Amplitude FormulaΒΆ
The amplitude calculations follow Chapters 13 and 17 of Ammon et al. [AVLW20]. Equation numbers referenced below correspond to those in Ammon et al. [AVLW20]. The displacement amplitude of a seismic phase is computed as:
where each term is calculated as follows:
Source moment (Nm/s) (Eq.~7.24): The source term is the moment-rate function,
\[\dot{M}(t)=M_0\,\dot{s}(t),\]where \(s(t)\) is the normalized source-time function. Here, we only use the scalar seismic moment, computed as:
\[M_0 = 10^{\,1.5M_w + 9.1} \; \mathrm{N\,m}.\]
Radiation pattern (dimensionless), \(F\) (Eqs.~17.71--17.73):
\[\begin{split}\begin{aligned} F_P ={}& \left[\cos(\lambda)\sin(\delta)\sin2(\phi_{r}-\phi_{f}) - \sin(\lambda)\sin(2\delta)\sin^2(\phi_{r}-\phi_{f})\right] \sin^2(i_h) \\ &+ \left[\sin(\lambda)\cos(2\delta)\sin(\phi_{r}-\phi_{f}) - \cos(\lambda)\cos(\delta)\cos(\phi_{r}-\phi_{f})\right] \sin(2i_h) \\ &+ \sin(\lambda)\sin(2\delta)\cos^2(i_h),\\ \\ F_{SV} ={}& \left[\sin\lambda\,\cos(2\delta)\sin(\phi_{r}-\phi_{f}) -\cos\lambda\,\cos\delta\,\cos(\phi_{r}-\phi_{f})\right]\cos(2i_h) \\ &+ \frac{1}{2}\cos\lambda\,\sin\delta\,\sin2(\phi_{r}-\phi_{f})\sin(2i_h) \\ &- \frac{1}{2}\sin\lambda\,\sin(2\delta) \left[1+\sin^2(\phi_{r}-\phi_{f})\right],\\\\ F_{SH} ={}& \left[\cos\lambda\,\cos\delta\,\sin(\phi_{r}-\phi_{f}) +\sin\lambda\,\cos(2\delta)\cos(\phi_{r}-\phi_{f})\right]\cos i_h \\ &+ \left[\cos\lambda\,\sin\delta\,\cos2(\phi_{r}-\phi_{f}) -\frac{1}{2}\sin\lambda\,\sin(2\delta)\sin2(\phi_{r}-\phi_{f})\right]\sin i_h.\\ \end{aligned}\end{split}\]where,
\[ \begin{align}\begin{aligned}i_h = \text{takeoff}, \qquad \phi_r = \text{azimuth},\\ \phi_f = \text{strike}, \qquad \delta = \text{dip}, \qquad \lambda = \text{rake}.\end{aligned}\end{align} \]
Radiation term ( \(\text{kg s}^{-3}\) ), \(R_s\) (Eq.~17.70):
\[R_s = 4\pi \rho_s V_s^3 \times 10^{12},\]where, \(\rho_s\) and \(V_s\) are density and velocity at source, respectively. Conversion factor of \(10^{12}\) for changing density from \(\text{g } \text{cm}^{-3}\) to \(\text{kg } \text{m}^{-3}\) and velocity from \(\text{km } \text{s}^{-1}\) to \(\text{m } \text{s}^{-3}\).
Geometrical spreading (1/km), \(G(\Delta,h)\) (Eqs.~13.9):
\[G(\Delta,h)= \sqrt{\frac{E(\Delta)}{E_h}}\]\[E(\Delta)= E_h \left(\frac{\sin i_h}{r_0^2 \cos i_0 \sin\Delta}\right) \left(\frac{di_h}{|d\Delta|}\right),\]where \(E(\Delta)\) is the seismic energy density (energy per unit area) at epicentral distance \(\Delta\), \(E_h=K/2\pi\) is the source energy density on the initial hemispherical wavefront with total radiated energy \(K\), \(r_0\) is the receiver radius from the Earth's center, and \(i_h\) and \(i_0\) are the takeoff and incidence angles.
Reflection and transmission coefficients (dimensionless), \(C_k\).
The product is over all discontinuities encountered over the phase path. The four interface types considered are summarized in Tables S1--S4.
\[\prod_{k=1}^{n} C_k\]
Attenuation (dimensionless) (Eq.~B13.2.2):
\[\text{Attenuation} = e^{-\pi f t^*}\]where the integrated attenuation along the ray path is
\[t^* = \int_{\mathrm{path}} \frac{dt}{Q} = \sum_{i=1}^{N} \frac{t_i}{Q_i},\]with \(Q_i\) and \(t_i\) denoting the quality factor and travel time through the \(i\)-th layer, respectively.
Free-surface correction (dimensionless), \(FS\) (Eqs.~13.66--13.68):
\[FS_r^{(P)} = \frac{4p\,\alpha\,\eta_\alpha\eta_\beta \beta^{-2}}{A}, \qquad FS_z^{(P)} = \frac{2\alpha\,\eta_\alpha E\beta^{-2}}{A}\]\[FS_r^{(SV)} = \frac{2\eta_\beta E\beta^{-1}}{A}, \qquad FS_z^{(SV)} = -\frac{ 4p\,\eta_\alpha\eta_\beta\beta^{-1}}{A}.\]where,
\[\eta_\alpha=\sqrt{\frac{1}{\alpha^2}-p^2}, \qquad \eta_\beta=\sqrt{\frac{1}{\beta^2}-p^2}, \qquad A=4p^2\eta_\alpha\eta_\beta+\left(\frac{1}{\beta^{2}}-2p^2\right)^2, \qquad E=\eta_\beta^2-p^2.\]\(\rho\) is the density, \(\alpha\) and \(\beta\) are the P- and S-wave velocities, respectively.
Eq. 13.66 in Ammon et al. [AVLW20] should not have the term \((\eta_\beta^2-p^2)\) and Eq. 13.67 is missing a a factor of 2 times for the ray parameter.
S1. Solid-solid
Solid-solid displacement reflection and transmission coefficients.
The P--SV System
\[ \begin{align}\begin{aligned}R_{PP} &= [(b\eta_{\alpha1}-c\eta_{\alpha2})F -(a+d\eta_{\alpha1}\eta_{\beta2})Hp^{2}]/D\\R_{PS} &= -2\eta_{\alpha1}\left(ab+cd\,\eta_{\alpha2}\eta_{\beta2} \right)p({\alpha_1}/{\beta_1}) /{D}\\T_{PP} &= { 2\rho_1\eta_{\alpha1}F \left({\alpha_1}/{\alpha_2}\right) }/{D}\\T_{PS} &= { 2\rho_1\eta_{\alpha1}Hp \left({\alpha_1}/{\beta_2}\right)}/{D}\\R_{SS} &= {-[ \left(b\eta_{\beta1}-c\eta_{\beta2}\right)E - \left(a+d\eta_{\alpha2}\eta_{\beta1}\right)Gp^{2} ]}/{D}\\R_{SP} &= { -2\eta_{\beta1}\left(ab+cd\,\eta_{\alpha2}\eta_{\beta2}\right) p\left({\beta_1}/{\alpha_1}\right) }/{D}\\T_{SP} &= { -2\rho_1\eta_{\beta1}Gp({\beta_1}/{\alpha_2}) }/{D}\\T_{SS} &= { 2\rho_1\eta_{\beta1}E ({\beta_1}/{\beta_2}) }/{D}\end{aligned}\end{align} \]The SH System
\[ \begin{align}\begin{aligned}R_{SS} &= { (\mu_1\eta_{\beta1}-\mu_2\eta_{\beta2}})/ ({\mu_1\eta_{\beta1}+\mu_2\eta_{\beta2} })\\T_{SS} &= { (2\mu_1\eta_{\beta1}})/ ({\mu_1\eta_{\beta1}+\mu_2\eta_{\beta2}})\end{aligned}\end{align} \]where,
\[ \begin{align}\begin{aligned}\eta_{\alpha} &= \sqrt{\frac{1}{\alpha^{2}}-p^{2}}\\\mu_1 &= \rho_1\beta_1^2\\a &= \rho_2\left(1-2\beta_2^2p^2\right) -\rho_1\left(1-2\beta_1^2p^2\right)\\b &= \rho_2\left(1-2\beta_2^2p^2\right) +2\rho_1\beta_1^2p^2,\\c &= \rho_1\left(1-2\beta_1^2p^2\right) +2\rho_2\beta_2^2p^2,\\d &= 2\left(\rho_2\beta_2^2-\rho_1\beta_1^2\right)\\\eta_{\beta} &= \sqrt{\frac{1}{\beta^{2}}-p^{2}}\\\mu_2 &= \rho_2\beta_2^2\\E &= b\eta_{\alpha1}+c\eta_{\alpha2}\\F &= b\eta_{\beta1}+c\eta_{\beta2}\\G &= a-d\eta_{\alpha1}\eta_{\beta2}\\H &= a-d\eta_{\alpha2}\eta_{\beta1}\\D &= EF+GHp2\\\rho, \alpha,\beta: \text{density, P- and S-wave velocities}\end{aligned}\end{align} \]Corrections to Table 13.1 of Ammon et al. [AVLW20]:
In the P-SV system, the \(R_{SS}\) expression in Ammon et al. [AVLW20] contains \(b\eta_{\alpha2}\eta_{\beta1}\), which should instead be \(d\eta_{\alpha2}\eta_{\beta1}\).
Similarly, the \(T_{SS}\) expression contains \(2\rho_2\eta_{\beta1}\), which should instead be \(2\rho_1\eta_{\beta1}\).
S2. Fluid-fluid displacement reflection and transmission coefficients.
Incident P
\[ \begin{align}\begin{aligned}R_{PP} &= {{ [\rho_1\eta_{\alpha2} - \rho_2\eta_{\alpha1} }]/{D_{ff}}}\\T_{PP} &= {2\rho_1 \left({\alpha_1}{\alpha_2^{-1}}\right) \eta_{\alpha1} }/{D_{ff}}\end{aligned}\end{align} \]where,
\[ D_{ff}= \rho_2\eta_{\alpha1} + \rho_1\eta_{\alpha2}.\]
S3. Solid-fluid and Fluid-solid displacement reflection and transmission coefficients.
Incident P--SV From Solid (Medium 1)
\[ \begin{align}\begin{aligned}R_{PP} &= {[\eta_{\alpha1} \left( 4\beta_1^{4}p^{2}\rho_1\eta_{\alpha2}\eta_{\beta1} +\rho_2\right) - \rho_1\eta_{\alpha2} C_{sf}^{2} }]/{D_{sf}}\\R_{PS} &= {-4\alpha_1\beta_1 p\rho_1 \eta_{\alpha1}\eta_{\alpha2} C_{sf} }/{D_{sf}}\\T_{PP} &= { 2\left({\alpha_1}/{\alpha_2}\right) \rho_1\eta_{\alpha1} C_{sf} }/{D_{sf}}\\R_{SS} &= [{ \rho_1\eta_{\alpha2} C_{sf}^{2} + \eta_{\alpha1} \left( \rho_2 - 4\beta_1^{4}p^{2}\rho_1 \eta_{\alpha2}\eta_{\beta1} \right) }] /{D_{sf}}\\R_{SP} &= {4\alpha_1^{-1}\beta_1^{3}\rho_1 p \eta_{\alpha2}\eta_{\beta1} C_{sf} } /{D_{sf}}\\T_{SP} &= { 4\alpha_2^{-1}\beta_1^{3}\rho_1 p \eta_{\alpha1}\eta_{\beta1} } /{D_{sf}}\end{aligned}\end{align} \]Incident SH From Solid (Medium 1)
\[R_{ss} = 1\]Incident P From Fluid (Medium 1)
\[ \begin{align}\begin{aligned}R_{PP} &= {{ \left[ -\rho_2\eta_{\alpha1} \left( 4\beta_2^{4}p^{2}\eta_{\alpha2}\eta_{\beta2} +C_{fs}^{2} \right) +\rho_1\eta_{\alpha2} \right] (\alpha_2^{2}\beta_2^{-1}\rho_1^{-1}) } /{D_{fs}}}\\T_{PP} &= { -2\alpha_1\alpha_2 \eta_{\alpha1} C_{fs} \beta_2^{-1} } /{D_{fs}}\\T_{PS} &= {4\alpha_1\alpha_2^{2} p \eta_{\alpha1} \eta_{\alpha2} /D_{fs}}\end{aligned}\end{align} \]where,
\[ \begin{align}\begin{aligned}D_{sf} &= \rho_2\eta_{\alpha1} +\rho_1\eta_{\alpha2} [4\beta_1^{4}p^{2} \eta_{\alpha1}\eta_{\beta1} +(C_{sf})^{2}]\\C_{sf} &= 1-2\beta_1^{2}p^{2}\\D_{fs} &= -{\alpha_2^{2}}{(\beta_2\rho_1)^{-1}} \Big[ 4\rho_2\beta_2^{4}p^{2} \eta_{\alpha1}\eta_{\alpha2}\eta_{\beta2} +\rho_2C_{fs}^{2}\eta_{\alpha1} +2\rho_1\beta_2^{2}p^{2}\eta_{\alpha2} +\rho_1C_{fs}\eta_{\alpha2} \Big]\\C_{fs} &= 1-2\beta_2^{2}p^{2}\end{aligned}\end{align} \]The fluid-to-solid coefficients here are significantly different from Table 13.2 in Ammon et al. [AVLW20]. We show our calculations in the Jupyter notebook refltrans.ipynb in the Zenodo repository citep{crotwell_2026_22086196}.
S4. Free-surface displacement reflection and transmission coefficients.
Incident P--SV From Solid
\[ \begin{align}\begin{aligned}R_{PP} &= {\Big[ -\left({\beta^{-2}}-2p^2\right)^2 +4p^2\eta_{\alpha}\eta_{\beta} }\Big]/{D_{sfree}}\\R_{PS} &= {{ 4\left({\alpha}{\beta^{-1}}\right) p\,\eta_{\alpha1} \left({\beta^{-2}}-2p^2\right) }/{D_{sfree}}}\\R_{SP} &= {{ 4\left({\beta}{\alpha^{-1}}\right) p\,\eta_{\beta} \left({\beta^{-2}}-2p^2\right) }/{D_{sfree}}}\\R_{SS} &= {\Big[({\beta^{-2}}-2p^2)^2 - 4p^2\eta_{\alpha}\eta_{\beta} }\Big]/ {D_{sfree}}\end{aligned}\end{align} \]Incident SH From Solid
\[R_{ss} = 1\]Incident P From Fluid
\[R_{pp} = 1\]where,
\[D_{sfree} = \left({\beta^{-2}}-2p^2\right)^2 + 4p^2\eta_{\alpha}\eta_{\beta}\]Corrections in Table 13.3 Ammon et al. [AVLW20] \(R_{ss}\) is missing a multiplication factor of -1.