Mechanics of Earthquakes and Faulting · 11/14/2019 · Seismic Spectra & Earthquake Scaling laws. Aki, Scaling law of seismic spectrum, JGR, 72, 1217-1231,1967. Hanks, bValues and

Mechanics of Earthquakes and Faulting

www.geosc.psu.edu/Courses/Geosc508

Lecture 18, 14 Nov. 2019

• PS 5, questions?• Planning for project presentations on Dec 3, 5, 10 & 12

• Moment, Magnitude and scaling laws for earthquake source parameters• Brune Stress Drop• Seismic Spectra & Earthquake Scaling laws.

Magnitude and Seismic Moment. Moment is a more robust measure of earthquake size than magnitude because magnitude is a measure of size at only one frequency.

Mo = µ A u, where µ is shear modulus, A is fault Area and u is mean slip.

Relation to magnitude: Mw = 2/3 log Mo – 6 or Mo = 3/2 Mw +9 (for Mo in N-m)

N

Rupturearea, A

Slipcontours, u

W

L

Surface wavesresult from the interaction of elastic waves with the free surface.

Their wave motion is confined to the surface of the body

Magnitude is a measure of earthquake size base on:• Ground shaking• Seismic wave amplitude at a given frequency

N

Rupturearea, A

Slipcontours, u

W

L

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

ML = log10uT! "

# $ + q (Δ,h ) + a

Magnitude is a measure of earthquake size base on:• Ground shaking• Seismic wave amplitude at a given frequency

ML = log10uT! "

# $ + q (Δ,h ) + a

Systematic differences between Ms and Mb --due to use of different periods.

Source Spectra isn’t flat. Saturation occurs for large events, particularly saturation of Ms.

e.g: http://neic.usgs.gov/neis/nrg/bb_processing.html

ML (Richter --local-- Magnitude) & MS, based on 20-s surface wave

MB, Body-wave mag. Is based on 1-s wave p-waveMW, Moment mag. (see Hanks and Kanamori, JGR, 1979)

Seismic Moment.

Mo = 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): Mw = 2/3 log Mo – 6 or Mo = 3/2 Mw +9 (for Mo in N-m)

N

Rupturearea, A

Slipcontours, u

W

L

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 minor league baseball team http://www.rcquakes.com/ _____________________________________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 1960 1000 100 >10 9.7Landers, CA 1992 70 15 5 7.3San Fran 1906 500 15 10 8.5Alaska 1964 750 180 ~12 9.3

N

Rupturearea, A

Slipcontours, u

W

L

N

Rupturearea, A

Slipcontours, u

W

L

Brune Stress drop

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?

Seismic Spectra & Earthquake Scaling laws.Aki, Scaling law of seismic spectrum, JGR, 72, 1217-1231, 1967.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, 2235-2242, 1979.

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 area2 Physical self-similarity: stress drop, seismic strain, scaling of slip with rupture dimension3 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

Source spectra for two events of equal stress drop: omega square model

Large and Small Eq

log

u at

R ω-2

log freq. (ω)

ωοS

ωοL

L

S

High-freq. spectral properties: produced by rupture growth, represent nucleation and enlargement

log

a at

R

log freq. (ω)

fmax

fmax

ωοL

ωοS

L

S

Circular ruptures (small)

Scaling and Self-SimilarityAre 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

Hanks, 1977 Abercrombie & Leary, 1993

Gutenberg-Richter frequency-magnitude scaling.

Earthquake Scaling: Size-frequency of occurrence

Ms

log Nb

GR scaling, with constant b implies self-similarity of earthquakes (rupture physics, fracture process, fault roughness, etc.)

b ~ 1

B ~ 2/3

Observed for the world-wide eq catalog



b ~ 1

B ~ 2/3Observed for the world-wide eq catalog

Scholz, 1990

Cum

ulat

ive

num

ber


Scholz, 1990

Cum

ulat

ive

num

ber

note dimensions


What about scaling breaks?

Ms

log N

Scaling break

It could imply that stress drop is not independent of size orIt could imply a preferred or characteristic size

note dimensions



Ms

log N

Scaling break


Ms = 7.3

Ms = 7.3Characteristic Earthquake model


Ms = 7.3

Characteristic Earthquake model

Scholz, 1990

Cum

ulat

ive

num

ber






Read Scholz, BSSA 1982andPacheco, J. F. Scholz, C. H. Sykes, L. R. (1992). Changes in frequency-size relationship from small to large earthquakes, Nature 355, 71- 73.




Hanks, 1977 Abercrombie & Leary, 1993

note dimensions


Rectangular ruptures (large)

Slip determined by W:

Slip determined by L





Shimazaki, 1986

Transition from small to large eq’s

Scholz, 1982, 1994Romanowicz, 1992, 1994




http://seismo.berkeley.edu/annual_report/ar01_02/node22.html

Scaling of Large Earthquakes: Is slip determined (limited) by W or L?











Earthquake Source Properties, Spectra, Scaling, Self-similarity

Displacement and acceleration source spectra. Spectra: zero-frequency intercept (Mo), corner frequency (ωo or fc), high frequency decay (ω−γ), maximum (observed, emitted) frequency fmax

log

u at

R

ω-n

ωο

log freq. (ω)

ω-square model, ω-2

ω-cube model, ω-3

Far-field body-wave spectra and relation to source slip function

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

Aki, 1967

Earthquake Source Properties, Spectra, Scaling, Self-similarity lo

g u

at R

ω-3

ωο

log freq. (ω)

ω-cube model, ω-3

Similarity condition Mo α L3

ωo α L-1

Ω(0) α ωo-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 (Mb decreases with Mo)

Seismic Source Spectra.

Saturation occurs for large events, particularly saturation of Ms (T=20 s)

Aki, 1967

Corner frequency, Brune Stress drop.

Earthquake Source Properties, Spectra, Scaling, Self-similarity lo

g u

at R

ω-2

ωο

log freq. (ω)

ω-square model, ω-2

Two possible explanations

1) !Similarity condition (not-similarity)Mo α L2

ωo α L-1

Ω(0) α ωo-2

2) Have similarity condition in terms of nucleation, but high-freq. asymptote is produced by “stopping phase” if rupture stops very abruptly


Displacement and acceleration source spectra. Spectra: zero-frequency intercept (Mo), corner frequency (ωo or fc), high frequency decay (ω−γ), maximum (observed, emitted) frequency fmax

log

u at

R

ω-n

ωο

log freq. (ω)

log

a at

R

ωο

log freq. (ω)

fmax

Aki, Scaling law of seismic spectrum, JGR, 72, 1217-1231, 1967.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, 2235-2242, 1979.

Source spectra for two events of equal stress drop: omega cube model

Large and Small Eq

log

u at

R ω-3

log freq. (ω)

ωοS

ωοL

L

S

High-freq. spectral properties: produced by rupture growth, represent nucleation and enlargement

log

a at

R

log freq. (ω)

fmax

ωοL

ωοS



Relation between source (a) displacement (b) velocity(c) acceleration history and asymptotic behavior of spectrum

Aki, 1967

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 103 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 Mb is the same for both eventsMb is independent of Mo.

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 Mb scales directly with Mo.

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 Mb scales inversely with Mo.

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 eventsMb is independent of Mo.


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 103 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.



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.



From data: cases (c) and (b) are clearly wrong: Mb does not decrease with Mo and Mb 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.


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 103 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.



log

u at

R ω-2

log freq. (ω)

ωοS

ωοL


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 103 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.



log

u at

R ω-3

log freq. (ω)

ωοS

ωοL

Documents

Mechanics of Earthquakes and Faulting · 11/14/2019 · Seismic Spectra & Earthquake Scaling laws. Aki, Scaling law of seismic spectrum, JGR, 72, 1217-1231,1967. Hanks, bValues and