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[SSS06-10] Geometrical spreading and radiation pattern for the near-field term of point source ground motion
Keywords:near-field term, geometrical spreading, radiation pattern
Since the 2016 Kumamoto earthquake, ground motion in the near-field has been attracting attention. However, among the point source ground motion obtained from the solution of the point force ground motion in Love (1906), we could not find the literature that clarified the geometrical spreading of the near-field term, though the geometrical spreading and radiation pattern of the far-field term have been well investigated for a long time. In addition, the radiation pattern of the near-field term was obtained by Aki and Richards (1980, 2002), but it is for an uncommon point source of horizontal strike-slip fault and is somewhat impractical. Therefore, we here reexamine the geometrical spreading and radiation pattern of the near-field term.
Since it is considered that a vertical one is more common than a horizontal one among strike-slip faults, we will examine ground motions caused by a point source of vertical strike-slip fault in the infinite medium. Koketsu (2018) gave their near-field term as Un=(30γnγxγy–6δnyγx–6δnxγy)/4πρR4∫τM0(t–τ)dτ (integration range [R/α,R/β]). It is assumed that ground motion is represented by ground velocity ∂Un/∂t, and the moment time function M0(t) is the product of the constant final moment M0 and the origin-shifted ramp function U0(t) with rise time τr. Since the time function in ∂Un/∂t is ∫τ∂M0(t–τ)/∂tdτ=M0∫τ∂U0(t–τ)/∂tdτ, the peak ground velocity (PGV) is ∂Un/∂t when (1) ∫τ∂U0(t–τ)/∂tdτ (integration range [R/α, R/β]) is maximum.
In the case of τr=1 s, P wave velocity α=5.5 km/s and S wave velocity β=3.0 km/s of the medium, and distance R=3 km (R1) or 12 km (R2), Eq. (1) is plotted in the figure on the next page. From this figure, it can be seen that the waveforms at R1 and R2 are different, depending on whether R/α+τr is after or before R/β. Therefore, the boundary is Rτ=τr/(β-1–α-1) obtained by solving R/α+τr=R/β, and when R≧Rτ, the central part of the waveform becomes a straight line with a positive slope, and the apex of the maximum value appears (R2 in the attached figure). Since ∂U0(t–τ)/∂t is 1 when t–τr≦τ≦t and 0 otherwise, the integration range changes with t. Considering the range of R/α+τr<t<R/β, the lower limit of integration must be changed from R/α to t–τr because of t–τr>R/α, and the upper limit of integration must be changed from R/β to t because of t<R/β.
From the above, Eq. (1)=∫τdτ (integration range [t–τr,t])=τrt–τr2/2, and it can be seen that the straight line in the central part of the R2 waveform is this straight line with a slope τr. Furthermore, the maximum value of Eq. (1) at t=R/β is obtained as τrR/β–τr2/2, so if this is substituted into ∂Un/∂t, it can be seen that the peak ground velocity (PGV) of the near-field term mainly decays along 1/R3 except in the vicinity of the point source.
Finally, we move on to the radiation pattern of the near-field term, but since the formulas are very long, only the results are shown here. When Un, n=x,y,z at the beginning is converted to the spherical coordinate system n=R,θ,φ and the direction cosines are replaced with trigonometric functions of θ, φ, An in Un=An/4πρR4∫τM0(t–τ)dτ (integration range [R/α,R/β]) are given as follows. AR=9sin2θ sin 2φ, Aθ=–3sin 2θ sin 2φ, Aφ=–6sin θ cos 2φ. These are the results of a vertical strike-slip fault and do not agree with Eq. (4.33) of Aki and Richards (1980, 2002) in the case of a horizontal strike-slip fault.
Since it is considered that a vertical one is more common than a horizontal one among strike-slip faults, we will examine ground motions caused by a point source of vertical strike-slip fault in the infinite medium. Koketsu (2018) gave their near-field term as Un=(30γnγxγy–6δnyγx–6δnxγy)/4πρR4∫τM0(t–τ)dτ (integration range [R/α,R/β]). It is assumed that ground motion is represented by ground velocity ∂Un/∂t, and the moment time function M0(t) is the product of the constant final moment M0 and the origin-shifted ramp function U0(t) with rise time τr. Since the time function in ∂Un/∂t is ∫τ∂M0(t–τ)/∂tdτ=M0∫τ∂U0(t–τ)/∂tdτ, the peak ground velocity (PGV) is ∂Un/∂t when (1) ∫τ∂U0(t–τ)/∂tdτ (integration range [R/α, R/β]) is maximum.
In the case of τr=1 s, P wave velocity α=5.5 km/s and S wave velocity β=3.0 km/s of the medium, and distance R=3 km (R1) or 12 km (R2), Eq. (1) is plotted in the figure on the next page. From this figure, it can be seen that the waveforms at R1 and R2 are different, depending on whether R/α+τr is after or before R/β. Therefore, the boundary is Rτ=τr/(β-1–α-1) obtained by solving R/α+τr=R/β, and when R≧Rτ, the central part of the waveform becomes a straight line with a positive slope, and the apex of the maximum value appears (R2 in the attached figure). Since ∂U0(t–τ)/∂t is 1 when t–τr≦τ≦t and 0 otherwise, the integration range changes with t. Considering the range of R/α+τr<t<R/β, the lower limit of integration must be changed from R/α to t–τr because of t–τr>R/α, and the upper limit of integration must be changed from R/β to t because of t<R/β.
From the above, Eq. (1)=∫τdτ (integration range [t–τr,t])=τrt–τr2/2, and it can be seen that the straight line in the central part of the R2 waveform is this straight line with a slope τr. Furthermore, the maximum value of Eq. (1) at t=R/β is obtained as τrR/β–τr2/2, so if this is substituted into ∂Un/∂t, it can be seen that the peak ground velocity (PGV) of the near-field term mainly decays along 1/R3 except in the vicinity of the point source.
Finally, we move on to the radiation pattern of the near-field term, but since the formulas are very long, only the results are shown here. When Un, n=x,y,z at the beginning is converted to the spherical coordinate system n=R,θ,φ and the direction cosines are replaced with trigonometric functions of θ, φ, An in Un=An/4πρR4∫τM0(t–τ)dτ (integration range [R/α,R/β]) are given as follows. AR=9sin2θ sin 2φ, Aθ=–3sin 2θ sin 2φ, Aφ=–6sin θ cos 2φ. These are the results of a vertical strike-slip fault and do not agree with Eq. (4.33) of Aki and Richards (1980, 2002) in the case of a horizontal strike-slip fault.