Spherically Averaged Earth Structure
The determination of elastic parameters and density throughout the Earth using observations of seismic waves and other constraints is the prototype inverse problem in geophysics. Like many inverse problems, it is formally nonunique and suffers from practical difficulties such as incomplete sampling and errors in the data. In spite of this, it is remarkable how much is now known about mantle structure, and this is particularly true of spherically averaged properties.
The major divisions of the Earth's interior are shown in Figure 3.1. The crust, mantle, and core were recognized in the early part of this century following Mohorovicic's 1909 discovery of the crustmantle boundary (Moho) and Gutenberg's (1913) determination of the outer core radius. Lehmann (1936) inferred the existence of the inner core in 1936 and by 1939 Jeffreys (1939) had produced compressional and shear wave velocity profiles featuring a transition zone between about 400 km and 1,000 km depth. Thus, a nearly complete picture of the firstorder spherical structure of the Earth was obtained prior to 1940. This period
Figure 3.1. The major components of the Earth's interior are the crust, the mantle, and the core. The oceanic crust has a basaltic composition and a mean thickness of about 6 km. The continental crust has a more silicic composition and a mean thickness of about 30km. The mantle has an ultrabasic composition; the compositional boundary between the crust and mantle is the Mohorovicic seismic discontinuity (the Moho). The mantle has major seismic discontinuities at depths of about 410 and 660km. The core is primarily iron; the outer core is liquid and the inner core is solid. The depths and radii of major boundaries are shown assuming spherical symmetry.
Figure 3.1. The major components of the Earth's interior are the crust, the mantle, and the core. The oceanic crust has a basaltic composition and a mean thickness of about 6 km. The continental crust has a more silicic composition and a mean thickness of about 30km. The mantle has an ultrabasic composition; the compositional boundary between the crust and mantle is the Mohorovicic seismic discontinuity (the Moho). The mantle has major seismic discontinuities at depths of about 410 and 660km. The core is primarily iron; the outer core is liquid and the inner core is solid. The depths and radii of major boundaries are shown assuming spherical symmetry.
culminated with the publication of standard travel time curves for major seismic body wave phases, the JB tables (Jeffreys and Bullen, 1940), which are still in use today.
A method for obtaining radial density variations from radial profiles of compressional and shear wave velocities was developed by Williamson and Adams (1923). It provides a simple equation of state for describing compressibility in chemically homogeneous layers. Within such a layer, the radial variation in density p can be expressed in terms of any other two state variables. In particular we consider the pressure p and the entropy s and write dp _/ dp /dp (321)
dr \dp Js dr \ds Jp dr where r is the radial coordinate and the subscripts s and p refer to isentropic and isobaric variations, respectively. An isentropic process is a reversible process in which there is no heat transfer (adiabatic) and an isobaric process is a process at constant pressure. Applications of thermodynamics to the mantle will be discussed in greater detail in Section 6.8.
The thermodynamic derivative on the right side of (3.2.1), (dp/dp)s, is related to the adiabatic compressibility xa of a material. The definition of xa is
P \dPJs Ka where Ka is the adiabatic bulk modulus. The connection with elasticity comes from the relation between Ka and the velocities of the seismic compressional and shear waves, VP and VS, respectively, where
and p is the shear modulus or rigidity of the solid. Elimination of the shear modulus from (3.2.3) and (3.2.4) gives
where $ is the seismic parameter. From (3.2.2) and (3.2.5), we can write the thermodynamic derivative (dp/dp)s simply as dp \ 1
The radial profiles of VP and VS determined from seismology also give $(r) and (dp/dp)s as a function of r.
For a homogeneous layer that is well mixed, e.g., by convection, it is appropriate to assume that the layer is isentropic and ds/dr = 0. With this assumption and (3.2.6), the variation of density in (3.2.1) can be written as dP 1 dp
The radial pressure derivative dp/dr in (3.2.7) is given to a good approximation throughout the mantle by the hydrostatic equation dp dp = pg (3.2.8)
dr where g is the acceleration of gravity. Substitution of (3.2.8) into (3.2.7) gives the AdamsWilliamson equation for the variation of density with radius in the mantle:
The subscript s on the left of (3.2.9) indicates that the process is isentropic. It should be emphasized that this result is valid only if the composition is uniform.
The acceleration of gravity g in the mantle also varies with radius. For a spherically symmetric Earth model g(r) satisfies the Poisson equation where G is the universal constant of gravitation. Integration of (3.2.9) and (3.2.10), with $(r) known from seismology, gives a spherical Earth model, consisting of the radial variation of spherically averaged density and gravity. Integration of these coupled firstorder differential equations requires two boundary conditions or constraints which are provided by the Earth's mass and moment of inertia; the densities of crustal and upper mantle rocks are also used to fix the densities at the tops of individual spherical layers. With p(r) and g(r) determined, the variation with radius of the adiabatic bulk modulus and rigidity can readily be found from (3.2.5) and (3.2.4), for example. The spherical Earth model also consists of fi(r) and
Bullen (1936, 1940) first used the above procedure to obtain a sixlayer Earth model, consisting of the crust (layer A), from the Earth's surface to the Moho (at a mean depth of 6 km beneath the oceans and 30 km beneath the continents), an adiabatic upper mantle (layer B), from the Moho to a depth of 400 km, an adiabatic lower mantle (layer D), from a depth of 1,000 km to a depth of 2,900 km, an adiabatic outer core (layer E), from a depth of 2,900 km to 5,100 km, and an adiabatic inner core (layer F), from 5,100 km depth to the center of the Earth. He found that the adiabatic approximation was not appropriate for the transition zone (layer C), from a depth of 400 km to 1,000 km, and instead used a polynomial function of radius to represent the density variation in this layer.
The resulting model A, as it was called, was an immediate success. Later, Bullen published model B, in which he added the assumption of continuity in the bulk modulus and its pressure derivative across the coremantle boundary. This allowed him to further subdivide the lower mantle into layers D', to a depth of 2,700 km, and D'', between 2,700 km and 2,900km depth. Bullen's models A and B are masterpieces of inductive science. They have been superseded by Earth models derived from larger data sets, and among all of his layer notation, only D'' has survived. However, these models were remarkably accurate. Perhaps more importantly, they introduced a new subject into geophysics  the use of spherical Earth models to infer mantle and core composition.
Modern Earth models are based on vastly larger seismic body wave data sets than were available to Bullen, and we now have normal mode frequencies from both toroidal oscillations (the mode equivalent of seismic Love waves) and spheroidal oscillations (equivalent to seismic Rayleigh waves) of the whole Earth. Normal mode frequencies are sensitive to the density distribution as well as the distribution of VP and VS, and incorporating them into Earth models substantially tightens the constraints on density, particularly in the lower mantle and outer core. The mathematics of the inversion procedure can be found in a number of review papers and texts, including Gilbert and Dziewonski (1975), Aki and Richards (1980), and Dziewonski and Anderson (1981). Spherically symmetric Earth models such as Dziewonski and Anderson's 1981 Preliminary Reference Earth Model (PREM) are derived using 2 x 106 Pwave (compressional wave) and 2 x 105 Swave (shear wave) travel times and approximately 103 normal mode frequencies. The technique used in constructing these models is to apply body wave data to resolve fine structure and normal modes to determine the average density in each layer of the model. Uncertainties in density are quite low for averages over a finite depth interval. In the lower mantle and outer core, the uncertainty is less than 2% for averaging intervals of 400km or more (Masters, 1979). This is somewhat
misleading, however, because the uncertainty increases rapidly as the averaging interval is reduced. In particular, the spherical Earth models are not able to accurately resolve discontinuities. The procedure for treating seismic and density discontinuities is to prescribe their depth a priori and allow the inversion procedure to determine the bestfitting values for properties in the layers on either side. Because precision decreases as the averaging interval decreases, it is difficult to resolve accurately the fine structure in important regions like the transition zone, where there are several closely spaced discontinuities. This limitation should always be kept in mind when interpreting the fine structure in spherical Earth models.
Profiles of Vp, Vs, and density p from model PREM are shown in Figure 3.2 for the whole Earth, and in detail for the upper mantle in Figure 3.3. The variations in bulk modulus
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