Differential wave equation and seismic events

Sean Ford & Holly BrownBerkeley Seismological

Laboratory

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

Introduction• Seismic sources

• Earthquake

• Explosion

Slip on a plane

Pressure pulse on a sphere

• Start with Newton

• Add ‘constitutive equation’ to relate stress to strain to displacement

Seismic wave equation

€

F = ma

F = ρ ∂2u∂t 2

• Can decompose to P and S wave solutions of the form

• Wave equation can be solved by plane-wave

Seismic wave equation

€

∇2u− 1α 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)]

• 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

Possible project: Sumatra earthquake

Time (sec)

Possible project

Time (sec)

Possible project

T

Time (sec)

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

From

Wal

ter e

t al.

(200

8)

North Korea Nuclear Test

From Walter et al. (2008)

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

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

Moment tensor inversion

in matrix formd = Gm

m = vector of 6 independent moment tensor elementsm = (GTG)-1GTd

Western US

Western US

Western US

Western US

Source-type plotWe calculate source-type plot parameters (Hudson et al., 1989)

Source-type plotWe calculate source-type plot parameters (Hudson et al., 1989)

Explosion

Implosion

Source-type plotWe calculate source-type plot parameters (Hudson et al., 1989)

Explosion

Implosion

DC -CLVD+CLVD

Source-type plotWe calculate source-type plot parameters (Hudson et al., 1989)

Explosion

Implosion

DC -CLVD+CLVD

HOYA

Little Skull

Source-type plotWe calculate source-type plot parameters (Hudson et al., 1989)

Explosion

Implosion

DC -CLVD+CLVD

HOYA

Little Skull

HOYA

Little Skull

Source-type plotWe calculate source-type plot parameters (Hudson et al., 1989)

Explosion

Implosion

DC -CLVD+CLVD

HOYA

Little Skull

HOYA

Little Skull

Western US

9 Oct 06 North Korea test

Crandall Canyon, Utah

Site of 6 Aug 08 Mine Collapse

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

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

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

Use method for nuclear test explosions to find source at Crandall Canyon

Crandall Canyon event plotted near where a closing crack or collapse source should plot

Dominantly implosional

Crandall Canyon

Crandall Canyon event plotted near where a closing crack or collapse source should plot

Dominantly implosional

But also some shear portion

Hayward Fault

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