Exploration geophysics has a dual scientific role: one is the continuous search for new petroleum and mineral reserves, the other is the continuity of research which provides better exploration tools and techniques and a more complete knowledge of the physics of the earth. Recognizing, and developing, the interdependent nature of the search and research roles is essential for progress in exploration geophysics. In the past, geophysical field work and geophysical research often have been quite removed from each other, even though the interplay between them has been central to a comprehensive view of exploration geophysics. Indeed, the interchange of ideas is essential for processing seismic data with computers. Any idea that can be embodied in a computer program can be utilized immediately in the processing of field data. In order to further develop this potential of the interchange of ideas, every geophysicist must have a broad objective within which to operate, but they also must be given full freedom to use their initiative, imagination, and creativity. New ideas must come from both the search and research aspects of geophysics.

To create a map of the subsurface, a geophysicist converts received seismic traces, which record events as a function of time, into a format that records them as a function of depth. In other words, a time function recorded at the surface must be transformed into a depth function. Unlike radio waves, the velocity of seismic waves is very dependent on the medium through which they travel. Thus, the velocity changes as the waves travel into the earth. Generally, velocity increases with depth, although occasionally there may be layers in which a decrease in velocity occurs. For a given surface point, the velocity plotted as a function of depth is called the velocity function. Thus, in reflection seismology, there are two equally important variables: reflection time and velocity. Knowing these variables allows the depth to the reflecting horizons to be determined. Because there are important lateral changes in velocity, that is, because the velocity function varies from one location to another, a given velocity function cannot be assumed to be valid for an entire prospect. As a result, the velocity depth function must be corrected from place to place over the area of exploration. The problem of velocity estimation is not easy in the case of isotropic rocks. In the case of anisotropic rocks, it is, in fact, even more difficult.

An important step in the processing sequence consists of making traveltime adjustments, or corrections. Time adjustments are characterized either as static corrections or as dynamic corrections. For a static correction, a time shift (or translation) is applied to an entire trace; that is, regardless of record time or reflector depth, a constant time correction term is added to or subtracted from all reflection times. In contrast, dynamic corrections vary with record time and therefore depend on reflector depth.

Waves can be classified as either standing waves or traveling waves. A standing wave, also known as a stationary wave, is a wave that remains in a constant position. A traveling wave is not confined to a given space in the medium, but propagates through the medium. The waves on a string of a musical instrument are examples of standing waves. Such waves are the result of interference of traveling waves propagating in opposite directions.

Let us suppose that a point source is emitting light waves of wavelength λ in all directions. These waves may be represented by a set of spheres (with radial spacing λ) spreading from their center source. Because every point on each sphere is equidistant from the source, it can be thought of as representing the crest of a wave. If straight lines are drawn outward from the source, each line (ray) represents the direction along which the wave is advancing.

Let us turn to traveltime t . Traveltime represents the time required for seismic energy emanating from the source point to reach a given depth point ( x, z ). Traveltime has magnitude but no direction; thus, traveltime may be represented by the scalar function t ( x, z ). The traveltime surface is a plot of traveltime t against ( x, z ). The traveltime function can be depicted by a surface plotted against horizontal coordinate x and vertical (depth) coordinate z . An imaginary terrain, depicted by a topographic map, can be used to visualize the traveltime configuration. Topographic maps provide information about elevation of the surface above sea level, representing elevation with contour lines (level lines).

A common assumption of ray theory is that energy transport occurs along the ray paths, not across them. This assumption is violated when the ray passes a caustic or when diffraction at a model discontinuity occurs. The three-dimensional theory of classical ray tracing is given in the works of Cerveny, Hubral, Hubral and Krey, and many others. In this chapter we will present a simplified two-dimensional version. In isotropic media, a raypath is a line everywhere perpendicular to the wavefronts. In the situation of horizontal layering, a raypath can be characterized by its direction at the surface. The ray parameter p = sin ui/vi is constant along any ray, where ui is the angle with the vertical and vi represents velocity at the given point.

Seismic waves are mechanical waves, meaning that these waves involve the actual motion of the rock particles. Seismic waves that travel through the entire body of the rock are called body waves. In contrast, seismic surface waves travel close to the surface rather than through the rock. Seismic body waves are of two fundamental types: longitudinal and transverse. In a longitudinal wave, the oscillating particles of the medium are displaced parallel to the direction of propagation (i.e., the direction of energy transmission) of the wave. In a transverse wave, the particles are displaced in a direction perpendicular to the propagation direction. When a steel rod is struck on one end by a hammer, a wave pulse in the form of a longitudinal compression of the rod travels down its length. If the rod is struck periodically, a succession of such pulses, known as a wave train, travels down the rod. Sound propagates through the air as longitudinal waves. In contrast, the familiar water waves that come from the point where a stone is dropped into a quiet pond are transverse waves. Furthermore, because these waves are confined to the water layer close to the surface, they are surface waves rather than body waves. Importantly, various types of seismic surface waves also can be identified on seismograms. Surface waves usually travel more slowly and have larger amplitudes and longer wavelengths than body waves.

The Dirac delta function δ( t ) and the Heaviside unit step function θ( t ) are known as singularity functions. By definition, a stable function has finite energy. A stable time function h ( t ) and a stable space function g ( x ) have well-defined Fourier transforms, designated by H (ω) and G ( k ), respectively. However, the study of wave propagation involves functions that are not stable. For example, the function sin k 0 x does not have finite energy, so it is not stable. The Fourier transforms of such unstable functions are not well defined. Thus, in wave propagation, we must consider using quantities that involve a certain kind of infinity.

A traveling wave is created because a deformation (in the material medium or in the field-medium) causes the medium to snap back toward the equilibrium state. However, the medium overshoots the equilibrium state, and ends up oscillating back and forth, all the while bringing neighboring regions into the same motion. The wave speed, therefore, is determined by the medium, not by an external agent (such as the stone thrown into the water) pushing on the medium. Pushing harder on the medium makes the amplitudes of the wave larger. It does not make the wave travel faster. This picture of a traveling wave is the same for all waves.

The wave equation is called nonhomogeneous because of the non-zero source term f ( r, t ) on the right side of the equation. Because, we are working in a homogeneous medium, velocity v remains constant. The 1D, 2D, and 3D solutions all can be derived from the three-dimensional wave equation, if we define initial conditions at a plane (for 1D), line (for 2D), and point (for 3D). The line solution for two dimensions is Hadamard's solution.

Let g ( r, t ) be the Green's function (i.e., impulse response function) for the wave equation. Let the source function f ( r, t ) be causal. Of course, the Green's function is also causal, so both g and f vanish for t , 0. A black box is any complex piece of equipment (for example, a unit in an electronic system) with contents that are mysterious to the user. In our case, the black box is the subsurface of the earth. On the surface, we can measure the input signal that we send into the subsurface, and we can measure the resulting output signal received at the surface. Here, we also are considering an input-output problem, but is a much simpler case. It is the case of oneway transmission through a homogenous isotropic medium. The black box is represented by its impulse response (also known as the Green's function). The output w ( r, t ) is the convolution of the impulse response g ( r, t ) and the input f ( r, t ).

The purpose of this book is to provide the information required for understanding the fundamental aspects of the elaborate computer processing schemes prevalent in exploration geophysics. Basic Wave Analysis has three parts. Part 1 addresses velocity analysis. The correct determination of velocity is the most important problem in seismic exploration, and an understanding of velocity analysis is a valuable asset for a geophysicist. Part 2 discusses raypath analysis. Raypaths provide a geometrical picture of how waves travel, so that a person can visualize raypaths in their imagination. Geometrical pictures are as important in seismology as they are in optics. Part 3 addresses wavefront analysis. A person cannot easily visualize traveling wavefronts in their imagination; however, a computer can follow their motion, and give the geophysicist the final outcome. Knowledge of wavefront analysis helps a geophysicist understand many modern computer methods.