Directivity of linear sources
In Figure 7.5a, a linear vertical source (such as a column of explosives) of length , being a constant and the wavelength, is activated at all points simultaneously at time . Taking the initial waveform as , show that the effect at point is
What is the array response?
A distributed source can be thought of as an array. The array response is the ratio of the output of an array to the output when all of the elements are concentrated at the midpoint of the array.
Detonating cord is an explosive cord with a constant velocity of detonation; it is used to connect two charges in order to delay detonation of the second charge. Detonation of the first charge initiates detonation in the cord, which in turn detonates the second charge. By varying the length of cord, detonation of the second charge can be delayed a desired amount.
Although the explosive is exploded instantaneously, energy from different parts of the column arrive at at different times because they must travel different distances. Denoting by , the total effect at at time will be
To integrate we must get a relation between and . We assume that ; then
Noting that and , this becomes
where sinc .
If the linear source is replaced by a concentrated source of equal strength at the center , the effect at would be . Dividing the right-hand side of equation (7.5c) by this quantity, we get for the array response
An explosion initiated at the top of the explosive column in Figure 7.5a travels down the column with velocity . Show that the array response is
where is the velocity in the rocks. Under what circumstances does equation (7.5e) reduce to equation (7.5d)?
In part (a), the entire column exploded at . We now consider the case where the explosion starts at point and travels down the column with velocity , that is, the explosion starts at at , where . Writing , , the wave generated by the element dz arrives at with the phase Using equation (7.5b) the phase becomes
Assuming a harmonic wave function , we can write
where amplitude, , . Integrating, we obtain
If we locate the same amount of explosive at and explode it at , we get at
hence the array response is
Omitting the first factor, which is independent of and and hence is merely a scale factor, we have
[The minus sign for sinc occurs here and not in equation (7.5d) because the direction of integration is opposite to that assumed in deriving equation (7.5d)]. For an instantaneous explosion the result is given by equation (7.5d), namely
Equation (7.5c) reduces to equation (7.5d) whenever . For , i.e., for rays traveling almost vertically downward, the required condition is that . For most explosives, km/s, so should be no more than about 1.5 km/s, the velocity of water, for the two equations to give nearly the same result.
Calculate the array response for a column 10 m long, given that m, km/s, km/s, and , , , .
Differences in directivity are negligible when the charge length is much smaller than the wavelength.
If the column in part (c) is replaced by six charges, each 60 cm long and equally spaced to give a total length of 10 m, the charges being connected by spirals of detonating cord with detonation velocity 6.2 km/s, what length of detonating cord must be used between adjacent charges to achieve maximum directivity downward?
Let be the length of detonating cord between successive charges; maximum directivity downward is achieved when the traveltime through the explosive column is the same as that in the adjacent rocks. In part (c) we were given km/s, km/s, so , hence m.
What are the relative amplitudes (approximately) of the waves generated by the explosives in part (d) at angles , , , and when m?
An approximate solution can be obtained by assuming that the average velocity is equal to ; this means that the traveltime down through the 10 m column of explosives is the same as that for a wave in the adjacent 10 m of rock. In this case, , and
|Effective penetration of profiler sources
|Characteristics of seismic events
|Reflection field methods
Also in this chapter
- Radiolocation errors because of velocity variations
- Effect of station angle on location errors
- Transit satellite navigation
- Effective penetration of profiler sources
- Directivity of linear sources
- Sosie method
- Energy from an air-gun array
- Dominant frequencies of marine sources
- Effect of coil inductance on geophone equation
- Streamer feathering due to cross-currents
- Filtering effect of geophones and amplifiers
- Filter effects on waveshape
- Effect of filtering on event picking
- Binary numbers