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.

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

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.