Measurement of the ray parameter

2.11.1 Measurement of the ray parameter

Given a seismic source record, the ray parameters for the upcoming waves arriving at the receiver spread may be estimated by measuring the horizontal apparent velocity of each wave (see section

2.9). As shown in Figure 2.16, the emergence angle, θ 0 of a ray arriving at a particular pair of receivers is

where v 0 is the instantaneous velocity immediately beneath the receivers, ∆r is the receiver spacing, and ∆t is the time delay between the wavefront arrival at r and r+∆r. In arriving at this expression, any wavefront curvature has been assumed to be inconsequential at the local site of measurement.

According to equation (2.51), the ray parameter is given by sin θ 0 /v 0 so

where v r is the horizontal apparent velocity of the wave at the measurement location. Generally, it is expected that p changes with position due to changes in emergence angle and due to lateral variations in the instantaneous velocity. Horizontal events, i.e. waves traveling vertically when they reach the receiver array, have a ray parameter of 0. Waves traveling horizontally across

the array have a ray parameter of 1/v 0 and on a seismic (x, t) plot have the maximum possible

48 CHAPTER 2. VELOCITY

Figure 2.17: The physical considerations that place limits upon ray parameter manifest as limits on the maximum time-dip in (x, t) space (A) and segment (k x , f ) space into allowed and forbidden regions (B).

slope. Even though the wavefronts are vertical, the slope on the time section cannot exceed 1/v 0 . Taking sign into account, the range of possible values for the ray parameter is −v −1 0 ≤p≥v −1 0 .

This maximum possible steepness in (x, t) space is a fundamental property of a seismic time section. Assuming a value for v 0 in the course of data processing means that any events with slopes steeper than v −1 0 are interpreted as non-physical and must be rejected. Since apparent velocities can also

be expressed in (k x , f ) space as f /k x (equation (2.40)), this phenomenon means that (k x , f ) space is segmented into allowed and forbidden regions. Figure 2.17 shows this situation for both (x, t) and (k x , f ) spaces. In (x, t) space, the fan of allowed time dips forms a bow-tie when plotted at a specific point, though such events may exist all (x, t) locations. In (k x , f ) space, the zero ray parameter plots vertically down the f axis while it is horizontal in (x, t). Thus the fan of allowed p values in (k x , f ) forms a symmetric shape about the f axis. (Only one half of the bow tie is shown here because negative frequencies are generally redundant for real-valued data.) In (k x , f ) space, p values are found only along radial lines emanating from the origin, not along lines of the same slope emanating from some other point. The portion

of (k x , f ) space corresponding to |k x /f | < v −1 0 (i.e. outside the shaded region in Figure 2.17B) is “forbidden” to simple rays and is called the evanescent region. (It will be seen later that certain exponentially decaying wave types can occur here.) The shaded portion of Figure 2.17B is called the bodywave region and is the portion of (k x , f ) space that is available for seismic imaging.