupture occurs on a pre-existing surface. Faults are finite features the Earth does not break in half every time there is an earthquake. Earthquakes represent failure of a limited part of a fault. Most earthquakes within the crust are shallow Definitions of Focus, Epicenter NOTE: Epicenter is also the Rancho Cucamonga Quakes stadium they are single-a team in S. CA: Earthquake Size (Source Properties) Measures of earthquake size: Fault Area, Ground Shaking, Radiated Energy Fault dimensions for some large earthquakes: L (km) W (km) U (m) Mw Chile > Landers, CA San Fran Alaska ~ N L W Rupture area, A Slip contours, u
9 Wave resulting from the interaction of P and S waves with the free surface. Their wave motion is confined to and propagating along the surface of the body
10 Magnitude is a measure of earthquake size base on: Ground shaking Seismic wave amplitude at a given frequency
11 N W L Rupture area, A Slip contours, u Magnitude is a measure of earthquake size base on: Ground shaking Seismic wave amplitude at a given frequency Magnitude accounts for three key aspects: Huge range of ground observed displacements --due to very large range of earthquake sizes Distance correction to account for attenuation of elastic disturbance during propagation Site, station correction small empirical correction to account for local effects at source or receiver M L = log 10! " u T # $ + q(δ,h) + a
12 Magnitude is a measure of earthquake size base on: Ground shaking Seismic wave amplitude at a given frequency M L = log 10! " u T # $ + q(δ,h) + a M L (Richter --local-- Magnitude) & M S, based on 20-s surface wave M B, Body-wave mag. Is based on 1-s wave p-wave M W, Moment mag. (see Hanks and Kanamori, JGR, 1979) Systematic differences between M s and M b --due to use of different periods. Source Spectra isn t flat. Saturation occurs for large events, particularly saturation of M s. e.g:
13 Earthquake Source Properties, Spectra, Scaling, Self-similarity ω -2 log u at R ω ο ω-square model, ω -2 Two possible explanations log freq. (ω) 1)!Similarity condition (not-similarity) M o α L 2 ω o α L -1 Ω(0) α ω o -2 2) Have similarity condition in terms of nucleation, but high-freq. asymptote is produced by stopping phase if rupture stops very abruptly
14 Earthquake Source Properties, Spectra, Scaling, Self-similarity ω -3 log u at R ω ο log freq. (ω) ω-cube model, ω -3 Similarity condition M o α L 3 ω o α L -1 Ω(0) α ω o -3 This defines a scaling law. Spectral curves differ by a constant factor at a given period (e.g., 20 s), but they have the same high-freq. asymptote This behavior is expected when the nucleation phase is responsible for the high-freq. asymptote --but consider problem of time domain implication for amplitude (M b decreases with M o )
15 Earthquake Source Properties, Spectra, Scaling, Self-similarity Source spectra for two events of equal stress drop: omega cube model Large and Small Eq L log u at R ω ο L ω -3 S ω ο S f max log freq. (ω) High-freq. spectral properties: produced by rupture growth, represent nucleation and enlargement log a at R ω ο L ω ο S log freq. (ω)
16 Source spectra for two events of equal stress drop: omega square model Large and Small Eq L log u at R ω ο L ω -2 S ω ο S L f max log freq. (ω) High-freq. spectral properties: produced by rupture growth, represent nucleation and enlargement log a at R ω ο L ω ο S S f max log freq. (ω)
17 Earthquake Source Properties, Spectra, Scaling, Self-similarity Relation between source (a) displacement (b) velocity (c) acceleration history and asymptotic behavior of spectrum
18 Earthquake Source Properties, Spectra, Scaling, Self-similarity Hanks (1979) 0. Consider two events that differ in size by 10x and assume self-similarity of rupture so that their moments differ by a factor of 10 3 and durations differ by a factor of 10 (rupture velocity is the same and constant for each event.) Take the events to be large enough so that their corner frequency is well below 1 sec. 1. Four cases for time-domain interpretation of spectral source models, 2 for each model. ω-square (spectral amplitude at 1 sec period is 10x greater for larger event) a) If: 1-s. energy arrives continuously over the complete faulting duration. Then: 1-s time domain amplitude is the same and M b is the same for both events M b is independent of M o. b) If: all 1-s energy radiated at the same time (arrives at the same time) Then: M b scales directly with M o. 1-s time domain amplitude is 10x larger for the larger event ω-cube (spectral amplitude at T=1 sec is the same for each event) c) If: 1-s energy arrives continuously over the complete faulting duration. Then: M b scales inversely with M o. 1-s time domain amplitude is smaller for the larger event d) If: all 1-s energy radiated at the same time (arrives at the same time) Then: 1-s time domain amplitude is the same for both events M b is independent of M o.
19 Earthquake Source Properties, Spectra, Scaling, Self-similarity Hanks (1979) 0. Consider two events that differ in size by 10x and assume self-similarity of rupture so that their moments differ by a factor of 10 3 and durations differ by a factor of 10 (rupture velocity is the same and constant for each event.) Take the events to be large enough so that their corner frequency is well below 1 sec. 1. Four cases for time-domain interpretation of spectral source models, 2 for each model. ω-square (spectral amplitude at 1 sec period is 10x greater for larger event) a) If: 1-s. energy arrives continuously over the complete faulting duration. Then: 1-s time domain amplitude is the same and M b is the same for both events M b is independent of M o. b) If: all 1-s energy radiated at the same time (arrives at the same time) Then: 1-s time domain amplitude is 10x larger for the larger event M b scales directly with M o. ω-cube (spectral amplitude at T=1 sec is the same for each event) c) If: 1-s energy arrives continuously over the complete faulting duration. Then: 1-s time domain amplitude is smaller for the larger event M b scales inversely with M o. d) If: all 1-s energy radiated at the same time (arrives at the same time) Then: 1-s time domain amplitude is the same for both events M b is independent of M o. From data: cases (c) and (b) are clearly wrong: M b does not decrease with M o and M b does not increase beyond about 6.5. Either (a) or (d) could be right, but very simplified approach. Data tend to support ω- square model (See Boatwright and Choy 1989) but also see ω -5/2 and lots of scatter. Propagation effects very hard to remove in practice.
20 Earthquake Source Properties, Spectra, Scaling, Self-similarity Hanks (1979) 0. Consider two events that differ in size by 10x and assume self-similarity of rupture so that their moments differ by a factor of 10 3 and durations differ by a factor of 10 (rupture velocity is the same and constant for each event.) Take the events to be large enough so that their corner frequency is well below 1 sec. 1. Four cases for time-domain interpretation of spectral source models, 2 for each model. ω-square (spectral amplitude at 1 sec period is 10x greater for larger event) a) If: 1-s. energy arrives continuously over the complete faulting duration. Then: 1-s time domain amplitude is the same and M b is the same for both events M b is independent of M o. b) If: all 1-s energy radiated at the same time (arrives at the same time) Then: 1-s time domain amplitude is 10x larger for the larger event M b scales directly with M o. log u at R ω ο L ω -2 ω ο S log freq. (ω)
21 Earthquake Source Properties, Spectra, Scaling, Self-similarity Hanks (1979) 0. Consider two events that differ in size by 10x and assume self-similarity of rupture so that their moments differ by a factor of 10 3 and durations differ by a factor of 10 (rupture velocity is the same and constant for each event.) Take the events to be large enough so that their corner frequency is well below 1 sec. 1. Four cases for time-domain interpretation of spectral source models, 2 for each model. ω-cube (spectral amplitude at T=1 sec is the same for each event) c) If: 1-s energy arrives continuously over the complete faulting duration. Then: 1-s time domain amplitude is smaller for the larger event M b scales inversely with M o. d) If: all 1-s energy radiated at the same time (arrives at the same time) Then: 1-s time domain amplitude is the same for both events M b is independent of M o. log u at R ω ο L ω -3 ω ο S log freq. (ω)
22 Earthquake Scaling: Size-frequency of occurrence Gutenberg-Richter frequency-magnitude scaling. log N b b ~ 1 B ~ 2/3 Observed for the world-wide eq catalog Ms GR scaling, with constant b implies self-similarity of earthquakes (rupture physics, fracture process, fault roughness, etc.)
23 Earthquake Scaling: Size-frequency of occurrence Gutenberg-Richter frequency-magnitude scaling. b ~ 1 B ~ 2/3 Observed for the world-wide eq catalog Cumulative number Scholz, 1990
24 Earthquake Scaling: Size-frequency of occurrence Cumulative number note dimensions Scholz, 1990
25 Gutenberg-Richter frequency-magnitude scaling. What about scaling breaks? Scaling break log N It could imply that stress drop is not independent of size or It could imply a preferred or characteristic size Ms note dimensions
26 Gutenberg-Richter frequency-magnitude scaling. What about scaling breaks? It could imply that stress drop is not independent of size or It could imply a preferred or characteristic size log N Scaling break Ms = 7.3 Ms Characteristic Earthquake model Ms = 7.3
27 Gutenberg-Richter frequency-magnitude scaling. What about scaling breaks? It could imply that stress drop is not independent of size or It could imply a preferred or characteristic size Cumulative number Ms = 7.3 Characteristic Earthquake model Scholz, 1990
28 Scaling and Self-Similarity Are small earthquakes the same as large ones? 1 Geometric self-similarity: rupture aspect ratio 2 Physical self-similarity: stress drop, seismic strain, scaling of slip with rupture dimension Circular ruptures (small) Read Scholz, BSSA 1982 and Pacheco, J. F. Scholz, C. H. Sykes, L. R. (1992). Changes in frequency-size relationship from small to large earthquakes, Nature 355,
29 Scaling and Self-Similarity Are small earthquakes the same as large ones? 1 Geometric self-similarity: rupture aspect ratio 2 Physical self-similarity: stress drop, seismic strain, scaling of slip with rupture dimension Circular ruptures (small) Hanks, 1977 Abercrombie & Leary, 1993
30 Circular ruptures (small) Rectangular ruptures (large) Slip determined by W: Slip determined by L note dimensions
31 Circular ruptures (small) Transition from small to large eq s Rectangular ruptures (large) Slip determined by W: Shimazaki, 1986 Slip determined by L
32 Scaling of Large Earthquakes: Is slip determined (limited) by W or L? Rectangular ruptures (large) Slip determined by W: Slip determined by L Scholz, 1982, 1994 Romanowicz, 1992,
33 Scaling of Large Earthquakes: Is slip determined (limited) by W or L? Rectangular ruptures (large) Slip determined by W: Slip determined by L
34 Scaling of Large Earthquakes: Is slip determined (limited) by W or L? Rectangular ruptures (large) Slip determined by W: Slip determined by L
35 Some Topics in the Mechanics of Earthquakes and Faulting What determines the size of an earthquake? What physical features and factors of faulting control the extent of dynamic earthquake rupture? --Fault Area, Seismic Moment What is the role of fault geometry (offsets, roughness, thickness) versus rupture dynamics? What controls the amount of slip in an earthquake? Average Slip, Slip at a point What controls whether fault slip occurs dynamically or quasi-statically? Nucleation: How does the earthquake process get going? What is the size of a nucleation patch at the time that slip becomes dynamic? How do we define dynamic versus quasi-dynamic and quasi-static? Nucleation patch: physical size, seismic signature What controls dynamic rupture velocity? How do faults grow and evolve with time?
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