1. Earth Structure: The Preliminary Reference Earth Model (PREM)1
Earth Physics Autumn, 2017
1 Dziewonski and Anderson, 1981

2. Travel-time curves
Earthquake travel-time curves are consistent with a structural model of the mechanical properties of the Earth's interior

3. P-wave velocity

4. S-wave velocity

5. Adams-Williamson 𝑣 𝑝 = 𝐾+ 4 3 µ ρ 𝑣 𝑠 = µ ρ
If we knew density as a function of depth, we could obtain the mechanical properties, µ (rigidity) and k (bulk modulus of incompressibility) given the seismic wave velocities deconvolved from the travel-time data:
𝑣 𝑝 = 𝐾+ 4 3 µ ρ 𝑣 𝑠 = µ ρ
Williamson and Adams, in 1923, first developed a method to estimate the density profile.

6. Adams-Williamson Equation
From the deconvolved seismic velocities, form:
We know that
Pressure locally changes with density in the gravity field as
then
and

7. Adams-Williamson Equation
The Adams-Williamson equation has, here, been developed assuming a hydrostatic Earth and adiabatic compression from the surface to depth. It doesn’t account for mineral phase changes, for the compositional changes at the core-mantle boundary or for the solid inner core. In fact, the inner core and mineral phase changes were unknown at the time of their development.
Early density estimates were obtained by numerically integrating the equation from the surface (subscript 0) to depth.

8. Density
Now knowing the Earth’s moment of inertia, something of interior mineralogy and with better definition of seismic velocities, we have a much improved density profile.
Adam-Willamson
PREM A&D

9. Modulus of incompressibility

10. Modulus of rigidity

11. PREM
We normally don’t take PREM to be a definitive model of the variation of the mechanical parameters of the Earth with depth and averaged over spherical shells. Still, it is a very useful reference model because it does seem to fit these parameters quite well and any deviation from the PREM does or can provide detailed information about the interior.
PREM does not offer error bounds on its estimates. It is “definitional” or “referential”. It remains the favoured reference model but there are many others: See IRIS data services.