ALBERT

Description / Table of Contents: PREFACE Seismic imaging is the process through which seismograms recorded on the Earth's surface are mapped into representations of its interior properties. Imaging methods are nowadays applied to a broad range of seismic observations: from nearsurface environmental studies, to oil and gas exploration, even to long-period earthquake seismology. The characteristic length scales of the features imaged by these techniques range over many orders of magnitude. Yet there is a common body of physical theory and mathematical techniques which underlies all these methods. The focus of this book is the imaging of reflection seismic data from controlled sources. At the frequencies typical of such experiments, the Earth is, to a first approximation, a vertically stratified medium. These stratifications have resulted from the slow, constant deposition of sediments, sands, ash, and so on. Due to compaction, erosion, change of sea level, and many other factors, the geologic, and hence elastic, character of these layers varies with depth and age. One has only to look at an exposed sedimentary cross section to be impressed by the fact that these changes can occur over such short distances that the properties themselves are effectively discontinuous relative to the seismic wavelength. These layers can vary in thickness from less than a meter to many hundreds of meters. As a result, when the Earth's surface is excited with some source of seismic energy and the response recorded on seismometers, we will see a complicated zoo of elastic wave types: reflections from the discontinuities in material properties, multiple reflections within the layers, guided waves, interface waves which propagate along the boundary between two different layers, surface waves which are exponentially attenuated with depth, waves which are refracted by continuous changes in material properties, and others. The character of these seismic waves allows seismologists to make inferences about the nature of the subsurface geology. Because of tectonic and other dynamic forces at work in the Earth, this first-order view of the subsurface geology as a layer cake must often be modified to take into account bent and fractured strata. Extreme deformations can occur in processes such as mountain building. Under the influence of great heat and stress, some rocks exhibit a taffy-like consistency and can be bent into exotic shapes without breaking, while others become severely fractured. In marine environments, less dense salt can be overlain by more dense sediments; as the salt rises under its own buoyancy, it pushes the overburden out of the way, severely deforming originally flat layers. Further, even on the relatively localized scale of exploration seismology, there may be significant lateral variations in material properties. For example, if we look at the sediments carried downstream by a river, it isclear that lighter particles will be carried further, while bigger ones will be deposited first; flows near the center of the channel will be faster than the flow on the verge. This gives rise to significant variation is the density and porosity of a given sedimentary formation as a function of just how the sediments were deposited. Taking all these effects into account, seismic waves propagating in the Earth will be refracted, reflected and diffracted. In order to be able to image the Earth, to see through the complicated distorting lens that its heterogeneous subsurface presents to us, in other words, to be able to solve the inverse scattering problem, we need to be able to undo all of these wave propagation effects. In a nutshell, that is the goal of imaging: to transform a suite of seismograms recorded at the surface of the Earth into a depth section, i.e., a spatial image of some property of the Earth (usually wave speed or impedance). There are two main types of spatial variations of the Earth's properties. There are the smooth changes (smooth meaning possessing spatial wavelengths which are long compared to seismic wavelengths) associated with processes such as compaction. These gradual variations cause ray paths to be gently turned or refracted. On the other hand, there are the sharp changes (short spatial wavelength), mostly in the vertical direction, which we associate with changes in lithology and, to a lesser extent, fracturing. These short wavelength features give rise to the reflections and diffractions we see on seismic sections. If the Earth were only smoothly varying, with no discontinuities, then we would not see any events at all in exploration seismology because the distances between the sources and receivers are not often large enough for rays to turn upward and be recorded. This means that to first order, reflection seismology is sensitive primarily to the short spatial wavelength features in the velocity model. We usually assume that we know the smoothly varying part of the velocity model (somehow) and use an imaging algorithm to find the discontinuities. The earliest forms of imaging involved moving, literally migrating, events around seismic time sections by manual or mechanical means. Later, these manual migration methods were replaced by computer-oriented methods which took into account, to varying degrees, the physics of wave propagation and scattering. It is now apparent that all accurate imaging methods can be viewed essentially as linearized inversions of the wave equation, whether in terms of Fourier integral operators or direct gradient-based optimization of a waveform misfit function. The implicit caveat hanging on the word "essentially" in the last sentence is this: people in the exploration community who practice migration are usually not able to obtain or preserve the true amplitudes of the data. As a result, attempts to interpret subtle changes in reflector strength, as opposed to reflector position, usually run afoul of one or more approximations made in the sequence of processing steps that makes up a migration (trace equalization, gaining, deconvolution, etc.) On the other hand, if we had true amplitude data, that is, if the samples recorded on the seismogram really were proportional to the velocity of the piece of Earth to which the geophone were attached, then we could make quantitative statements about how spatial variations in reflector strength are related to changes in geological properties. The distinction here is the distinction between imaging reflectors, on the one hand, and doing a true inverse problem for the subsurface properties on the other. Until quite recently the exploration community was exclusively concerned with the former, and today the word "migration" almost always refers to the imaging problem. The more sophisticated view of imaging as an inverse problem is gradually making its way into the production software of oil and gas exploration companies, since careful treatment of amplitudes is often crucial in making decisions on subtle lithologic plays (amplitude versus offset or AVO) and in resolving the chaotic wave propagation effects of complex structures. When studying migration methods, the student is faced with a bewildering assortment of algorithms, based upon diverse physical approximations. What sort of velocity model can be used: constant wave speed v? v(x), v(x, z), v(x, y, z)? Gentle dips? Steep dips? Shall we attempt to use turning or refracted rays? Take into account mode converted arrivals? 2D (two dimensions)? 3D? Prestack? Poststack? If poststack, how does one effect one-way wave propagation, given that stacking attenuates multiple reflections? What domain shall we use? Time-space? Time-wave number? Frequency-space? Frequency-wave number? Do we want to image the entire dataset or just some part of it? Are we just trying to refine a crude velocity model or are we attempting to resolve an important feature with high resolution? It is possible to imagine imaging algorithms that would work under the most demanding of these assumptions, but they would be highly inefficient when one of the simpler physical models pertains. And since all of these situations arise at one time or another, it is necessary to look at a variety of migration algorithms in daily use. Given the hundreds of papers that have been published in the past 15 years, to do a reasonably comprehensive job of presenting all the different imaging algorithms would require a book many times the length of this one. This was not my goal in any case. I have tried to emphasize the fundamental physical and mathematical ideas of imaging rather than the details of particular applications. I hope that rather than appearing as a disparate bag of tricks, seismic imaging will be seen as a coherent body of knowledge, much as optics is...