Seismic moment tensor ($f_i=-{\partial }_j[M_{ij}(t)\delta (\mathbf{x}-\mathbf{y})]$)

A localized zero-resultant seismic moment tensor represents the leading force-dipole part of an elastic wave source through the displayed equivalent body force. For a prescribed displacement jump $b_i$ across a surface with normal $n_j$, its moment density is $c_{ijkl}{b}_k{n}_l$, where $c$ is the elastic stiffness tensor. Spatial integration gives the point-source moment when the source is small compared with the wavelengths.

For uniform slip on a shear fault, scalar seismic moment is $M_0=\mu Ad$, with shear modulus $\mu$, fault area $A$ and slip magnitude $d$. More generally it integrates rigidity-weighted slip. It measures source strength rather than radiated energy alone.

A double-couple seismic source has a moment tensor proportional to $\mathbf{b}\otimes \mathbf{n}+\mathbf{n}\otimes \mathbf{b}$, with the slip direction $\mathbf{b}$ perpendicular to the fault plane's normal $\mathbf{n}$. Its eigenvalues are proportional to $(1,0,-1)$. A general symmetric seismic moment tensor need not have this restricted form.

For mass-normalized normal modes, an instantaneous step in a point seismic moment tensor excites mode $\alpha$ with $C_{\alpha }=\mathbf{M}:{\mathbf{e}}_{\alpha }^{\ast }({\mathbf{x}}_s)$ and $\tau =t-t_s$. Convolving the causal oscillator Green's function with the step gives $a_{\alpha }=C_{\alpha }[1-\cos ({\omega }_{\alpha }\tau )]H(\tau )/{\omega }_{\alpha }^2$. The expression contains a static response and undamped oscillations; attenuation requires an extension of the conservative model.

A stress glut tensor describes a seismic source through an equivalent body-force density $\mathbf{f}=-\nabla \cdot \mathbf{\Gamma }$. For a symmetric stress glut tensor supported away from the boundary, integration by parts makes its coupling to a normal mode equal to $\int \mathbf{\Gamma }:{\mathbf{e}}^{\ast }dV$, where $\mathbf{e}$ is that mode's strain.

An ideal compact volume source has $M_{ij}(t)=M(t){\delta }_{ij}$. In an unbounded homogeneous isotropic solid its far-field body radiation is P only. Boundaries and heterogeneity can create converted SV and surface-wave energy; a real explosion can have nonisotropic components.

A description of earthquake source geometry, often strike, dip and rake for a double-couple shear fault. First-motion polarities constrain the radiation sign pattern, with a fundamental interchange ambiguity between fault and auxiliary planes.

The dependence of radiated P wave or S wave amplitude and polarity on direction from an earthquake source. For an ideal double couple it has nodal directions, so amplitude differences between stations do not measure distance or directivity alone.

The negative and positive principal directions of a double-couple moment tensor. They bisect the nodal-plane quadrants, but are not automatically the principal axes of regional stress. A stress inversion needs additional faulting assumptions and multiple mechanisms.

Interchanging the slip direction and fault normal leaves a double-couple moment tensor unchanged. First-motion polarities therefore cannot identify the physical fault plane without information such as mapped rupture, aftershock geometry or geodetic displacement.

An imagined sphere around a seismic point source on which source-to-station ray takeoff directions are plotted. A lower-hemisphere equal-area projection commonly displays first-motion polarities and nodal planes.

The sign of the initial direct P wave motion after instrument, component and ray-direction conventions are calibrated. It constrains the sign of ${\hat{r}}^{\mathsf{T}}M\hat{r}$ at the source takeoff direction. Near-nodal arrivals and uncertain takeoff angles need special treatment.

An isotropic point moment is a dilatational gradient source. Its radial projection gives direction-independent P wave radiation, whereas its transverse projection vanishes. Thus it produces no far-field S wave in a homogeneous isotropic solid; conversion at an interface can change that conclusion.

Tangential fault slip $\mathbf{b}$ in an isotropic elastic material has the displayed moment density. It is symmetric and trace-free, with eigenvalues $\mu |b|,-\mu |b|,0$. The two off-diagonal force dipoles form equal couples with opposite torques, producing zero resultant force and torque. For a uniform patch of area $A$, the scalar seismic moment is $M_0=\mu A|b|$. Traction continuity removes the single-force layer in the elastodynamic surface-jump representation.

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 $n_1^2+n_2^2-4n_1^2{n}_2^2$.

For orthogonal slip and fault-normal unit vectors $s,n$, the P radiation amplitude is $2M_0(\hat{r}\cdot s)(\hat{r}\cdot n)$. Its zeros are the two planes normal to $s$ and $n$, producing four alternating polarity quadrants.

A small planar crack of area $S$ opening normally by $d(t)$ has volume change $V=Sd$ and the displayed seismic moment tensor. Its principal dipoles are $\lambda V,\lambda V,(\lambda +2\mu )V$: two act in the crack plane and the largest acts normally for positive opening and ordinary positive Lamé parameters. The definition follows by contracting the isotropic elastic stiffness tensor with two copies of the crack normal.

For a planar tensile-crack seismic source, $\theta$ is measured from its normal. The P wave displacement is radial with the first displayed amplitude. The S wave displacement is tangential in the plane containing the observation direction and crack normal, with four alternating signed lobes and nodes at the poles and equator. Both radiative fields depend on the opening rate and vanish once opening stops; the static displacement field depends on the final opening.