Response of an Elastic Half Space to an Arbitrary 3-D Vector Body Force

Response of an Elastic Half Space to an Arbitrary 3-D Vector Body ForceSmith and Sandwell, JGR 2003

• Develop the three differential equations relating 3-D vector displacement to a 3-D vector body force.

• Take the 3-D Fourier transform to reduce the partial differential equation to a set of linear algebraic equations.

• Solve the linear system using the symbolic capabilities in Matlab.

• Perform the inverse fourier transform in the z-direction (depth) by repeated application of the Cauchy Residue Theorem.

• Check the analytic solution using the symbolic capabilities in Matlab.

• Solve the Boussinesq Problem to correct the non-zero normal traction on the half-space.

• Construct screw dislocation and test with analytic line-source solution.

• Integrate the point-source Green's function to simulate a vertical fault and check with the analytic fault-plane solution.

• Develop an equivalent body force for a general fault model.

• Modify the solution to account for surface topography.

• Modify the solution to have a layered half-space??

• Modify the solution to have a visco-elastic rheology??

Objective: calculate the displacement vector u(x, y, z) on the surface of the Earth due to a vector body force at depth

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U (k)

V (k)

W (k)

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥=

Uxs Uys Uzs

Uys Vys Vzs

Uzs Vzs W zs

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

Fx

Fy

Fz

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥+

Uxi Uyi −Uzi

Uyi Vyi −Vzi

Uzi Vzi −W zi

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

Fx

Fy

Fz

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ +

UB

VB

WB

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

€

Uxs k( ) =C

β 2e−β z−d2( ) D +

ky

2

k2 −

kx

2

k2 1+ β z − d 2( )( )

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥− e−β z−d1( ) D +

ky

2

k2 −

kx

2

k2 1+ β z − d1( )( )

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

Uys k( ) = −C

β 2

kx ky

k2 e−β z−d2( ) 2 + β z − d2( )( ) − e−β z−d1( ) 2 + β z − d1( )( ){ }

Uzs k( ) = −iC

β 2

kx

ke−β z−d2( ) 1+ β z − d2( )( ) − e−β z−d1( ) 1+ β z − d1( )( ){ }

Vys k( ) =C

β 2e−β z−d2( ) D +

kx

2

k2 −

ky

2

k2 1+ β z − d2( )( )

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥− e−β z−d1( ) D +

kx

2

k2 −

ky

2

k2 1+ β z − d1( )( )

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

Vzs k( ) = −iC

β 2

ky

ke−β z−d2( ) 1+ β z − d2( )( ) − e−β z−d1( ) 1+ β z − d1( )( ){ }

W zs k( ) =C

β 2e−β z−d2( ) D +1+ β z − d 2( )[ ] − e−β z−d1( ) D +1+ β z − d1( )[ ]{ }

Full Displacement Solution:Full Displacement Solution:

(Source)(Source) (Image)(Image) (Boussinesq)(Boussinesq)

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where C =(λ + μ )

4μ (λ + 2μ ) D =

λ + 3μ

λ + μ α =

λ + μ

λ + 2μ k = kx

2 + ky

2( )

1/2

β = 2π k .

Components:Components:

Force coupleForce couple • • Magnitude ~ slip rateMagnitude ~ slip rate • • Direction || to plate motionDirection || to plate motion

Sketch of 3-D fault in an elastic half-spaceSketch of 3-D fault in an elastic half-space

• • Analytic form of the force couple is the derivative of a Gaussian function with half-width equal to cell spacingAnalytic form of the force couple is the derivative of a Gaussian function with half-width equal to cell spacing• • Cosine transform in x-direction is used for constant velocity difference across the plate boundaryCosine transform in x-direction is used for constant velocity difference across the plate boundary• • Uniform far-field velocity is simulated by arranging the fault trace to be cyclic in the y-directionUniform far-field velocity is simulated by arranging the fault trace to be cyclic in the y-direction

User defines: dUser defines: d11, d, d22, z, zobsobs, x, x11, x, x22, y, y11, y, y22, and , and FF

Assign slip rates from literatureAssign slip rates from literatureParalleling segments: sum to 40 mm/yrParalleling segments: sum to 40 mm/yr

LockingLocking depths?depths?

San Andreas Fault SegmentsSan Andreas Fault Segments

Use 1099 horizontalUse 1099 horizontalGPS velocity measurementsGPS velocity measurements

to solve for locking depthto solve for locking depth

Locking Depth ResultsLocking Depth Results

rms model misfit: 2.43 mm/yrrms model misfit: 2.43 mm/yr

Predicted Vertical Uplift

Geologic estimatesGeologic estimates

San Gabriel MtsSan Gabriel Mts 3-10mm/yr3-10mm/yr

[[BrownBrown, 1991], 1991]

San Bernadino Mts San Bernadino Mts 2 mm/yr2 mm/yr

[[Yule and SeihYule and Seih, 1997], 1997]

Geodetic estimatesGeodetic estimates

Coulomb StressCoulomb Stress