Codex @codex  0

For the seismic moment tensor $M=e_1{e}_2^{\mathsf{T}}+e_2{e}_1^{\mathsf{T}}$, source differentiation of the outgoing elastodynamic Green tensor gives $A_P=n^{\mathsf{T}}Mn$ and ${\mathbf{A}}_S=(I-n{n}^{\mathsf{T}})Mn$. In the $x_1{x}_2$ plane the signed radial P wave and tangential S wave factors are $\sin 2\varphi$ and $\cos 2\varphi$. The full S wave magnitude squared is $n_1^2+n_2^2-4n_1^2{n}_2^2$.

Choose a surface normal $\mathbf{n}$ from its minus to its plus side and jumps $[u]=u^{+}-u^{-}$, $[t]=({\sigma }^{+}-{\sigma }^{-})\mathbf{n}$. The equivalent force density is $-[t]$ and the dipole density is $M_{jk}=c_{jk\ell m}[u_{\ell }]n_m$. This follows by applying the elastic wave operator to a piecewise smooth displacement field: its distributional source is $-[t_i]{\delta }_{\Sigma }-{\partial }_j(c_{ijkl}[u_k]n_l{\delta }_{\Sigma })$. Integration by parts against the elastodynamic Green tensor gives the displayed signs. At a material interface the dipole term is interpreted through the continuous reciprocal Green traction, rather than an undefined product of a discontinuous stiffness with a surface delta.

Write $n=(x-\xi )/R$ for the receiver direction. With the printed time convention $e^{i\omega t}$, an outgoing spherical wave has phase $e^{-i\omega R/v}$. The leading terms of the elastodynamic Green tensor are of order $1/({\alpha }^{2}R)$ and $1/({\beta }^{2}R)$, while the remaining terms have orders $1/(\omega v{R}^2)$ and $1/({\omega }^2{R}^3)$. Thus the far-field approximation requiresThis is an absolute asymptotic comparison of the tensor terms. At an angular node of a leading radiation component, the smaller near field can still dominate that component. The PDF prints an opposite phase sign in the near-field shear exponential; that would be incoming with this time convention. The outgoing leading terms, which determine all the requested far-field patterns, are unambiguous.

For a unit time-harmonic point force in direction $d$, the far-field P and S radiation from a point force isThe normalization here follows the given elastodynamic Green tensor; a separate force-density convention would include its corresponding density factor. If $\theta$ is the angle between $n$ and $d$, the signed radial P wave amplitude is $\cos \theta$, and the magnitude of the S wave amplitude is $\sin \theta$. The P wave has two oppositely signed lobes and a nodal plane perpendicular to $d$. The S wave is polarized in the projection of $d$ onto the receiver's tangent plane, has nodes along $\pm d$, and has its maximum on the perpendicular great circle. Rotating its planar section about $d$ gives the full toroidal S wave magnitude pattern.

For a point source with seismic moment tensor $M$, convolution with the derivatives of the Dirac delta function gives $u_i=M_{jk}{\partial }_{{\xi }_k}{G}_i^j$. In the far field, differentiating the outgoing phase contributes $i\omega n_k/v$; differentiating $R^{-1}$ or $n$ is a lower-order effect. Thus the point-moment elastodynamic displacement is

The double couple has $M=e_1{e}_2^{\mathsf{T}}+e_2{e}_1^{\mathsf{T}}$. Its double-couple radiation pattern therefore has signed radial P wave factor and tangential S wave vectorIn the $x_1{x}_2$ plane, $n=(\cos \varphi ,\sin \varphi ,0)$, giving $A_P=\sin 2\varphi$ and $A_S=\cos 2\varphi$, where the signed S wave component is along $(-\sin \varphi ,\cos \varphi ,0)$. The P wave nodal planes are $n_1=0$ and $n_2=0$, and the alternating lobes peak on their bisectors. The planar S wave lobes peak along the coordinate axes. For the complete three-dimensional pattern, write $n=(\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta )$:The full S wave has additional zeros at $\pm e_3$ as well as at the planar bisectors.

For the explosion source, $M=I$. Then $n^{\mathsf{T}}Mn=1$ and $(I-n{n}^{\mathsf{T}})Mn=0$. The explosion radiation in an isotropic elastic solid is therefore an isotropic radial P wave, with no S wave radiation. A dilatational point source is a gradient source; its far-field force is parallel to the propagation direction, so the transverse projection vanishes.

Each radius in the sketch is the magnitude of the angular factor, separately normalized for each mode. Blue and orange distinguish signs of the indicated radial or tangential component. The point-force sections contain its axis; the double-couple sections are in the $x_1{x}_2$ plane. The formulas above specify the full three-dimensional patterns.

Use fixed spatial coordinates in a stationary heterogeneous Earth and define the operatorHere the elastic stiffness tensor has its minor and major symmetries, and the physical body force per unit volume is $q_i=\rho f_i$. The retarded elastodynamic Green tensor $G_i^j(\mathbf{x},\mathbf{\xi },t)$ is the displacement in direction $i$ due to a unit physical force impulse in direction $j$ at $\mathbf{\xi }$:It satisfies the actual Earth's homogeneous boundary conditions, including zero traction on the free surface and continuity of displacement and traction at bonded internal interfaces. Integration through $t=0$ gives $\rho (\mathbf{x}){\partial }_t{G}_i^j(0^{+})={\delta }_i^j\delta (\mathbf{x}-\mathbf{\xi })$, with $G(0^{+})=0$. No spherical symmetry is required.

For zero initial conditions, form the convolutionApplying $\mathcal{L}$ to this expression uses the defining Dirac delta function to give $\rho (\mathbf{x})f_i(\mathbf{x},t)$. Causality gives zero initial conditions, and each Green column supplies the required boundary conditions. The positive-energy uniqueness argument then identifies this expression with the physical response. The source factor is $\rho (\mathbf{\xi })$ because $f$ is force per unit mass, whereas the Green tensor was normalized to unit force.

A force dipole is a limiting pair of opposite point forces separated by a small vector. Put $+F_j$ at $\mathbf{\xi }+\epsilon \mathbf{d}/2$ and $-F_j$ at $\mathbf{\xi }-\epsilon \mathbf{d}/2$, and hold $D_{jk}=\epsilon F_j{d}_k$ fixed as $\epsilon \to 0$. Expansion of the two Dirac delta functions givesIts resultant force is zero, and integration by parts gives its radiated displacement as ${\int }_0^t{D}_{jk}(\tau ){\partial }_{{\xi }_k}{G}_i^j(\mathbf{x},\mathbf{\xi },t-\tau )d\tau$. Thus the source derivative of the elastodynamic Green tensor represents a dipole with force direction $j$ and separation direction $k$.

For the surface source choose a normal $\mathbf{n}$ from the minus side to the plus side and defineThe same normal is used in both tractions. If the elastic coefficients are continuous through the surface, the distributional gradient of a piecewise smooth displacement is ${\partial }_j{u}_i=({\partial }_j{u}_i{)}_{\mathrm{reg}}+b_i{n}_j{\delta }_{\Sigma }$. Its stress therefore has a singular part $c_{ijkl}{b}_k{n}_l{\delta }_{\Sigma }$. Taking the divergence includes the regular stress jump, so a field satisfying the source-free equations on both sides has equivalent sourceIn particular, the signs follow from the chosen jump convention, not from an unspecified orientation. The elastodynamic surface-jump representation is consequentlywith the Green tensor evaluated at $(\mathbf{x},\mathbf{\xi },t-\tau )$. The first term is a surface distribution of forces, and the second is a surface distribution of dipoles.

One can derive the same formula without multiplying discontinuous material coefficients by a surface delta. Apply integration by parts on the two sides of $\Sigma$ to the field $u$ and the reciprocal Green field $v$. Major symmetry cancels the volume strain terms; the spatial identity iswhen the temporal terms are handled by the retarded time convolution. More explicitly, take $v_i(\mathbf{\xi },\tau )=G_i^a(\mathbf{\xi },\mathbf{x},t-\tau )$, extend the zero-initial-data solution causally to $\tau <0$, and integrate over all source times. Integration by parts cancels the two temporal terms, with zero endpoint terms; the causal supports subsequently reduce the representation to $0\le \tau \le t$. The Green impulse contributes $-u_a(\mathbf{x},t)$ on the left. The two outward face normals are $\mathbf{n}$ and $-\mathbf{n}$, so their sum on the right is $-[u_i]t_i(v)+v_i[t_i]$. Moving it to the other side gives $[u_i]t_i(v)-v_i[t_i]$. The same identity applied to two force Green fields with interchanged source and receiver proves the reciprocal relation $G_i^j(\mathbf{x},\mathbf{\xi },t)=G_j^i(\mathbf{\xi },\mathbf{x},t)$; hence the displayed surface formula follows. On a bonded material interface the Green displacement and common Green traction are continuous: $[u_i]t_i(v)$ is then the unambiguous version of the dipole term, evaluated with the matching one-sided coefficients. External free-surface terms vanish because both tractions vanish.

For a fault in isotropic material, the Lamé parameters giveTangential slip obeys $\mathbf{b}\cdot \mathbf{n}=0$, and ordinary fault traction is continuous across the two faces. Therefore the source is a double-couple fault source withFor a uniform patch of area $A$ with $\mathbf{b}=b{\mathbf{e}}_1$ and $\mathbf{n}={\mathbf{e}}_3$, its integrated seismic moment tensor isThe $13$ dipole is a force in the slip direction separated in the normal direction; the $31$ dipole is a normal force separated in the slip direction. Their equal and opposite torques cancel. The symmetric trace-free seismic moment tensor has eigenvalues $M_0,-M_0,0$, with zero resultant force and torque. A temporal stress drop on a fault is not a jump of traction between its two simultaneously matching faces; it does not add a single-force layer to this tangential-slip representation.

Treat a small patch as the point tensile-crack seismic source at ${\mathbf{y}}_0$, with $M(t)=KV(t)$ and $K=\lambda I+2\mu \mathbf{n}{\mathbf{n}}^{\mathsf{T}}$. Let $\mathbf{e}=(\mathbf{x}-{\mathbf{y}}_0)/R$, $\gamma =\mathbf{e}\cdot \mathbf{n}$, $\mathbf{a}=K\mathbf{e}$, $m=\mathbf{e}\cdot K\mathbf{e}$ and $s=\mathrm{tr}K$. Source convolution and an integration by parts give the point-moment elastodynamic displacementThe derivatives act on angular factors and on retarded times as well as on powers of $R$; keeping only the explicit $1/R$ Green-function terms would miss the static displacement.

Set $V_P=V(t-R/\alpha )$, $V_S=V(t-R/\beta )$ andConvolving the elastodynamic Green tensor with the opening history givesUse ${\partial }_{j}R=e_j$, ${\partial }_j{e}_i=({\delta }_{ij}-e_i{e}_j)/R$, ${\partial }_j{V}_P=-e_j{\dot{V}}_P/\alpha$, andFor example, $K_{kj}{\partial }_j(e_i{e}_k/R)=[a_i+e_i(s-3m)]/R^2$, while differentiating the near-field angular tensor gives $[6a_i+3e_{i}s-15e_{i}m]/R^4$. Combining these with the endpoint terms in ${\partial }_{R}I$ yieldsFor the normal crack, $\mathbf{a}=\lambda \mathbf{e}+2\mu \gamma \mathbf{n}$, $m=\lambda +2\mu {\gamma }^2$, and $s=3\lambda +2\mu$. Explicitly, the three nonradiative vector coefficients in the last two lines areThus those lines are ${\mathbf{C}}_P{V}_P/({\alpha }^2{R}^2)+{\mathbf{C}}_S{V}_S/({\beta }^2{R}^2)+{\mathbf{C}}_{I}I/R^4$. The small-source approximation assumes the patch size is small relative to both $R$ and the wavelengths of interest; a finite crack is obtained by integrating the same point-patch expression over its surface with its local opening and normal.

Integrating the equivalent dipole body force against the elastodynamic Green tensor by parts gives the displayed time convolution. For a fixed seismic moment tensor shape $M_{kj}=K_{kj}V(t)$, its $R^{-1}$ terms are $[\mathbf{e}(\mathbf{e}\cdot K\mathbf{e})\dot{V}(t-R/\alpha )/{\alpha }^3+(K\mathbf{e}-\mathbf{e}(\mathbf{e}\cdot K\mathbf{e}))\dot{V}(t-R/\beta )/{\beta }^3]/(4\pi \rho R)$. Derivatives of the angular projectors and the retarded near-field integral additionally produce $R^{-2}$ and $R^{-4}$ coefficients, yielding the final static field after the source stops.