fracspy.mtinversion.greensfunction.pwave_greens_comp#

fracspy.mtinversion.greensfunction.pwave_greens_comp(cosine_sourceangles, dists, src_idx, vel, MT_comp_dict, comp_idx, omega_p)[source]#

Particle velocity component of the P-wave Green’s function

Compute Green’s function for a given-component (x, y, or z) of the P-wave between a source and a set of receivers

Parameters:
cosine_sourceanglesnumpy.ndarray

Cosine source angles of size \(3 \times n_r \times n_x \times n_y \times n_z\)

distsnumpy.ndarray

Distances of size \(\times n_r \times n_x \times n_y \times n_z\)

src_idxnumpy.ndarray

Source location indices (relative to x, y, and z axes)

velnumpy.ndarray

Velocity model

MT_comp_dictdict

Dictionary containing Moment Tensor parameters

comp_idxint

Index of component at receiver side

omega_pfloat

Peak frequency of the given wave

Returns:
G_znumpy.ndarray

Green’s functions of size \(6 \times n_r\)

Notes

This method computes the amplitudes of a given component of the particle velocity Green’s functions associated to the first P-wave arrival, assuming a known source location based on the far-field particle velocity expression from a moment tensor source in a homogeneous full space (eq. 4.29, [1]):

\[v_i^P = j \omega_P \left( \frac{\gamma_i\gamma_p\gamma_q}{4\pi\rho\alpha^3} \frac{1}{r} \right) M_{pq}\]

where:

  • \(v\) is the particle velocity measurements (seismic data) at the arrival of the wave, in other words the P-wave peak amplitudes;

  • \(M\) is the moment tensor;

  • \(\theta\) describes whether we are utilising the P-wave information;

  • \(i\) describes the i-component of the data, aligning with the below p,q definitions;

  • \(p\) describes the first index of the moment tensor element;

  • \(q\) describes the second index of the moment tensor element;

  • \(\omega_P\) is the peak frequency of the given wave;

  • \(\gamma_{i/p/q}\) is the take-off angle in the z/p/q-th direction (for a ray between the source and receiver);

  • \(r\) is the distance between source and receiver;

  • \(\alpha\) is the average velocity (currently we assume a homogeneous velocity);

  • \(\rho\) is the average density;

[1]

Aki, K., and Richards, P. G. “Quantitative Seismology”, University Science Books, 2002.

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