Fast Computation of Global Sensitivity Kernel Database Based on Spectral-Element Simulations

Abstract

Finite-frequency sensitivity kernels, a theoretical improvement from simple infinitely thin ray paths, have been used extensively in recent global and regional tomographic inversions. These sensitivity kernels provide more consistent and accurate interpretation of a growing number of broadband measurements, and are critical in mapping 3D heterogeneous structures of the mantle. Based on Born approximation, the calculation of sensitivity kernels requires the interaction of the forward wavefield and an adjoint wavefield generated by placing adjoint sources at stations. Both fields can be obtained accurately through numerical simulations of seismic wave propagation, particularly important for kernels of phases that cannot be sufficiently described by ray theory (such as core-diffracted waves). However, the total number of forward and adjoint numerical simulations required to build kernels for individual source–receiver pairs and to form the design matrix for classical tomography is computationally unaffordable. In this paper, we take advantage of the symmetry of 1D reference models, perform moment tensor forward and point force adjoint spectral-element simulations, and save six-component strain fields only on the equatorial plane based on the open-source spectral-element simulation package, SPECFEM3D_GLOBE. Sensitivity kernels for seismic phases at any epicentral distance can be efficiently computed by combining forward and adjoint strain wavefields from the saved strain field database, which significantly reduces both the number of simulations and the amount of storage required for global tomographic problems. Based on this technique, we compute traveltime, amplitude and/or boundary kernels of isotropic and radially anisotropic elastic parameters for various ($P$, $S$, $P_{\mathrm{diff}}$, $S_{\mathrm{diff}}$, depth, surface-reflected, surface wave, S ₆₆₀ S boundary, etc.) phases for 1D ak135 model, in preparation for future global tomographic inversions.

Appendices

Appendix A: Rotation of Elastic Tensor and Kernels

For any arbitrary point $\mathbf {x}_\mathrm{p}$ at longitude $\phi$ and colatitude $\theta$, the rotation from the global coordinate system $(\hat{x}_1, \hat{x}_2, \hat{x}_3)$ to the local spherical coordinate system $(\hat{\theta }, \hat{\phi }, \hat{r})$ can be accomplished in two steps. First rotate the coordinate system with respect to $\hat{x}_3$ and then about the new $\hat{x}_2$, i.e.,

$$\begin{aligned} (\hat{x}_1,\hat{x}_2,\hat{x}_3)&\xrightarrow {\phi \text { angle w.r.t. }\hat{x}_3} (\hat{x}_1',\hat{x}_2',\hat{x}_3') \nonumber \\&\xrightarrow {\theta \text { angle w.r.t. }\hat{x}_2'} (\hat{x}_1'',\hat{x}_2'',\hat{x}_3'') = (\hat{\theta },\hat{\phi },\hat{r}). \end{aligned}$$

(34)

By defining $C_t = \cos \theta$, $S_t = \sin \theta$, $C_\mathrm{p}=\cos \phi$, and $S_\mathrm{p}=\sin \phi$ for shorter notation, the product of the above two rotation matrices gives the transformation from global to local spherical coordinates

$$\begin{aligned} {\varvec{M}}= {\varvec{M}}_2{\varvec{M}}_1=\begin{bmatrix} C_tC_\mathrm{p}&C_t S_\mathrm{p}&-S_t \\ -S_\mathrm{p}&C_\mathrm{p}&0 \\ S_tC_\mathrm{p}&S_tS_\mathrm{p}&C_t \end{bmatrix}. \end{aligned}$$

(35)

Therefore, the elastic tensor elements ${\mathsf {C}}_{i''j''k''l''}$ in the local spherical coordinate system can be calculated from the elastic tensor elements ${\mathsf {C}}_{ijkl}$ in the global coordinate system as

$$\begin{aligned} {\mathsf {C}}_{i''j''k''l''} = M_{i''i} M_{j''j} M_{k''k} M_{l''l} {\mathsf {C}}_{ijkl}. \end{aligned}$$

(36)

In the local spherical coordinate system, we can rewrite the ${\mathsf {C}}_{ijkl}$ tensor with the Voigt notation, following the notations in (Chen and Tromp 2007, Appendix A.1), as

$$\begin{aligned} {\mathsf {C}}^{\text {Voigt}}_{\text {spherical}}= & {} \begin{bmatrix} {\mathsf {c}}_{\theta \theta \theta \theta }&{\mathsf {c}}_{\theta \theta \phi \phi }&{\mathsf {c}}_{\theta \theta rr}&{\mathsf {c}}_{\theta \theta \phi r}&{\mathsf {c}}_{\theta \theta \theta r}&{\mathsf {c}}_{\theta \theta \theta \phi } \\&{\mathsf {c}}_{\phi \phi \phi \phi }&{\mathsf {c}}_{\phi \phi rr}&{\mathsf {c}}_{\phi \phi \phi r}&{\mathsf {c}}_{\phi \phi \theta r}&{\mathsf {c}}_{\phi \phi \theta \phi } \\&{\mathsf {c}}_{r r r r}&{\mathsf {c}}_{rr\phi r}&{\mathsf {c}}_{rr\theta r}&{\mathsf {c}}_{rr \theta \phi } \\&&{\mathsf {c}}_{\phi r \phi r }&{\mathsf {c}}_{\phi r \theta r}&{\mathsf {c}}_{\phi r \theta \phi }\\&&{\mathsf {c}}_{\theta r \theta r}&{\mathsf {c}}_{\theta r \theta \phi } \\&&&{\mathsf {c}}_{\theta \phi \theta \phi }, \end{bmatrix} \nonumber \\= & {} \begin{bmatrix} {\mathsf {C}}_1&{\mathsf {C}}_2&{\mathsf {C}}_3&{\mathsf {C}}_4&{\mathsf {C}}_5&{\mathsf {C}}_6 \\&{\mathsf {C}}_7&{\mathsf {C}}_8&{\mathsf {C}}_9&{\mathsf {C}}_{10}&{\mathsf {C}}_{11} \\&{\mathsf {C}}_{12}&{\mathsf {C}}_{13}&{\mathsf {C}}_{14}&{\mathsf {C}}_{15} \\&&{\mathsf {C}}_{16}&{\mathsf {C}}_{17}&{\mathsf {C}}_{18} \\&&{\mathsf {C}}_{19}&{\mathsf {C}}_{20} \\&&&{\mathsf {C}}_{21} \end{bmatrix}, \end{aligned}$$

(37)

where ${\mathsf {C}}_I$, $I=1,\ldots , 21$ represent the 21 independent elements of the elastic tensor in the spherical coordinate system. Note the symmetry of the elastic tensor is assumed for the above expressions and only the upper triangle elements are listed. Also, note that ${\mathsf {C}}_I$’s are different from C, one of the five independent elastic parameters for transversely isotropic media.

We define the sensitivity kernel for ${\mathsf {C}}_I$ as $K_I, I=1,\ldots , 21$ ($K_I$ and $K_{ijkl}$ can be distinguished easily based on the number of indices). By considering the symmetry in $K_{ijkl}$, it can be shown that

$$\begin{aligned} K_A&= K_1 + K_2 + K_7 \end{aligned}$$

(38a)

$$\begin{aligned} K_C&= K_{12} \end{aligned}$$

(38b)

$$\begin{aligned} K_F&= K_3 + K_8 \end{aligned}$$

(38c)

$$\begin{aligned} K_L&= K_{16} + K_{19} \end{aligned}$$

(38d)

$$\begin{aligned} K_N&= - 2K_2 + K_{21}. \end{aligned}$$

(38e)

Therefore, for any given point $\mathbf {x}_\mathrm{p}$ in space, a rotation ${\mathsf {M}}$ about the center of the Earth that transforms it from the global coordinate system to the local spherical coordinate system is first applied to the global elastic tensor kernel $K_{ijkl}$ to obtain $K_I, I=1,\ldots\,21$, from which the kernels for the five transversely isotropic elastic parameters can be then constructed based on Eq. 38.

Appendix B: Rotation of an Arbitrary Source–Receiver Path onto the Equator

For an arbitrary source $\mathbf {x}_\mathrm{s}$ at longitude $\phi _\mathrm{s}$ and colatitude $\theta _\mathrm{s}$, and receiver $\mathbf {x}_\mathrm{r}$ at longitude $\phi _\mathrm{r}$ and colatitude $\theta _\mathrm{r}$, we seek rotations that transfer a general source–receiver great-circle path onto the equator. The same set of rotations can then be used to populate the sensitivity kernel values in the original global coordinate system by those extracted for equatorial source–receiver path from the strain database. We first rotate the global $\hat{x}_1$ axis (at the intersection of prime meridian and the equator) onto the unit-normalized source position $\hat{\mathbf {x}}_\mathrm{s}$ ($=\mathbf {x}_\mathrm{s}/\Vert \mathbf {x}_\mathrm{s}\Vert$) similar to the procedures in Appendix A,

$$\begin{aligned} (\hat{x}_1,\hat{x}_2,\hat{x}_3) \xrightarrow {\phi _\mathrm{s} \text { angle w.r.t. }\hat{x}_3} (\hat{x}'_1,\hat{x}'_2,\hat{x}'_3) \nonumber \\ \xrightarrow {-(90^\circ -\theta _\mathrm{s}) \text { angle w.r.t. }\hat{x}_2'} (\hat{r}_\mathrm{s},\hat{\theta }_\mathrm{s},\hat{\phi }_\mathrm{s}), \end{aligned}$$

(39)

corresponding to two rotation matrices. By assuming $C_t=\cos \theta _\mathrm{s}$, $S_t=\sin \theta _\mathrm{s}$, $C_\mathrm{p}=\cos \phi _\mathrm{s}$, $S_\mathrm{p}=\sin \phi _\mathrm{s}$, the product of these two rotations gives the rotation matrix from the global to a local spherical coordinates such that the unit-normalized source position $\hat{\mathbf {x}}_\mathrm{s}$ becomes the new $\hat{x}_1$,

$$\begin{aligned} {\varvec{R}}_\mathrm{s} = {\varvec{R}}_2 {\varvec{R}}_1 = \begin{bmatrix} S_tC_\mathrm{p}&S_t S_\mathrm{p}&C_t \\ -S_\mathrm{p}&C_\mathrm{p}&0 \\ -C_tC_\mathrm{p}&-C_tS_\mathrm{p}&S_t \end{bmatrix}. \end{aligned}$$

(40)

Thus, by assuming $C_\mathrm{r}=\cos \theta _\mathrm{r}$, $S_\mathrm{r}=\sin \theta _\mathrm{r}$, $C_q=\cos \phi _\mathrm{r}$, and $S_q=\sin \phi _\mathrm{r}$, $C_{q-p}=\cos (\phi _\mathrm{r}-\phi _\mathrm{s})$, and $S_{q-p}=\sin (\phi _\mathrm{r}-\phi _\mathrm{s})$, the unit-normalized receiver position $\hat{\mathbf {x}}_\mathrm{r}$ after this rotation will have coordinates,

$$\begin{aligned} {\varvec{R}}_\mathrm{s} \hat{\mathbf {x}}_\mathrm{r} = \begin{bmatrix} S_tC_\mathrm{p}&S_t S_\mathrm{p}&C_t \\ -S_\mathrm{p}&C_\mathrm{p}&0 \\ -C_tC_\mathrm{p}&-C_tS_\mathrm{p}&S_t \end{bmatrix} \begin{bmatrix} S_\mathrm{r}C_q \\ S_\mathrm{r} S_q \\ C_\mathrm{r} \end{bmatrix}=\begin{bmatrix} C_t C_\mathrm{r} + S_t S_\mathrm{r} C_{q-p} \\ S_\mathrm{r} S_{q-p} \\ S_t C_\mathrm{r} - C_t S_\mathrm{r} C_{q-p} \end{bmatrix}. \end{aligned}$$

(41)

We can then rotate this new coordinate system with respect to $\hat{r}_\mathrm{s}$ by an angle $\Theta$ such that the receiver ends up on the equator, similar to those in Appendices C and D,

$$\begin{aligned} {\varvec{R}}_\mathrm{r}=\begin{bmatrix} 1&0&0 \\ 0&\cos \Theta&\sin \Theta \\ 0&-\sin \Theta&\cos \Theta \end{bmatrix}, \end{aligned}$$

(42)

which requires $\Theta ={\text {arctan2}}(S_t C_\mathrm{r} - C_t S_\mathrm{r} C_{q-p}, S_\mathrm{r} S_{q-p})$. By assuming $C_\theta =\cos \Theta$, $S_\theta =\sin \Theta$, the full rotation matrix that needs to be applied to the original global coordinate becomes

$$\begin{aligned} {\varvec{R}}={\varvec{R}}_\mathrm{r}{\varvec{R}}_\mathrm{s}=\begin{bmatrix} S_tC_\mathrm{p}&S_t S_\mathrm{p}&C_t\\ -C_\theta S_\mathrm{p} - S_\theta C_t C_\mathrm{p}&C_\theta C_\mathrm{p} - S_\theta C_t S_\mathrm{p}&S_\theta S_t \\ -C_\theta C_t C_\mathrm{p} + S_\theta S_\mathrm{p}&-C_\theta C_t S_\mathrm{p} - S_\theta C_\mathrm{p}&C_\theta S_t \end{bmatrix}. \end{aligned}$$

(43)

To compute the kernel values on any grid point $\mathbf {x}_\mathrm{p}$ for a general source–receiver pair, its coordinates in the rotated coordinate system, $\mathbf {x}_\mathrm{p}^{\text {new}}$, with source–receiver along the equator can be calculated by multiplying the ${\varvec{R}}$ matrix. Hence, the kernel value at point $\mathbf {x}_\mathrm{p}$ in the original coordinate system is the same as the kernel value at $\mathbf {x}_\mathrm{p}^{\text {new}}$ for the equatorial source–receiver path which can be extracted and interpolated from our existing strain database.

Appendix C: Rotation of Moment Tensor Strain Field

We derive the set of rotations used in Sect. 3.1 to transform from the original coordinate system to a coordinate system aligned with the best double-couple orientation of a general moment tensor source, and then rotate the arbitrary point $\mathbf {x}_\mathrm{p}$ to the equatorial plane. We assume a general moment tensor ${\mathsf {M}}$ is placed at the surface, and transform the original coordinate system such that the moment tensor becomes diagonal,

$$\begin{aligned} ({\mathsf {M}})^{\varvec{A}}= {\varvec{A}}{\mathsf {M}}{\varvec{A}}^\mathrm{T} = \begin{bmatrix} M_1&\\&M_2&\\&&M_3 \end{bmatrix}, \end{aligned}$$

(44)

where $M_2 \le M_1 \le M_3$ are the eigenvalues of the symmetric moment tensor. As most earthquakes have predominantly double-couple source mechanisms (Aki and Richards 2002), it may be advantageous to further rotate the coordinate system such that the new $\hat{x}_2$ and $\hat{x}_3$ axes align with the fault plane normal $\hat{n}$ and slip direction $\hat{D}$ of the dominant double-couple component of the source (Fig. 17a),

$$\begin{aligned} ({\mathsf {M}})^{{\varvec{A}}{\varvec{B}}} = {\varvec{B}}{\varvec{A}}{\mathsf {M}}{\varvec{A}}^\mathrm{T} {\varvec{B}}^\mathrm{T}, \end{aligned}$$

(45)

where

$$\begin{aligned} {\varvec{B}}=\frac{1}{\sqrt{2}}\begin{bmatrix} \sqrt{2}&0&0 \\ 0&1&1 \\ 0&1&-1 \end{bmatrix} \end{aligned}$$

(46)

and

$$\begin{aligned} ({\mathsf {M}})^{{\varvec{A}}{\varvec{B}}} = \begin{bmatrix} M_1&0&0 \\ 0&(M_2+M_3)/2&(M_2-M_3)/2 \\ 0&(M_2-M_3)/2&(M_2+M_3)/2 \end{bmatrix}. \end{aligned}$$

(47)

For pure double-couple sources, we have $M_1=0$, $M_3 = -M_2 = M_\mathrm{s}$ where $M_\mathrm{s}$ is the scalar moment of the moment tensor ${\mathsf {M}}$, and the moment tensor is reduced to

$$\begin{aligned} ({\mathsf {M}})^{{\varvec{A}}{\varvec{B}}} = M_\mathrm{s} \begin{bmatrix} 0&0&0 \\ 0&0&1 \\ 0&1&0 \end{bmatrix}. \end{aligned}$$

(48)

After the rotations ${\varvec{A}}$ and ${\varvec{B}}$, the original point $\mathbf {x}_\mathrm{p}$ has been transformed to $(\mathbf {x}_\mathrm{p})^{{\varvec{A}}{\varvec{B}}}={\varvec{B}}{\varvec{A}}\mathbf {x}_\mathrm{p}$. Based on symmetry of the 1D background model, we plan to only store strain fields on the equatorial plane. Hence, we further transform $(\mathbf {x}_\mathrm{p})^{{\varvec{A}}{\varvec{B}}}$ through a rotation ${\varvec{C}}_\mathrm{s}$ with respect to the $(\hat{x}_1)^{{\varvec{A}}{\varvec{B}}}$ axis with an angle $\Phi _\mathrm{s}$ such that it ends on the equatorial plane (shaded areas in Fig. 17b), i.e., $(\mathbf {x}_\mathrm{p})^{{\varvec{A}}{\varvec{B}}{\varvec{C}}_\mathrm{s}} = r_\mathrm{p}[\cos \Delta _\mathrm{s},\sin \Delta _\mathrm{s},0]^\mathrm{T}$, where if we define $(\mathbf {x}_\mathrm{p})^{{\varvec{A}}{\varvec{B}}} = [x_1,x_2,x_3]$, $r_\mathrm{p}=(x_1^2+x_2^2+x_3^2)^{1/2}$ is the radius of point $\mathbf {x}_\mathrm{p}$, and $\Delta _\mathrm{s}$ is the epicentral distance of $\mathbf {x}_\mathrm{p}$ from the source. It can be shown that the epicentral distance $\Delta _\mathrm{s}$ and the rotation angle $\Phi _\mathrm{s}$ can be computed from

$$\begin{aligned} \cos \Delta _\mathrm{s}&= x_1/r_\mathrm{p}, \end{aligned}$$

(49)

$$\begin{aligned} \cos \Phi _\mathrm{s}&= x_2/(r_\mathrm{p}\sin \Delta _\mathrm{s}),\end{aligned}$$

(50)

$$\begin{aligned} \sin \Phi _\mathrm{s}&= x_3/(r_\mathrm{p}\sin \Delta _\mathrm{s}),\quad \Delta _\mathrm{s} \in [0,\pi ] \end{aligned}$$

(51)

and the rotation matrix is given by

$$\begin{aligned} {\varvec{C}}_\mathrm{s}=\begin{bmatrix} 1&0&0 \\ 0&\cos \Phi _\mathrm{s}&\sin \Phi _\mathrm{s} \\ 0&-\sin \Phi _\mathrm{s}&\cos \Phi _\mathrm{s} \end{bmatrix}. \end{aligned}$$

(52)

Under this transformation the moment tensor source becomes

$$\begin{aligned}&({\mathsf {M}})^{{\varvec{A}}{\varvec{B}}{\varvec{C}}_\mathrm{s}} \nonumber \\&\qquad = {\varvec{C}}_\mathrm{s}({\mathsf {M}})^{{\varvec{A}}{\varvec{B}}} {\varvec{C}}^\mathrm{T}_\mathrm{s} \nonumber \\&\qquad = \begin{bmatrix} M_1&0&0 \\ 0&\frac{M_2+M_3}{2}+\frac{M_2-M_3}{2}\sin 2\Phi _\mathrm{s}&\frac{M_2-M_3}{2}\cos 2\Phi _\mathrm{s} \\ 0&\frac{M_2-M_3}{2}\cos 2\Phi _\mathrm{s}&\frac{M_2+M_3}{2}-\frac{M_2-M_3}{2}\sin 2\Phi _\mathrm{s} \end{bmatrix} \end{aligned}$$

(53)

as in Eq. 14.

Fig. 17

Coordinate transformations applied to compute the strain field at point $\mathbf {x}_\mathrm{p}$ due to a general point moment tensor on the surface (the parentheses used in coordinate transformation is omitted to reduce clutter): a The original coordinate system (black) is rotated by ${\varvec{A}}$ as in Eq. 44 and ${\varvec{B}}$ as in Eq. 46 such that the moment tensor aligns with the fault plane normal $\hat{n}$ and the slip direction $\hat{\mathbf {D}}$, producing the $(\hat{x}_i)^{{\varvec{A}}{\varvec{B}}}$ coordinate system (blue). b The $(\hat{x}_i)^{{\varvec{A}}{\varvec{B}}}$ coordinate system is rotated about the $(\hat{x}_1)^{{\varvec{A}}{\varvec{B}}}$-axis by angle $\Phi _\mathrm{s}$ such that the point of interest $(\mathbf {x}_\mathrm{p})^{{\varvec{A}}{\varvec{B}}}$ now exists on the equatorial plane (shaded area) as $(\mathbf {x}_\mathrm{p})^{{\varvec{A}}{\varvec{B}}{\varvec{C}}_\mathrm{s}}$. In the new $(\hat{x}_i)^{{\varvec{A}}{\varvec{B}}{\varvec{C}}_\mathrm{s}}$ coordinate system, the epicentral distance of the point $(\mathbf {x}_\mathrm{p})^{{\varvec{A}}{\varvec{B}}{\varvec{C}}_\mathrm{s}}$ from the source is $\Delta _\mathrm{s}$

Appendix D: Rotation of Point Force Strain Field

We derive the set of rotations used in Sect. 3.2 to transform the receiver onto the prime meridian and rotate the arbitrary point $\mathbf {x}_\mathrm{p}$ to the equatorial plane. We first transform the original coordinate system such that the receiver $\mathbf {x}_\mathrm{r}$ falls on the new $\hat{x}_1$ axis (Fig. 18a) through a rotation matrix

$$\begin{aligned} {\varvec{D}}=\begin{bmatrix} \cos \Psi&\sin \Psi&0 \\ -\sin \Psi&\cos \Psi&0 \\ 0&0&1 \end{bmatrix}, \end{aligned}$$

(54)

where $\Psi$ is the epicentral distance of the receiver from the source. We further transform the $(\mathbf {x}_\mathrm{p})^{\varvec{D}}$ point onto the equatorial plane (Fig. 18b) through rotation matrix ${\varvec{C}}_\mathrm{r}$ similar to Eqs. 51 and 52 in Appendix C,

$$\begin{aligned} {\varvec{C}}_\mathrm{r}=\begin{bmatrix} 1&0&0 \\ 0&\cos \Phi _\mathrm{r}&\sin \Phi _\mathrm{r} \\ 0&-\sin \Phi _\mathrm{r}&\cos \Phi _\mathrm{r} \end{bmatrix}, \end{aligned}$$

(55)

where

$$\begin{aligned} \cos \Delta _\mathrm{r}&= x_1/r_\mathrm{p}, \end{aligned}$$

(56)

$$\begin{aligned} \cos \Phi _\mathrm{r}&= x_2/(r_\mathrm{p}\sin \Delta _\mathrm{r}), \end{aligned}$$

(57)

$$\begin{aligned} \sin \Phi _\mathrm{s}&= x_3/(r_\mathrm{p}\sin \Delta _\mathrm{r}), \quad \Delta _\mathrm{r} \in [0,\pi ], \end{aligned}$$

(58)

$\Delta _\mathrm{r}$ is the great-circle distance between $\mathbf {x}_\mathrm{p}$ and the receiver $\mathbf {x}_\mathrm{r}$, and $\Phi _\mathrm{r}$ is the rotation angle about $(\hat{x}_1)^{\varvec{D}}$ axis (where the receiver resides) to transform $(\mathbf {x}_\mathrm{p})^{\varvec{D}}$ onto the equatorial plane.

Fig. 18

The rotations applied to calculate the strain field due to a point force on the surface: a The original coordinate system (black) is rotated by ${\varvec{D}}$ as in Eq. 54 (i.e., $\Psi$ angle about the $\hat{x}_3$-axis) such that the receiver at $\mathbf {x}_\mathrm{r}$ coincides with the $(\hat{x}_1)^{\varvec{D}}$-axis. b The $(\hat{x}_i)^{\varvec{D}}$ coordinate system (blue) is rotated about the $(\hat{x}_1)^{\varvec{D}}$-axis by angle $\Phi _\mathrm{r}$ such that the point of interest $(\mathbf {x}_\mathrm{p})^{\varvec{D}}$ now exists on the equatorial plane (shaded area) as $(\mathbf {x}_\mathrm{p})^{{\varvec{D}}{\varvec{C}}_\mathrm{r}}$. In the new $(\hat{x}_i)^{{\varvec{D}}{\varvec{C}}_\mathrm{r}}$ coordinate system, the great-circle distance of the point $(\mathbf {x}_\mathrm{p})^{{\varvec{D}}{\varvec{C}}_\mathrm{r}}$ from the receiver $\mathbf {x}_\mathrm{r}$ is $\Delta _\mathrm{r}$

Appendix E: Algorithm Summary

Here we summarize the algorithms we proposed in Sect. 3.3.

• 1. Choose the cut-off period the Gaussian source time function, h for numerical simulations (e.g., 16 s), and the cut-off period l (e.g., 20 s) with which kernels can be constructed such that $2l^2>3h^2$.

• 2. Compute the forward strain fields $({\mathsf {\varepsilon }})^{\text {equator}}[t,r,\Delta ; {\mathsf {M}}^q,d,h]$ for a series of depth $d=d_{\text {min}},\ldots , d_{\text {max}}$ and at every depth four fundamental mechanisms ${\mathsf {M}}^q$, $q=1,\ldots ,4$, as well as the adjoint strain fields $({\mathsf {\varepsilon }})^{\text {equator}}[t,r,\Delta ;\hat{x}_i, h]$ for three unit force vectors $\hat{x}_i$, $i=1,2,3$, based on 3D SEM simulations. Store both the strain and velocity fields only on the equatorial plane.

• 3. Loop over spatial points in the 3D volume. For each point $\mathbf {x}_\mathrm{p}$, compute the kernel values based on the following steps:

  • Calculate rotation ${\varvec{A}}$, ${\varvec{B}}$ and ${\varvec{C}}_\mathrm{s}$ to transform the general moment tensor into a coordinate system associated with the best double-couple ($\hat{n}$, $\hat{D}$) directions and then further rotate $\mathbf {x}_\mathrm{p}$ onto the equatorial plane.

  • For the moment tensor forward strain field, extract the strain values at radius $r_\mathrm{p}$ and epicentral distance $\Delta _\mathrm{s}$ for the four fundamental mechanisms, $({\mathsf {\varepsilon }})^{{\text {equator}}}[t,r_\mathrm{p},\Delta _\mathrm{s}; {\mathsf {M}}^q,d,h]$, $q=1,\ldots , 4$, from the precomputed strain database. Compute the strain time series at $\mathbf {x}_\mathrm{p}$ due to an arbitrary moment tensor source ${\mathsf {M}}$ based on the superposition in Eq. 16. Similarly, construct the corresponding velocity time series $\dot{s}(t,\Psi ;{\mathsf {M}},g_\mathrm{h})$ at the receiver location.

  • For the adjoint strain field, first compute the rotation matrix ${\varvec{D}}$ and ${\varvec{C}}_\mathrm{r}$ to align the receiver with (0$^{\circ }$ latitude, 0$^{\circ }$ longitude) and then transform $\mathbf {x}_\mathrm{p}$ onto the equatorial plane. Compute the corresponding components $f_i'$ of the transformed force vector $(\mathbf {f})^{{\varvec{D}}{\varvec{C}}_\mathrm{r}}$. Extract the stored strain time series due to individual unit forces and sum them up as in Eq. 19 to obtain the adjoint strain field $({\mathsf {\varepsilon }}^\dagger )^{{\varvec{D}}{\varvec{C}}_\mathrm{r}}$. Transform it back to obtain the adjoint strain time series at $\mathbf {x}_\mathrm{p}$ due to an arbitrary force vector $\mathbf {f}$ in the original coordinate system based on Eq. 20.

  • Convolve the two strain time series calculated from the last two steps. Window the velocity time series at the specific phase we are interested in, normalize it by $N_t$, and convolve it with a Gaussian function with cut-off period of $\sqrt{2l^2-3h^2}$ (e.g., 5.6 s). Further convolve these two resulting series and Fourier transform it back to the time domain.

  • The kernel values for the elastic tensor elements for point $\mathbf {x}_\mathrm{p}$ is given by the values of the transformed time series at time $t=T$ (i.e., the end of the simulation).