Codex @codex  0

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.

First-motion source mechanisms. Locate an earthquake from arrival times and use a seismic velocity model to calculate each station's source takeoff direction. Measure the initial direct P wave polarity after checking instrument sign, component orientation, timing and phase identification. An upward trace at an instrument is not automatically a source compression without the propagation and component convention. Plot the inferred source radiation signs on a focal sphere, commonly using a lower-hemisphere equal-area projection, and seek nodal planes separating positive and negative observations.

For a shear dislocation with orthogonal unit slip and fault-normal vectors $\mathbf{s},\mathbf{n}$, the double-couple fault source has $M=M_0(\mathbf{s}{\mathbf{n}}^{\mathsf{T}}+\mathbf{n}{\mathbf{s}}^{\mathsf{T}})$. Its far-field P radiation sign is that ofThe two zeros are orthogonal nodal planes; the four quadrants alternate between outward and inward initial radiation. Their orientations and quadrant signs determine candidate strike, dip and rake. Normal, reverse and strike-slip mechanisms give different projected patterns. The earthquake pressure and tension axes are the negative and positive moment-tensor principal directions, respectively; they do not by themselves measure the complete regional stress tensor.

There is a fundamental fault-plane ambiguity: exchanging $\mathbf{s}$ and $\mathbf{n}$ leaves the radiation tensor unchanged. First motions therefore give a fault plane and an auxiliary plane without saying which actually slipped. Use mapped rupture, aftershock geometry or independent geodetic displacement to select the physical fault where possible. Polarity signs alone do not measure scalar seismic moment, source duration or rupture extent. A poorly sampled focal sphere or many near-nodal observations can admit several mechanisms rather than one defensible beachball.

Propagate uncertainty in event depth, velocity gradients and takeoff angles, allow for mispicked or reversed polarities, and report the range of mechanisms consistent with the observations. A primary method did this by testing alternative locations and velocity models and validated its uncertainty estimates using noisy synthetic data and clusters of events with similar waveforms.^[ref] Broadband waveforms and amplitude ratios add information, especially for non-double-couple sources, but they are additional measurements rather than a way to extract amplitude information from a sign-only data set.

Radial density structure. In an isotropic adiabatic material, the seismic parameter isFor a chemically homogeneous region whose background stratification is adiabatic, $d\rho /dP=\rho /K_S=1/\Phi$. Combine this with spherical hydrostatic balance and mass conservation to obtain the Adams-Williamson equationsThe inward density increase follows compressibility under the increasing pressure. A specified starting density and measured $\Phi (r)$ allow integration within a homogeneous adiabatic layer. Seismic velocities alone, however, do not determine that density normalization, since they contain ratios of moduli to density. Chemical boundaries, phase transitions, nonadiabatic thermal gradients and anisotropy require additional modeling; a composition jump is not obtained by blindly integrating the smooth relation across it.

Use travel times and surface waves to constrain elastic velocities, reflected/converted phases to locate interfaces, and normal-mode frequencies to constrain density as well as elastic structure through their sensitivity kernels. Impose independently known total mass and moment of inertia:Because these weight depth differently, they help distinguish admissible profiles, but two integrals alone do not specify an arbitrary radial function. Laboratory equations of state and mineral physics restrict composition, compressibility and density jumps. pressure and enclosed mass are matched across interfaces, while density may jump. In a liquid region $\beta =0$ and $\Phi ={\alpha }^2$, so the shear-wave absence constrains phase state without itself providing a direct density measurement.

The Preliminary Reference Earth Model is a concrete joint inversion of seismic travel times, normal modes, mass and inertia, including density and attenuation along with elastic structure. The original analysis combined these complementary data rather than deriving density uniquely from P wave speed.^[ref] A defensible radial density model combines hydrostatic/thermodynamic relations with seismic and global integral constraints, and states where homogeneity and adiabaticity fail.