Seismic Moment and Moment Tensor
The scalar seismic moment quantifies rupture size:
\[M_0 = \mu\, A\, D\]where $\mu$ is the shear modulus of the source region, $A$ is the rupture area, and $D$ is the average slip. The moment magnitude is $M_w = \tfrac{2}{3}\log_{10}(M_0) - 6.07$.
The full source is described by the symmetric, traceless moment tensor $\mathbf{M}$ (a $3\times 3$ matrix). Its eigendecomposition separates the double-couple (shear faulting), CLVD (compensated linear vector dipole), and isotropic (volume change) components. The far-field displacement from a double-couple source in direction $\hat{\mathbf{r}}$ is
\[u_i^P(\mathbf{x}, t) = \frac{1}{4\pi\rho v_P^3 r} \mathcal{R}_P^{ij} \dot{M}_{ij}(t - r/v_P)\]where $\mathcal{R}_P^{ij}$ is the P-wave radiation pattern factor.
Focal Mechanisms
The beach-ball diagram projects the P- and T-axis orientations onto the focal sphere, dividing it into quadrants of compression and dilatation. The strike $\phi$, dip $\delta$, and rake $\lambda$ of the fault plane are recovered from the two nodal planes that separate these quadrants. Moment tensor inversion from waveform data uniquely determines $\mathbf{M}$ and hence the focal mechanism.
Gutenberg-Richter Recurrence
The frequency–magnitude distribution of earthquakes in any region follows
\[\log_{10} N(M) = a - bM\]where $N(M)$ is the cumulative number of earthquakes with magnitude $\geq M$, the $b$-value is typically close to 1, and $a$ describes regional seismicity rate. This power-law implies that $N$ scales as $M_0^{-2b/3}$, reflecting the self-similar geometry of fault networks. Deviations from $b = 1$ signal stress heterogeneity or fluid pore-pressure changes.