= Solution
\b[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 of
$$
A_P(\hat{\mathbf r})\propto\hat{\mathbf r}^{\mathsf T}M\hat{\mathbf r}=2M_0(\hat{\mathbf r}\cdot\mathbf s)(\hat{\mathbf r}\cdot\mathbf n).
$$
The 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>.
\Image[/past-exam-of-the-physics-of-the-earth-as-a-planet-course-of-the-university-of-cambridge/2016/paper-1-focal-mechanisms.png]
{title=Original lower-hemisphere equal-area P-radiation polarity examples for ideal strike-slip, normal and reverse double couples}
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.<A New Method for Determining First-Motion Focal Mechanisms by Jeanne L. Hardebeck and Peter M. Shearer>[pp. 2264–2266 and abstract, first-motion construction, polarity/takeoff uncertainty and independent validation tests]{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.
\b[Radial <density> structure.] In an <isotropic> adiabatic material, the <seismic parameter> is
$$
\Phi(r)=\frac{K_S}{\rho}=\alpha(r)^2-\frac43\beta(r)^2.
$$
For 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 equations>
$$
\boxed{\frac{d\rho}{dr}=-\frac{\rho g}{\Phi},\qquad\frac{dm}{dr}=4\pi r^2\rho,\qquad g=\frac{Gm}{r^2},\qquad\frac{dP}{dr}=-\rho g.}
$$
The 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>:
$$
M=4\pi\int_0^R\rho(r)r^2\,dr,\qquad I=\frac{8\pi}{3}\int_0^R\rho(r)r^4\,dr.
$$
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>.<Preliminary reference Earth model by Adam M. Dziewonski and Don L. Anderson>[abstract and pp. 297–300, data classes and coupled radial reference-model inversion]{ref} \b[A defensible radial <density> model combines hydrostatic/thermodynamic relations with seismic and global integral constraints, and states where homogeneity and adiabaticity fail.]
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