The propagation of energy via waves is a familiar phenomenon in our everyday life. The particular waves to be studied here are seismic waves which are intentionally created to image the interior of the earth [1], [8] and [21]. Our three dimensional earth consists of more than the geological structures we are accustomed to thinking about, much of the earth is fluid or fluid-like. Here the principal fluids of interest are hydrocarbons. The other essential fluid to consider is water. The fluid-like materials include the many gases trapped in earth, gases like carbon dioxide, helium, and natural gas. Actually, we may not be aware of the less visible subsurface structure, but all of the surface geology you can observe, and more, exists in some form under the surface.
To find accumulations of petroleum requires an intimate knowledge of the subsurface geology, the history of the material source and the structure of the subsurface. A reservoir requires porosity, a sealing mechanism, and a hydrocarbon source. The storage capacity is dependent on the porosity, the seal prevents leakage of the hydrocarbons, and the source generates the hydrocarbons. Note, the source rocks are not always the same as the reservoir rocks.
To produce hydrocarbons the reservoir must be found and be capable of producing fluids. The interconnections of the porous spaces, the permeability, permits flow of the gases and liquids. Tightly connected porous spaces are difficult producers, but well connected spaces have good permeability and are productive. The oil exploration process finds possible drilling locations, and the actual drilling of a well is used to test the geological hypothesis of hydrocarbon existence.
We shall study the seismic method for determining the subsurface structure. The geophysical technique is to generate artificial seismic waves and record their reflections from impedance differences within the earth. Echoes come from the reflections created by relatively hard surfaces where the impedance changes.
Mathematically, the simplest hyperbolic partial differential equation is the constant density acoustic wave equation. The basis for using this particular wave equation [2] will be developed further. Because the computational effort of solving three dimensional problems ([16] Chapter 10 and [15] Chapter 1) exceeds most computing environments, this study will primarily focus on one and two dimensional problems. All the techniques and algorithms presented here can be directly extended to three dimensions.
The constant density assumption simplifies model representation: only a sound speed is required. The earth density variation is important for modeling and imaging. However, neglecting density variation will still provide a useful wave equation. One additional comment about earth parameters: surface measurements using physical methods use potential fields, gravity for example. These physical measurements are of a different scale compared to reflection seismology. The seismic data wavelength has sufficient resolution for structural imaging. Resolution of the seismic experiment is a direct function of the wavelength; the shorter the wave length the higher the resolution.
The acoustic assumption is also in contrast to the elastic assumption. A fluid medium supports the propagation of a pressure wave. An elastic medium supports both shear and pressure waves. Marine seismic data are collected using water borne receivers or receivers which rest on the ocean bottom. Propagation of shear energy is restricted to solid media, and this makes land seismic measurement the principal generator of elastic seismic data. However, it is possible to find mode converted elastic information within an acoustic marine seismic dataset. In an elastic medium when waves impinge on a reflector, both shear and pressure waves are created, i.e. there is a mode conversion.
Seismic pressure waves are recorded by using a geophone, a microphone on a spike. The geophone has a small weight which is spring mounted with a magnet and a coil of wire. The pressure wave from the vibrating earth generates a vertical displacement moving the coil while the weight tries to resist this motion. The magnetic field generates a voltage proportional to the earth acceleration. Elastic waves have three components of displacement. Three geophones are aligned in orthogonal directions to measure the pressure wave and the two shear waves.
The seismic wave is man-made and requires a significant amount of energy to propagate any distance in the earth. Dynamite charges are used to generate the primary pressure wave (P-wave) and some shear wave energy (S-wave).