Overview

Gravity observations include the combined effects of instrumental, surface, terrain, and planetary sources in addition to the subsurface mass variations that are the objective of an exploration gravity survey. To isolate the effects of subsurface sources, extraneous effects which include both temporal and spatial variations are removed from the data using theoretical considerations, geological information, and empirical observations. Some are considered universally, but others only in specific geological, surface, and observational conditions. There is an increasing need to eliminate a broader range of extraneous effects more precisely as the objectives of gravity surveying require higher precision and accuracy and are focused on both long- and short-wavelength anomalies. Unwanted effects are removed by calculating the gravity anomaly, which is the arithmetic difference between the observed vertical acceleration of gravity and the predicted or theoretical acceleration at the observation site. Theoretical gravity is based on a conceptual model of the sources of gravity variations. This model varies depending on the intended use of the gravity anomaly and the conditions of the survey.

Three classes of anomalies are recognized. The primary class, planetary anomalies, incorporates only analytically determined planetary considerations in the theoretical model, e.g. the Bouguer gravity anomaly. A second type, geological anomalies, applies additional effects from known or postulated subsurface geological conditions in the model, e.g. the isostatic residual gravity anomaly. The third type, filtered anomalies, is calculated by removal of arbitrary gravity effects caused by unknown sources, empirically or analytically determined by filtering that enhances particular attributes of the spatial pattern of the anomalies, e.g. wavelength-filtered gravity anomalies. The latter type seeks to isolate or enhance those of interest at the expense of other anomalies.

Introduction

It is unlikely that many seismologists given a choice between seismic magnitude and some physical measure of seismic-source size would choose magnitude. As Thirlaway (1973) notes, some of the founders of seismology regarded seismic amplitudes as not worth serious study and since then others have argued that magnitudes should no longer be used (Jones, 2000). The main disadvantage of the magnitude scale is that it is not based on the physics of the source. It is an arbitrary scale; there is no wrong answer. Seismologists are therefore free to propose modifications to the scales. Scales defined on one type of recording system or component of ground motion, for example, are simply applied without change to new systems or components, which often then leads to attempts to relate the different scales.

Further, how A and T should be measured is often only loosely defined. Consequently, what is measured depends on analyst interpretation of the rules, and rarely are corrections applied to allow for: departures in the propagation path from some average structure; station effects; or the effects of earthquake radiation patterns. So it is easy to make the argument for abandoning magnitude as a measure of the size of a seismic disturbance and turning to physical measures such as seismic moment, which is based on the source physics and in theory at least, is independent of radiation patterns and Earth structure.

Overview

The gravity method of geophysical exploration is based on the measurement of variations in the gravity field caused by horizontal variations of density within the subsurface. It is an important technique for many problems that involve subsurface mapping, and it is the principal method in a number of specific types of geological studies. The method, which has its roots in geodetic studies from the seventeenth to the twentieth centuries, has developed rapidly over the past century as a result of significant technological advances: primarily high-accuracy gravity measuring instruments for use on land, sea, and air, and in space, as well as the increasing computational and graphics power of digital computers. These advances are backed up by continuing improvements in data processing, interpretational schemes, practical experience, and surveying methodologies.

Traditionally the gravity method has been used primarily in regional characterization of the Earth for determining the architecture of the crust, identifying potentially favorable regions for resource exploration, and developing conceptual exploration models. This is made possible by the millions of gravity observations now available worldwide in both public and commercial data sets. The accuracy now available in gravity observations and the relative ease of measurement have made the method viable for exploration objectives such as assessing subsurface changes over time and combined interpretation with seismic reflection mapping.

Sometime in the early 1970s, Dr Ronald (Ron) Girdler, University of Newcastle, UK, made a plea for someone in the seismology group at the Atomic Weapons Establishment (AWE), UK, to write a book on earthquake seismology based on the AWE's research in forensic seismology, that is, seismology applied to the verification of nuclear test bans. The principal purpose of the research was to provide comprehensive advice on forensic seismology on which the UK government could base its policy for the control of underground nuclear tests. The work at the AWE, however, has applications to seismology in general which was what interested Ron Girdler.

The UK's forensic seismology programme began at the AWE in the late 1950s. Work was carried out at both AWE Aldermaston (principally on the design and manufacture of seismological recording systems) and at the outstation AWE Foulness (on decoupling, seismogram analysis and seismometry). In 1961, much of the forensic seismology work moved to an unclassified site at Blacknest, a country house near AWE Aldermaston. Over the next ten years forensic seismology research at the AWE was increasingly concentrated at Blacknest.

I and my colleagues at AWE Blacknest, Peter Marshall and Frederick (Fred) Key, produced a rough draft of a possible book but it was clear from the draft that: some of the subjects covered, such as the application of seismometer arrays in seismology were then still undeveloped; the analogue recording and processing systems then used were obsolescent; and overall, the range of subjects covered was too limited.

Overview

Raw gravity observations are reduced for their non-geologic effects to one of a variety of anomalies which in turn are processed by isolation and enhancement procedures into residual anomalies that map the gravity effects of interest in interpretation. Anomaly interpretation, which models the gravity anomalies for the nature and processes of the subsurface, is a relatively straightforward process compared with the measurement, reduction, and residual-regional separation phases of the gravity method. However, anomaly interpretation is never unique, owing to the ubiquitous presence of data errors and the inherent source ambiguity of the gravity potential. Thus, ancillary geological, geophysical, and other constraints on the subsurface are essential to help limit the ambiguity.

Gravity interpretations can be qualitative where the analysis objectives are satisfied by the mere presence or absence of an anomaly. Interpretation also can be highly quantitative with comprehensive modeling of the geometric and physical properties of the anomaly sources. Effective interpretation requires knowledge of the key geological variables that influence the anomaly's amplitude and geometry. It also requires an understanding of the key geophysical variables that control the inverse problem of estimating source parameters from the anomaly.

Gravity interpretation generally is initiated using simplified techniques to estimate preliminary source depths, depth extents, margins, density contrast, and mass. These estimates are often enhanced by more comprehensive analyses that include forward modeling using trial-and-error inversion methods if only a few unknown modeling parameters are involved. For more unknowns or where they must be estimated by least squares or some other error norm, inverse modeling is commonly implemented by matrix inversion. Both inversion methods compare the predicted anomaly from an assumed forward model with the observed gravity anomaly.

Introduction

The statistical analysis of reported arrival times of seismic body waves carried out by Jeffreys and Bullen to derive travel-time tables is a landmark in the development of seismology. This is evident from the longevity of the tables – first published in 1936 – it was only in the 1990s that a new set has been judged sufficiently superior to the J-B tables to warrant their replacement. Before the CTBT negotiations, seismic amplitudes had not been subject to the same rigorous analysis. Yet when the negotiations began almost all the discussion on explosion seismograms seems to have been on P amplitudes and station magnitudes. The magnitudes (principally recorded at local and regional distances) are scattered and how these should be combined to give a magnitude for an explosion was a matter of sometimes bitter debate.

As the number of nuclear tests of significant yield and hence the number of amplitude observations increased, it became clear that some stations tend to record above average amplitudes and others below average. As amplitudes and magnitudes, particularly their relationship to yield, was of such interest in the early negotiations, AWE Blacknest began to try to make sense of magnitude observations and look for systematic effects.

The Earth's dynamical behaviour is a complex and fascinating subject with many practical ramifications. Its description requires the language of mathematics and computation. In this book, we attempt to make the theoretical foundations of the description of Earth's dynamics as complete as possible, and we accompany the theoretical descriptions with computer code and graphics for the implementation of the theory.

Scalar, vector and tensor analysis

We will make extensive use of scalars, vectors and tensors throughout the book. In this section, we will summarise the properties most often used. It is assumed that the reader is familiar with the elementary results of vector analysis summarised in Appendix A.

Scalars

Physical quantities determined by a single number such as mass, temperature and energy are scalars. Scalars are invariants under a change of co-ordinates, they remain the same in all co-ordinate systems. They are sometimes simply referred to as invariants. A scalar field is a function of space and time.

Vectors

Vectors require both magnitude and direction for their specification. They may be described by their components, their projections on the co-ordinate axes. An arbitrary vector then associates a scalar with each direction in space through an expression that is linear and homogeneous in the direction cosines.

In general, a vector can be defined in a space of arbitrary dimensions numbering two or greater. Our applications will be confined to a space of three dimensions and we will adopt this limitation.

The study of Earth's figure and gravitation goes back to the very roots of the physical sciences. We begin with a description of its very interesting historical roots.

Historical development

Although ancient Greek and Egyptian philosophers and astronomers believed the Earth to be spherical and had made rough measurements of its size, the modern theory of the Earth's figure originates with the work of Newton. Using his newly developed laws of dynamics and gravitation, and making the remarkable assumptions that the Earth's figure is nearly an oblate spheroid (the surface generated by an ellipse revolved about its minor axis) and that the Earth behaves as a fluid, he was able to show that the ellipticity of figure is directly given by the ratio of centrifugal force to gravitational force at the equator. As a result of the neglect of the concentration of mass towards the Earth's centre, Newton's estimate of the ellipticity was more than 30% too large. However, the oblate spheroid continues as the figure of reference for geodesy, and the assumption of hydrostatic equilibrium remains as the basis of much of the modern theory. Perhaps Newton's most important contribution, though, was his unequivocal demonstration that any theory of the Earth's figure must take account of both its gravitation and its rotation.

Newton published his calculation in the first edition of the Principia in 1687, but the meridian arc length per angular unit of geographical latitude, as measured on European baselines, appeared to increase with increasing latitude, implying that the Earth was prolate rather than oblate. It was not until expeditions had been dispatched by the French Academy to Peru and Lapland that the matter was settled in Newton’s favour in the 1740s.