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![Page 1: Differential wave equation and seismic events Sean Ford & Holly Brown Berkeley Seismological Laboratory.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f005503460f94c15b13/html5/thumbnails/1.jpg)
Differential wave equation and seismic events
Sean Ford & Holly Brown
Berkeley Seismological Laboratory
![Page 2: Differential wave equation and seismic events Sean Ford & Holly Brown Berkeley Seismological Laboratory.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f005503460f94c15b13/html5/thumbnails/2.jpg)
Outline
• Holly and Sean 101
• Quick intro: Wave equation
• Monitoring nuclear tests
• Predicting ground motion for a future event on the Hayward Fault
• Hayward Fault tour
![Page 3: Differential wave equation and seismic events Sean Ford & Holly Brown Berkeley Seismological Laboratory.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f005503460f94c15b13/html5/thumbnails/3.jpg)
Introduction
• Seismic sources
• Earthquake
• Explosion
Slip on a plane
Pressure pulse on a sphere
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• Start with Newton
• Add ‘constitutive equation’ to relate stress to strain to displacement
Seismic wave equation
€
F = ma
F = ρ∂ 2u
∂t 2
![Page 5: Differential wave equation and seismic events Sean Ford & Holly Brown Berkeley Seismological Laboratory.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f005503460f94c15b13/html5/thumbnails/5.jpg)
• Can decompose to P and S wave solutions of the form
• Wave equation can be solved by plane-wave
Seismic wave equation
€
∇2u−1
α 2
∂ 2u
∂t 2= 0 or in 1− D α 2 ∂
2u
∂x 2=∂ 2u
∂t 2
€
u(x, t) = Acos[ω(t − x c)]
![Page 6: Differential wave equation and seismic events Sean Ford & Holly Brown Berkeley Seismological Laboratory.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f005503460f94c15b13/html5/thumbnails/6.jpg)
• The wave equation is solved on a computer by using a discrete representation of the differential equation
Finite differences
€
∂2u
∂x 2=ui+1 − 2ui + ui−1
dx 2
![Page 7: Differential wave equation and seismic events Sean Ford & Holly Brown Berkeley Seismological Laboratory.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f005503460f94c15b13/html5/thumbnails/7.jpg)
Possible project: Sumatra earthquake
Time (sec)
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Possible project
Time (sec)
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Possible project
T
Time (sec)
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Seismic sources
• Earthquake - slip on a plane
• Explosion - pressure pulse on a sphere
Compressional (P-wave)
radiation pattern
Shear (S-wave)radiation pattern
No volume change
Compressional (P-wave)
radiation constant
No Shear (S-wave)radiation
Volume change
![Page 11: Differential wave equation and seismic events Sean Ford & Holly Brown Berkeley Seismological Laboratory.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f005503460f94c15b13/html5/thumbnails/11.jpg)
Fro
m W
alte
r et
al.
(2
008)
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North Korea Nuclear Test
From Walter et al. (2008)
![Page 13: Differential wave equation and seismic events Sean Ford & Holly Brown Berkeley Seismological Laboratory.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f005503460f94c15b13/html5/thumbnails/13.jpg)
Seismic moment tensors
M represents all possible force couple components due to a seismic source in a cartesian coordinate systemNecessary to have two force couples (double couple, DC), so that angular momentum is conserved in the source sphere, which leads to Mij = Mji and the moment tensor is symmetric
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Seismic moment tensors
Double-couple (DC)
y z
x
Compensated linear vector dipole (CLVD)
Isotropic
y z
x
y z
x
Model Source M Couples Focal
Ring Fault
Explosion
Strike-slip
Mechanism
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Moment tensor inversion
in matrix form
d = Gm
m = vector of 6 independent moment tensor elementsm = (GTG)-1GTd
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Western US
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Western US
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Western US
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Western US
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Source-type plot
We calculate source-type plot parameters (Hudson et al., 1989)
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Source-type plot
We calculate source-type plot parameters (Hudson et al., 1989)
Explosion
Implosion
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Source-type plot
We calculate source-type plot parameters (Hudson et al., 1989)
Explosion
Implosion
DC -CLVD+CLVD
![Page 23: Differential wave equation and seismic events Sean Ford & Holly Brown Berkeley Seismological Laboratory.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f005503460f94c15b13/html5/thumbnails/23.jpg)
Source-type plot
We calculate source-type plot parameters (Hudson et al., 1989)
Explosion
Implosion
DC -CLVD+CLVD
HOYA
Little Skull
![Page 24: Differential wave equation and seismic events Sean Ford & Holly Brown Berkeley Seismological Laboratory.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f005503460f94c15b13/html5/thumbnails/24.jpg)
Source-type plot
We calculate source-type plot parameters (Hudson et al., 1989)
Explosion
Implosion
DC -CLVD+CLVD
HOYA
Little Skull
HOYA
Little Skull
![Page 25: Differential wave equation and seismic events Sean Ford & Holly Brown Berkeley Seismological Laboratory.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f005503460f94c15b13/html5/thumbnails/25.jpg)
Source-type plot
We calculate source-type plot parameters (Hudson et al., 1989)
Explosion
Implosion
DC -CLVD+CLVD
HOYA
Little Skull
HOYA
Little Skull
![Page 26: Differential wave equation and seismic events Sean Ford & Holly Brown Berkeley Seismological Laboratory.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f005503460f94c15b13/html5/thumbnails/26.jpg)
Western US
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9 Oct 06 North Korea test
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Crandall Canyon, Utah
Site of 6 Aug 08 Mine Collapse
Crandall Canyon, Utah
Site of 6 Aug 08 Mine Collapse
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QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
![Page 30: Differential wave equation and seismic events Sean Ford & Holly Brown Berkeley Seismological Laboratory.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f005503460f94c15b13/html5/thumbnails/30.jpg)
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
Can be implosion
![Page 31: Differential wave equation and seismic events Sean Ford & Holly Brown Berkeley Seismological Laboratory.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f005503460f94c15b13/html5/thumbnails/31.jpg)
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
Or shallow normal event
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Use method for nuclear test explosions to find source at Crandall Canyon
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Crandall Canyon event plotted near where a closing crack or collapse source should plot
Dominantly implosional
Crandall Canyon
![Page 34: Differential wave equation and seismic events Sean Ford & Holly Brown Berkeley Seismological Laboratory.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f005503460f94c15b13/html5/thumbnails/34.jpg)
Crandall Canyon event plotted near where a closing crack or collapse source should plot
Dominantly implosional
But also some shear portion
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Hayward Fault
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Hayward fault
• GoogleEarth tour
• 1868 Hayward event M6.8
• Scenario events from USGShttp://earthquake.usgs.gov/regional/nca/simulations/hayward/
• San Pablo epicenter
• Fremont epicenter
• View from Concord
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