Three Lectures on the Role of Theory in Science
1. CLOSING THE LOOP
Testing Newtonian Gravity, Then and Now
2. GETTING STARTED
Building Theories from Working Hypotheses
3. GAINING ACCESS
Using Seismology to Probe the Earth’s Insides
George E. Smith
Tufts University

vs.
Theory-mediated measurement vs.
“Theory-mediated observation”
Areas of science in which theory is indispensable to having empirical access to the subject matter at all
Microphysics: atomic and subatomic
Internal structure of the Earth

• Some historians and philosophers contend that science is a construct constrained on its boundaries by observation
• What evidence is there then that unobserved “theoretical entities” like electrons really exist – vs. mere constructs?
• Questions of this sort gain their maximum force when the evidence for theory has to come from “data” that presuppose the very theory in question
• Seismological research over the last century is no less an example of this than research since 1850 in microphysics
• What sort of corroboration has there been for the conclusions from seismology about the internal structure of the Earth?

I.
Introduction: the issue
II.
Seismological research from 1900 to 1960
III.
Seismological research since 1960
A. From 1960 to “Preliminary Reference Earth Model”
B. The years since “PREM”
IV.
Concluding remarks

“All these things will be so on the hypothesis that the earth consists of uniform matter…. If [, however,] the excess of gravity in these northern places over the gravity at the equator is finally determined exactly by experiments conducted with greater diligence, and then its excess is everywhere taken in the ratio of the versed sine of twice the latitude, then there will be determined … the proportion of the diameters of the earth and its density at the center, on the hypothesis that the density, as one goes to the circumference, decreases uniformly.”
Isaac Newton, Principia, 1687

Deviation of surface gravity from Newton’s ideal variation implies the value of (C-A)/Ma 2 and hence a correction to the difference (C-A) in the Earth’s moments of inertia, and the lunar-solar precession implies the value of (C-A)/C and hence a correction to the polar moment
C ; these two corrected values constrain the variation

(r) of density inside the Earth by implying it is notably greater toward the center, but they do not suffice to determine the variation

(r) .
Hypothetical models of

(r) :
• Legendre (
1793 )
• Laplace (
1825 )
• Roche (
1848 )
• G. Darwin (
1884 )
• Radau (
1885 )
• Wiechert (
1897 )
• Georg Kreisel (
1949 ):
Gravity measurements at or above the surface of the Earth can never uniquely determine the variation of density below the surface.

Observational advances
Early pendulum seismometers e.g. Palmieri (1856) e.g. Ewing (1881)
Networks of observing stations
Italy
Japan
Increasing sensitivity
Milne (1892)
Wiechert (1903)

RICHARD DIXON OLDHAM
1899: Report on the great earthquake of 12 June 1897
1900: On the propagation of earthquake motion to great distances
1906: The constitution of the earth as revealed by earthquakes

Theoretical foundations
Transmission of compression ( p ) and transverse shear ( s ) waves
Poisson (1829, 1831)
Stokes (1849)
Surface waves
Rayleigh (1885)
Love (1911)
Free oscillation modes of a sphere
Lamb (1882)
Love (1911)
Assumptions elastic linear isotropic
2 stress-strain parameters vs. as many as 21 in the general case of anisotropy homogeneous
….

•
Poisson: Addition to Mémoire sur l’équilibre des corps élastiques
Mémoire a classic in continuum mechanics
Mathematical consequences of Navier-Stokes equation
• Basic equations of continuum mechanics
Fundamental principles of physics, e.g. F=ma
•
Constitituve equations for individual media
Solid vs. fluid, elastic vs. plastic, isotropic vs. ….
• The question of evidence : Do the proposed constitutive equations hold for the medium?

“Of all regions of the earth none invites speculation more than that which lies beneath our feet, and in none is speculation more dangerous; yet, apart from speculation, it is little that we can say regarding the constitution of the interior of the earth….
The object of this paper is not to introduce another speculation, but to point out that the subject is, at least partly, removed from the realm of speculation into that of knowledge by the instrument of research which the modern seismograph has put in our hands .”

•
Crust-mantle boundary
Mohorovičić 1909
•
Core
(Oldham 1906)
Gutenberg 1914 at 2900 km below surface
•
Core is liquid
Jeffreys 1926
•
Inner Core
Lehman 1936

• from
Arrival times of seismic waves from earthquakes at many locations around the Earth
• to
Travel times (Δt vs. Δθ) for a spherically symmetric
Earth for p and s waves – reflected and diffracted as well as refracted within a medium of varying density
• to
Velocity variation of p and s waves in a spherically symmetric Earth, via ray theory and the Herglotz-
Wiechart integral (1907) for an isotropic medium

Need to identify phases
(different pathways) of waves reaching a single point at different times

Assumptions:
• Arrival times of principal phases distinguished from each other
• Times and source locations of wave-origin identified, including focal depth
• Systematic errors corrected for
– Ellipticity of Earth
– Double quakes
– Late readings due to weak p, pkp
• Averaging for spherical symmetry makes sense

Assumptions:
• Fractional change in v gradient over one wavelength small compared to v
• Velocity increases slowly with depth or
– Decreasing velocity zones identified and provided for
• Numerical derivatives of Δt vs.
Δθ are well behaved
• (Isotropic, linear elasticity with continuous properties except at identified discontinuities)

A FURTHER PROJECT: INFER DENSITY vs. RADIUS
P velocity in isotropic elastic medium


[( bulk-mod + 4 shear-mod / 3 )/ density ]
S velocity in isotropic elastic medium


( shear-mod / density )
Two equations in three unknowns:
( bulk-modulus / density )
( shear-modulus / density )
From gravity constraints, lab experiments at high pressure, and assumptions
(equations of state), infer density vs. radius in symmetric Earth
Bullen, 1940-42

• Precision: error bands?
• Resolution: scale of detail?
• Idealization: uniqueness?
• Corroboration: assumptions?
Form of evidence: coherence, as judged by magnitudes and absence of systematicity in residual discrepancies
Inference to best explanation

I.
Introduction: the issue
II.
Seismological research from 1900 to 1960
III.
Seismological research since 1960
A. From 1960 to “Preliminary Reference Earth Model”
B. The years since “PREM”
IV.
Concluding remarks

• Nuclear testing yields evidence supporting travel times
• Nuclear detection → U.S. finances open-data network
– World Wide Standardized Seismographic Network (1960)
– International Seismological Centre (1964)
• Advent of digital computers, of increasing power
• Satellites → improved values of mass, moments of inertia
• Improved and new instrumentation
– Including long period, electronic strain-based seismometers
– Fast Fourier transform: spectra (Cooley & Tukey, 1965)
• Burgeoning number of people entering the field
•
Detection of natural modes of vibration of the Earth
– Proposed 1958, confirmed following Chile (1960), Alaska (1964)
– Initiating advanced efforts on “inverse methods” (late 1960s)


New data , independent of travel times (& ray theory)

Each mode of oscillation samples the whole Earth, but differently

Long period modes give direct information about density variations
 Conclusive evidence for solid inner core

Differing amplitudes give information about action in individual earthquakes

Initial Earth model: densities & material properties
Calculate natural frequencies for model
Find array of discrepancies vs. observed frequencies
Use array of discrepancies to revise Earth model

“1066” inverse solution
:
Start from two prior models
Use 1064 natural modes + mass, moments of inertia
Obtain new Earth models
Results:
• Reconstruct two quakes
•
Systematic discrepancies between calculated and traditional travel times

Low frequency waves more highly attenuated, producing anelastic wave dispersion
Outer mantle is anisotropic , with different velocities horizontally and vertically

• 1000 normal mode periods,
500 summary travel times,
100 normal mode Q-factors, mass, moment of inertia
• Mantle includes anelastic dispersion and anisotropy
(transversely isotropic, yielding two velocities)
• In spite of other models and known shortcomings, still preferred as textbook model

• Multiple spherically symmetric models
• Question:
What exactly do they represent?
• Interest turns to details, including tomography using compact arrays of seismometers to identify lateral density variations

“The early satellite results yielded anomalies that exceeded expectations and led to the conclusion that significant lateral variations in the density of the mantle occurred.
These departures from isostatic and hydrostatic equilibrium imply either a finite strength for the mantle or convection within it. With the finite strength interpretation, the gravity field reflects a long-past condition of the planet, while the convection interpretation implies an on-going evolutionary process. The inability to distinguish between two extreme alternative hypotheses emphasizes once again that Earth models based on gravity observations alone are no better than the assumptions made to render a non-unique problem tractable.”
Lambeck, Geophysical Geodesy: The
Slow Deformations of the Earth, 1988
Van der Hilst et al., 1997

Inner Core Differential Motion Confirmed by Earthquake Waveform
Doublets , Zhang et al., 2005
Crustal Dilatation Observed by GRACE
After the 2004 Sumatra-Andaman
Earthquake , Han, et al., 2006
Gravity changes in μgal

• The highly redundant data have been sufficiently well-behaved to be yielding reasonably unequivocal answers to questions
• Systematic discrepancies between observation and theoretical models have proved informative, e.g. in answering questions
• Complementary sources of data have converged on the same conclusions rather than opposing one another
• Theoretical models have enabled advanced research to develop evidence for details that reach well beyond those models

• Without the theoretical basis supplied by continuum mechanics, seismology would not have given us empirical access to the interior of the Earth
• While this theoretical basis has been indispensable to turning seismographic data into evidence, that basis has itself been tested in the process, providing corroborative evidence
• Seismology has given us, in particular, an enormously more strongly confirmed answer to Newton’s question about the density variation than we had in 1900
• Seismology has done this even though the constitutive equations it used throughout much of the last century were over-simplified and hence were made “more exact or liable to exceptions.”

• Theory-mediated measurements vs. theoretical entities
– Do electrons really exist?
–
Does the Earth really have a liquid outer core 2891 km below its surface and an anisotropic solid inner core of radius 1221.5 km?
• The evidence for these entities consists of gross differences we have concluded that they make in our measurements
• For which is the evidence stronger, that we should take electrons to exist or that we should take the liquid outer and solid inner core to exist?

• Science viewed from inside is an endeavor to turn data into compelling evidence, something that is difficult to do and for which theory is invariably needed
• Success in doing so has generally presupposed theoretical claims that were first adopted when little evidence was available for their truth
• Knowledge pursued is not merely theory, but also, even more so, which details in the domain make a difference and what differences they make
•
How, if at all, has the theory presupposed in turning data into evidence while establishing such details itself been tested in the process?