# Velocity analysis - book

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Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 3 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

The wave equation governs seismic wave propagation. Because of the mathematical difficulties inherent in the exact wave equation, simpler concepts often are used instead. One is Fermat’s principle, which lets wave propagation be reduced to the study of raypaths that have minimal traveltimes. As a consequence of Fermat’s principle, at an interface between two media, the raypath is bent in accordance with Snell’s law.

In reflection seismology, we have two equally important quantities: the times of reflected events and the velocities of seismic waves as they travel through the earth. Knowledge of those quantities allows us to determine depth to the reflecting horizons.

Seismic waves travel with medium-dependent velocities. The assumption that the velocity is the same for two forms of the same type of rock, such as two sandstone formations, generally is not valid. Seismic velocity in different kinds of sandstone can vary over a wide range. Each new rock layer traversed by the seismic wave has its own characteristic velocity. The challenge for the geophysicist is to determine those velocities. Generally, velocity increases with depth, although occasionally, layers can exist in which a decrease in velocity (or a velocity reversal) occurs.

Seismic processing methods must take into account the fact that wave velocity changes as the waves travel through the heterogeneous earth. Because velocity depends on the position of a wave in a given subsurface volume, a velocity function dependent on the spatial coordinates must be determined. If x and y are the horizontal dimensions and z is the vertical dimension, then $v\left(z\right)$ denotes a velocity function in one dimension, $v\left(x{\rm {,\ }}z\right)$ in two dimensions, and $v\left(x{\rm {,\ }}y{\rm {,\ }}z\right)$ in three dimensions.

Velocity estimation refers to finding empirical values for the velocity function. One method of measuring the velocity function v(z) is to conduct measurements in an existing oil well or borehole. An instrument consisting of a seismic pulse generator with two attached detectors separated by a fixed distance is lowered into the hole. This fixed distance is of the order of a few meters or less. As the instrument gradually is pulled back up the borehole, changes in the transit time across the fixed distance between the two detectors are recorded as a continuous curve. This curve is known as either a continuous velocity log (CVL) or a sonic log. The seismic wave velocity is the reciprocal of the interval transit time.

Velocities obtained from CVLs are reasonably representative of the velocities of seismic waves through the corresponding formations, although that correspondence is by no means exact. Differences arise from several causes. Among them is the invasion of a porous formation by the drilling fluid so that the velocity is not representative of the formation’s true velocity. Discrepancies also result from borehole geometric effects as well as from the fact that seismic-exploration frequencies are substantially lower than those employed in recording the CVL.

To clear up such discrepancies, various methods can be used. One method is the use of check shots. A check shot uses a surface source and a detector down the well. The time recorded for the check shot can be compared with the total traveltime obtained by integrating (down the borehole) the interval transit times on the CVL. A better but more expensive method is the VSP.

In many cases, it is necessary to estimate subsurface velocities by measurements confined to the surface because boreholes are not always conveniently available. In such cases, the actual seismic data must be used to estimate velocities. A well-established method estimates by considering the time differentials for the same event as it is received by a lateral array of detectors. Any such estimate always depends on a ceteris paribus (other things being equal) assumption.

Computers can determine velocities by carrying out calculations based on many intricate time-distance relationships, and the results can present empirical velocity as a function of traveltime (or depth) in a display called the velocity spectrum (Taner and Koehler, 1969). Empirical velocities so determined are called stacking velocities, and the problem is then to relate those velocities to a mathematical relationship from which the thicknesses and velocities of the subsurface layers can be estimated.