The radiation-pattern section showed that the far-field P amplitude of a double…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Earth, Climate & Energy Physics · Seismology, tectonics and geodynamics

The Solid Earth, Ch The earthquake source: moment tensor, radiation and spectra

Problem

The radiation-pattern section showed that the far-field P amplitude of a double couple is proportional to

\hat{\mathbf r}\cdot M\hat{\mathbf r}=M_0\sin2\theta\cos\phi

, with

\hat{\mathbf r}

the unit vector from the source to a station,

\theta

the angle from the fault normal

\hat{\mathbf n}

and

\phi

the azimuth about

\hat{\mathbf n}

measured from the slip direction

\hat{\mathbf d}

. Finding

\theta

and

\phi

afresh for every station is awkward. This problem works with the vector form instead, on a fault that is pure dip-slip rather than the vertical strike-slip case worked in the text. ((a)) Show that

\hat{\mathbf r}\cdot M\hat{\mathbf r}=2M_0(\hat{\mathbf r}\cdot\hat{\mathbf n})(\hat{\mathbf r}\cdot\hat{\mathbf d})

for the double couple

M_{ij}=M_0(\hat n_i\hat d_j+\hat n_j\hat d_i)

, and that this expression is unchanged when

\hat{\mathbf n}

and

\hat{\mathbf d}

are exchanged. Say what that exchange does to the fault plane, and hence what first motions alone can and cannot decide. ((b)) A thrust fault has strike

\phi_f=350^\circ

, dip

\delta=12^\circ

and rake

\lambda_r=90^\circ

(pure reverse slip). Find

\hat{\mathbf n}

and

\hat{\mathbf d}

in the frame

x

north,

y

east,

z

down. ((c)) A ray leaving the source with take-off angle

i_h

(from the downward vertical) and azimuth

Az

(clockwise from north) points in the direction

\hat{\mathbf r}=(\sin i_h\cos Az,\ \sin i_h\sin Az,\ \cos i_h)

. Two stations report

(Az,i_h)=(080^\circ,30^\circ)

and

(260^\circ,30^\circ)

. Using the result of part (a), say whether the first P motion at each station is a push (compression) or a pull (dilatation).

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