Source radiation patterns

Only an explosion sends the same amplitude in every direction. Every other source pushes harder along some directions than others, and all of them but the explosion send S as well, two to five times their P peak in a Poisson solid, though not always where their P is weak. For acquisition the direction that matters is the takeoff angle $\\theta$, measured from straight down: with straight rays, a flat reflector at depth $z$ returns the ray that left at $\\theta$ to a receiver at offset $x = 2z\\tan\\theta$. So the radiation pattern tells you, before a single receiver is laid out, how much P each offset can expect from the source.

In the figure below, pick a source from its table and move $\\theta$. It opens on a vibrator at 50°; find the angle at which its P falls to half, then compare it with the other four sources.

The five idealised sources

• Explosion (a monopole). A sudden increase in volume pushes outward equally: P is the same at every $\\theta$, always a compression, and there is no S. Dynamite in a shot hole and an air gun in water come close, until the free surface above them adds a reflected ghost of opposite polarity, which gives the pair a vertical, frequency-dependent pattern (Section 1.5).
• Vertical force buried in the rock. A single force, a dipole in acoustic terms, sends P as $\\cos\\theta$: a compression below, a rarefaction above, and a null along the horizontal. It sends S as $\\sin\\theta$, $(V\_{\\mathrm P}/V\_{\\mathrm S})^2$ times larger than the P peak and strongest sideways. It is not a force dipole (two opposed forces), whose P goes as $\\cos^2\\theta$.
• Vertical force on the free surface. A vibrator or a weight drop pushes on the surface, not inside the rock, and the free surface reshapes the pattern (Miller and Pursey, 1954). In a Poisson solid, $V\_{\\mathrm P}/V\_{\\mathrm S} = \\sqrt 3$, the P falls to half at 60° and to nothing along the ground. The S has a node where $\\sin\\theta = V\_{\\mathrm S}/V\_{\\mathrm P}$ and peaks just beyond it, near 38°, at 2.4 times the P peak. Most of the energy leaves as neither (Miller and Pursey, 1955): 67 % goes into the Rayleigh wave along the surface (ground roll), 26 % into S and only 7 % into P.
• Horizontal force buried in the rock. A force that pushes sideways inside the rock, such as a downhole shear source, sends P as $\\sin\\theta$, with its null straight down, and S as $\\cos\\theta$, strongest straight down. That is why a horizontal source is used when you want S at the target. A shear vibrator on the surface is reshaped by the free surface, as the vibrator is, and its P also vanishes along the ground (Cherry, 1962); the figure draws the buried force only.
• Slip on a fault (a double couple). Two opposed force couples with no net torque, the model of an earthquake or a microseismic event. P goes as $\\sin 2\\psi$, with $\\psi$ measured from the fault plane: four lobes of alternating polarity, with nulls along the fault plane and along the plane perpendicular to it. S goes as $\\cos 2\\psi$, strongest where P is null.

Reading the figure

At the opening state the vibrator sends P toward 50° at −4.2 dB of its straight-down peak, and a reflector at 2000 m returns that ray at 4770 m offset. At 60° (6930 m) the P is down to half, −6 dB, and along the ground it is gone. The same source’s S peak is 2.4 times its P peak, at 38°. Loosen the ground ($V\_{\\mathrm P}/V\_{\\mathrm S}$ of 4) and the P curve in (c) hardly moves, while the S node closes in to 14.5° and the S peak grows to 14 times the P peak.

Why this matters for geometry

A vertical source’s P null lies along the ground, at 90°, which no reflection reaches: even a far receiver records a reflection that left the source well below the horizontal, at 60° for an offset of 3.5 times the target depth. What the null does is keep the direct P weak along the surface, where the ground roll carries most of a surface source’s energy instead. The real limit at far offsets is the steady loss of P with angle, 6 dB at 60° for a vibrator, which adds to the spreading loss and to the change of the reflection coefficient with angle. For slip on a fault the nulls cut through the recording array itself: above a vertical fault, receivers on either side record P of opposite polarity, and stacking them without correcting the polarity cancels the event.

References

• Aki, K., Richards, P. G. (2002). Quantitative Seismology (2nd ed.). University Science Books.
• Cherry, J. T. (1962). The azimuthal and polar radiation patterns obtained from a horizontal stress applied at the surface of an elastic half space. Bulletin of the Seismological Society of America, 52, 27–36.
• Miller, G. F., Pursey, H. (1954). The field and radiation impedance of mechanical radiators on the free surface of a semi-infinite isotropic solid. Proceedings of the Royal Society of London A, 223, 521–541.
• Miller, G. F., Pursey, H. (1955). On the partition of energy between elastic waves in a semi-infinite solid. Proceedings of the Royal Society of London A, 233, 55–69.
• Sheriff, R. E., Geldart, L. P. (1995). Exploration Seismology (2nd ed.). Cambridge University Press.
• Pritchett, W. C. (1990). Acquiring Better Seismic Data. Chapman & Hall.